228
i I l l .j ! J ! l .. ' CPSK TPJl,NSMISSIOO T'.dR:)UGI NCNLINEAR O:@ .. Ni.'ifilS . ' ... / \ I I I i 'l • t ,· ') l ,. rr; ' t } 1 · t f I • i . \ l. u ' '
i I l
l .j ! J ! l
..
'
CPSK TPJl,NSMISSIOO T'.dR:)UGI NCNLINEAR O:@ .. Ni.'ifilS .
' ~
...
/
\
I
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,.
rr;
'
t } 1 ·
~
t f I
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•' t'.
PERF()Rf,WqCE ANALYSIS OF q'SK TRANSMISS!al
'.t'HIUJGI NOOJNFAR CHANNErS
By
PRITI HEl'JW<OL, B.E., M.E.
A.t:h:!sis
Subnitted to the Schooi of Graduate. Studies .. . .
in Partial_. Fulfilnent" of the P.eqlli.ren_l:mts
for the Degree
Ik>ctor of Philosophy
I I
McMaster University
March, 1976
@ PRITI HETRAKUL 1977 J ••
I
...
-. . . ' ,-. '. ':.,., / ( ~ , .... ~
; .. )..·,
.:;' i -~:1 :'t~}
;
' 1 · 1· l • {
I
... { .. .
. ' ,,•
•,,,
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' '
IX'.Cl'OR OF PH!U60PHY (1976) (Electrical Engineering)
M:MAS'I'ER UNIVERSITY Hamilton, Ontario
TITIE:
AUl'HOR:
J
(
PerfoI!llal1te Analysis of a>sK Transmissicn through Ncnlinear Cllannels.
Priti Hetrakul B.E. (University of New South Wales, 1971) M.E. (University of Newcastle, 1972)
Dr. D.P. Taylor
xvii, 210.
..
; ..
ii
/
, I
t I
' • I
t .
' . I
Virtually all satellite repeaters use a traveling-wave tube ('lWI')
as their mrln p<:M:!r arc;;,lifier. Because en-board power is a limited can
mxlity, it is highly desirable that the TWr be q,erated as efficiently as
possible, narrely in or near saturaticn where it is highly non1i;near. These
ncnlinear effects manifest themselves as an anplitu:le conpression (#VAM
a:nversion) effect and an anplltude <'JeFendent phase rocrluJaµon (Al-VPM con
version) effect; In this thesis a number of investigaticns have been
nade in ~lation to the TWI' nonlinearities and their effect on th,tper-1
· formanoa of camunication system:J.
A novel quadrature nodel of the 'lWl' has been clevel.qai. This·
mxlel is nnst useful in that it is analytic and requires~ choite of
only four paranete.rs to obtain an excellent fit to the TWr diaracteristics.
~ optirral bandpass nonlinear trans.fe;" characteristjc that maxi
mizes its output signal to interference~ ratio has also been derived.
By making use of this c:ptinal transfer characteristic and the quadrature
' . JIDdel of the tube, a ronputer-aided design prc>Cl:!dure h.3$ ~ described
for obtcµ.ning a predistortion co1E5isaticn network for the 'lWl'. This
~ oonsists of a sinple arrange.nent of attenuators and ~-law
devices and has been shown, by caipUte.r s.f.nulaUai, to yield about 1 dB
.i.nproven5rt. in system performance fo:t: the case when ooly a single ca.n:1er
is present in the 'lWl'.
iii
'; ,,
..... '
. •, ..... ~·
r
·.
. ~·
. •,
... ·'
.. ~. '
In the case when a single sanple detection and najo.rity logic
decision cii:cuit is assuned at the .reaaiver, it has bee: possible to de
rive anal~cal expressions for the probability of error for M-ary CPSK
signals transmitted ~gh a piecewise-linear envelope limiting repeater.
An infinite series expression for~ bit error rate of bi.naty CPSK
transnrl.ssion t:J:,.rough an actual TWl' channel has also been derived.
A perfo~ analysis of a ro.rrelatiai rerei~r with a linear . . iritegrate and &mp circuit has been carried~ for the case of binary . CPSK transmi.ssi/ through a ~s ~~ exhibiti.ng. AM,IPM con-
versicn. · I I I
I For the case of ptlrely anplitude-limiting channel.$, an optimal
(maxf.mum-likeliho::xl) receiver structure and its approximate. ~orrnan~
has also been investigated. .. .
iv
.•
\
,,
·' '· ., .
.... ,,'.
\· •...
.. . ~ :, .
·,.
..._, . .. ·' . . ·- :
•.
'l'he author gratefully acknowledges the help and guidance given to
him by his supervisor, 0r.' Desmmd P. Taylor throughout the preparatioo of
the thesis. Sped.al thanks are due to Dr. $. S. Haykin for his suggestion
of the problem and many valuable discussions and to Dr. M. A. Stephens for
serving on the superv.f.s9ry oomrrl.ttee.
'lhe aut:h?r would like to thank Dr. ~.c. Chan' for his pez:mf.ssioo
to include f.;gures ~3 to 4-8 in the thesis and his many enjoyable and ·
he~ discussions. Thanks are also due to the research J;ersonnel of the
camunications Research Lal:xtratoxy at M:Master University for participating
in many lively discussions, and to Ms. Bemi~ Johnsen for her assistance
in the typing.
Finally, the author is grateful for the financial support fran
Md-faster University through thg award of a Benefactor Scholarship.
'
..
v
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.. ~
,.,,_ ·.,i.:,¥.<t
. . ... -; .. ', " I
... I";.: ' ' .... (:-;.~ i"' ~ '.':
~;.~/; L ; ~
.. . . . f
•. • ..
,.
\
LIST OP !ILUS'I'PAT!OOS
LIST OP TABIES
LIS'!' OF PRINCIPAL SYMBJl6
dIAPI'ER 1 - ~00 MID BACKGF.OtND 1.1 Introductioo 1.2 'llie Traveling Wave Tube ('I'Wl') 1. 3 Sa:,pe of the Thesis
. OIAPTER 2 - PRBLIMINARY DISOJSSICN AND~ M)OEILING
2.1 Perfannance of Cohenmt Phase Shift Keying (CPSK)
2.2 f.bdelli.ng of Satellite Ncnlinearity·
OiAPl'ER 3 - a:M'ENSAT:rrn OF S.MELL!TE NCM.INFARITY 3. i Optimum Bandpass Honlineari ty ~ TWI' Linearlzation Techniques
3. 2 .1 Butler Matrix. Transponder (BML') 3.2.2 Feed-forward Technique 3.2.3 Signal Predistortion Tedmique
OIAPTER 4 - ~CE MIALYSIS OF CPSK SYSTEMS THIOJGH NCMJNEAR OWJNELS 4 .1 Sirnulaticn Study of Bandllrnitation Effects 4. 2 Effect of Theimal Noise Disturbances
4.2.1 Piecewise-linear Envelq,e Limiting~
· 4. 2. 2 Actual '1Wl' Channel
OmFTER 5 - RECEIVER srRUCI'URES roR NCNLINEAR WIDEBAND BINARY CPSK amNNELS 5 .1 Correlation P.ecei ~-5 .2 Maximum Likelihood Ieceiver
vi
Page
iii
v
viii
xi
xii
1 1 12 17
20 20
26
38 I 39 46 47 49 52
66
68 84 89
) 117
131
132 142
'
4 •'
.. , .
. ,.
,'.;
'
Tables ot c.ont.ents (Continued}
CHA?l'ER 6 - CXNCWSICNS AND SUa;ESI'!CNS FOR FU1'tJRE STUTJ'l 6.1 CCnt.ribut:tons of ~ Thesis 6.2 Suggesticns for FUture Study
APPENDIX A: eurve Pitting the Quadrature M:xiel of 'IWI'
APPENDIX B: The Cl!ebyshev Transform ·
Page
·164
164 166
169
171
APPENDIX C: The Noise Statistics and calculation of 175 A' for Piea:M.1.se-linear Arrplitooe Limiting (Soft-limi. ting) Channels C.l Derivation of the Noise Statistics 175 c. 2 Calculation of A' 184
APPWDIX D: calculaticns of A' , S and Various Noise z.t::trents for Satellite Channels D .1 Calcu.1.atioo of A' and 8 D.2 Nuxrerical Evaluation of M::m:!nts of .. In-phase Noise Calponents
APPENDIX E: Absolute Corwerg,"...noo Property of the Bit Error Rate Expression
v.i.i
.,,
185
185 188
198
203
.. <
..
LIS'!' OF ILll.JSTFATICNS
Figure caption ~
1-1 Capacity ccnparison. 6
. 1..;.2 Traveling wave tube • 13
1-3 The single-carrier characteristic of~ 16 1
Intelsat DJ 'IWI', Hughes 261-H. \
2-1 Block diagram of a correlation recei, 23
for CPSK systems.
2-2 ~ error probability w ~O' M-ary 27 .: '"':
+ ~ ' ~
2-3 cascade node! of '!WI'. '. ~·~ ... 29 :~ .. ·,; ,,. .. ,
< .. 2-4 General quadrature~ ,model of 'IWI' 30 •";, ...
2-5 The nonlinear dlaracteristics of TWl' 33 ,, "
I (Hughes 261-H) and the Bessel functicn' approximation •
. " 2-6 Bl.ode diagram of the camunicaµm system. 36
I I 3-1 The optinum envelope transfer dlaracteristic. 45
3-2 m-<llannel Butler Matrix Transpooder 48 (
,, 3-3 Open-loop feedforward anplifier rn:x:lel. 50 ~ ..
~--.. ·, 3-4 Predi.stortion ooapensator for TWI'. 54 f.; •.. ·" ... 3-5 The ca1t,~n.sator characteristics. 59 ..... . ' ": .. ,
3-6 The single-carrier characteristics of the. ' '• 60 I~ ,,,
. . '-" ·~ CCJll)E!llSated and uncn1pansated tube (Hughes 261-H) •
~ ~ , • I
'•' > ... J' 3-7 Nonlinear inplementation of tre (X)J1i:,ermator. 63 I~~.
t ,. 3-8 Probability of error vs EiJNo, 2-phase CPSK. 64 . >: "-~.) ~- ..
,!•
' -,.,.. ~
: . .. , viii ··~ .. ;,I •'1
' ·,
•[
......_ I
' ' " : . .,. ~ ..... . "'
~,... l
.. .. . ·.
I ;
,, I
\
IJ.st of Illustrations (Ccntinued)
Figure
4-1
4-2
4-3,4-4
4-5,4-6
4-7,4-8
4-9
4-10
4-ll to 4-16
4-17 to 4-22
4-23 to 4-25
4-26 ,;
4-27
4-29
4-30
Caption
Bandl.imi ted M-aty CPSK sys~ transmissicn through a nonlinear channel.
Co'tplex plane representation of the received signal at t 0, (M::8).
Probability of error vs ~O' 2-phase CPSK.
Probability c,f error vs ;JN0, 4-phase CPSK.
Probability of error vs ~O' ,8-phase CPst<.
CPSK system transmission through a nmlinear channel.
A ooherent read ver for CPSK system.
Shifted pdf of in-phase noise fran limiter, Pn' (x-A'). ,
1 • Pdf of quadrature noise frcm limiter, Pn' (x). 4 ,
2 r
Probability of error vs ~ 0, 2-phase CPSK.
Probability of error vs ~O' 4-phase CPSK.
PJ;obabi_lity of e~r vs ~O' 8-phase CPSK.
MJdel for binary CPSK transmission through a satellite type charmel. ·
V ~ of equivalent in-phase and quadrature noise . ratio of output to input <NR (2.s· dB t power badroff) •
·J Bit errpr ra for a binary CPSK system as a function of. re ved <NR CA: majority logic rereiver, B: matched filter receiver) • ..
... ix
~ I
69
74
77,78
79,80
81,82
85
87
95-100
101-106
111-113
115 '
I 116
118 s ' J i
124 ~ ,. ' f ,,
'"" ' I' t
129 ;. , . .
·,
' .. ~ •4-:
I \
.... .
. ,
.•.
,·
. •' \, ~ ,, . ' r •,
. . I
'
....
"
List of Illustratums ! (Continued) • ,v..,
FiCJ%:e Caption
5-1 a Actual cx,rrelation re0=iver, O<-l)T ~ t ~ kl', 133 ... -5-1 b Approximate correlation reroiver, O<-l)T ::. t ~ kT. / 133
5-2
5-3 to 5-10
Probability of error vs up-link CNR, 2-phase PSK, CA
~ likelihood nonlinearity for soft-limited channel.
5-ly_, .Maxinum likelihood receiwr, ()<-l)T ~ t :: kT. I 5l2 Perfoxmance of binary CPSK transrnisaioo th.rough
hard-limited chaimel.
5-13 ~rfo.rmanoe of binary CPSK transmission through 5-15 \,~t-limited channel.
C-1 //1 Integration regiC!'19 as used in the detennination L /· of the in-phase noise statistics •
C-2 Integration regions as used in the detennination of the quadrature noise. statistics.
D-1 Coovergence of the second m:rrent of in-phase noise to the final value as a function of M.
D-2 Convergence of the third m:rrent of in-phase.noise to the final value as a ftmctiai of M. ,.
D-3 Convergence of the fourth rranent of in-phase noise to the final value as a function of M.
I, D-4
D-5
D-6
D-7
Convergence of the fifth m::xrent of in-phase noise to the final value as a function of M •
Ccnverge.nce of the sixth m:rrent of in-phase noise, to the final value as a functicn of M.
Conver9=!11~ of the ~th m:xrent of in-phase noise to the final value as a ftmction of M •
Convergence of the eighth m:rrent of in-phase noise to the final value as a function of M.
x
14],
147-154
156
158
160-162
178
182
191
192
193
(
194
195
196
197
l l I
.I
Lisr OF TA.BLE.5
Table ~en ~ , . 1-1 Ccnparison of~ .'and FDMA for satellite 7-9 b"
cx:mnunications.
" 2-1 M:>dels of T'l'lr. 34
3-1 The·cx::efficients ~and!\: for the 57-
a:xrpensator (M=4).
(1 3-2 The coefficients 'le and~ for the 58
c.urp:msator (M=E?) •
4-1 Hernrl.te :EXJlynanials. 121 ,
4-2 Nurrerical calculation of m:::rrents of 128
in-phase.noise. l
:· j /I ,)
......_
I
t ' /
I l
t'
xi
~.. f' ·, . ':~~::t.
... •
.... ; .. .,. .....
•'.
' . .... ~"'
.. :~ .... -. -·· -,.,. ';'.'~
---------~·
Syrrbol
A
BMI'
C/I
CNR
c (.) n
CPSK
(CNR) d
(CNR)u
DC
dB
d8m
dBW
··-·--· ---'-o.-
LIST OF PRINCIPAL SYM30IS ).
~resentation
Single carrier signal anplit~
F.quivalent signal anplitude at the output of the
ncnlinear devire
· kth coefficient of .N ( • ) p
kth o:>efficient of G ( • ) .,, p
Anpli~ m:xlulaticn. to ~lit:uci= m:xlulation .
Anp:µtooe nodulatian to phase m::x:lulaticn ~
Anplitooe phase shift keying
kth coefficient of N ( -") q
kth coefficient of G ( ·) q
Butler matrix ,transpcnder
carrier to interlerence po.,.,er ratio
carrier to noise paver ratio
J
I
Oiaracteristic functicn of randan variable n .
Coherent phase shift keying
Do.'111-link CNR
Up-link CNR
Direct rurrent
Decibel
Decibel relative to 1 milliwatt :paver
D9cibe1 relative tGl 1 watt paver
xii
..
·'
J
...
' I ~
• !
f
'°'", .... ', .(
'·. ,, •,
Lisbi of Principal Synrols (Continued)
Synbol
VNo ~[·]
E___ (.] ~,e:
erf(·)
Representation '·
Bit energy to noise pc:Mer spectral density ratio
Expectation with respect to the probability distribution
function of randan variable R
E>:pectaticn with respect to the joint probability distri
bution functicn of R and e:
Error functirn
erfc( ·) Ccnplenentary error function
li [a, b; x] Cmfluent hypergearetric functicn
FCM
FDMA
FM
f (R}'
Gssl
G (.) p .
G {.) q
'\ (.)
I
I'
Im(x)
IMP
'
F.tequency divisim multiple access
Frequency nodulaticn
Pierewise-linear envelcpe LiJni ting nonlinearity
Small signal gain of TWI' l
In-phase _e1;1velcpe flOI;}linearity of the rorrpensator,
Quadrature envelcpe noolinearity of the o:::npensator
Hermite polynani.al of degree k
Countable index set
Countable index set excluding zero
Imagina}:y part of the catplex mmber x
Inte.nrodulation products
xiii
' j
-I
,-
--------------~------- - -------------
Li.st of Principal Syrrbols (Ccntinued)
Synbol
I ( •) n
J
kl
I.PF
1n (.)
Miz
ML
N
NO 2
N[µ,a ]
N I· J q
N u
N' u
~tation
Mxli.fied Piessel functioo of the first kind of order n
Una:nstrained objective function
Factorial of integer k
k! = m! (k-m) !
ION pass filter
Log-likelihood rat'i.o nmctional.
Natural logarithm
Maga.cycles per sea:nd
Maxim.Im likelihcxxl
kth central m::rrent
Noise paver
Noise power spectral density
Nonnal density ftmction with nean µ., and varianre a2 Down-link noise~
In-fhase instantanecus volta~ ncnlineari.ty of the
cx:itpen.Sator
Quadrature instantaneoos wltage ncnlineari_ty of the
ccmpensator
Up-link noise pc:,y.er
Effective in-phase noise~ at the output of the
nonlinear derire
xiv
·t,.
5
I i I
List of Principal Synbols (Continued}
Syrrbol
n' (t)
P01
PDP
Pb(M)
p (M) e
PSK
QPSK
R
Re(x}
RF
s * 0 s·
SSW\
s' (t)
sgn(x)
~tation
Effective noise a:nponent at the output' of the nonlinear '1
device
F.quivalent in-phase noise carponent at the output of the
noolinear devia:
F.quivalent quadrature noise c:xnponent at the output of
the nrnlinear deviae
Pulse rode rcodulatim
Probability dist.rib.lticn functi.oo
Probability density ftmction
Bit error probability for M-ary CPSK
Synool error pro~ility for M-ary CPSK
Phase shift keying
Quatemacy phase shift keying
Envelope of signal plus nar:rowband noise
Real part of the ca:rplex nurrter x
Radio frequency
s1gna1·por,,er
Opt:imal,signaI power
Spread spectrum nultiple ao:ess
Effective signal ~tat the output of the nonlinear
device
Sign of real- nurnrer x
xv
~··
.. ,..--
List of Principal Syrrbols (Continued)
Syrrbol
sup
T
T (.) m
TWl'
w
lxl
zP C. >
* zP c. >
zq.<. >
* z (.) q * z (.)
e:
* ).
... >.
Pepresentation
Sup.renun
Synbol duraticn
Olebyshev polynan.ial of order m
Traveling wave tul:e
System banct.r.i.dth
M:>dulus of real or a:nplex m.mll::er x
Integer part of real rn.nnber"x
In-phase envelope ncnlinearity
I
~timal in-phase envelq;,e ncnlinearity
Quadrature envelope nonlinearity
Cptizral quadrature envelope nonlinearity
Cptizral overall envelope transfer characteristic
Effective phase shift at the output of the nmlinear
devire
Rancbn phase of signal plus narro..Jband n,oise
In-phase and ~ture a::npcnents of d0,,/11-link Gaussian
noise
t.p-link carrier to noise pa,.ier ratio
~-link carrier to noise J?O'A=r ratio
Limiter softness factor
Nomli.z(?(i limiter softness factor
Lagrange Il1l1 tiplier
xvi
• . . '
----- ---------~-------------------.-~ ---
List of Principal Syrrbols (CCntin~)
Synbol
~ (x)
~(x)
p
=
Iepresentaticn
= 3.1415926 ..... .
= NfO, 1]
= ! erfcf-x//2]
Effective receiver carrier to noise.power ratio
Transmitted phase during the nth synt,ol intenral .,
kth cumulant
carrier frequency (in radians per serond)
Defined as ·
Approximately equal to
' I \ ...-'
" ,,.,
xvii
';,, . .._ .. . ..,,_. ... ,.,. .. , A- :;~.'
-:,-. -~ •, ;,_.. ..
'::
I - ·. I •
! ·::.:~
CHAPTER 1
INTRODUCTION AND BACKGROUND
1.1) INTRODUCTION
Since the launch of the first, successful geo-
stationary satellite, SYNCOM II, in July 1963, satellite
connnunications have expanded at a phenomenal rate. Within
the last d~cade, communications traffic has been growing
at an approximate rate of 15 to 25 per cent per year.
This growth is expected to continue for the next 10 to
20 years [62].
At an orbital altituije of 34,863 km above the
equator, i.e. about six times the earth's radius, econ-
omical means can be provided to keep a satellite stationary
with respe~t to the rotating earth. This is known as
syn~hronous orbit. The angle subtended by the earth from
the satellite is about 18° and the resulting earth coverage
for conununicationrurposes is approximately four-tenths of
the earth's surface. Hence only three SUGh satellites are
sufficient in principle to provide connnunication between
any two points on the earth's surface [26].
Early satellites were sharply power-limited but
had more than sufficient bandwidth for the small number of
1
.. ..
.. , .....
I:>.· ( .,,
,• ' . ' ~-:~
· .. · . ...
"' .
. 2 available users (ground station accesses)~ In order to
trade off bandwidth -for signalling power, wideband fre-
quency modulation (FM) techniques were used in conjunction
with frequency division multiplexing (FDM) in order to
accommodate a large number of simultaneous messages from
each user. An additional factor influencing this choice
was that the FM/FDM technology was highly developed and
widely used in terrestrial microwave systems.
In today's systems it is desirable that a large num-
ber of earth terminals (each with a different volume of
message traffic) simultaneously access or use a given satel-
lite cpannel. This is known as multiple access conununications
and there are several methods of achieving it [65]. Of main
concern to us in the subsequent analysis are the following
multiple access methods:
(1) frequency division multiple access (FDMA)
(2} time division multiple access (TOMA) .
(3) spread spectrum m~ltiple access_ (SSMA)
Conunercially, the most widely used methods are fre
quency division mulliple access (FDMA) and time-division
multiple access (TOMA). In frequency-division multiple access,
the~repeater bandwidth ~s divided into a number of nonover
lapping frequency bands which constitute ac,cess channels. The
transmitting stations are tnen assigned to each of these acc~ss
channels. Either frequency-division multiplexing (FDM) or
time divisi~n multiplexing may be employed in order to .
assemble the individual baseband
,\ 3
channels prior to the transmission. This multiplexed
. baseband is then modulated onto the main radio frequency
(RF) carrier either by analog means, such as, wideband
frequency modulation (FM) or phase modulation (PM), or by
digital methods, such as, phase shift keying (PSK). When
each user is transmitting an analog modulated FM/FDM signal,
the only viable multiple access method is FDMA with pre-
assigned channels. In such systems, several simultaneously
transmitted signals with different carrier frequencies are
passed through the traveling wave tube (TWT) amplifier
transponder on board the satellite (see section 1.2). As
a result of the TWT nonlinearity, these multicarrier sigr,als
interact with each other to cause intermodulation products
(intermodµlation noise) to appear at the output of the (
satellite repeater. This intermodulation effect' in<;..rease?
with the number of simultaneous users. It may result in
out-of-band noise which is subtracted from the output power ,
available to the desired signal or it may be within the
frequency band of a d_esired signal causing severe signal
distortion. This undesirable effect may be mitigated to
some extent by properly spacing the frequency bands of the
varicus transmitted signals so that most of the intermodul-
ation noise falls outside the bands of the desired signals.
However, this causes a significant portion of the satellite
bandwidth to be unused and unusable and hence a significant
.. •, . . ...:
/•
.. ~~-:-·:-:
·rtt <·~< -".,.- -·~·
• ~- ' t ~ .. ~ ....
... ~ l" ~
j ;-;
decrease in available transmission capacity and efficiency
of utilization of the satellite (74].
Another major problem of the FM/FDM/FDMA system
is the need for complicated and stringent up-link power
control among the transmitting stations. In gene~al, the
ratio of powers of any two component signals at the out-
put of the TWT will differ fr.om the power ratio of the
same two components at the input, the change favoring the
stronger signal at the expe~se of the weaker one (weak
signal suppression effect). This .situati8!':l may require
extra margin in the power requirement for each weak-signal
transmitting earth station and, in the absence of power
control, reduce the total number of stations able to use
the repeater simultaneously.
An alternative multiple access communications
method is time-division multiple access (TOMA) in which
earth stations or users communicate with each other by
using ~hort non-overlapping bursts of signal. In the case
of PCM/PSK/TDMA used for voice transmission each user
4
first converts the analog signals (e.g. voice) into digital
signals via pulse code modulation (~CM) encoding. These
digital signals are converted into burst signals by using
compression buffers. These burst signals are then used
to digitally modulate the assigned carrier by means of
phase shift keying (PSK) and these are transmitted over
I
....... ,· .~ .. •.,..\ ~· ~. ·/
.· ..
_____________ \ __
the satellite link[60]. Because only one signal is pa sing \
through the satellite at any instant, intermodulatio'n \
noise problems and accurate power control requirements at:e
eliminated. Thus, a substantial improvement in useful
channel capacity may be obtained. Fig. 1-1 (reproduced
from [69]) illustrates the transmission capacity of an
Intelsat IV transponder as a function of the number of
active earth stations ( those simultaneously accessing
the satellite transponder) in a network using the different
multiple access systems. It is readily evident from the
plots in Fig. 1-1 that when only a small number of earth
stations is accessing the channel in either its assigned
frequency or time partition, then the available transponder
capacity is high and efficient use is made of the satellite
transponder in both the TOMA and FDMA modes. However, as
the number of simultaneous accesses increases, the trans-
ponder capacity drops sharply for the FDMA mode while that
of 'l'DMA decreases much m::>re slowly. Table 1-1 compares the
general capabilities and requirements of TOMA and FDMA
satellite systems (extracted and modified from [58]).
Either FDMA or TOMA is well suited for a high
capacity system with few traffic fluctuations with TOMA
5
having an advantage as the number of accesses increases. •
However, if there exists a larg~ number of users ( in the
order of several hundreds), each with small 'but changing
conununications traffic requirements, other fonns of multiple
~ ;
kt r
I V) _, .....
l z z
I <( . I :x: v
>-!:::: v <( a.. <(
u 0.:: U,J
0 z 0 0. V)
\ z ·: ' . l <{
~:....· ;'": :..: 0.::
'~ •' ,.• t-
6
..,.
900 --e,~-
PC MI P S K I T [>MA PERFORMANCE
800 ---- ----- ----~---- -----
700
.. ,·
600
500
~ FM/fDMA PEl<f'ORMAN([
'4 00 -- -
300 --~~~~...._~~~~-'-~~~~~~~~~-'-~~~~-'-~~~~...J ·o 4 8 17 16
STATIONS PER TRANSPONDER
(SINGr.E CARRIER PER QIA:'HlEL)
Fig. 1-1. C.apacity c:x:,nparisai.
20 24
v
..
I i ,·.-. ~~ !
, . -_., I
.,
7
Table 1-1' . _.-~
· cx:MPARISCN OF TIMA AND Fr1-!A FOR SATELLITE ~IG\TICNS
\
Total system
capacity
Effective ~ak.
transmit ~r
requirenEnt at
earth statioo
l'-Essage
rrodulation
Up-link p:,~r
rontrol
Frequency
stability
Interferenre to
adja0=nt RF
channels
. - ·----- ·------ 'I.
()
Drops ~lo,.;ly with nurri.:er Undergoes a rapid drcp
of acresses
~ach statioo may satu-
'·t.;r satellite transp:n-
Digital
I
Not critical
Short-tenn frequency
stability may te
critical to carrier
recx,very circuit
A band-limiting filter
follCMing the satellite
'lWI' may be required to
redure the effect of ·
energy,spread.ing
fran one to four a~sses
Po.,;er requirerrent 1s pro-
portiooal to traffic
density
Malog or digital
'May be critical if high
capacity is required
u:,ng-term frequency
control is critical for
small channel bandwidths
Out of band intermxhlla
tion products do hot ge
nerally limit system
design
.. ( oont.inued)
~
8
r I
Table 1-1 (continued}
'lU1A. fU,1A
.M-:xl2rrs Irentical wideband J Operating bandwidth of
burst m::xJems for all rrodem varies with -
stations statioo' s traffic
requirerrent ill
Timing and Burst syndlronizatim Not .required
a::ntrol with m::xlem ccntrol
interface is required
at earn station for
t:.ine multiplexing at
the satellite repeater
Data buffers !€qui.red Not required
* ·Frequency Sirrple because of the Conplex frequency
planning fer sarre, single transmit multiplexing equiµrent is
earth station and receive carrier required
freqooncy for all ,
earth stations .,,
{ continued) 'i ·, ' l ~ 1
.., i * Cnly awlies to~ case.
l
• I ) •
I l 'I
t
- ---- -· ,~- ------ -- ---- -- ..
Po.-~r bandwidth
efficiency
• -...i..---· ---~ •-•• ___ ..........,...._ ~-· V M ..__ ' ~.. - -- , ....
Table 1-1 (rontim.ed)
TDM\ FD.MA.
* Bandwidth utilizaticn Bandwidth is inten-
efficiency can te tiooally tr~ off for
increased by using p::1,-;er.
higher order m:xlula-
tion. In pcwer-limited
case forward-acting
/ \ * Cost .<?~~sent Expensive tine rrulti- Can use the existing
technology (up plexing equiprrent is I terrestrial mic:ro.-,ave
to 1975}.
\_
required. equiµrent.to reduce the
/ overall o::>st.
9
".
* Only applies to FM/FI!.f/FDMA case.
I
' j
• I l
l
l I
·;. j' '' ~~· . .. .
• 0
... ..
· ... ,, ''°:;
10
access, e.g. spread spectrum multiple access {SSMA), may
be more approrriate.
In SSMA, the carriers from each earth st~tion are
frequency or phase modulated in such a way that their
transmitted spectra occupy the whole of the available trans-
mission bandwidth. The message modulation bandwidth in
such a system (SSMA) is normally small relative to the system
bandwidth. The message modulation can take the form of an
analog frequency or phase modulated signal or it can take
a digital form such as phase or frequency shift keying.
The usual method of generating a spread-spectrum signal,
known as the direct-sequence method, is to assign to each
* user a distinct pseudonoise (PN) sequence . This is known
as pseudonoise carrier. The clock rate of this PN sequence
{equal ~o half the transmitted signal bandwidth) is
norrnally several orders of magnitude (=1000) larger than
the message bandwidth. Each active user then modulates
' his message onto this pseudonoise {PN) sequence and trans-
mits the resulting PN·carrier through the satellite
repeater to the receiving ti!rminals. Both the spectrum spread-
ing and the function of addressing the transmitting station
are achieved by this PN carrier. Upon receiving this spread-
,,
spectrum signal, each earth station employs a phase coherent
* A sequence of 2'11-1 binary numbers (zeros and ones) that is gen~ated fran a lirrear feedba~ shift register of length n .
' /
-- -- - - ·-------
correlator capable of locking onto any one of the trans-
mitted signals while rejecting the others. Once the
receiving station is locked onto one of the PN carriers,
the message can then be recovered by correlation detection.
The PN sequence used to spread the spectrum of
the"i'nultiple-access carrier also serves to address the
transmitted carrier to the desired receiver automatically.
In general, owing to the length of the PN sequence normally
used, an extremely large number of distinctly different
11
addres~es are available, many more than.the number of active
links a satellite repeater can support. However, this
multiple access method is not widely used commercially at
present because it-requires fast symbol synchronization to
the transmitted PN sequence, and because it usually can
handle only slow to medium speed messages. Furthermore,
it requires more complex terminal equipment, more accurate ...,.
power control at the transmitter ( in order to avoid the .. incidence of weak signal suppression), and most important
of all it yields very poor transponder bandwidth utilization.
1.2) THE TRAVELING WAVE TUBE (TWT)
Most satellite transponders to date and in the near
future will employ traveling wave tube (Tl-rt') amplifiers in
the filgh power amplifier stage of the transmitting section
of the satellite repeater. This choice has been dictated
: .... ~· '
.... , .....
_, . .
--·--·~- -- ...... ------~-- -- -~ - - -· ---- ,. __ ----...... -- _,,....._. __ ..,..~-- -~-· -- - ,.._. ---
by the needs for broad bandwidth, high gain, relatively
light weight, long life, high reliability and high direct-
current (DC) to radio frequency (RF) power conversion
efficiency all of which are supplied by the TWT.
In order to facilitate an understanding of its
behaviour, a brief and elementary physical description of
the traveling wave tube is now given. The discussion in
this section essentially follows that in [57], and can be
found in most texts on electromagnetic devices [64].
The basic elements of a traveling wave tube are an
electron beam and, surrounding the benm\ an RF slow wave
12
structure (64] which supports the propagation of a traveling
electromagnetic wave. The RF structure, an electron gun
(cathode, anode), electron focusing structure and a col~
lector comprise the components of the TWT as shown in Fig.
1-2.
In the TWT an electron beam generated by the
.electron gun traverses the cylindrical axis of an evacu-d
ated glass tube about 30 cm. long and about 25 mm. wide.
The beam, which travels through the centre of the tube, is
enclosed by a spiral wire coil (the helix). The coil and
the electron collector are both given a positive bias. In
order to keep the beam focused, the tube is surrounded by
an axial magnetic field. This focusing field, which is•
produced by an external magnetic coil, is also.known
- . ".
~~" :..:· ' ... .
I 1
--~---- -------- ------~-~-.... ,._ ... ....,.. 11,-"•- ,- .... ~-~ ---, -,,.,. ....
13
ELECTK:N BEN1 I
l>NO[E BEAM FOCUSING! STRU:'IUI£
CA'IliOCE ~ ~l~#$/J1,£W#&&ldWl@AJ
O)LILCfOR
~ o/lll/~ff//////////111$/I, ml rf :C-lPur rf O!Jl'PUT
,.__ n,rn;AACI'ICN REr;IO~ ....
----'l,1r--i-----------_J
Fig. 1-2 • Traveling wave tme .
.(
. ..
• i '
i t
:. I I
..
as the beam directing field.
The high frequency signal(RF input) that is to be
amplified is fed in at the beginning of the helix. It I
becomes increasingly amplified in its passage ~~ong the
helix and is ·then uncoupled at the end (RF output). The
continuous and progressive interaction of
the fields of the electron beam and the signal leads to a
bunching of the beam electrons. These bunched electrons
form electric fields and consequently induce electric
fields on the helix. These induced fields are 90 out of
phase with the initially present field~ The result-
ant field present on the helix appears greater in com-
parison to the original field, and thus exhibits an
amplification effect. Essentially there is a transfer of
energy from the electrons to the signal field. Under
correct operating conditions, the total energy of the
electrons decreases along the beam and the total energy
of the high frequency signal increases along the helix.
At low RF drive levels a faithful reproduction of
the input signal is found at the cutput of the TWT, except
14
that there has been a considerable increase in power due to
beam-wave interaction as explained earlier. The TWT in
this case is a linear device, where the output signal
grows in direct proportion to the applied input signal.
Above a certain power level, however, an increase in RF
-.. - l ~,.·" '
•• .. 1 ~ .
.. ~. ~. l;.f
,:{-...
"'
::;::r---···.
15
inputp::wer will no longer result in a corresponding
increase in output power. The TWT's amplification process
is then said to be in saturation. This relationship between
input and output power of the TWT is generally known 2s
the amplitude modulation to amplitude modulation (AM/AM)
conversion effect.
Apart from this ,AM/AM effect, the input drive power
level also effects the velocity distribution of the electron
bunches and therefore results in an amplitude-dependent
phase change at the output of TWT. This phase shift as a
result of input drive level is known as amplitude modula-
tion to phase modulation (AM/PM) conversion effect. At
small input signal level this relative phase shift is small
but it increases as the saturation level is approached.
This is to be expected, since a large velocity change of
the electron bunches occurs wie1 the input is high which
in turn results in a large phase shift at the output of the
TWT. The plots of phase shift and output power versus
input power level for a typical TWT (Hughes 261-H tube) I
is as shown in Fig. 1-3.
In general, both AM/PM and AM/AM conversion
effects cause nonlinear signal distortion at the output
of the TWT. When a single carrier is present at the ~
input, unwanted harmonic signals, whlch are frequency multiples
of the fundam~ntal signal at the input, are generated at
'
..
~ '·
________ , ---·- ------
-;;-re 0::
! ~ § 0
10
PO-lER 'l'Rl\NSFER aJJM; 5.
0.
OUIPUI' PHASE SHIFT
-5.
-40. -35. -30. -25. -20.
INPl1l' POlER { dBm)
Fig. 1-3. 'l1le single-carrier dlaracteristic of an
Intelsat r.v 'IWI', Hu:jles 261-H.
16
35.
30.
........ (I)
25. [ -8
20. ~
t H
15. 55 ~
~ 10. 0:
~
5. § o.
t
. .
...... -.......
':: ... ... ; .. -....... ' -........ ' . /1 ~--·:-;" ,:
i7
the output. In the case of multicarrier input signals,
the output signal usually contains the intermodulation
products that are displaced in frequency on the high and
low side of the original carriers. Both these harmonic
signals and intennodulation products can severel~distort
the output signal especially when th~ TWT is operating
near its saturation point.
1. 3) SCOPE OF THE THESIS
The analysis of the effect of TWT nonlinearity on
the performance of coherent phase shift keying (CPSK) is
examined in some detail in the thesis. Other impairments
caused by multipath propagation, amplitude fading, carrier
recovery, bit timing recovery and frequency translation in
the satellite will not be considered, however, as each of
these defects does create a number of new problems. In
most cases, provided the nonlinear distortion effects are
properly accounted for these other problems can be handled
by well known means [9, 22, 36, 53, 55-56].
The satellite repeater is assumed to be nonregener-. ative and its main function is to amplify the signal, trans-
rnitted from an earth station, prior to retransmitting it to
another receiving earth station. Therefore in this thesis,
the satellite conununications channel will be modelled simply by:
I l
-~ ~-~ -- ...... ___________ ---- .. - - -- - .. ~
18
(1) additive Gaussian noise sources on the
up-link and the down-link paths.
(2) a wideband TWT exhibiting a nonlinear and
frequency independent gain and phase
shif~ characteristics.
This proposed communications model approximates the PSK/
TOMA case with wideband transmitting and receiving filters.
Furthermore it also represents the PSK/SSMA case where
the channels, other than the one being1monitored, can be
regarded as independent equal power white noise processes
[ 4].
A review of the analysis of coherent phase shift
keying (CPSK) systems through the linear additive noise
channel is' given in Chapter 2. 'Ithis analysis, though
not directly applicable to the system of interest here
yields a bound on the attainable performance that can be
used to compare with that of the nonlinear channel. In
the latter part of Chapter 2, different existing models
of wideband TWT's are described and compared concurrently
with the development of a new and simple quadrature model.
This novel model is most useful in that it is well behaved
for all input drive levels and requires the choice of only
four parameters to obtain a good fit to the actual TWT
characteristics.
In Chapter 3 the optimum bandpass nonlinearity that
maximizes its output signal to noise ratio for a certain
I . f
.· .. ·-
I
' ' '
.. ,-
... '
class of input signals is derived, ·and different methods
of compensating the TWT to approximate this optimum band-
pass nonlinearity are investigated.
The performance of C~SK systems through nonlinear
charnels is examined in Chapter 4 for ~oth the bandwidth:
limtted and the power-limited channel. For the latter case
a single sample detection and majority logic decision
receiver is assumed, and the bit error rate is determined
as a function of up-link and down-link bit energy to noise
spectral density ratio.
0 In Chapter 5 the performance of a binary CPSK r
signal transmitted through a p~rely amplitude - limiting
channel is considered for two other types of receiver.
These are:
(1) the linear integrate and dump correlation
receiver
(2) the nonlinear maximum - likelihood receiver.
19
•
Finally, Chapter 6 presents· conclusions and suggest-
ions for further researdl.
..
1
.l
I J I i
J
l 1 J f
I r
.I t
-J I
• t
l
' f
1 I
•
CJ-IAl>TER 2 ... ---P>RELIMINARY DISCUSSION AND CHANNEL MODELLING
/' In this chapter we fixst, briefly review the per-
formance obtainable when M-ar~ coherent phase shift key-
ing (CPSK)signals are transmit~ed through a linear channel.
This performance will be used, in the subsequent chapters,
as the basis of comparison for the performance of a similar
CPSK system transmitting signals through a nonlinear channel.
In the latter part of this chapter a novel quadrature model
of the TWT is developed and compared to other existing TWT
models.
2.1. PERFORMANCE OF COHERENT PHASE SHIFT KEYING (CPSK)
At high transmission rates, coherent phase shift
keying (CPSK) techniques are the most widely used of all
digital modulation methods. This occurs because they are
efftcient from the point of view of ,
(1)
(2)
conservation of bandwidth ...____,,,.,
the possibflities of using very simple
techniques for transmission and reception.
The performance of a CPSK system in an additive
thermal noise channel is well known and can be found in
most texts on digital communication [20, 53, 76, 79]. I
20
-,-,
•. ·~ . ~· ·,
21
Nevertheless, we shall include the review here for the
sak~ completeness.
In the case of an additive noise channel, the truns-
mitted signal in the time interval O ~ t ~ T, where T is
the symbol interval,can be written as
where
+ 6.) l.
0 < t < T
s. (t) = l.
\ O elsewhere
i == 1, 2, 3, .... , ,,!. ( 2 • 1)
1) E is the transmitted signal energy per symbol
2) g(t) is a unit energy pulse shape, which in the
3)
4)
additive noise case may be conveniently taken
to be rectangular
~ is the angular carrier frequency which is 0
assumed to be a fixed integer multiple of
Where T-l . th b 1 t is e sym o ra e or 2 rr w >>-
0 'l'
e. is the transmitted phase and assumes one of l.
h 1 21T i · 1 2 . . h b 1 t e va ues ~· 1 = , , •••• ,Min eac sym o
interval.
Assuming that the ith symbol has been transmitted, the
received signal may be written as
x(t) = s.(t} +n(t) l.
0 < t < T. ( 2. 2)
where n(t} is narrowband white Gaussian noise which may be
written as
( 2. 3}
J r
1 (
t I
t •
~ . .. \
... . _ __/' .
where n1
(t) and n2
(t) are both zero mean, statistically
independent white Gaussian noises, with single sided
power spectral density N watts/Hz. 0
Within one symbol duration the received signal
may be written as
- ~(t) sin wt 0
(2 .4)
Assuming that e~ is equally likely to have any one
22
of the M possible values, demodulation is readily accomplished
by forming the quantities
~ fl T x f x{t) cos (l) t dt
0 ( 2. 5)
0
ff T and y = I xct> sin wt dt
0 0
( 2. 6)
which.are then sampled and passed, at time T, to a decision
device which makes a decision on the value of the transmitted
e .• A block diagram of this receiver is shown in Figure 2-1. ).
Conditioned on the knowledge of 6., the variables l.
X and Y are statistically independent Gaussian random
vari:ies with means~ cos,, e . .;_n~ sin e. and common vari-i ). ' .
a nee wat·ts. The conditional joint probability density
function (pdf} of X and Y given ai may be written as
= 1 . ,rN
0
/
/
CY -.JE sin o1 /]
No
(2. 7)
.;,J,( :.; ~· '
/'.,.'~-: --·- _____ _,,_,___ I •• , 4 lh A •.flillfil1's M.¥ .......... ==i
!.J,
J ros(wot)
. Si (t) + n(t)
IOCAL OSCIUA'IOR
~
~ ei
a 90. ~
sin (wot)
J
SAMPLE AT t c kT
Fig. 2-1. Block diagram of a correlatioo receiver for CPSK systens.
i ~yi; ... ,-.l'-,a,;.1r4:~a,1.11r:·,.,.,,,,~u re-··.;.~Ji ;rr ... ,'i1';,:~·~--.,.... • • ..,..~.,. .,,...,~ .. ~ .. .-~ ...... --~ '•~ ~ ,,._. ~ ,, ' ~
N w
I
• !
I·
.I
·, ,.·1 .. ~ -~· ""~:;.:..~ .. '' ~ ... .:. ,,:-: . j
. .
'·,,.
Now define the new random variables Rand e: as
X = R N cos e: 0
Y =RN sine: 0
(2.8)
In terms of Rand e: equation (2.7) then becomes
p ( R, e: I e . ) = B. exp { -[R 2 - 2 Ro cos ( e: + e . l + / j } ( 2 • 9 )
l. 11 l.
E where p = N is the symbol energy to noise.power spectral 0
density ratio.
24
Since the values of the transmitted phase e. are equally l.
probable, we can take e. to be zero without loss of generality l.
' and the symbbl probability of error is given by the probab-
il.1. ty tf\at I e: I>~ which can be written as
P (M) = le f 0
"' p(R,e:je.=O)dR de:
l. (2.10)
Substitution of equation (2.9) into (2.10) yields after some
manipulation
2 Pe(M) = 1- n
0 0 (2.11)
It was shown by Lindsey [54] that equation {2.11) can be
reduced to a readily co~putable expression as
p {M) = M-l - !. erf (p . 1T1 e M 2 Sl.IlM-
l p sin(tr,M) r 2 11 r;-0
exp [ - y ] erf [y cot M) dy (2.12)
. . - / ----,.. --.--
-~ ·1 ' -,., ~~,~.: .... ~--·
I ·1
,
----------------- -------· -- .
-t2 e dt
x f is the well known where erf (x) = 0
error funct:ion.
In the case of binary phase shift keying (M=2),
equation (2.12) reduces to the well known result
(2 .13)
where erfc (x) = 1 - erf (x)
It is also possible to evaluate equation (2.12)
in closed form for 4-phase CPSK (M=4). The result ·is
given by [53].
p
p (4) = erfc [ - 0- ] - .!. erfc 2 [ 12 ]
e ~ 4 (2.14)-..
In the region where p>>l, equation {2.14) can be
approximated by
= erfc [ ..£_ J 12
(2.15)
In the M~ary case, if we encode the equiprobable,
source symbols using a Gray code [56], which has the
property that only.one binary bit is changed in going
from one symbol to an adjacent symbol, then the average
bit error probability Pb(M) is related to the average
~ymbol error probability, Pe (M) by [ 77] · ...
p (M) e (2.16)
25
•
'y .. .,.
•,
where each symbol conta~ns log2M bits of in~~r:mation.
Hence for M = 4 and from equations (2.15· - 2.16)
1 2 erfc [ _P_]
12 (2.17)
But the bit energy for 4-phase PSK is half that of binary
PSK. Hence for equal bit energy to noise power spectral
density ratios, 4-phase CPSK and binary CPSK have
identical bit error performance. This result is signifi-
cant since 4-phase CPSK requires only half the bandwidth
of.binary CPSK for the same transmitted power. Fig. 2-2
illustrates the symbol error performance for M-ary CPSK as
a function of the bit energy to noise ratio defined as
p
=
2.1 MODELLING OF SATELLITE NONLINEARITY
In order to analyse the nonlinear effects of a TWT
on the.performance of conununication systems or to com-
pensate for this nonlinear distortion, it is first
necessary to develop a relatively simple, analytical model
of the tube.
In recent years considerable.effort has been made
to develop analytic expressions that app~oximate the TWT
nonlinearity. In an attempt to investigate the effect of
ipt~rmodulation no?-se caused by TWT nonlinearity or the I
26
i~tersymbol interference ef~ect (caused by adjacent symbols),
.~ ..... ~.·,·. -.:-r. -. -. ':"',;-.,':"'. r";-''f':(------~:""':"'-"."""s' ........ ------ - - ...
,· ,. ·,
. ',
1
p (M) e
10-2
-5
27
M=2
·,
·.
O 5 10 15 20 dB, [Fb/No) .: "
Fig. ·2-2. Word error probability vs ~~O' M-ary .CY~.
-. . ,
----------
28
different models of the tube have been developed [11-13,
25, 31 33-34, 51]. Basically these models are either the .
cascade combination of phase and amplitude nonlinearities
as shown in Figure 2-3 or some form of quadrature envelope
nonlinearity model as depicted in Figure 2-4. However,
the models developed earlier [11-13, 25, 31, 33-34, 51]
only approximate the tube characteristics over certain,
restricted regions.of tube operation, as dictated by the
amount of the peak-power-limited interferences at the input
of the tube, and the characteristics of the tube beyond
saturation are not critical. However, in the case when
up-link noise is present at the input of the tube, these \_
' models can lead to an anomalous be~aviour of_ the equivalent
noise statistics at the output. This occurs because the
noise is not peak-limited or bounded. The cascade model of
the tube in the presence of noise at the input has been
considered in reference [80].
In this section we introduce another approximation
for the env~lope nonlinearities in the quadrature model of
the TWT as originated in [31, 51] 4nd depicted in Figure 2-4.
The two envelope nonlinearities Z {R) and~ {R} in the .• p q
in-phase and quadrature paths are assumed here to take the
form 2 2 Z {R) = c
1Re-C2R I
0[C 2R.]
p (2.18)
2 2 Z {R) = s
1Re-S2R I
1[s
2R]
q (2.19)
.\
• \ . ., i 1
... . • 1 .. ,
., '
}. ' 1
,y. '4·~~,,, ~~--- I -{,,.1:·.~,. ,. , .t · , ..:~ , , ,., unJILl,ft.;,#,#Slez~• ... •wa: , sa1Z!Sd! ... ec .... .«:~ ff.-... .... « ..... -...~ ............ '"!lf,
NARR:mWID NOISE
,·-·-·;;-~~~VE ~-·-·-1 I
CX1IPUT SIQZAL
AWPM AM/AM
I I f ( R) cos ( w0 t + e + £ + g ( R) )
L ______ _ ·-·-·-.J ~
TRANSMIT.CEO SIQq\I, INPl1l' SICW\I, Rcos(w0
t + e + t; + g(R))
Acos (C110 t + e> Rcx>s_(coot + 6 + &)
Fig. 2-3. Cascade 11XXEl of '!WT
~:r~~--:-.....--•,r• •5"•• r.• ltiiwGA~-· ... ,w•+a«J ,, :? K:«Mt:txfj'( '• id,~;-~·~··, .......... -~·'-+.1~f&il-~r___.......... .. ~,....--.-·-"'~ ·~ -·- --- > .. ,,. .......... ......
,',(:·,:. ~··:".::, :\·, ' /: :, ,• , ' : ,, :~. ,' :·,: · .. :.; ,.-· --·"I ... ,,_ ' -
rv '°
J.,., ," -~: '•' • 1:· , '• .ti ·,I•
, ... 1,/ 1 (1j': "' ' WJ.::UC ... £2$ Wt a ""'-" Jllil WJIII reA o 4 - • t ___....... _ "· ---
'•I .,
NARR:MBNID ~ISE
INPUl' SICNM..
('
AcnsC1110t+e> Roos(wot+e +£)
90·
z ( .) p
~-..
Fig. 2-4. General ~ature m::>001 of 'lWl'
~ .~~
- ~
,:; F. ..·~ ..,,.,.,. .. ~ .. r:C,r :aJll$1$f¢ ~~;;;;.--wt·(:"1&:"'-.a-Jtllfl'"Miilwf-Mitrii'~~~~Mr¢11111••~ 91,i;.-+--,
.-;; _,,.,.~.;",<;"' -· . .,.,~ •.
z (.) q
( ',
Zp(R)c:os(w0
t+e +£)
OU'IPt.rr SIQW.
f(R.)oos((IJ0
t+e +c +g{R) l
Zq(R)cos(w0
t+e +c)
' ' ...... -'*::. ....,_,.r ... ~- .......
(.;J 0
. I
. ..
•
where In(.) is the modified Bessel function of the first
kinct of order n and R is the envelope of the TWT input
signal .
The coefficients (C1,~,s1 ,~2) are computed from a
conventional optimization subroutine as described in
Appendix A, so as to yield a least squares curve·fit to
the actual tube nonlinearities and are found to be
c1 = 1. 6.J.245 c2 = .053557
s1
= 1. 71850 s 2 = .242218
31
for a Hughes 261-H tube as used in the Intelsat IV satellite.
Unlike the other quadrature models [25, 31, 33 - 34,
51], this Bessel function approximation only needs a few
coefficients (four as compare9 with sixteen in [3l])to give '
a good fit to the TWT nonlinearities up to and bey-0nd
saturation. In addition the model is well behaved for all
Rand permits straightforward evaluation of the TWT output
noise statistics. For very large R, equations. (2.18) and
, (2.19)
and
can
lim
R -+
lirn Z (R) q
R -+ co
to give [2].
= (2.20)
Each Z (R) and Z (R), as shown in equation (2.20), p q
\
I .t \
t 1 J
,.,
. ! l
' '• ,,
32
is asymptotically constant and therefore well behaved for
large R. This distinct characte7istic o' our model circum
vents any anomaly that may arise in eval~ating the statis-
tics of the interference terms at the output of TWT, subject
to the up-link noise at the input.
For small R, Z (R} and Z (R) can be expressed as a p g
power series in R. Retaining only the first term in such
power series expansion, we get (2).
Z (R) q
S R2
1 -2- (2.21) ' f
I
The nonlinearities in equations (2.18)• and (2.19)
are plotted and compared with the measured characteristics
of the tube (Hughes 261-H) in Fig. 2-5. It is obvious from
this figure that the model proposed provides an excellent
approximation to the actual, tube characteristics in the
region of interest for tube operation.
This quadrature model of a 'IWI' as described by equations
{2.18 - 2.19) can also be extended to include othe~ types
of nonlinear amplifiers exhibiting gain compression and
AM/PM conversion effects. Examples of such amplifiers
include Klystron power amplifiers and crossed field amplifiers.
Table 2-1 summarizes the existing models of the
traveling wave tube (TWT) I that have been used so far in the
~ .,-~'
L l
! I I
( I
33
4
l ./
0 Zp(R) ,.._ en 3 ~ "'l
0 > .........
>-f-0:: <t ..... w z ::i
i z 0 ... z 2 l
j
w
I .-. I 0:: ·, ',.. ::::>
-~ 0::: Zq{R) 0 <( 1 ::::> 0 0 0 .. 0 .,
:j 0 ,;\ z
1 <(
w U) <(
o MEASURED VALUES I 0. I z - BESSEL FUNCTION
APPROXIMATION :
0 0 2 3 4 5 6 7
INPUT PEAK VOLTAGE R (MILLIVOLTS)
Fig. 2-5. '!he noolinear dlaractoristics of 'non' (Hughes 261-H)
c!!ld the Bessel fmcticn approrimaticn.
; '\?,·,. - ,'
•,.: .:
f~;_u~~l~?,;~-~ :n
-,
··.
:.; ,··
M:ldel type (originators)
Cascade
(Beman & M:ilil) l _.. .;'• }:
.< Cascade ,••,
'· (Thanas et al)
Cascade
(Schwartz et al)
Quadrature
(Kaye et al)
<:uadrature
(Eric)
Quadrature
(Hetrakul &
Tavlor)
j •,., ,-:· ' i ~,,. ~ .. ~.
Table 2-1: lvODEL.5 OF TWr
z (R) or f(R) p
z (R) or g(R) q
Fourier sine series E,q:x:,nential plus.quadratic
functicn
-Exponential and linear Er:pcnential plus quadratic
functioo functicn
Hard- or soft-limiting Zero phase shift
envelope nonlinearity .
Series of Bessel functions Series of Bessel functions
of the first kind of order of the first kind of order
cne one
Odil. polynanial of R Odd polynomial of R
1-bdified Bessel functicn M:xlified Bessel function
of the first kind of order of the first kind of order
zero one
---- _..,,,._, . '
Peferences
(11-13)
(80]
(4, 48-50,
59 / 70]
_[34 t 51]
! [25 I 31, 33)
I I ( 42]
I
I
w .:::.
j
I I
' · 1 · ... ~ '
I
I
-----¥ ·------ --- ... ___ .___ ·------ --- -- -·
analysis of satellite communications systems, and provides
som{ indication of the advantages 9 nd disadvantuges of the
various models.
35
The specific overall channel rrodel for the ccmnunicution system
considered in the thesis is shown in Fig. 2-6. The input
to the satellite repeater consists of a single carrier,
M-ary phase-modulated, constant-envelope signal and zero-
mean, stationary Gaussian noise. The bandpass filter
bandwidth, unless it is stated explicitly otherwise, is
assumed to be wide enough to pass the signal with negligible
distortion and to limit the up-link noise tb a bandwidth
that is small compared to the center frequency of the filter.
The traveling wave tube amplifier on board the satellite
exhibits two kinds of nonlinearity, namely the AM/AM and
AM/PM conversion effects as described earlier. The TWT is C followed by an ideal zonal bandpass filter that confinesthe output
spectrum to essentially only the fundamental band of the
signal. After passing through the iatellite repeater, the
signal is transmitted to the receiver and independent thermal
noise is added to it on the down-link. The receiver processes
the composite signal a·nd extracts the information-bearing
signal coherently using a locally generated carrier refer-
ence signal. It is assumed throughout the thesis that
carrier and bit timing recovery at the receiver have been
achieved .
. The system model depicted in Fig. 2-6 is also valid
-·-··--~---:-,---.~"':"""-~~-:""'~-------""'"!"..,...,_.,_....,....,_ _________ _ -,·,,,.,._
' • .J ... ~ ,, --~....:.·~~ .... ~~~~~ ·1 !
·1
·Ii •
1 ' • •• .J
·l 1
u
OP-~ OOISE
MTA IN' M:>lXlrmOR
M-acy a>SK SIGW,,
"' ...,,.-..........,............-- ....... ,_. . . .. "' -----
\ ...
~-LINK NO!Se
BANCPASS 3A'.IEILI'IE BANDPASS I:EM)Il.JIATOR & ..
FIIJIER ~'IER FIL'IER IECISIW CIFCUIT
~
~RelKE SICNAL
Fig. 2-6 ~ Block diagram of the ccmnt.m.ca~ syst.em.
""""--·-
o:it"z~cr~~~?:,artc .. ~·ra:,m;,..,,_::;,;.,-:;:;;;~.;:::;;4:~,1·····--'!'·• .. '4-. .. rtf • r,rlrii ...... ..:.~. ;~a, ..... ,. ;.4 -"••• o' • ... _ "·---.... M ~ijif -~ • L. ·~~- •
--- --
~ ... ~
a,r
ti
w O'I
~·....-.:
•' .-
.. ' ' ,, •,
' .)
for the analysis of M-ary CPSK signals transmitted
through a linear channel with a limiting front end in
the receiver. In some practical applications, insertion
of a limiter in the receiver front end may be desirable
from consineration of dynamic range requirements .. In .
such applications, we only need to disregard the down-' link noise component in our model and replace the satel-
l4te repeater nonlinearity.by a hard-limiter.
(
..
l
'"~,f ~ -;' '"I ~, .._ - ~ .• t ~ • ' '·--:-, °':',--":"-"'"'l""_,,_. _________ _ .•
37
1 ( ..
!'. i ' }
''
0
" .
r ., . ~(,:
' :,z,; .. ' tl, -,",',ta s ,,, :• ·: .. ,. .... ., ...
' \,: :,, / ·y,·': .. •, :
,. -,',,
•'
CHAJ?TER 3
COMPENSATION OF SATELLITE NONLINEARITY
The signal distortion resulting from the non
linearity of the trav~ling wave tube (TWT} amplifier has
presented a number of problems to designers of various
multiple access satellite communication systems and has
been a 'significant factor in determining communications
system performance. The ~volution of multiple access
-techniqu,es from. essentially an~log frequency-division
multiple access (FDMA) to essentially digital time-division
muft.b;le access (TOMA), though in no ·small measure mot~vated
by tbe problem of nonlinear distortion caused by the TWT,
has not res~lted in complete elimination of the problems .
but merely in a shift in emph~sis. For example, the basic
characteristics of nonlinear power transfer and amplitude
dependent phase modulation (AM/PM) conversion producea by
the TWT operating close to maximum power output result in
the px:oblerns associated with intermodulation and cross-
modulation distortion, gain suppression, large signal
capture (small signal suppression), and harmonic distortion
in systems employing multicarrier FDMA techniques. However, ' . these same nonlinear characteristics of the T~ also cause
.single carrier waveform distortion, co-channel interference
38
l
l 1
i ),
) ,, !I ~ ,l
i1
l t
.: I I I
• I
(
as a result of spectrwn-spreading, additional inter-
symbol interference and reduced error rate performance
in systems employing single-carrier TOMA techniques.
In this chapter d~fferent means·of compensating
the TWT, in order to alleviate the degradation caused by
TWT nonlinearity, are investigated. In section 3.1 an
expression for an optimal bandpass nonlinearity that
maximizes its output signal to interference power ratio
(C/I} for a certain class of input signals is derived.
This is followed by a qiscussion of three main compens
ation techniques which can be used in conjunction with
the TWT so that the overall transfer characteristic of
the tube approximates that of the optimal bandpass non-
linearity •
3.1 .a.
OPTIMUM BANDPASS NONLINEARitY .
39
To .simplify our notation without loss of generality
we shall. consider the case of an unmodulated sinusoidal
signal of amplitude A, plus narrow-band Gaussian noise at I
the input to the general quadrature model of the TWT
(depicted in Fig. 2-2). In terms of the envelope,R, and
phase, ·t, the input signal can be written as
x(t) = R(t) cos I«»o t + , (t> l (3 .1)
The joint probability density function (pdf} of·
the envelope, R, and the phase, c, of the input signal is
.;: i
;:_!
' .. ' ' ,,
...
... . •
then [67]
[ -A
2-_2A_Rco_s e:+l] exp -
202 (3.2)
p{.R,e:) =
for O s R < "" and O S. e: S. 2n
h 2 . th . ' . were a is e 1~put noise variance.
The pdf of the envelope R can be obtained by
integrating equation (3.2) with respect toe: and can be
shown to be (67]
p(R)
(3. 3)
where r0
(.) is the modified Bessel function of the first
kind of order zero.
Simil'arly, the integration of equation (3.2) over R
would yield the pdf of the phase, e; , as [ 6.7 J
-p IP° . 2 p(c) = ~,r + coi t I 1r e - pSJ.n c erfc {-./p cos £}
p = = equivalent input CNR and the complement-
ary error function erfc (x) is defined as:
erfc(x)
40
• <, \
,,
,,
..
41
For our purpose, we define the. optimum bandpass
nonlinearity as the nonlinear device that maximizes the
output signal power subject to a constant signal plus noise
power at the output. This criterion is equivalent to ~
maximizing the output SNR. The joint pdf of the envelope,
R, and phase, c, and the pd~ of the envelope, R, are assumed
to take the forms shown in equations (3.2) and (3.3). This . criterion di optimality, i.e. maximum output SNR, has been
shown by Jain and,Blachman [18, 49] to correspond to the
minimum probability of error for the case of a binary CPSK
signal transmitted through a nonlinear, purely a'mpli t.ude
limiting c:nannel. ·
In th~ case of a nonlinear channel with AM/PM con
version as depicted in Fig. 2-2, the signal power and the total
signal plus noise power at the output can be defined as [ls]'
(3.4)
and
where ZP and zq are ·the in-phase and quadrature env~lope
nonlinearities in equations (2.18) and (2.19), and E~ (.) ~. c
denotes the statistical average over Rand c.
* * 'l'he objective is to determine Z'P ( •} and zq ( ·) ,
,\ . ... _:-~:, -,-,~-------. -.-.. ---...... ,-1 --------------~
'.
I
' . .... , '~ ,~
,, •, j .. · ,,'
42
the optimum in-phase and quadrature envelope nonlinearities
such that Sis maximized subject to a prescribed, constant
value, C, of S + N. From equations (3.4 - 3.5), we may
then define an unconstrained objective function as
"' 2n co 2n J = {/ f zP (R) cos tp(R,e:) dRrl£}2 + { f I
0 0 0 0 ... z (R) cos q tp(R,£) dRd£ }2 + A {2C - f rz; (R) + Z2(R)]
q 0
p(R) dR}} (3. 6-)
... where>- is the well known Lagrange multiplier.
By taking the integral in equation (3.6) as the
Riem:mn sum and equating the differentials of J with respect
to z (R) and z (R}, for fixed R, to zero, we obtain the p q
following necessary conditions defining the optimum
* • envelope nonlinearities, ZP (R), and Zq (R) as
c 2u }· #,. • = f fZ * 2 ). zP (R) p(R) (R) cos E: p (R , c ) d R • d e: 0 0 p
{ r cos c p (R 't) d ~} (3.7)
and - . & r 2n dR dJ " * * 2 ). Zq · (R) p(R) = f" I zq (R) cos c p (R, E:)
0 0
{ 21T
·} I cos c p (Rt E:) d (3.8) 0 . .
1
! ..
') . ·,
. ,,,,.
\
43
where
* s = = { ER [ Z * (Rl cos c] }
2 + {ER [ Z * (R) cos e: ] } 2 ,t p ,t q
2C 4C
(3.9)
From equations (3.7} and (3.8} it is readily .. evident that
= (3.10}
for any real constant B including zero. The case when
B = 0, however, yields a trivial solution for equation
(3.7} and merely indicates that the characteristic of an
* optimum bandpass nonlinearity is that of zg (R).
In general it is then nec~ssary.for the optimwn
bandpass nonlinearity to have
(1) an amplitude characteristic satisfying
equations (3.?) and (3.8) to within a multiplicative
constant, and
(2) a constant ~utput ~hase shift, independent
of the input envelope, R, which may therefore -be set to zero.
Substituti9n of (3.9) and (3.10) into (3.8) yields
* { 211' ~a f ~OS c p (R, E:) l+S O
(3.11)
\ /•~ '
Equation (~.11) is now identical to [eqn. 31 of tis)],
except for a nonzero constant B. There fora, _by using the
same argument as in [18), the optimum quadrature amplitude.
* response ·c~aracteristic, Zq(R). can be expressed as
J ,.
J \
\ '(
1
. I )
j
l ! I
·' • ~ ,,1 ",.,
,::::~ •' . .., : :~s ~' '• '"l
' ~· . · ..
.. -"
...... ' ... ~ .
:;~~··}:~~ .:.:: "'
' . ' ,.,
'' J
', .. ~ " ' .
', .·\ . . \ : ... :.,
..
.... ·~. '' ... - '
44
(3.12)
where K =
From equations (3.10) and (3.12), the overall optimum
* envelope transfer characteristic Z (R), is then obtained as
z*(R)'= yrz;(R)] 2 + [Z:(R)l 2
= c I 1 [ :~ J /Io[~ J c = Kf l+S
2
(3.13)
where
* The function z (R) increases linearly from zero and
rises asymptotically toward the value c for large R. This
behaviour closely 'approximates that of a piecewise linear
limiter. Also the larger the up-link carrier to noise ratio,
A2 , the more closely this limiter characteristic. approaches ""2 €Rat of a hard limiter. * Fig. 3-1 shows a plot of z (R) as a
function of!.... for different values of input carrier to noise A
ratio define~ as
fCNR) 1 .,. ;:z p
The results obtained ~n this section confirm the pre
vious conjectures (44, 51, 82] for the optimum transfer
characteristics of banupass nonlinear saturation devices with
AM/PM coversion. To facilitate the compensator design
presented in the subsequent analysis, it will be assumed that
(l) amplitude dependent phase shift at the output is
·:1 •. ~ .. '* _ .. -~ -- -----------
:-} .. .. •.;
i· .. ,
1~
;J
z•cR)
:-~. ;":~~/}~:~
1.2,------------------------
1.0 ,,..------ -
( ---.--·-· . ,,,,.,.,. ·-
{ / ~--··-··:..==-==:.-
/ _,.. .. ~ ··-.8-ll / ··-·· ,.-
/ i // ___ ,,.,,,,----------···-···-···-···-···-.6-H I I ... / I ~ ~-,~
.Hf.; ,I /, I ........ ./ .2-11.·/11 .......... ·····
I: ......... ·· .. . ..,.
•' . ,, a## •• ,·· # ..
o~········· ...
0 5 JO
.. ..,..,,.·. ....
15
- R/A
.... •" r•• •
······ -.... ···· .... -... ····· .-·· ·-·······-· ,. ...
20
(CNR)1 = -15 dB
-10 dB
- 5 dB
O dB
5 dB
10 dB
c=I.O
25
Fig. 3-1. T'ne optinum envelope transfer characteristic
....,_ ....... _,.. • ,.. • •,•.,.~ ... ,~.~-_..,._.,o,..._,_,,.,...._ ·~ ,.._, /.,~."' •v _.
30 ~ (Jl
,·''
" .·,.,.
' ,,
completely eliminated.
(2) the envelope nonlinearity should npproximate
that of a piecewise linear limiter.
This choice of a design objective avoids the
problem of designing a compensator whose characteristic
depends on the input CNR, p. Although it is only an
approximation to the optimum transfer characteristics, it
has been shown to yield very good results for the case of
rnulticarrier signals [51, 82]. 1'<J
3.2 TWT LINEARIZATION TECHNIQUES
In order to compensate for the signal distortion
caused by TWT nonli'nearities, various linearization tech-
niques have been attempted in the past. There exist,
fundamentally, two different approaches.
The first approach takes advantage of the fact
46
that the satellite usually contains a number of independent '
TWT's which together form a multiport transponder of n
inputs and n outputs. By the proper design of an n x n
Butler matrix phase-shifting network precedi_ng and follow
ing these n nonlinear devices, the effect of intermodul-
ation noise generated by each TWT nonlinearity can be sub
stantially reduced [19, 68, 82].
The second approach is to alter the nonlinear
characteristics of the overall compensated TWT by means of
j 1
' ,(
''
'l
~-
I, ' .
47
input-signal predistortion or feedforward correction of
the output signal. Other methods include the equalization
of AM/PM conversion effect by controlling the helix voltage
modulation [74, 86] and the control of basic frequency and/or
spatial characteristics by proper design of the slow wave
structure of the TWT [81].
In this section we shall first briefly review the
maln compensation methods, namely, Butler matrix, feed-
forward correction and signal predistortion method. A
procedure and novel implementation of the signal pred·istort
ion circuit is then suggested in section 3.2.3.
3.2.l Butler matrix transponder (BMT)
The basic configuration of the BMT amplifier network
is shown in Fig. 3-2. It consists of two complementary
(n x n) Butler matrix networks which precede and follow a
set of nonlinear amplifiers. These networks perform a
special phase shifting o-peration on the signal at the input
and output of the amplifiers and cause a substantial frac-
tion of the intermodulation products (IMP)appearing at out
put ports to fall outside the frequency band of the trans
mitted signal. Consequently a significant portion of the
IMP can be reduced by proper filtering at·the output ports
and the amplifiers can be operated closer to saturation for
a specified output C/I ratio. More details on the mathemat
ical description of a Butler matrix transponder (BMT) can be
)
------- - ' --- - ------- .... __
..
t : ' ~
'~~. • ,I
". l.·~, .. T>..~ •;>,. , . . ....
.,£,
l
2
0 0 0
INPlJIS
m
rum
BUTLER
MI\TIUX
1------1 'IWT
0 0. 0
BUTIER
MA.TRIX
Fig. 3-2. m-Olrume l Butler Matrix 'l'ransponder
• l ', '• ~ ~. • i
l
2
0 0 0
Oll'IPU'fS
m
48
.l 1 ' • :I
1 ) ' ·~ 1,
: J { .I
i , . l
..
,. 'It,:~ . '
· .. ..
• ". \~.. _, ,I;
found in [68]. In addition to its main feature of reject-
ing the out of band IMP, the BMT also provides a means of
increasing the RF power per channel in the case when the
number of inputs to the BMT exceeds the number of existing
channels. Furthecmore, under non-peak loadin9 condition,
the BMT offers a power sharing capability which·allows
some channels to operate with a greater output power than
others without C/I degradation in any channel.
However, in the case of single-carrier operation
as in TOMA systems, i} has been pointed out in [68] that
the BMT loses its power saving qualities and more distort-
ion could occur. The other disadvantages of the BMT
include a required increase in the transmission bandwidth
as well as the obvious one of requiring multichannel
operation.
' 3.2.2 Feed-forward technique
This method of linearization makes use of a sub-
sidiary TWT to help in the compensation of the operating
~~mplifier tube. A schematic block diagram of the feed
forward linearization technique is shown in Fig. 3-3. The
input signal to the transponder i~ divided into two
portions. Part one goes through the main amplifier, TWT1 ,
and a sampling is taken through a directional coupler with . a coupling factor cl __ whore
ssl signal in-band gain of TWT1 .
)
, .. ,, ;
G981
approximates the small
This sample is compared with .,.
49
~\ t
·..• ~--\::~;~:;~~~:-:(;' '. . .
0
ll:2l1l l "-
'• •
't"l
. A+1 · 81 a
~_.........,_. .. _ - .... --~--~
~~--cz
,. -~~-~ ...... ~
k3
I ... ' 4-- . "'2 ll.+z -----,------·----~ Ern>rT~lr-~-.- ~-------------·-·-·-'·--?::>
L
Varia>le piase shifter
Error Suppx:essial Loop ..
Attenuator
! ! I !
b I
I
[7~] . Varl.able attsruatx>r
}
I • --Lr
Pig. 3-3., ~loop feedfo:tWaJ:d anplifier _rrr::,cel.
~::~~~,,;~.......-liillll\)Ui~-~~~· JJ9Ji:N, •• ,, l • t~~~_.t.....~??~~.., .. ~..._ .... ~~~----M·l--------· •·•
Lossless delay line
V1 o.
I I :,
(;
'\ j, ....
,.
the reference path signal whic~ has b~en delayed by a
time t 1 , equal to the propagation delay of TWT1 • The 1~ ' phase and amplitude of this delayed reference signal is
adjusted by_ an amount 6~1
and s1 by means of a variable
phase-shifter an~Jn attenuator. _ T~e time ... shifted error
obtained is then paesed through another attenuator, s2 ,
and a phase-shifter and amplified by a subsidiary tube,
TWT2 , whose small signal gai.n, ~ss 2 , is such that the
error is restor~d to the appropriate level to c~ncel the
error of the main tube. The linearity of the compensated
tube i~ achieved by the proper adjustment of the variable
phase:shifters· and attenuators C71 J; "An appropriate time ' ,,.
delay t 2 equal, to. the propagation dGlay of TWT2 must also
51
be provided in the main amplifier path in orde~ to achieve
synchro~ization~f the two paths." The fixed attenuators
in,the main and reference paths in the final stage represent .... th~ iosJ in th~ directti>nal couple~s and the delay line.
Th.is method allows the output signal and the sample of the
input signal to bo compared and corrected to any desired
degree of acquracy [11-73).
The main advant~ge of tho food-forw~rd tochn~quo . ·io fta simplicity. ·Tho subsidiary tube, 'l'WT2 ia usually
operated at very low level and c~n be uood as a standbf
amplifier in tho sat.ollite. Furthormore,. as distinct
·from feedback tochniquoa the foad-forw~rd TW'l' amplifier
'does not contain ~ny closed loopo and honoe is an inhorontly
' ):,-·
. '
•'• ,, ' ,, .~ ... "': ' . ~-.f.
.. •'• '.
,•
' ' .. .. '•' .. .
stable sistem.
The main disadvantage of the feed-forward tech-. . ".
nique is that of requiring the second TWT and two extra
low-loss delay waveguides. This may put a restriction on
the weight o...f the compensated tube. Furthermore, the
degree of reliability of this technique depends on the
balance of the two.loops which may be impaired by several
long term effects, such as, aging and temperature effects,
temperature difference of the dalay lines and shift in
TWT helix supply voltages. The latter factor is critical
because the TWT characteristics are sensitive to changes
in its helix voltage and the feed-forward technique there
fore requires a highly regulated power supply [7).
3.2.3. Signal orodistortion techniqu~
The technique of signal pre~istortion has recently
been applied to the compensation of traveling wave tube .
amplifier nonlinearities •. Tho design.method requires
accurate knowledge of tho tube • Kayo et ~ [51) have
initiatod tho dovolopmbnt of two difforcnt ap~oache~ to
tho synthesis of a prodistortion compenaatot in cascade
with the TWT. -r;s· •
Tho first ~proao~ consists of separate
QnV~OpO ·and.ph~s~ predistortion notwo~kS connoc~eQ in
oascado. '!'ho prodistortod onvolopQ of\~ho input signal . , . .
firot dotoctod and then applied to tho input of a phaao-• l
corroction notwork. "A oimplo !mpt~montntion of thi~
t
is
52
I
{l r, ~ t l ' ~ '
~
:'~ :, '• .J ·.~ ,,, ' ·'-
,; ... 1: ..
} ~
(
.~ ·"
technique has been reported in [75].
An alternative approa~h is to simultaneously pre
distort the envelope and phase of the input signal by
using two separate envelopo nonlinearit.'ies, one in the
in-phase path and the other in the quadrature path. In
53
the 'subsequent analysis we shall describe a design pro
cedure and a possible, novel implementation of a quadrature
rnodel compensator and the performance improvement attainable
for the case of single carrier, binary PSK transmission.
Fig. 3-4 depicts a predistortion compensator in
cascade with the TWT. The quadrature model of the TWT is
as described in section 2.1, in which Zp(.) and Zq(.).are
assumod to take the forms.
..
• -s R2
S1R o 2 I [S ~2] . l 2
(3.14)
(3.lS)
For our purposos wo shall seek a sot of envelope
nonlinearities, GP(~) and Gq(R), tho in-ph~so and quadrature
components of tho componsation notwork, suoh that .tho over~i1, r
transfe~ oharactoristios ~pproximato that of a piocowiso
linear onvolopo limitor .with zoro onvolopo .. dopondont pha·ao
ohift as doocribed in ,cction 3,l, In torrns qf Gp(R) and
Gq(R), ·tho doslrod onvolopo· and phaso shift at tho output
•, ,,
·:.\ "' ') . ' .. '
~~ ~? ~- .1 \. ~
ltt~.\ • ·.~ 1 ~· ;
'
' I -.i Zp
. Gp(R)cos(co0t+E) ! G'(R)cos [a,J+€ +9><R>] {G'CRJ~os[
Reos \CO~'i"EJ \ -'
Gp(R) l: Zp(R} - • /.. I -... . , !4of+E) -· -
' .
~ ' 900
· . .
, - Gq{R) Zq(R) I , \
Gq{R)sln{ca0t+E) I I
Zq{G'(R)}sin["'of+(+cf,(R)]
- -_J_ T\VT
.--....-..-.....--.. - --- ... ·--.--.-----·-, --·--C0!1PENSATOR
Fig. 3-4. Predi.storti.m a::npensator for Tt'1l'
, (--.,.........----~~~- --·~- ..... ~ £;::!:Stl4l((f'6 ' ~r==..:A ~~~~--~~-~~-,--J ~ < &•-
VI ~
.. '
::.""
'.
:; ':' .:: : 1
~ ' ,'
~ ', _.
' .. ' ... ,,, .
" ..
55
of the compensated TWT can be written as
-C [G2(R} +G2{R)] f (R) = ClG (R) e 2 p q I [C {G2{R)-t<;2 (R)} J
p " 0 2 p q
and
-C2[~ (R)-+G! (R)] 0 • ClGq(R) e Io[~{~(R)-+G~(R) }].
(3,17)
The objective here is to determine G (R) and G (R) • p . q
such that equn~ions (3. l.6) and (3.1·7) are qatisfied where
,f(R) i~ a pioaewise line~~ limitot .oharaduo~istio dO"fined by ' . '
& ·Ro R 0'!1\!Ri ~ '
" . f(R), • (3.18}
"' Ro R > !\·· .. .
whore .i0 ia tho TW1l' output satur~tion'volt~go.and Riis tha
~nimurn volttig~ ·~orr~sp~ndi"'ng to~ aaturo.tion of tho. limiter.
Tho oxoot aolution of thoao oquationa i~ ~ vory difficult .. . . problem, and in ordor to obtain a tracta~lo problo~ from
' .. whioh maaningful rooulta c~n bo ohtainod,· ·wo t\pproximato
. .
· .• !
..
.. ~.
·, : 'j
... \< I
'.r -. . , . . '" •"'... . ...
·'· ·(
' '
"'
Gp(R) and Gq(R} as· the polynomials~
) -
(3.19)
(3.20)
where the coefficients ak and bk are to be determined so
as to satisfy equations (3.16) and (3.17) in some optimal
sense. .
The coefficients ak and bk can be determined from
a conventional optimization subroutine [e] such that
equations (3.16} and (3.17) are satisfiod in a least
squared error sense. The required coefficients ak and
~ for two different values of Mare tabulated in.Tables
56
3-1 and 3-2 for the Intelsat IV tube (8ughos 261-H).
Corresponding to these sots of coofficionts tho resulting
?Ptimal compensator charactori~tios aro as shown in Fig. 3-5
and the overall compensated onvolope · transf.or .and phase
Jhift characteristics aro shown. in Fig. 3•6. It io ovidont
from tho plot in Fig. 3-6 that tho over~ll en\fi?lope ~ransfor - " .
ohar~ctor!stic o~n be mado to closoly approxim~to that qf a . . pfocowiso - linoar onvolopo - limiting dovioo by incre~sing
. M, i,o. moro comploxity in tho componaation network.
rurthormoro~ ~ho amount ot AM/PM conversion offoot ia ..
ovidon~ly roducod at tho oxpanao of inoroa~ing phaoo•olopo
I
i rl
,,
. ' :,',
,',
TABI.i&. 3-1.
The cooffiai.EmtS l\_ and ~ for the ~nsators (~t-4) *
k ~ ~ millivolt/(millivolt)2k-l
1 1.04931 ';'l .29116 (-1)
2 1,03959 (-1) -9.40475 (-2)
3 .l -8.30888 (-3) s. 99492 (-3)
4 1,44441 (-4) -9.81332 (-5)
* Too. nogativo integer in ~thElses follCM.i.ng each entry in t;nQ tuble represents the powor of ten by which the entry shoU].d be multiplied.
57
..
..
:· .~·: ·:. ' ..... ... 1
•. I
' '• ',
.. '.,.. ·.~·.:
... 1 ~
''1 .. 11. ... ·.• , .. ~ .. , .. ,_ •\..,:;
'· .. " ' .. ,·' I
c'
' '.
. ,_,
! / ~ ; ~ ... : ;1-_ .: . ;,
: ... :~ /. -i,•, ...
" \\ ' fv
..
'l'AULE 3 ... 2 •
. The coofficients of ~ nnd ~ for the carpensators (f-\-6) *
*
k
l
2
3
4
s
6
~ l\ mUlivoltl {millivolt) 2k-l
l .OQl8l1 -1.06837 (-2)
1.53746 (-1) -1.76412 (-1)
-1. 64945 (-2) 1.88326 (-2)
4, 53263 (-4) -7.70929 (-4)
1.8116514 (-8) 1. 31865 (-5)
-9.50101 (-8) -7. 35811 (-8)
'l'he negative integer in eare,ntheses following ea.ch entry in the table represents the power oft~ by which the entcy
. should be multiplied.
8
..
58
! i i
I i i I l I
' i ! I l l
i J i
·l 59 '!
r•, ,,, ',· I•· .... ". '\ ,., ~-·
.. . , ·.
,. ,:
3.5
3.0
en 2.5 ti g
w 2.0 ~ tj g ~ <t 1.5 w 0..
5 f: ::, 1.0 0
.5
'
ENVELOPE NONLINEARITY
At, SHIFT
A· UNCOMPENSATED
o COMPENSATED M aa 4
a COMPENSATED M • 6
60
24
-(f). w LtJ
20 er (!) w 0 -
16 t: :i: Cl)
w 12 ~
8
4
-4
:c a..
t::> a.. ...... :::, 0
----.~-~---r---.---...,.._--.,---,----,----,.~-.----8 o .G 1.2 · ,.a 2.4 3.0 3r6 4.2 4.8 ~4 s.o INPUT PEAK : VOLTAGE R ( mllllvolts)
Fig. 3-6. The Elinglo C'1rrior charactoristiai ot tho ~na~tod
... l
l I ·' i ,\ f
f
I
I•
j
11 ' ....
variations. These phase-slope effects are undesirable as
they t~nd to spread out the output spectrum. However, they
can be mitigated by determining a set of coefficients ak
and bk such that equation (3.16) is satisfied in the
minimax sense subject to appropriate inequality constraints l
on equation (3.17). ~
An implementa~ion of the compensation ~k based
on equations (3.19) and (3.20) would require an envelope
detector preceding a set of power law devices. An altorn
ative ~ethod that doea not require the envelope detector
is to pass the RF signal directly through nonline~r
If ~P(u)nnd Nq(u)denote the in-phase and quadrature
nonlinearities acting on the instantanoous signal u, wo can
express Np(.) and Nq(.) as tho first order invorse Chebyshev
transforms of Gp(R) nnd Gq(R) [l~. This results in instant-.
aneous signul nonlinoarJtios of the form (Appendix C).
M 2k-l t\, (u) • l i\ u ,
~l
M 2k-l _Nn(u) • r l\U
'11 k-l
• J~· (-l) k-j-1 [ ·2l<~l l ~
22(k-l) [2(k-j)-l]
;,
(3.21)
(3.22)
.. ,•:
,f
., I ;,
' ,, c
1 ,·
r J I
I
l \
k-1 k-j-1 [ 2k-l l l (-1) k1\_
~ a _j•_O~~~~_...j..._~-
22 (k-l) r ~ (k-j) -1 ·1 \
• A schematic diagram representing a possible imple-
ment~n ot the compensator is de.picted in Fig. 3-7. This
co~ponsator may be, implemented with a. set of power-law
devices or aiternatively with RF multipliers together with
appropriate ~ain elements.
62
The performance of this.compensated satellite channel
has been investigated by means of computer simulation using
a general purpose satellite simulation program, [231. The
results of the simulation· are shown in Fig. 3-8 for the case
of a binary CPSK systom operating in a single carrier per
channel mode and in the absence of up-link noise. Tho trans
mitting and rocoivin9 filters are ohoson to be idontical 4 . pole, ~d~ ripple Chebyshev filtors with as MHz bandwidth.
Throughout the simulation, porfoct car9ior recovery and bit
timing are aseumod at tho recoivor. For differen~ valu~o of
tranomiooion ratQ, it io ovidont from Fig. 3~8. that soma mar
ginal gain, about 0.3 - o.s dB, in ayatem porformanco can bo
obtainod no a rooult of tho componaation. Thia ordor of improvo
mont comparoo wall with moaouromonts obtainod b; ot.h~ro (21~·
For tho caoo of multicarrior oignalo it io wall known [Sl, 82]
that thio overall transfor oharaotoriot!oo of tho tubo oup
proasoo tho intermodulation offoot and can provide up to 10 dB
l ' •' !
j . 1
<•' '·
' t
. , ' ~
,,
I ,,
,.I
,. '
.. •.
-w + -0 s -Cl)
8 0::
INPUT
r--·- ·---·--·-·-·---
(. )~
... + • • • • • •
( • )2M·1
9Q• i-·-·-GP~ .. -·-·
-~ + -0 s ...... c ·rn 0::
( . ) , ~ I
.
I + •• • .
I • • • • .
-}-- ( • )2M·1 B~ . G ( • ) L._._.L._._._
-
Zonal Bandpass
Fiiter
-to
TWT
Zonal Bandpass
Filter
63
II
.. '
,l ,,
;
]
r l", '
.
i ~ : •,
,1
,! J l I
\'
l p ' ', , ' !
i
' . l ' f
..
·t !~ i'·: :.,..{ ~,~.:~ . J.r , . '• '/""I.
\ .. .J •
.. ·'
·, .
. " ...
,' . .
r,. ~·
10·1
,o-
64
Chobyshov flltor. ,, TWT oporatlng ot ·1 dB lnpu, pt>wor backoff
di •• 1 • • ,.
' . • • 0
.. . · 4 . , · · . e a. -. · .. · 10-
. . ' \: :·. Bit' .o~or.Qy/ N0 :·. Jda) : .
' ~°' ; • • • i'iiJ,. 3-0, : . . ., ot '. . ' ·. . . ,,, .
,,.. .
l l f .) . . I :J•• •
' . ,. . . I
t-i.
I I· ,•'
t i l l
' l - . t
It
l ·, 65 .
improvemoht in carrior to interference power rtltio (C/I) r
i whon the 'l'WT iD opor.(lting At 6 dB input power backoff /~
J
Q'"
l f i
' ~ ! 1 ! .) . 1 1 ~
. l .,.
'" .. !~~
., ~ . •: ;
I >:,
... ~-;.
:\,'<>\ , .. t· '
111" '\.~
... .. -; ~ -. .. \
~· I t
;• .. I } ~.:
.. l ~ l .. , 1L,
' f: . r:
,, .. :j i>;
{) .. : , .,.,_: . . · . .
ti ,• . \ ~ . '
.. . .. a
l· . '
\ ...
' .. . ' • . ,
'i .,. ...... , . , ~ . ..
/.. . . ..
' . . ' ..... . . , .
,, -.. . , .. ,. . . . \:
J .... \ . i: •, ·, :~ ·'
i:,.. \ ,_ .;, ~ .. ~' ... .. ·- . ,;..i :
' . •' . .. . .. '' .. . . .. ' :,.: . I .. . .. . ~ ..... ",. \ ;:.~ \( . . "" ,\'' .-. • I ~,"' I ., •, . ... ',,, ' • .
:· " ' ~* "/ .
; ~' ·. ,• .. ' ~ ... ·~ ~ ' ., . ,... ... . . . ..
-~
l • f
Q;
.. ,, ','
/
..
' ,'
'11\Q offoot. of U<\tallito roptX\tor nonlinoM'itioa on tho por~or ..
imnoo of digitnl cam~iCAtion oyatcmrt hM bcezl tho oubjeot of oxtontdva . .
otuay [4·6, lO, 23, 29-301 4l, 43, · 47; 49-SO, 59,. 63, 70], 1'B <mrly
eatollite tranopondo:o woto pc:MOr limttodt it wM·no~eu:y to oporato
tho tmveling ~vo tubo (\Wr) anplitior near m\tura.\:ion ® M to obtain .. ~ trMal\\ittcd PQ.'l'OX', UndQr nucn cx:indi~ona « hw.'d-limi tor. witho\\t . , . . . 'cnwlopa d,ponoont phaDO.-ohift:. w:ta found to lt¥.Xbl ,:ro~a.bly ncc:uratoly
' ~ th'.l aa.tollito ttanopondor [4-S, lO, 47, 49, Si. 701, ~~. in· . . . .... . ~n \t\.q,ll°"'tiel)o [70), tho ~Wl' WW'l pro~ hr ft ~o A~ limi--
tor in o=r t:o ko.op tho en\'Qlopo of tho input ~;na:i to tho ffl at. o. .
~tant lovol Mel thotoby to olirnt.nate tho \1.n~airod affect of'~N .. .. . . • .'I".
\
o:m~oh ... ~tne~ ~Qt~ -that favcm .. thi.a n{\Xd--Un\iting' trodol in tho· .
W\lyOi~' of eeSK af~lo tianomit.~· thi:o~il\ tho G~to~i;t.o trMC~~. ',· . . ,;. . . io i~'m..i.then\1tiex\l ~~atabi:lity, lt\\r~ it'·ol0,.;~ ~tea
' ' . tho QPtim.'ll. ~~a nonUnoori~ ~ tho :a.:v:a ~ ·~ih ·~link. ~icr
tp noieo ·~ mtio (~R) CQ ·c.uaQUQCOO ,Ul. ~~~· 3, . , ' . . ' ~ ... \ '
. - .lm'Q.vor, -~.~~t; tbvo~'t Of ~'Jhor ~ a~t.ol.Uto)wr~ct, .. / . . . \ . . , . ,• ' '•
And tho in~~ in k~·demind :to- <\=co..th;;' .. c;tal.Uto le~ to tl . :. , \
..
_/
' . . .· ....... ~ . . . ~
b."ln.~iat:n-·r~thor than·.~ ~Nimi.te<S :channoi~ : · o~ .n-oauiatif;n, · · ... · · . . ' •• 'j ' ' '.. ' • • • :. .. :: • ' • ~.:. - ·:: ·,,~ '.·, ./ •• _;' > .: ·.': ·.;:_.\··. < . ' ·.
\' ......
~ . ~ ~ .. ' . •; !•
~ .t ~ ,• •
:: .·, ;··Ga . ' .· .. ~· ' ,' .• \ . . . . '
.. ~: . : . .:
l f
l l
. ! . 1 ' , f ':
I l ·1
••
I,
,•, ,..,;, . '•
,.•.: . ·\~
; ' ,: .... ,,.
~ tcchniqooa th.at utilize band.-/idth m:>rQ offootJ. voly thM binary CPS!<
(for Q.'CMPle,' ~ o.>SK (~co oootion 2,l)) than nocd to bo c:onaidered,
In this oono tho h~·limi.ter no longer t\QCUX'(ltely n'Odola tho a~tollitc . '
tranopondor sinoa tho offoot of l\W'PM convorsion M dioeu8sed in Chei,.oter
l bocc:m:io ll'Ot'O nignifio.,nt (l~, 23, 32-33, 41, 46, 63), Unfortunately, . .
67 I
tho ana~oio of p..,ndwidth-~~tod syutoll'o involvoo o~u~tini tho ·atatia-
ti.CXl of the into.rsymbol intorforenCX> (ISI) , . Evon for tho c~o of a Unoar
cho.nnol it io var; diffic:ult. to <btoxmi.no tho ~~ty don_aity funo-
ticn of tho into~yrrQOJ. interferon~ in ol~ form OXQOpt f~ a vori
f<M ~oial C."U';QfS (46] t Md ~B grc"tly CQlplicatoo tho ~alyaio, Tho
~on«) of a ~dpo:lo nQlllnoari.ty ~ tho chC\MOl furthQr QalPli~tea
tho Mt\l.yo1o and too'· uou.~ ll'Othqcl of M(l).Y,Ul\g ouch n ccq,lox pJ:Oblem ia . . . '
~ ll'OMO ct oonput.or oirrulation [23, 41, 63] or in OaTI.') ®Cea tho' cau-. . ~. ~ WJO of m,uror~CK\l .and bnal.ytj.~l rrethccla .(lQ, 32-33), .
. .
ou.r · objoativa in thio chapter 1a to in.vooti.v;to too affoot of
tl¥l ~~o~ ditlwtian °"'u:cd by ~ TWr on tho PQr~eo of ~
~t ph.~oo-obif~ ~ini . (~SR). &ya~,· _Saro. carp\\~· ~la~cn. ' . .
• • ' '1 •
i:eoulta, ~~od. by ct.horn,. .en thQ i,ortott\'MO'l d...~nck\ti.cn, ~ .a :oow.t
· of bMdUmit.elti~. ~ob. ~uo~ xsi> ~~ 'llt.r ~~e~ity, al'Q din~;.od
· . ~ ccct4on 4}1, Thi~ ia. th~ fol~ by ~~;~~ ot ~O ~-ea
· · · Of C).lal.( ai9ntllo tronQmit~ -~· Cl ~r-~tod ca.tcllito d\Ml\Ol
~ol~i· tm o«oct ot no.tea in ~th tho ~Unk Q.nd ~-link path~~ . . . . .. ~ :· . ' . , . . . ,• . . . . .
. O\U' obj~vo·1n·ccc~9n_4,a io. tq cn~;~m)int\ly~~,1 mo~-to.
·. hcn419: ~ ~~i nonlinot\r ~tol.~t.q':¥®~:-~~-.inrA~o· both ca~ ':~:,:•',\: • ,·,· • ' • • • • ': ,o \:-:': • ,/<,,:: ... ', ' .. : • .:. ' •, I \o '
', , • "' h 't' • • ~· ,' •• ' ;,\ - "•• . : ,' ". . . .:: . ' ..... ' . . ... . . ... \ .... ~ ·,; . ., • ,: • + I: ,
-.~•, ,. • • • " r ,I • 't :,,::·,,,:',:..• •,. .... j ' ; Ii(• '/.t 4 I'•'.,,,, ' ...,. • ' • •
'. ': •• "', •• • : •• :·. ... .. ' ' ••• ~·. • ... ·:' • •• t -l ••• . . •,, •, . ,": ~~ ,:·.:,... •' .... ;.··;-· ~ . ... ... · .. '-~·' . , ,• · .... : .... ·'.·~·-:. ·~ ·. 1:_:.". ~ •: ' .. ·•·. .. .• ......... , ' .·• t ..... ·• ~..,_,~.~:,.. 1,, t' . .. -~""; . .:.. .... ... .. ~ .... ·; ... -
1":_.;.\'
... "
.~.
l 68
4, l) SIMUml'IOO STl)l).'tES OF Bl\ND t.IMITllt'I~ m'EC'l'S
'l'ha oatellito CXUll\1.U\i~tiona link of intoxcot in thio ooet:ia, io
ll'CXbllo.d as ohc:Mn in Fig, 4-1, ln ontellito 0Qffl\\1llio.'ltiono, tho mgular .... ~
=rrior·~o:( Wo is much l~ thM the traoomittod oymcol ro.t.Q, ~,
* and c:aTplex bMcbN\d cnvolopa funcUona ( 83] CM be uood to rcpro~o.nt
both tho trMCll\'l~t.od Mel rceoivcd oignnl, ~t tho Qa\\>lox ~ onvo-. .
lcpo of tho trMoroit.te.d oignal Pl ·,
l)
o.(t)"' • l &. ca (t-l'W) I t ct.-nr) o n .. ncl ··
. f(t) • l
• o atno.twico . ' .
( 4,-l)
(
. 2) · 9(t) . ia ·tho W{WO~ of tho ni;nalinv- pw.co WJcd. in th.o'I ~ .
oyutcm to tmt'~ tfiQ ph.aco vnria.tiQ'>. ·I
* A bol\~tl aignalt -~(t), .With. ~i\\ltlr o:ntot~Q'i, WQt
end .it4 Will?~ cnVQ~~po f\\not:t..~. O·ttl, O,J.\1 l"Qlatcd by · · ,.. •' ' T ... ' ' ,
.. . / · · · , · Q·tt> _.. ~ . ~·ta.<t> a j,.,,0'\ : c · ... :
·, ~.~}~l\O~!l ~ ~al.part ~tl\Q·.~ .. ~\\J\~~,· .... • • ' ( • • , ' • \ < ,. : • ''II : ' ' ••• • f ' '
•• 0 • T
.. -\. . •• • T
' '.". ' .. '
•' ~ .. ,, .· ... ... . : " . . . . . . . .. . :.· ·':' .'~ · .. ;· .. ::.· ... · .. ;'. ~ '. .
... •, .. •: \ ~ ,•'•, ' •, .,, r ,/''
' ' : ·'' .~ : ,".' '• ' ....
' ... ..
'. ,.. • •• l • ' -. ~ • ~ ·, ••
-· . , ... t \,··.· /·· , .... . "•
.; .. . r " ' ,' ,. ..
'
-·
..
.• '
,....._,_,.__ ...
/.
. , ..... '•.
~ '; .... ..... ~. ....
·· . -.... :
. .. ; . . ,.
-· ,---., ....
~
.f. f
' , ..
(
'!>\
j·
• 69
l
tl .@ I f. (U
I r"-
O J ~-
l ,,,
...
. ' '
. . '
'f .;,\
.,,, ..
I 1
I { . 4
I "" I
. < .. .,.
' ..
.. ---·· t
• ~ ~ • Jt rtlJ 7 E U!;"f., •
I _3) on ia tho trMamitt.cd,ph.aco during tho nth s~l interval . .
Md'ia Mflum:x'i to~ tl'¥l cqual.ly likely V~l\\Otl
on Ci) • [3.i - .ogn (i) J R
M M i • ·r, .. ••\f -1.1, ....... ., r. foJ:'M.o~ ...
ond
Cc.Pl (i) • . sign of i
lot h(t) ®nota half ·tho cc.arpl~ boOclxu\d M\!Qlopo of tho ~ '
o.i.o~ trancmitt.ing and channol filter ~co =~Q given by l .
70 •
whozo tho oub:loripta o md a ~a~ tho in·~~ end quodNtum CXX\l?Ol\OnttJ, ~ . .
· h. (,t) ia \1:lu:tlly 'obt{\1.noc.\ mil\ ~ invarca roui'l.or t.rontlfolm ol .. ' , . ' . it:;l ·~opon~g ~tOX' tun~on. . lt((ll)_ · . ~Rn~ a.:( '
,~ ~ ' .. /,
- . . . .it~~o.>· .. ·.:·: · .. :·. =: <. ,l~L < ·~-:
, . U(C\lf • , · ... · • o. _ · a1ea-: · · ·(4.4> .
. '. - .... ~ ' ' . ·" ··.:~. :l·,.:".··. ." .. \:, .. · .... : .... ,· -~ ~ ta~. ~l. ~~ cqµly"'1~t:lC!~ -~·44.tl\,- ~~: U(~). ~O· .. ·.
~- ~ ~-i~~~ ~~on ;f:~-~;~~ ·~t~ ch~l· .· ~. · ·· · : fi~~r/.: ; '.;, '/ t ·: .r . .. : : :: : :,> . ; ~ . · , :· :· ·:::· f ;~> <\:· ...
· ·.-~ ~~ QX.\~W9 ·~·:.tn_i:fei~~t.:e;,.u:~ ~~c. Qt ~.ng\~. ·. ·' ... ·
. . ... .,-
I I' • l
• l
j . t }
t \ I ;'5:;
l
I
\1
11
,•
... :
tQ
u <t> • / o (t-t) h. h > ch - -· ~ -
~
• 11\, (d t (t.•n'l'-t > GOO 1 Ong (t-n'l'-'f' > l dt ~
... - _£ ~(tl _f(t·nT-t) ~in!Gni(t.-~'l'-t)l dT
••• • I hc/d ~(t.-~t) e.in[Ongtt-n'l'--r) l' ~t · . -- . . .
. . . ' . . + J ~tr) f (t.•nT-t) tco:i [On9·(t•nT-t) l dt .... . ' .
'.·
•
H tha l\Q\&Qt\r ~vi~ ia ~cent.cc\ by '?bO ~4\tw:o rrccbl
· 41.c~od in ~tar. 2, tho -~iox ~.cnVQl.C{JQ ei~l. nt. ita , .. . .
' . --- -
(4, ?~.
71
. ',
.
l I
'
If h.l:\lf tho ocnf?lox ~d onvolopo of tho illlJulso reoponoo
ot tho. reCQiv;t.ng filter io donote~ by
(4.9) \ ..
then tho carplox bacoh."ul.d cnvolC{X) of oignru., v(t), pluo cbwn-link.1 .. I
noico, n(t), QftQr ~oin9 ~ugn tho reco!Vin9 filter io .. ~
r(t) • f {V(t) + n{t)l p(t•t) dt .. - .. .. ..
• A{tton 1) + jD{t1·on 1) + n0
(t) + jl\a (t.) (4.10) c c .
whoro Gel
.A<tt°ntl) • ~{ I V(T) p(t~t) ~T} -.. .. •
B(t,on~~) • Im( I V{t) p{t•t) d,} .. .. . ... . end
• nd(tl • · &1< I nh> ptt~t> ·ctt> ~·- ..
~ -- ,
l\i (t)' . lm( I -ntt) Pl~'t) dt l (4,ll) •• . . . .......... , qj. . -.
,I • • ' , ~ • •
. ~ ~ Md :tm d:;nQto thQ l'9{l.l. 01\d ~g~ -~ ~ tho. ~lox
~intiti.O!J, M<1 n(t} .:I.a ~ · ~~ ~.n~ cn~i~ of t.hQ ~ffi~JJ.nk. ~ '\ , . . ,•
If ..
~in~· thil in~\\\ · o ~- -~ ·s .1.: ~ litJo\ltQ at ~ roQcii\W' that
72
.. .' tJ\O ~~.ncmt1;~_·ph~o i.a. o0• ~4 thA\ i~ ic ~t.oa~ ~ ·a~9 ·~ · .. ,, ' I , - ' . . .· • . , .
..
}
! \
I !
i
· ;Q~~~~~~\~ .. \~. o·.< \~., ~·~.i.ni ~ ro~yc"-Qi~Ql ·
. ~~- 4, i4o: ~~t. w~~ ~. ·~~ ~~· · - .1:.1.~ (4,lQ) I : ::::. . • ' : \; ::: • .,: '/ : < /,: • ' : (: : ,- :, \/.. • • • : . • ' • ,::· : : . • • : • :. l
· c, ·,:.,' \) ::: t: · , ) . t\\(i){:' ;/.:):~ c,/{ / ... ,·.... : ·· \. ;::- ... , . '/ . . . .
' \
! ,
73
(4.12)
Thon tho phaco anglo at tin'o t 0 M'cbtotml.nod nt tho reooivor io
given by
(4.lJ)
kl _tno a.iczn~l. ia ~ecd through thQ fil~ in the oyatcro it.
~ccmw cbl~ in t:irro and diotottod in ~web and Mi~ paoooa . '> th1'cWi1h tho trnvo-lin; W:lW t.ubo it uncbrgQOO an ~itucb Ml)lifi<Xltioo.
Md oimultonool.Ulq·nn on~lcpo ~pcn~nt rotatic:n of tho O\ltput. ph"ao·
with ::capaot to ·tno. input pnMO, · G:1Gl'Otri~l]¥, tho 1'lCOivod oi(Jllnl t\t.
ean-pling ·titro t 0 ~ ba ~proo~ntod Qtl ~ ·~ Fig, 4•2, . In Fig, 4~~ . . tho mQOi ved phr.oo • (to)' '.if1 '(\QQ~c.\ to fall into tho co=cac dooioioo
lQ9iCl'l_ 91~ t:helt. o0 ~ i i~ ~,nomt.ttcd {f«' tho cacQ of oi;ht~t1co
m (M-&l) ,. · . '
~~ tho O.n'o: ~ _cco~ to.be 1:MdcJl\ v~i~co.uni~
. diotrib\l.~d ~~ ~ V.Al\\Otl (14 - - ~ ·tho ~~U~ of QQttQQt ., .• ...
<btoc;i~ ;p.·ii~ ~ tooi~U1:y tb.a.t ·+tt0> lieo ~- ~~dQci•_ ... ' "" • , ," ' ' , > • ' , • ' ~ '
~al.:. thl"Ool\Old.1 ~ ~ ~~ttcd pfio.co o0, .t..o;i ft,ll ~~Q 0010~ ~-
:.,~,t.t.~ ~ .·c..in !» ·~~04 ~ . . . . . ' . , ' . .,
' ' ·: ·.
' ' .... . . . ~ . . ' f.
·.' .' · .... . . ·' ' ...
•, , ~ . '.,_' '' ....
,, ... ' .• . . . . . ., ' . \ .. '
,•. 1:
~ . . '• ' . ..
' ' ' • v ...... ••• • • i, : . ·. • ' • JJ
•' < •••
. . . ·, .. ' '
: .'·. I • ,t . . ' . . . ·"' ~ .. t · ••
•. I ~ : ... ~ . . '
l ... ' ~. •
. ·' : .. ,
' ·,,
,. ,. i
' :, l
" : ' . '
·, ;,
{, • '
f l t
j I
! I
i
I l ?
I
J
• • n
74 r •
Im I I no I I
no
~
,,
I,'
_,_\_ .... ______ ,.
!'' .
. · om ~ ~DLXNI~'l'lttf &
NOICE~~~·
'. fiCJ, 4-a,, ~lOlC t,ltM,. =~~ta.~ai Of 1:hc>. ~coiwd
ei\,i\a~ ('t.. \ , ·(K-0~ i .
',i.· .. · .... "'
• ~ I •
. • ,·.
.,
.. ;: /,
'l
•' :, \'
=· . . • ,. J:
l'
li I ~ i
~ I ,; '
I ! '
~ ,, '
r ( 1''
~ ' ' ;' !
I I
I I I
. !
I \ ; t • • j
,1
\ "
7!5'
W2 • ft l Pr{Oo(i)/ - ~ < ,<t0)< o0(i) + ~ IOo(i) l
i•-M/2 {4.14)
un<br tho hypoth.ooit1 that tho throohold lowla ~ (i) arc
(4.l5)
(4.16)
.. 'lbo di!fiwl~ in thG an.alydcal ovaluatic:n ot tho ~lit:y
of 01:ror in cqut\titn (4.16) ic lnrgol.f duo to tho fAot tlu\t tho probabili
ty diot.ributicn funot:.icn of tho intoroyl\'CQYintorfotenco io Wr'J hw:d to
obt.oin oxcx:ipt. for G VOrJ tew opecit\l CXICQO (46), md ovon if it ia !Olewn - .
a\Q ~ to c.btomino tho ~l.1.~ diat;:1bution ~oti'X\ after tho · . ' .
CQTpl.ox nallinam: ~~Ua\, In i»:inoiplo, it tl¥l ~ nunix.w of . ,.. . • J .
rendan variaoloa CX\U'lin9 tho in~l intorfo:eno:2 io lU\~ and 1a . . ~ . .
ftnitQ, cquGUcn (4,14) QM ba OVQ.l\ltlto.d by ·_c,.n oxh<tUQt.iw, dimat. cnu• . .
~t\UQl _ovo.,: thcoo ~dell\ va.riablaa, Uowam, t1u catp.ltQtknal ~ort
ia ~Mblo Din=~ n\Jl\blr ot peooiblo o:xl'binaticna of ~co rMdCl\\ . . . . .
V<lri4blco ~o Q~cmtial.ly with their nUllixu:a, Xn. Cl,\ ~~tcnl?t. t.o, rcd\lco
tho M"CUnt. ~f nco:oom:y ~\l~AUcn, roioina [3aJ h~ o~woatod thQ . . . ,• • ' I
~<\tJ.on ot d.imct c.nunm:a.tion and. ~nUcn l\'C.thodo, 'l'h.it'l. l\'Cthcx\ , .
~~ awlic4 tQ. tho ~ of foU.r-phoco CPOK withcu~. ~ ~ccivin9 fi1tcJ: • • • ' 'o
. ' ~
/,
. " :. "
'
' :, J : ~ .·
-~
' J
·1 l
'·
a t s~i
~ \ •• lo'
}l . t
i ' l l I ' '
I
f ~ !
i ; I
l
{ j l
I I
' l t
.'; ( .-t
I t
.(;) l ..
J ' l I
' .
: I I i . f
I , I l
' .
•,
• \
N)d with no T\WPM caivoro.ton. Hcwover, tho rcculttJ nro obt41nct1 with
unknown aoouraoy Md tho nrrount of oorrputntion noooomuy in the direct
onuiroration pnrt ~o otill oubot.nntial. oopooinlly for tho · cane of tiewre
intorforenooa in which tro effoot ot int.oroyrrbol intorfcronoo oxtendo . O'A'll' many bit poricxltJ, Donodotto Ot al [10) h:&VO DU99Qllted tho UOO of
"\ tho Gnu:lo~nclrnture form,ln [ 9 J in ·ort'b.r to redu~ tho n~oooaa.ry oat't'U
tlltic:n. Tho ll'Othcxl, howovor, otill recJUirco tho '4'\laulatJ.oo ot variouo ,.
rraronta of tno randan variabloo uioino Aral\ tho nonlinollt' trwiofol.imtion
ot. tho intcroyni:x)l intorforencx, M ooown in CCil'-'ntion (4, 7), 'l'hia m:mont
QQJ.oulntion, aa it io na\rocurnivo, in ll\bO.riow, aopooit\lly when.high
nocur<\oy io required, 'l'ho loot rouo.r-t in to ovnl.u(ltc tho portorm.incc ot
ll bond·Umi~ CPSI< oyotcro through a nonl.inQAr Dfitollito c:mmnol l:l'J rroontl . . ot <XllpUtor 1Jinul&tiorw (al, Gll, Tho ll\.iill foatuxca ot t:heca uim.llatJ..on
' pqrwro-' {\J.'O oimiw to c~dl othol:, *"'°var, tho nmular torm ot tha ' .
dnuln~on program in [23] offort).nm.-o fl<»d.l).i.llty in QOOl?M"inCJ di!fw:cnt . :' ~ <
~a of m:>dulnt.1.on that m"O tippU~lo . to oa.t:olli to OQlll\UUC{lt.icaa, '!ta
l'CO~lta cota..tnod f~ tho ~utor oimllat.ian to: CPSl< oyatcm'l (dcocr.ibed ' ~
· ,i.n m,ro tbtatl in ·(all) am till ~ in fipotl 4.3 to 4-o· t= M-2, 4 Md ,J • ' • \) • •
o ~poet.t.wiy •. i»th tho ~ttin; encl rccoi~n; filtoro. m.-o M~urrod
u, bo i(b}tio..'1 4 po~, >t dI> t4,?lo CbQbyaho.v f~ltol:tl with 0!5 n1111, 3 40 · b:nd'tridth Md Oyttm)ttio:\1 toopCllCO With rQOpcot to tho midbMd an;u_laz:o.
' ' ' • " • l • •
a.m:.t.ar-~~ '°<>' 'lwo opo~t,i.no CQ\d.1.ti~a ~ tho 'lWl' M'C ~cd,
cm with ~ tubQ··~ting at catum.t.ton (l.. 4n· ~ut pc\','Ol'.bne.'tQ!t*) . 'I • • ~. ' a.l 1, I I
/ · •ieyut p)WQl' ~,cko# ia dof~4 M tho ~ff<ll'd\Q:). (in dD) batweon tb:l ~otUQ.1. 100m j.nput ~·of.tho ~«tQd ei~ol en~:·~tit l"CQ\U.M to oat~to tl1Q t~. . · ·.··.. · . .· :·
' .. ,.
. .
t t } . {.
l l l
· f
1 i .. ~ I
l I ··,
l t •
I
t l ) l
' i l ~ L
,
,..--)> ,.
:"!
,.
t
1 --l'"----------------....--------------------~----------,77
"'
l
\"o t!t
10-l
\.
. "' '!
o· ~
0
1 • ' 4•3,
IdcmtJc~l trNmmitting Mel n:1ooivinq filtom, 4 ·polatt, halt ell\ aquirit>r,ln Chf'..byohov.
05 ~tui, hMc1widt.h,
'lWl' opar<iUnq nt 12 <lU inrut ~r backof 1'.
Di t rnt:an. in ~nbi tn/000,
I
J. 10 ·u . I, of Mt'. YO JU. ..
20 ·40
60
70 90 90 !)!J.
~o ~ A .•
·-
~ .
I • • ..
as dLl ·,~ol ..,
#
'
J
-{ i
1 f
1 ..
\ . ·\
I~
Identical transmitting an<l reo.:?iving filters. 4 poles, half dB equiriople ,Cheb'fshev 85 ~:z: bandwidth.
'lWl' operating at 1 dB input pc,.-1er backof f.
\ . \ •
\ .
Bit rates in rregabits/~c.
------40 60
70 lL:U----- so
L----91)
95
\. \ . \ ' \ .. \ . \
Withoot I.S.I, 1
0
i:>ig.
\
\ \ . \
I
\ 5 \ 10 15
4 ... 4. Pro'1f"'ility cf error vs rht.-1,,, i
29 25 dB
' ' j ;
l ',
l I l 1 l
ri
f ! I' I
lq ii n r, I• I
'! l I
. t :1 I l ' l i
I: ' • I
' f
I
;
,.
I ,,
l
p e
10 ... 4
1 -s
-5
Fig .•
Identical transmitting and reoaiving filters.
4 poles, half dB equiripple Oiebyshev.
85 Mhz bandwidth.
operating at 12 dD input pa,,er backoff.
Bit rates in roogabits/sec.
60
ro 120
140
160
1 ro
....
Without I.S.I.· . \ • •
\ • \ . \ . \
0 5 10 ' 15 20 dB CVNo1 4-5. Probability of error vs ~O' 4-µhase C'PSK
79
.,,..: 1
·, ', ., ..
; ' ' '·
1
p e
10-3
1 -5·
-5
Identical transmitting and reoaiving filters. . 4 poles, half dB equiripple Olehyshev.
85 Mlz bandwidth.
'!WI' ooerating at 1 dB input power backof f.
Bit rates in negabits/sec.
~o 80
120
140
160
180
/ /
Without I.S.I.
t
0 5 10 15 20
··'' 80 '
· 1 , !
:· f .. .,
\ I . . f
. ~
·, .,
. {.
'
I I
I I
I-
1
p e
~
Without I. s. I.
0 5
. -\
'
·. \ .
Identical transmitting and reooivinq filters.
4 poles, half dB equirifl)le Chebyshev.
85·1-'hz bandwidth. '1Wl' q,erating at 12 clB input '"pc,,,er backof f ..-.
/
Bit rates in negabits/sec.
120
135
150
180
\
\ , \
10 . 15 20 dB [~0J
.. Fig. 4-7. PrdJability of error vs y-10, 8-phase O'SI<
81 ·
·, 'l > 1
; ' j ' \· ,, ,. :.~
I ' . \'
~'
' '. ,.
, . i I
' i l ' f l r ' l 1 I
i ~ ;i,,
• I i { I
iJ ! '! i ~
.,
~ ' 1 •
r i .. J. ·: J '~ !u ,;
I I
I ! I
l l I
t {
' 1
: f
,, '• '>
..
. ...
I f l
p e
' ·~ ''\
"\ . \ . \
Identical transmitting and receiving filters. ~4 poles, half dB equiripple Chebyshev.
\ . \ \ \ ' .\
85 Mlz bandwidth. '1Wr cperating_ at 1 dB input ~r bac.t<of f.
Bit r~s in megabits/sec.
120
135
1 so
...
. \
\ . \ . \ \ •
\ Without I.S.I. I
\ •
\ .
' \ 0 5 10 15 20
82
·) .,
i'
,, ,,,
{' .
f i
i; ,\
r i (
l i . ' '
, t
~· .,
•" ,
!
I
83
and the other with the tube operating in the linear regiai (12 dB input
power h>ackoff) of the pc:Mer transfer characteristic. Figures 4-3 to 4 .. 4
soow the probability of ex,:or of a biruu:y CPSK system (M::2) as a ftmctioo . . .
of bit energy to noise power spectral density ratio, Yo J , and for
various transmission rates. The performance degradaticn increases as the
transmission rates approach the filter bandwidth. However, at high
transmiss~ rates and high fEi!Nol, t.m perforrrance of the system with
the ~.q;,erating at saturaticn is slightly superior to that when TWr is
operating in the linear region. This .is due to the fact that, at satu
ration, the phase shift at the rutput of the tube due to input emielope
fluctuation is _relatively ccnstant and can be ccnpansated for by a
<Dhe:rent .reoeiver. on the other hand, the phase shift at the output of. the
'lW1', operating in the linear regicn, is rrore dependent en the input I
signal envelope fluct:uatiais. It then beoones m:>re difficult to trade
this phase variatiai with a cohemnt receiver. Similar results can be
obtained for the case of four- and eight-phase CPSK. In all the cases
rot:1Sidered it is ~t that the ~rfonnanoe degradati~ increases as
the transmission rates increase (as the bit rate approaches the filter
bandwidth) •
:i:t is well kna,m that the transmi.ssim rate (ireasui'ed in bits/sec)
of M-ary C1'SK is ]Dg,Jt tines that of biruny CPSK. However, the
~tted power .r:equi.red to attain the sama ~l error probability
· does not var:, 1:fnearly with the transmission rate.. ,or exanple, if_ we
. require a syntx,l 'error prcbability of 10-3 and a fixed synbol. rate of 60
· Megasynt,ols/seccmi with the '1'Wl' operating at 1 dB input pc::Mer ba~off,
~· •,
' ' '·
,;
\' ,•
. •, ,,
•, .. , •, '
" ' ,; '
,
~ ,· ..
' ' 1 f' ,,
( ~ ,. ''
' . ~ ' ' I
l •. ,)
. l
. l
; ' ' -~. l , 1
, ' i ·.1
t
1 1
84
cntparison of figures 4-4, 4-6 and 4-8 indicates that the foor-phase CPSK
system requires an extra 2 dB in the transmitted pa-Jer over that of the
binary CPSK case. However, the eight-phase CPSK requires an extta
transmitted p<:Mer of approximately 12.s dB.
To cx:nclude this section we point out that, to date, the analysis . of bandlirnited CPSK signals transmitted through a bandpBB's ncnlinearity
is still vei:y incxnplete and quite inCXl'lclusive. Alt:l)Ough oonputer sinu-
1.ation imthods may be used to assess the perfo.onanoe of such 9¥9tems, the
accuracy of the results obtained is generally unknown. rurthe.more, in
actual satellite ocmtl.ll'Ucatiais system:!, the effect of up-link themal
noise disturbanre nust be included j,Ji the analysis. This effect can not
-be readily :investigated by rreans of rorputer sinulatioo without the use
of veey large aroounts of oonputer tine. .
In the follcM.ng sectioos we shall disregard the effects of
intersyrrbol ~terferences and shall cx,nf.ine. our attention to the nailinear
analysis iri the presenre of only up- and down-link thennal noises. Such . .
a sinplified nodel not only yields a mathematically tractable prd:>lem ~t
also the solutioo leads to significant further insight into the effect
o1 bandpass ncn1.inearit1es on t11e performance of M-axy CPSK systems.
4. 2) EFFECT OF THE1M\L NOISE DIS'l'URBA,.~~
our objective in this sectiai is to investiga~ the effect of up-
link and dam-link thenna1 noise distutbances ai the performana! of a
~t phase shift keying (Cl?SI<) sys~. . A block diagram of ~
c:mm.mica~ai system oonsidered in this sectioo is shown in Fig. 4-9. On
-! 'I :1 -,i
., :,
I'
. :
,. f
,.
' ' ' . ; \ ; J
l i . f
f ii I t~
"
A cos (c.,0
t + ei)
Op-link noise
l\.i {t)
i & fl, •••••• ,M}
Bandpass
Nonlinearity
-----x(t) = R(t) Cos (<a>
0t + \ + e (t) ~
Ibm-link noise
nd(t)
~ ,...........,
+ a:tiERENr
R&:E1VER
'\.\, \c.
z(t) ca r(R) Cos{C) t + e. + c(t) +g(R)) .o J.
~g. ~-9. CPSK transmission~ an~~ . . .
,.
'l
A
6£
~
''1~~_......,~ __ ...._, __ +'<\~\I ',,:.J"'t; - .. " (' ....,.., •• ,..., 1 ""1;1'-....,. ~.._. ..-~"' ..... '., ... .,... "').,-'\.-... tr•-..,-,. ,..,.:. • •."'~'• ~ ~·-.:: ."-~ .. ..._ • "'°'!!,' ._. "'• •.., >,, ""'\. ,, < "· ::-... ,. ~ ... • _ ... ~ ... T, .. • • ,':..... ...... =,·!, ~:...: -~ -· ... , .-......"£:-.• .... ...':~_:_2,::. :· •. ~ ....... •
" ...
: ·, f.
..
I
I
I l , I
l I f
86'
the ~link path the transmitted <;J?Sl( signal ar)d the thermal noise are ,,
added to form the input signal to the satellite mpeater. The bandpass
filter preooding the 'IWl' is assurred to pass the sigrutl without'4storticn I
but to restrict the up-link noise to a finite bandwidth, w. The signal
after passage through the filter is. arcplifM:!d by~ 'lWl' andtretransmitted
to a ground station •
. On the dam-link the retransmitted signal is furthei; corrupted
by indeJ:tendent additive the~ noise. to foxm the receiver input signal.
An ideal coherent receiver with an appropriate CPSK.decisioo circuit is
assuned.. One pos~ible inl;,lerre.ntatioo of such a receiver is shc:Mn in
Fig. 4.10. At the xeceiver the RF signal is <.bm-convert:ed to baseband,
where a is the average of the AM/PM. effect in~ced by the 'lWl'. The .
baseband signal in eacn branch is then integrated and sanpled every bit
duratiai. 'lhls is the well-known il)tegrate and dunp (I&D) filter. '!his
reooiver structure is a specific fotm of a oon:elation reooiver. It will
be assurred throughcut our 'discussion that phase coo,erence and bit
timing xeroverj have been achieved at the receiver. . . In an attenpt to detexmine an upper bound on the probability of
error for biruu;y CPSK spread spectrum signals txansrnitted throogh a hard
limi~ satellite repeater, Aein [ 4-6] approximated the integral ~ dunp filter by a sanple and sum c4tector., It was pointed out~ [4] .
that for the case of an up-link noise limited channel with small down- .
link noise distw:bances the error probability oould be cx:nputed by the use . .
of the axiplem:mtal:y er.rqr functioo as if the cor:relator output statis-
tios obeyed the nonnal distrlbutioo law. By ~ting the in~ate1
and dunp filter by a san:ple and sum detector, Davissoo et al [29-30} have
I
'
, . • I
'• ,,
·, . '
!
, . )
l i' i } .
I I
} l I
I ' J
I: j,
t .[
l I
- ~ .. ,'t. . ' ... h'- ·""· , '," .. --.~:-;.. -,. ... ·, -~., ... ?"-,'!! ·, -~~,..;.,,,,._.,,~- .......... lo. ~ _,.·,.,"": ~"'-~' ,:- .... - -
• I --·~- ·,_,.~ ..... .....--~-'t( , ,,= • : )· •. ,' .. ,~.... -·~... £ -- ..,.__..~-.....
"
z(t) + nd(t) .._,,...
leoeived signal
. C)
't cos (ti10t - B)
sin c1Jl0t - e>
( \
' I i ·----t> l
sanple at t ::::::r kT
Fig. 4-10. A co.'lerent +eo=iver for CPSR system
- ~ --., . ...._ .. , ......... "~.~""'- ."'::."' ~ .. ·~. ~,. "'..~ll---,,.·'if.J'""'-"~---~--r.., .. -.._..._._ .... ~~- -· -
•
Phase
Detector A ei-1"> ...
Detected phase
~e
co ' -..J
',. ·.• 't•
·~ ...
..
calculated the probability of error for b.iruny CPSK signals transmitted
through a pieoowise-Unear envelope-limiting repeater. H~r, their . " ' '
rrethoc1 requi.ms the direct ntmarical cx:,nvofuticn of the probability
density functioos pf the sanple and sum detection outputs and is vecy
tedious recause a large nl!Jl'ber of sanples must be used in order to give
a~ approximation to the in~ and dunp filters.
In view of the difficulties associated with the integrate and
dunp filter or its aw~tion by a sanple and sum detectioo, Jacobs
(47] originated the substitution of this pre-detection integratia, by
88
a single sanple anc;l majority logic decision devi03. The probability of
error is then related to the single sanple detectiai error which is
cxnsiderably sin'pler to evaluate. Lycns [57,59] put forward a method of
charact:erj.zing the noise statistics at the output of the general bandpass
ncnlinearity·. Essentially, in his ~thod the equivalent (desired) signal . .
cxrrpcnent at the output of the noolinearity is defined as the statisticaj.
averase of the output signal'ccnditiaied oo the Jc:lCMledge of the trans
mitted informatioo. Onoe the equivalent signal ccnpment is defined the
effective noise cx:llp00e11t is sinply the_ signal at the output of TWl' less
tne equivalent signal ~t. Jain arid B~ [49] have expressed
the probability of error in tenns of infinite series of nodified Bessel
functioos of the first kind for the case of binary C!'SK signals transmit
ted through a hard-limited ~ter with a single sanple and majority \ .
logic receiver.
In section 4.2.1 \..e make use of Lyoos's approadl (57, 59] to
' ,.
, f ~
i
' r : l
l I
i l If . $ \ ~-J f. d,
J
. l ' I
!
I l l
· 1
.. ,.
..
. 89
. characterize the in-phase and quadrature noise and their statistic.a at
the output of a pie03Wi.se-llnear soft-limiter. Based on these statistics - I
the prooability of error for a rereive.r with a single sanple and irajority . logic detector is carputed; A similar approach· is applied in section
4.2.2 for the case of an actual 'lWl' channel (45]. HCMeVer, due to the
CXJrplexity of the 'lWl' n01linearities, the analytical e,q;,ressions for the
equivalent noise statistics are very difficult to obtain. In this case
wa expand the probability density functions in te.tJT\9 of Gram-Olarlier
series as us.ed by craner (28]. OUr enphasis in tltls section will be on
binary CPSK systerrs althrugh the nethod in general can be ext:enc1ed to the
M-ary CFSK case. . I
4.2.1) Piecewise-linear envelg?e limiting ~ter a
I.et us assune the transmitted M-ary CPSK signal, within each
synb:>l duration, to be of the form
' (4.17)
·A = signal anplitude
WO = angular carrier frequency
T syrrbol duration ,.,.,,,
=
ei = (21-1) !. i = o, 1, 2, •••• ," M-1 ' M
each value of e1 having equal probability of being transmitted.
l .1
J , 'i
.. '
I
·i l
'
l 1
f l
.. ,
'~ ' '
' ·.
. , .
' f
l
..
... ' . . Than the total ·signal at the input of the noolinear devib3 nay
be written as
' where n1 (t) ruy n
2 (t), representing the in-phase and quadratfu?e. noi~
~ts, are zero nean Gaussian randan variables with variance Nu
~ watts.
)
In tellllS of envelope and phase relationships, (4.18) can be
wrl 'ten as
~ x(t) = (4.19)
R(t)
and
c(t) a 1 [ n2(t) J tan... A + n
1 (t) (4. 20)
The signal at the output of the pieoow1.se-linear enve]q,e-µmi
ti.ng repeater then beCXJTes [43]
z(t)
where the limiter ncnlinearity is described by
f(R}
,('
R r.
1 R > >.
(
(4.21)
(4.22)
90
" .
~ t ' ' ' ' ' I l {
i i I
/
I ,. ,.
\ l ·. t .:,
>,
1 t i
1
~ 3 JISJ I Z
91
The cx:nstant A is known as the limiter softness factor. When
A is equal to zero a hard-limiting envelcpe nonlinearity as discussed in
[59] is obtained.
Follcw.ing the \'.Ork of Lycns [57] the effective signal cxnp::nent,
s' (t) at the output of the ncnlinear devi.oe is defined as the a::nditional
average of equaticn (4.21). Tl1e averaging of z(t), a::nditicned on tne
Jcno.,,,ledge of the transmitted phase a., is made over all the randan l.
fluctuatioos 4ue to the up-link no~ cntpenents. We can then write
s' (t)
wre:re A' is the effective signal anplitude defined by
A' == E...._ [f (R) oos e:] -R,e:
(4. 23)
As s!n-t;n in Appendix C.2, this value of A' for the piecewise-linear
envelope limiting repeater can re written as
A' == / ~e e-p/2 {I
0[p/2J + 1
1 [p/2n -
1 2 2ye -P f (1-x}x e -yx 1
1 [A>.x/Nu] dx (4.24)
0
where . ~,e: [ • J denotes the statistical ave.raging over R and E:.
p == A2/'1Nu is the ~link carrier to noise power ratio
1ri (. ) == nxxii.£ied Bessel ftmcticn of the first kind of order n
y = ·/12N u
., ,'
' ' : j
' . ?·\j
,.
' • l.. .•
:, ;, ' ,!: ,·
\ .
..•
92
The first tenn on the right hand side of equaticn (4.24) corres
p:nds to A' for the case of a hard-limited channel ( A = 0) and was origi
nally derived by Bladllnan [17),. the seCCX1d term represents the correcticn
tenn due to the 1imi ter softness factor, A.
The effective noise a:npcnent at the output of the noolinear
&wire is ili:m obtained by subtracting (4.23) fran (4.21) and is given by
n' (t) = ni (t) oos fwat+ei] - n2(t) sinfuut+ei]
where ni (t) = f(R) ros E - A' (4.25)
~(t) = f(R) sin£ (4.26)
NCM define
A+ n1 (4.27) . x = A
~ (4.28) y = A
The .\7ariables x and y, as defined are Gaussian rancbn variables with
means A/>. and zero :respectively, and both with the sane varianoo Nuf>. 2 •
In tenns of x and y we can write
x ... A'
n' 1 = .. x ,- A'
I x2+y2 and
y
n' = 2
y
/x2+,)-,·.
2 2 Oix+y ~1
(4.29)
(4. 30) ~
"
-~
(
,•'.
. '· .. "
' . . , ' ,.
'• ,, ::
. I
I '•
.,
"
l I r
' . { ! ~ I
l . > ; ' {
"'. ' l • l
i ~ ;
' >. f
.. : ... ;'
,· (
J
93
The prcbability distr.ill,tt:icn ftmcticn of ni can be ootained by
.iJltegrating these Gaussian randcin variables, x and y, over the appropriate
regioos in the x-y plane (see Appendix C. l). Once the prooability
distribution ftmction is obtained the rorresponding probability density
ftmction can re evaluated fran
p , (a) = dd rPr{n1' < a}] n1 a (4. 31)
Perfcmn:i.ng the .iJldicated differentiaticn .iJ1 ~ti~ (4. 31)
yields (Appendix C.1)
Pn' (a) 1
~(x}
=
=
IT ""P G ~ [Ha+A.->21]
/ l - (a+A 1)2
A(a+A')
IN u
t[ A(cri-A')- >. l hi u
I cri-A' I > 1
j c:rtA' I < 1
(4. 32) Ir--
' . ·; -i ·i :1 ;;
-·· I 1
.. ' ::
'• ',
; ~
/
,. '
,, t
t
' •' , .
.. ,•
' '
,_ ; ,, l , I , I
:, i .. I ' '.! / \
' ' -·-
.-.
94
and x
t (x} = 1 fexp I-a 2 /2] da = ! erf c I-x/ r'2]
ff.i
where erfc ( ·) denotes the carpl.enentary error ftmcticn.
Using the sane procedure the follc:wing exp:ressicn can be derived
for the,prd::>ability density functim of the quadrature noise ccnpa,ent n2
0
_>._
IN u
I al > 1
(4.33)
Figw:es 4-11 to 4 ... 22 illustrate the behavior of p , (a-A') and nl
p0
, (a) for different valtes of a no~zed limiter softness factor, - 2
* ). = >./A, and different up-link bit_energy to noise po.,.er spectral
* density ratios. The plots indicate that for ). = O, _oorresponding to
the hard-limited ~l, the ~valent quadrature noise a:npcnent
. \•
) 'i
" .. •' .,
•,
.. ,
,' ..
' , l
,..;..-... ;~··· i(' • ~ .• ......: . v· I 5 tf i" . )-' . ._,.· ,1!( ll!!:e£.SC... • .... "" 4 ~ .. M l>l:111 qr W l £± .... - - .... ~--· .... ~~." ... )'I ... ~ ..... - ...,.-·~"\>.£,... • ... -..-... ~- ... ~ .. ,..- .... '>\,"'-~,·~ ..... --...~~ ... ~-... ---"t.........,..._,.. ....... - ........................ ......-........... _ ..
5.64 ___________________ __:_ ______ 11
4.70.
3.76
2.82
P , (x-A') nl
1.88
0.94
~/Nolu = 1 dB
~ ::: 3
.•
). *
* >..
* ). • 0 ....
* >.. • 0.5 '1 I I \
* - 6
>.. • 1
* >.. • 2
J~ ~ --, o.oo_
1 ==*:::::, . 'C_~ -1
1.0 I I
I
-2/3 -1/3 0.0 1/3 2/3 1.0
x
Fig. 4-11. Shifted pdf of in-phase noise fran limiter, p , (x-A'} nl
~
, ...... ~-~ ... ~,. -~. ,,,, .. _.~~, .. _ • .,,.,..,,..,. .. ,. :- 1., .... ~·:-~~- "'","'··~ ... :>•,,.' ....... !' "~···'1 ... "'-'·'r'~,.., • .. ~ .. "~\t:."~ .-~;: __ ..,,\ .......... ~ .... _}",.~.)~ .._,;.:,} ......... ~~ .• -=...,-.-t. ... ,_...i,,;-~ ... ·.·:-..:\ .,~_.,.,..,....!!. "'-~~ • .-_.,... ., .....
~~~~":'--.!r'~~": .... !"'!'~~ ..... -~-
\0 u,
'
........... __
--_..--.-..... ' ,, ~ ·~;..1,:.· --~..,--------- . - - .,..,.,.c.,
~ . .....,_. ---......i--""'_"" __ 4'_,,~,.,._,..,.._., ,...._.. .... -~ .. ,,. ..... __ .,. ~~-.,.,....___...__.\...,. ....... $.-JU J t!!!l!.!!ll!tStt,).IS.S&.ll!fE. ... -..- .. .,. ... _~ .... "'"..,..;<. . ..__ .. ~.~....__.r---.J,-
1.44.T----------------------------
6,2 ~olu = 4 dB
wr = 3
4.9
* ). • 0
3. 72-i . I \
* * •9-f >. p (x-A ') I n' \ >. • 0.5
1
"'"" .. ~"""" .... ···· . .,.,,~~ · ......... ~~---
,·. 6-1-1 \\ * - l >. 2.4J * >.
* >. I I I ,,,. \ \ ........
1.2
o.oo 1~ · ~ ~ :=>2~. -~I -1.0 -2/3 -1/3 o.o 1/3
x
Fig. 4-12. Shifted pdf of in-phase noise fran limiter, p , (X-A' nl
2/3
.... -,~~" ... ,;:. .. ,- ..... t•'; ~,, .. ,-~·-\,;· ..... ·f~·-:'I<. .. ·-,'!>.~, .. ~, .. ~ ""' :·: ".:: !: ; .... ~: .... ~ , .. ', .... ,: "':.•.•, .-:.•! .... ~:.-\_•·~·..,,_,,,.,,.,.,,.!'l.,;,....UIII"~~·~.,.~ -,.,~ .,.,._,..,_,.1<.~•._,•..._ .. ,, ... ~~t,•._~•
1.0
\0 C1'I
f t, ,·
10.02
8. 35 .-1
6.68 ·1
5, 01 -/
Pn' (x-A') I
1
3. 34 -I
1.67 ~
o.oo
-1.0
Fig.
') ..
Yolu = 7 ~
WT = 3
(\
* I ' ). • 9
' I
I ~
* ). • 6
I I \ \ ;
*
I I I/ \ " -
-2/3 -1/3 o.o 1/ 3
x
4.13. Shifted pdf of in-:phase noise fran limiter, pn, (x-A') 1
< .,._~ .............. ~ .. ,... .. ,""" .. ~ ~ ... ,c, .. t.:: ........ ..., ..... ..1&-........ ~ .... ,,..._,. .... fi, •• ~'I...0,1'"'.,J.<1. .............. .,i .... ~ .... ,:1--_,,,..~, ... ;a;,,..Jc_,,.f~#~-~~.-...~-· .. ---::.1.,..• ..... ~A..,'R"',,· ... - ----- . - .... ..._,.1,• ........ n-'l'.• .. ,_~ ........... 4
• 3
A
. -
' ,:,
,,.
* }. • 0
* }. • 0.5
* >.
* • 2
I
2/3
• 1
Ji I
ll I
I
1. 0
\0 -..J
,,
I _.. .
1:,
, .. t -~· :,
-~
j._ "''I. t ~ ,n ""' - f ) J .. ZM I -------... . ,,.,. -:a• u z It a :c .. 1.dr'' ·· £_,,*'_ ...... ___. ........ _ ........ _ ........ .,,a. ......... .,, ... _..,._ .... ___ ~--- ................ ·~ ... ....... , -· ........... _~_.,..,,,....,, '"'""""~-·
13.so-·-r~~~~~~~~~~~~~~~~~--~~~~~~~~~~~~~~--
11. 25.
9.qo
6. 75
Pn' (x-A') 1
4.50
2.25
~/Nolu = 10 dB
wr = 3
..-..
9•
* A • 0-------
* A - o.s I * A • 9------'
* X • 6 I , A~- 1---+t
- 3
o. oo_J ~ J.?< \ s :s::-=::; ?::<:. ----1 I I I I • . .
-1.0 -2/3 -1/3 o.o 1/ 3 x
Fig. 4.14: Shifted pdf of in-phase noise fran limiter, p , (x-A'} nl
2/3 1.0
\0 00
- ___ .,. ...... .,,.,, ... -~ .... \ ... '~ '• ....... '\,, ... ~........ .. -·~·«, 4. '•,.... ~ .. ' ,-: r' 44 , ... ~,, :-•,..,;,•.._~'.~',, ..... _,. ~""' " ..... ! ..... ,,., '!- ·.i ....... ' , .. ·.#,:,.::"_.. ....... '-, '• •, ,,• './: "t.i:..:.~_•:._•,:_n~,.;:.,,"*1ft:'.J;!f_•._ . ;. _,..-~,c..,. ·~ .. ~~~~ ... ~.,:-·~ ....... ---...,~, ...... ,. .... •. ,:·~ ~ -
~·.~~~1~"}* ...... ~~'t .. 'r,',Fj',1,.--...----
~~~~~·~~~ ....................
18.00·-r~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~-
15. 00 ·-
12.00.
9.00.
~/N0Ju = 13 dB .,,,,,
wr = 3
* ). • 0
111 * }.. • 0.5
p (x-A') n'
A•.91 h * 1 }.. • l
6.00 * ). • 6 I I • ). • 3
3.00
0 00 . I // ~ \, ::::::::S ::::::>S::---- I • I l ! ' '
-1.0 -2/3 -1/3 .o.o
Fig. 4.15. x
Shifted pdf of in-phase noise f:rcrn li.mi ter, p , ( x-A' ) nl
1.0
s--..-.,: .. ~l" lr,r,-.-.,..,,.-.,.;. /..' ~'"/-"•~,''\_. >-~",y•- __ ,_.....,~o,;,.-:;, t 4, ...... ,, .. ,.......-a.~.,.._~ ...... AL,.,_...,_.._ •• -...,., ... £. __ ._ .... * ,-01. .,e -•~
'° "°
I .... '·
)
·.·~.
22.2---____;;.._ _________________________ l
18.5
14.8
11. l
Pn, {x-A'} l
7.4
3. 7_
0.0.
~ 0 J0 = 16 dB
wr = 3
'A* - 9~ I
* " - 6
* >._ .. 3
-I
-l.O -2i3 -1/3
' I
0.0
Fig. 4.16.
... ·~ '"-•~ , . r ... ~:~-:,.~ ·:· -,., ,- ,, * ~·,,.
x
Shifted pdf of in-J;:hase noise !rem limiter, p , (x-A') nl
•, .•'t .. ',..T\;\'·""'~ 1~'-!t/(~""!~:- -~, .... 1·, •, -~ .. ~ -~ ., ....
* >. • 0
1r 'A • 0.5
1r 'A .. 1
* "! 2
2/3 l. 0
..., 0 0
. -~-~~ ':_',., ,:--~ .. ____ .._., __ •. ~L.::.. ... ·t.~.:;.,...,,__~-"'u.~-· ,.,_ -~-·_,'.,..:J
':
::--· ·~ ~,
'
6
'
3,30~~~~~~~~~~~~~~~~~~~--~~~~~~~~~~~~~~~~~~--,
2.75 -
2.20 _
Pn, (x} 2
1. 65
1.10 -1\
I\~ 0.55
~;N0 J0 = 1 dB
wr = 3 ,
* ). - 3 "-
* "' / I .A ••
I I IA'
I
- 9
* ------A • 6
~ \ \ 1'.\
I \,
* ). - 0
* ). • 0.5
* ). - l
i----- r::::::::::: -<':" "'-- ---- --1 O • OO I I I _ __ I I I .
-1.0 -2/3 -1/3 0.0 1/3 2/3
x Fig. 4-17. Pdf of quadrature noise fran li.rni.ter, p~ (x}
.,,., ..... ,~···"!, ... .,./.,~i,...~,~, .. • ~ ... •• : ~ • • ~ 4,-,,. ·,.:'•,, -h :·,~·,~, ~·~·~~~::.~::~ .. ~:;..,',~~'-~~~~:,. ... ~~~~~--.::. __ :~~~·
b ......
.:
·'t: : '..; • f ,'"':'. • .: • .... l ~.:--.,,:~:~'\( . -•· ! E t p !!Wi.i!!!!S!!J£ZN&!IZE411 a ct & ... SZ!ZLCA tdCZ!~-· .. -·-4! ..... --~ ----"""-"'-·-·-·"' -·~· .... -....-
•
1 I
>
4.67-r~~~~~~~~~~~~~~~~~~...-~~~~~~~~~~~~~~~~~-,
3.8
3.0
2.3
Pn, (x) 2
1.5
0.7
~
~olu = 4 dB
wr = 3
* >.. • 9-----,
>.. * • 6 ----r-1
x* • 3
ft >.. • 0 ------..
0 00 ,--- ----- _/ ./ '>. ':-:---- ':::::::, ~ ---1 • l I I I I l 1 -1.0 -2/3 -1/3 0.0 · 1/3 2/3 1,, 0
x
Fig. 4-18. Pdf of quadrature noise fran lirni.. ter, p 1 (x} n2
...... 0 N
~ • y • ,,. ,~,. ; . ,._ . ·~ ~ .'.,.. . ~ ~-~':~-·-'I,.~~~---~..........._. ............... ~~ ··""~~._··~· t r · · · ·· ·• · .... ,,.. ·· · ,. · I ""'~!OIJ~ ..... ~ ..... ---...-~------
. ' ·~ ........ ,.. .... ~. ~ ..... '~·'> ........ ·. ..- ·~·"'
I I
...... , ... "t !;£ :cc:.::::::• !Ii! 1 ~ - , .,:;., •• .........__ __ ,..__,, ___ , ~,- .,..., '--...- .. , ... ~,~,.....,.,, ..... r-~ t
•
6.6
5.5
4.4 ·
3.3
Pn, (x) 2
2.2
1,1
o.o -, -1. 0
~JN0Ju = 7 dB
wr = 3
1t X • 0
-2/3 -1/3 o.o )C
.,. ___ ). - 9
.,. ). - 6
- 3
- 2
Fig. 4-19. Pdf of quadrature noise fran limiter, p ;• (x) . n 2
1. 0
I-' 0 w
--,.,...,.,,, •.-" :: < ,.,,. ,..,;:_, • ... • • ~~j,'-i,.,., r'~ ;,.~ t_.,,.... ,, \o,.• ..... ~. ·~·,,_;• -... ·~-•," ~ .. : ~: ~r:~:~~'>'~·;_;,,~-:';•(._..~:.;,, ,.~::... 'lo:~.._1\-'"'l',...,... ~~-~-... ,,~ :.~ • .._ .. ':. ,,., .. M • ..,,..:~ • ~ ~.1, -~--~~·
·~~~w.~~ .... , ..................... /' ~~....,. ~ '"'"111'"1'~t· .. ,'!tT-l'+i'°'--~- ~------~~~~ ';·~. • - • ·"'· I ... ·-·:... • • • • - • I•• • •• • It. "'" ,.. __
j
'
,:,'\J
• -~ • .t. • .......... ~ •
\ f
'{ .,
.-:-~~·"SJ.·_:;,~~.~~· ; l L....... - • .... , •,,${ .. ·£.:~-.!j" ::; _.,.. ...... _,, .. M._ ... , .......... .., ........... ,>J"- ••• <,,- ""''•'"''" • .,,, "-'""' -··-...----~ .... ~ .. - .... 0,0. - ............ 2:"-::=::5
9.30,------------~--------------
Pn, (x) 2
7.75
6.2
4.6
3.
1.
"
CEiJNolu = 10 dB
wr = 3
* >.. • 0
* >.. • 0.5
* >.. • 9
* ._). - 6
~ p~ u
""-
,; ; ~LI ~\.,.~ ;! O,OOj ' I 2/'3
-2/3 -1/3 0.0 1/3 1.0
Fig.
x
4-20. Pdf of quadrature noise fran liroi.ter, p , tx) n2
.,/'
c
t\• • <,,' ··"'~~"',, ,' ' ................... ',* •• r 1-t,'....'.'t'"•"• ',' ! ;,_,:!•• :.tJ. ·~ "~~'" t\ .< ... ·: "'.!''~t->J_~ , ·~~·· ~H _ ,,.,:_/ 1.-~';.._, -\,,_...,_; • .,.. .. _.',o..;l,,• .. -'f•'>_,""L,,.,z ... ~ ... ,J,'I,.~' ._-,rr~1 .... •.,,,».'~ ... \••••fl i-'"O • .,...,.-, • \,~ .. ·~ f.'..-"•t...~•"•• ]'~,,.. ..... ...,. -- F,. ~- ·-.,'!!,, ..... _......:...-...._.:~-., - "• - ,.,.. ., ·-• ~~ - -• •
..... 0 .i::,.
T -~ ~·-·
·::==---------------~ ~l_..-""' .... '"',......_.......-,..., ........ ~ ...... , ___ 11,1_•tt prq • sat ._, "'•"'•~~ .. -....,.....~:fl.Wf$,,......WM Wif __ _ e~5""'"9t ... ai.c •:sses ... ...-"!l''f4:'!.:~,.~ ...... --~~-"',~ .. , -J • t··-~--""'·-..,~.,~ ' S .. •.t'"" '" '·, • • ""T ... ' ..-~ '. • ~ l
' l I t
"
J
13.2
I ,
11.0 1 -.
a.~a-
6.6·
Pn, (x) 2
4.4
2,2 I
o.o· 1
I
-1.0
,.
~OJU = lJ dB
wr = 3 I I *
* >. • 0
* >. • o.s
~
-2/3
Fig. 4-21.
•
>- • 9
I I
·* >. • 6
_113 o.o 1/ 3 :x
Pdf of quadrature noise fran l.irniter, p , (x) n2
11f-<4 ,.;,,, f-.1,,_. ... ~,'l-··-:..~"-t'°F·:.!'..:'!'lol"' :,.!'~=- .. .-,,.,,. .... ;::,1.-~ ... ·,11~·1·,,:;~-. ,•t ........ ~ ... t•/-!t-,., ·:,~::;.,~"'"': 't.~~.; .. ~::.:_ i,,',····~:-~.~.;~':t."''~ ... .,.\.· ..._
2/3
.... ~ _.l>,,,..
1.0
>--' 0 u,
,.,. .. ,~•;..• .. - ,.t· .,,:.'lc.l !l •, ,,,.,!J..•~ .,1 •• 1.~! :..~1
~-~~---~;;; .. ,... ... _____ ,.. ~. - :::;:.;i:il'8'---¥.::...-,~ ..... --t~ ...__,., ........ _ ..... ,r :-~·'-~--~ .. ? ... ~ •t. _,, ,_ ---'-~S-J ,!',.,•._~ Jut: -~---d-·-«~ *!' ~~"".:._j-·
18.6.
15.5 .
12. 4
9.3
Pn, (x) 2
6.2
3.1
o.o I -1.0
.-
c~!N0J0 == 16 dB
wr == 3
,,· ,/
*
,,,.
* r---X • 9
* --->. - 6
- 3
>. •O---...
* >. • 0.5
.:2/3
Fig. 4-22.
- 2
x
I
1/ 3
Pdf of quadrature noise frcm limiter, p , (x) n2
- 1
2/3 I
l. 0
\-' 0
°'
:;::-~,..;.;;:~:=.:.~: ,{:,··/·•'"•" •' ·:·'•""<•V, •,,_, .. ,:,,,, • •' ,. '.. ,,,,,•,•> ~ ,.· ~ <"','• ,~, •;:,.~.,,;.-_./-,·-'·' •·-~ '• ··---#'-'••'".......,._,,,.,~_.,, '• •
, , • , J .- _,_.,., - • ,, ~-.-•'1"•-,,·-=--~-•-•~"..-'"'""'--a.,..._,,M,-..,.,,,._, ___ -, ...... --..-.~..,._,_..,.... . .,.-.,.,,..-. "'-->*•~·
..... , ....
I
I \ i 1 .. ! '! -,
·:!
IL~ {} ll i
' i 1,'
l !
i' ,, ', ,, , I
l i J
!
I lJ ·---
,."'11'!."'M!!l!'!-!!'!l!--!\I'*'"""~--!!!!!!E!'e!~~!!'t:!!'P'l."':~~!'!'!--~·--&o""';•"!l'!±!!!<!-tffl>""at-•----•---'"'h-•-w,..•-•:.,---. -'"'-~---·-111*---.. -- ~-,,._,_..,.__.,....-, · ...,,, -~-1r~ 15-""
behaves in a siJnilar manner to Gaussian noise at large up-link [Ei,!Nol,
!nvever, at srrall up-link I~0], both pni (x-A') and_pn2 (x) behave like
1
*
107
As ). increases, inplying that the channel beccnes nore and rrore
linear, 1both p 1 (x-A') and p 1 {x) tend toward Gaussian density functions nl n2
for all values of up-link bit energy to noise pcwer spectral density
ratio.
In oroor to obtain P , the receiver error rate, we rrust take into e
account both the noise a::inponent of the limiter output due.to.the up-link
Gaussian noise and additive cla-m-link Gaussian noise. Let us assurre that
(1 and t 2, repre~ting the in-phase and quadrature o:np:ments of the
aa.m.-link noise a:xtpe.nent, are zero nean Gaussian randan variables with
vari.anoo Nd watts. We may then refine the total in-phase and quadrature
cx:mpcnents of the noise at the receiver input as
u = ni + f;l (4. 34)
(4.35)
The probability density functions pu(x) and pv(x) of the in-phase
and quadrature noise cx::rrpcnents may then be cbtained by convolving the
oon:esponding density ftmcticns o~ ni and f;l or ni and t2 respectively • . For exarcple the in-phase prooability density functicn pu(x) is given by
'
,,
I 'I
l
.. .
,, I -~
..;; l
:
; '-'l . ' \ * !
l\ ~) "
. '
•,' ~·
I t
i ,
l
_ ""I l!!!!!W---•-....;;..---•ta-:e ... Je .. •-•""*4-+w-•...,---•-• _,,.,,..,,,,_, -·,---.... - .... ~~,..,.._.. .. _,.~·. -,,..,.._..r~-..t
1-A'
pu(x) = I Pn' (z) p~ (x-z) dz 1 1
(4.36}
'4) -1-A'
where
pf; (x) p~ (x) 1 2 = = exp I-x /2N dJ
1 2 /2,iNd (4. 37}
are the prd:>ability density functions of the in-phase and quadrature
noise cx:::llpOnents, ~land t 2 , in the dam-link. The probability density
fm.cticn of the quadrature noise, v, may l::e similarly ootained.
108
Sincl:! eadl sarrple of the signal at the input of the '!WI' is
statistically independent and the 'IWl' itself is assurred to be rreno:ryless·,
it then follows that the sarrples at the output of the 'IWl' are statisti
cally independent. Using this assurcptian the ronventicnal integrate and
dunp detector will be replaced by a single sanple detector and majority
logic decision device. The error prooabili ty is then given by the pro
babili ty that nore than half the, wr = 2.e.+l, baseband sanples qre detected
in error (47], when: Wis the system bandwidth.
Using the binanial distributicn, the error probability can then
be carputed fran·
.9. P = , (2.e.+l) ! 2t+ 1-j' (1 ) j
e l (2R.+1-j) ! j ! P -P (4. 38)
j=O
:En this expression, p, the probability of cne sarrple bemg in
error is given by the prd:>ability that u and/or v, as defined in equations
(4.34-4.35), fall outside the rorrect decision region. It can be shown
-, l .
j j i
\•'
·' ,. ,,, .•,
' ,.
I"'
i :
' i l
' l
•:
·: t~ '; · . ..:
that 159]
p = P1 + P2 (1-pl)
-A'
pl = Pr[u+A' < O] = f P (x) dx u .....
p2 = Pr{ lvl - u tan ~ > A'tan ~ lutA' > O]
To detennine p1
, wie first define
y = x + A'
'!hen substituting equaticns (4.36) and (4.41) into
0 1-A'
P1 = f f Pn' (z) pt (y-A' -z) dzdy __, -1-A' 1 1
(4.39) yields
Now define ilie new randan variables a and e as
y = 13-1 13+1
z. = sin a
(4.39)
(4. 40)
(4.41)
(4.42)
(4.43)
(4. 44)
Substituting equations (4.43-4.44) into (4.42) and in~ating
over 13, yields after sare manipulation
109
'
:I • .; ;,
i t } 1
\ ' .
; ;,
i ,,
I •' '
I
..
' ·-. .,.:./,.· ; ,.,...
... ~ !, .. ._!.~~" .. ,,': .~., ,;~.
.. • ..
{ r
i ! '
(4. 40)
A sin ("n) 2
l"N u
[
A sin(°lr) 4' 2
- {N u
(4.45)
\
Similarly,. substitutioo of equaticns (4. 34-4. 35) into equation
yields after sane manipulation 1 1
*
P2 '!::! ,r2 I I p [-A'-sin(mr)] 2 n' 2
0 -1 l P [sin (611')] • n' 2 2
~ cos(-) cos(-) [
sin (~ll') sin(~) + sin (~,r) cos(~) J cm 611'
/N: 2 2 d
(4. 46)
110
F.quatiais (4.38, 4.45-4.46) allow us to calallate the prcbab:i.lity
of en:or for ~- M-ary CPSK system operating over a soft-limited dlannel,
where the integrals in { 4. 45) and ( 4. 46) Im.1St be evaluated mmerically. I
In figures 4-23 to 4-25, the probability of ~, P , is plotted e
as a :flmction of davll-link bit energy to noise paver spectral density
ratio [ (A 1 ) 2wr;2Nd] , for different values of noma.lized limiter softress ·
. * factor, ). , and different up-link bit energy to noise ~ spectral den-
sity rati~ IA2rll'/2N ] . Also shcmn in these. figures is the plot of the . u .
correspcmding probability of error for the linear channel us_ing .a ma:trl1ed
filter reatlver, the optim3.1 ~ceiver for. linear channel. The results
in figures 4-23 to 4-25 indicate tHat for a majority logic receiver the
--- .... p-----·-------------------------* 'fiu.,; is v.n a'°:l"'l,. ro:d..1i.c1.ti01 b.:--~=.·1 r».1 ._ 1 --' ind ,L·,· "' • ., , -.c.· ,...,'°.. ·.r. .... e ,1SsU1=.. 1..!;Jen~,ce OJ. n I. a.i1u n 2 •
) )
' ',! . . ;
l '·
. ~
p e
10-3
[Y,,01.i = 7 dB
= 6
MAJORITY IDGIC ROCEivER , wr '= 2.') 1
• A ..., 1
MATCHED FIL'IER Rro:I\1£R
-·---,· SOFT-LIMITED OI1'NNEL
10 14 18 22 26
Fiq. 4-23. PrcbabiJity of. error vs EJ!N0
, 2-p.~a&! CPSK
111
l l $ l ;
t
) .,~
l '
) ~~
' • )
! ]
'i • ~ . j ,, ,! I;
' . .. i > ' l
p e
10 14
LINEAR OiA"JNEL
* ).. = 9 ).* :::: 6 ).* = 3 >. = o. 5
µNEAR CIA~ ).* = 1 . ).* = o.s ). = 0
LINEAR O@NNEL
=-
MA'.lOIED FIL'IER RECEIVER
18 22 26 30
Fig. 4-24. Prcbability of error vs ~O' 2-phase CPSK
112
\ .. l ·~ ,I,
~ ~ . ,!
I ' , f ' I <
l,
' · .. l
! . .
\'
> •
I . I _; I ;. I
~ J
u
.· ~ ·.
I Ji )
'
' >
! . . I
i
I I
p e
10-6
10-1
10
wr = 3 wr = 201
* * -·- ). = 1 ). = 4.5
* * ·- -· ). = 0.75 ). = 3
* * -··- ). = 0.5 ). = 1. 5
* * ---- ). = 0 ). = 0
Ml\JORITY WGIC ~IVER, Wl'=201
MAJORITI LOGIC J®:EIVER, WI' = 3
MATCHED FILTER ROCEIVER
_' 14 18 22
F:i.9: 4-25. :Prcbability of error vs EiJN0
, 2-phase CPSK
113
a
!: ... " .. . , .: •,
~ '
l
? \ .,
I •
presence of hard-limiter or any fonn of soft-limiter can .inprove the
perfonnance of the system. Ostensibly, this superiority in the petior-
mance of the limited channel may strike sare cx:xmunicaticn theorists as
ananaloos. Ha.,,,ever, we point out that our claim of this inproverrent in
the performanCE of the anplitude-1.im.:i.ted channel can only apply to the
case of the single sanple detection, najority logic reaciver. This sarre
ronclusion has also been rrade by others [ 49, 59 J for the case of a hard
limited channel. The generalizaticn to tpe piecewise-linear limiter is,
however, new. Heuristically cne can justify the afore-mentioned perfor-
nanre inproverrent by ccnsidering the anplitude-lirnited channel as a fonn
of regenerative repeater. The "soft-averaging" in the piecewise-linear
envelope limited channel tends to recover the signal phase fran the up-
l.ink in-phase noise, the only source of inpairment for the binary case, . 4
and thareby redures the degrading ef feet of the total noise. H~ver,
without cbv.n-link thennal distumanre, such inproverrent is cnly marginal
l:ecause in this case the limiter in the repeater will na-, be cascaded
~ly to t.'1-ie hard-decisicn devire in the reooi ver and the corrt,ined ~. '
effect is the sane as if we were· to ronsider the linear dlannel.
Figures 4-26 and 4-27 depict similar results for the case of 4-
and 8-phase CPSK where the up-link and dam-link carrier to noise paver
ratios are defined as
(CNR)u =
(<NR) d = (4. 47)
To cx:nclude this sectim we .restJ:ess the fact that a majo-
114
. '
4 115 3 4PHASE CPSK
\ 2 (SNR)u = 13 dB t
; 1 WT = 3
10-• fCR>
\ 1 \ \
> \. 1 \ \
X R \ ~ \ l
\ ' - ?
\ -I i i \ ',i }
~ \ \ \ j \ X=6 } \ ;~
'
Pe \ t \ '
,>
\ ., -·
_, \ X =4 ., \ ·--.,.
;
\ : ~ ~ ·•, 10- \ \ \ ··, \ ,:
\ ~ \ X =2 !
\ \ X =O f '
\ '(~ : \ • i
t l
\ ,l ' \ \ 10- \ \ 1 ' \ :
---- Without Limiter \ ' \. .,
,1
With Limiter ' ' :, ;,
' f l ' ' ·· 1 ' ' ' ' 10-~ ...............
0 5 10 15 20 25 30 35 (CNR)
0 , dB
~-Fig, 4-26. P.rd)ability of error vs ~ 0, 4-phase CPSK
I'- . - ----- --.......... ,. ' •,
.
"
.. '•
\ ~
' ' ' :
' ! i
; )
; ' 0
. 1
10-1
X=4 -r-
X=O
'·'
... , ' ' ' ' ' ' ' \ \
\. \.
\ \
-- With Limiter
Without Limiter
f{R)
\ \
\ \
\
R
\
\ \
\ \
\ \
\ \
' ' ' '
8 PHASE CPSK
(SNR)u = 16 dB
WT= 3
' ' ' ' ....... ........
.............. --
116
,o-4-+-~~~-,-~~~~~~~~--.-~~~--r-~~~--,.--~~~--.-~~~~
0 5 10 15 20 25 30 35
(CNR) 0 ; dB
P;ig. 4-27. P.rcbability of error vs ~O' 8-phase CPSK
----------------
\ ;
J
l
117
rity logic reooiver is not a good approxi.matim to a matched filter re-
ooiver for a linear drannel f 47]. The perfonuanre of the rrajority logic
rea:i ver is general}¥ _p:xrrer on the linear dlannel than that of the
matched filter which is the optimal receiver for the linear channel. The
main reasm for using the majority logic receiver in the present work is
to avoid the difficulties associated with evaluating the stochastic inte-
gral arising in tie analysis of the rratdled filter receiver. Any a:nclu-
sioos reached in the case of a majority logic rereiver shc:uld not be
assurred to translate to the situaticn when a matdled filter is used.
Further investigations 01 the perforrnanre of an cptimal rereiver (in the
naxinum likelihcxxl sense) will be deferred until Olapter 5.
4.2.2) Actual 'lWI' channel
In this section we shall apply the saIIE rrethoo to analyze the
performance of a Q>SK system through a nonlinear channel o:ntaining an
actual 'IWl' anplifier. A typical rrodel of sudl a cx:mn.mications system
is as shown in Fig. 4-28. The noolinear quadrature m::x:lel of the 'lWI' de-
' velqied in Cllapter 2 will be assurred throughout the analysis. Similar
to the argurrent leading to equatirn (4.21), tre signal at the output of ......___,,,
the TWl' anplifier then becanes
(4.48)
where Z (·} and z (·) are as defined in equatioos c21.1a~2.19).
p q .
,O,~--.... -.-~------~-----~-"""""--""!'""--..,...-..--.-----·------~----~ .... ~ ; ~ . ,: '
('
' ~ i
I,
{
~· ·-~ :~
;I
i
UP- LINK / DOWN-LINK
NOISE NOISE J r----------, PSK- r I
SIGNAL ZONAL I I A I COHERENT f ESTIMATE ~ BANDPASS AM/PM AM/AM
Acos tw0 t+8l FILTER I I I'-./ . l DETECTOR \ OF 8
B·~{ 0,1r} . I I L ___ TWT_ ___ _J
kT st<(k+l)T x(t) zlt)
Fig. 4-28. M::ldel for binary CPSK transmissioo through a satellite type channel . '
f I•
···~
/
... - ~~-
..... ..... CXl
( .. : ....
i i
<{"---' \..
119
. The effective signal o:npcnent, s' (t) , at the output of the
nmlinear device is defined as the axiditiooal average of (4.48) over all 'i ,,;
R and c • Hence V""
(4.49)
where A' and f3 are the effecti~ signal anplitude am. pha.se shift oofined
by
A' = A~ [Z (R) ros e::] }2 + {~,e: [Zq (R) ,e: p
and
-1 ~~ •• {zq{Rl =sd] f3 = - tan ~ {Z (R) cos e:}
, e: p
where E_ I. J denotes the expectatim over R and e: • --il, e:
(X)S e:]}2
C)
(4. 50)
The follcwing equality is ~ in deriving the ~ressicns for ,
A' and f3.
R [Z '(R) sin e: ] ...... ~,e: p = ~,e:[Zq(R) sin e:J = o ...
( 4. 51)
~ on the specific quadrature nodel of 'IW1' as described in
equat:icns (2.18-2.19), the values of A' and e as cefined in equatioo
(4.50) cari actually be evaluated in cl~ fonn as shCMn in Appendix D.l.
The effective noise at the output of the 'lWI' is then cbtained by
subtracting (4.49) fran (4.48) and is given by
I
a •:.w:. a '-t S •\, > .. ~r 4( .. ~ a , .!f ... J .. *' ~ ' (
.. , . , . ' •;.
-:: i ~ I °' I I I •,
} ' t ,, ~ ' ' , ' I ·,, I
J; '
' ' . , J
~
•' •
. ,
•
120
= Z (R) CX)S{e: + S} - Z (R) sin(e: + 13) - A' p q (4.52) ,
~{t) = Z {R) sin (t + S) - Z (R) cos Ct + tl) p q
(4.53)
txJe to the a:nplexity of z ( ·) and Z ( ·) it is very difficult to p q
find the prcbabili ty density function, p , (x) , in closed form. H~ver nl
ni as defined in ( 4. 52) is a bounded randan variable and zp (R) , Z q (R) are
snooth.ly cx:ntinmus noolinearities. Ccnsequently the prooability density
functicn of ni nust be of bounded variation and can be expressed in terms
of a Gram-Chartier series expansicn as [28] 2
Pn• (x) == l
1
I 2nN' u
x - 2N~ co 11ic
e I k! ~ k=O
[x/lNT] u (4.54}
where N~ is the variance of ni and 1\ (x) is the Hennite polyncxni.al of
degree k satisfying
Hic+1 (x) = x Hic·(x) - k Hic-1 (x) _(4.55)
Too He.nnite polynani.als up to the twelfth degree are as~ in ' ' ,,,
Table 4-1. ,, ,,
n : '
\l '
' t ·} ;, > ; ~ 1
tt0
(x)
H1
(X)
H2 (x)
H3 (x)
l-14 (x)
H5 (X)
HG (X)
H7(x)
H8
(x)
H9 (x)
HlO (x)
HU (x)
812<x> ...
Table 4-1: HEFMI.TE POLYNCMI:AI.S
1
.... x
2 - 1 x
3 3x x -
4 2 3 x - 6x +
5 10x3 + lSx x -
6 4 45x
2 - 15 x - lSx +
7. 5 3& x - 21:x + 105x - lOSx
8 - 28x~ + 210x4 - 420x2 + 105 x •
9 7 x -36x + 5 3 378x - 1260x + 945x
10 8 6 4 472Sx2 - 945 x - 45x + 630x - 3150x +
. 11 - 55x9 + 990x7
- 6930x5 + 1732Sx3 -).039Sx. x
x12 - 66x10 + 148Sx8 - 13860x6 +5197Sx4 - 62370x2 + 10395
..
121
' .·~
' .. . { I
' l
' .. < :....,;
.. i . 1 " ' \
f l
· .. \
~· . . , ' ~1; ""': ..
I ',
122
The ooefficient ~ in equation (4.54) is calculated as the inte-
gral
= f H. Ix/INT] p , (x) dx a· -x. u nl (4. 56)
where n is the range of x and correspc11ds to O ~ R < 00 and -n .:: £ < n •
Substituting (4.55) in (4.56) ~ obtain a recursive farm for\:_ as
= 1\-i [x/ ./ N~] pn, (x) dx -1
(k-1) I 1\-2[x/./ N~] Pn• (x) dx ,0 1
(4.57)
fran which it is readily seen that the evaluation of each successive~
requires only the evaluatioo of the kth narent of the random variable n1 as given in (4.52) sinoo all other terms have been previously calculated.
It should be noted that the first three coefficients may be evaluated by
inspecticn as
hi = 0
For the remaining coefficients l',c,, k ::-_ 3, numarical integratioo techni
ques (for &tails, see Appendix D.2) are used to evaluate the required
m::mmts of n1 and hence the coefficients 1\.· An identical procedure is .
foll~ to obtain the statistics of the quadrature noise a:np:>nent ~·
''kl<> _________ ...
.. Jl '~ )
..
r·"'
' \
I ),
'
. . The results cbtained fran the evaluation of the variances of ni
and n2 indicate that at large values of the input carrier to noise ix,.-.er
ratio (CNR} I' defined as
(CNR} I =
123
the equivalent in-phase noise is suppressed by the limiting acticn of the
'IWI', and henre there is an increase in the in-phase CNR at the 'lWI' output.
'lllis is shown in Fig. 4-29 which depicts the equivalent in-phase and
quadrature noise varianres as a functioo of the input carrier to noise
~ ratio (CNR)r Also shCMn in Fig. 4-29 is a plot of the ratio of
the in-phase 'lWl' output CNR to the input om where the 'output CNR is
_given by
<CNR>o =
I
This curve is ccnparable to .results obtained by Lesh [52) for the case of
an error functicn type of limi. ter. This i.nprovenent in in-phase CNR at
the 'IWl' output suggests that sarewhat i.nproved error-rate perfo.mance,
' cx:mpared. to the linear channel, may be obtained for signals which a:n-
tained no quadrature energy, as in the ca,se of binary CPSK signals.
In principle, the performance of arrJ M-ary CPSK system can be eva
luated, similarly to that in secticn 4.2.1, ooce the prcbability density '
functicn of ni and the joint probability density function of ni and n2
, . . ,
.,,'
L,
• !
. ~·~·
,. ,,
,,'
' ' ~
I ; • . /
'
\ ,, .+ f
0:: w
0
~ -10 a..
w en 0 z w 0:: :::::>
!i ~ -20 <( :::::> 0
0 z <(
w Cl) <( :c a.. ~ -30
-- NOISE , POWER
o o (CNR)0
I (CNR) 1
124
--0:: z u -......... _o 0:: z u -
10.
5
0
---,-~~~~-y-~~~~--,~~~~--,.--~~~~...-~~~~4--2
-20 -10 0 10 20 30
INPUT CARRIER TO NOISE RATIO (CNR)1
• dB
Fig. 4-29. Variance of equivalent in-i;ilase and quadrature noise and ratio of
output to input CNR (2 .. s, dB input ~ back.off) •
} i
' -l l t . I ;
• I ' ' .
•' ·• .
,.
. . ·~ • 1, ~~ .. ::-(., ,,• ,'
'· I
l
J i
,
a.re known, albeit in an infinite series e}(J?,ansirn fo:rm as in (4.54) •
Havever, the o:xiputation becares rather cx:rrplicated, for the case of M
greater than two, unless &m; fonn of bounding technique is used [ 85] •
In the subsequent anaiysis we shall cc:nsider cnly the performance of a
125
binary a>SK system. transmitted th.rough the actual 'IWl' channel. Folla<lng
the.rrethod ooscribed in secticn 4.2.1, the prcbability density function
of the total interference, ni,, is then obtained by ccnvolving the e::im
valent in-phase noise density functioo in (4.54) with the Gaussian density
f\lnction of the down-link noise t 1 (having variance Nd watts) to get
\_ Q)
pl'\r (x) 1 f 2 pn 1 (x-z) exp[-z /2Nd] dz I 2nNd _ 1 .
(4.58) =
Substitution of (4.54) into (4.58) yields after sare nanipulation .
[38, W• 837)
pl\r{x) =
(4.59)
where N.r = N~ + Nd is the to~ noise variance and h1 = ~ = O fran
· equaticn (4.57).
The fitst term in equaticn (4.59) is the Gaussian term which
would be obtained in.the case of a linear channel and the remaining terms
represent oo.n:ection factors which acoount for the ncnlinea:r effects of
the channel.
\l ~, I l
- l
. ' . .... \_. ..
(' ~· ,' . '
1 l • i I
~
126
'!'re probability of one sanple being in error, p, is then given by
-A'
p = f pl1.r(x} ax (4.60)
Substituticn of (4.59) into (4.60) yields after sare :rnanipulaticn
[38, pp. 837)
. p
n -p .., (-l)n 2
= } erfc( /p) + ~ l n"! 11n IN~ I N.rl Hn-l [ np] 12; n=3
(4.61)
~pis the effective .received carrier to noise pc:Mer ratio, define.d
as
p =
The absolute cx:nvergenoo property of the series in (4.61) is discussed
in sate &tail in Appendix E. Suffioo it t.o sey at tli.is point that the
error which arises as a result of truncating the series expansion of
equaticn (4.61) at the (Irl)st tenn can be made as small as d3sired,
provided that Lis sufficiently large and satisfies
(4.62)
---
·~
127
A CXl'!pUter program has been written to a::npute the bit error rate
of a ooherent bina.xy a>SK system fran equations (4.38) and (4.61) for the
case of wr = 3 and for different values of up-link and dc:Mn-link carrier
to noise pc7Ner ratios, (CNR}u' (CNR) d defined as
(Om)u =
(<NR) d =
/ In this program the cx:ntr.ib.ltions in the ncnlinear oorrection tenn iii
equaticn (4.61} up to and incluling the eighth m:JTJmt are included in the '
cx:nputatian. This was found ~rirrentally to yield quite p.recise .results ·t~
in the range of up-link CNR o:insideJ:ed. For exanple the ~tage
ootitributian of the eighth teDn in ~ticn (4.61) to the bit~; ra~.
in (4.38} was found to be less than 2% for all up-link CNR bela-., 5 dB.
This o:intribution is still within 1% for the case of 10 dB up-link CNR
provided that the cb.-m-link CNR is kept small (less than or equal to 10
dB) • HQ,.1ever, at higher values of up-link and da'111-link CNR' s , nore
tenn'3 are needed due to the increasingly nan-Gaussian nablre of ni. Ty-\....
pical num:rrical values of the noise m:irents are listed in Table 4-2 for
different values of up-link CNR.
In Fig. 4-30, the oatputed bit error rates are plotted and
cx:atpated with the oorrespooding resUlts for the linear dlannel. using a
~ingle sanple detecticn and majority logic rereiver. The bit error rate
· ..
'?:- • T 't~·- 5::·1· "•@ J' WWW6*1 b '"~·...,? \ I S!Ji1!# .... !S!!.l!£-"*'42• ;Jh44UW!Jt:ae,,f(,,)i!IOC$LSS ... ¥..WL ,,; ,...;:,..,.\ ~-~ ....... ~ .. ·-.-w--. .1.,;'" ,. ........... ! .w:: ,~
,.... -- ,
.,
•
.. Table 4-2: NtM::R!CAL CAtaJLA'.I'Ial OF M:'f,1EN'1'S OF IN-PHASE OOISE
A 1
, S r UP-LINK CARRIER TO NOISE RATIO (CNR) U' (dB) mth MOMENT
(VOLTS)m I -10 -5 0 S 10
A1
(volt) 8.7105xl0-l 1.50309 2.29829 2.89994 3.18779
6 (radian)· -.43658 -.40186 -.36057 -.33576 -.33155
m = 1 3.3242lxl-O-~ l.49927xl0-5 -4.4488lxl0-7 5.67843xl0-S 8.73919xl0- 6
m = 2 4.58953 3.62611 1.76501 4.71406xl0-l ~.64520xl0- 2
m = 3 -S.91645 -7.50001 -3.99085 -6.998llxl0-l -6.10510xl0-Z
m = 4 4.16007xl0 3.93109x10 l.7564lxl0 1.98796 ,, 8.41965xl0 -2
m = 5 -1.09157xl02
-l.46993xl02 -7.1909Sxl0 -5.93829 -l.24416x10-l
2 , 2 2 -1 m = 6 5,10399xl0 6.63206xl0 3.2433Sxl0 2.0l804xl0 2.20425x10
{, - 3 3 3 -1 m = 7 -1.73932xl0 -2.86816xl0 -l.51070xl0 -7.43842xl0 -4.40788xl0
3 4 3 2 -1 m = 8 7.24872x10 1.29189xl0 · 7.2725lxl0 2.93126xl0 9,77318xl0
~-- .. ~~~·~ ,,•o..~~.::.~~-;:~;pt~J'07i'~'!JJJ!f', .. 1~4·f!'~~'4,;~·.~.~J~'!? ~·~~~:v'"__~ >~-_,-.-..~ ... ~- • .~~ • :; ' :' •• '. _: ~>,. • ,_.j,..-,.~-
I-' N co
.,. ~?.::'" ~··.·--~~:...:..' 1• __ :...._.~,:-_M:~'·J. .. ,.::.,h........::-::......1"'..:;~"' ...... ·i 11 .... ,. .. ~;..., ....... <.. •• ,,;;,,,.:•:1 ...... ~\.::~
·~
,, I ,.'
J l
' ' '
' WT=3 6 129
UP-LINK CNR=-10 dB ~=-.-===-,,-,,.. --.-=---:..---· - -. =:-..=---. -
-- ------·--··---·--·--·-·--5d8
5
--·-5
-··-LINEAR A
--LINEAR B
5
,o-4
5 .""' .
IOdB\ -~.5dB
""'· "" ,o-~ -10 -4 2 8 14 20 26
DOWN-LINK CNR, dB
Fig. 4-30. Bit error rate for a binary CPSK system as a function of received
~ (A; majority logic reooiver, B; matched filter receiver}.
,; r. ~ ·l " '·
r ' • '!'
i:: ... ~ '""'' •,
1f ~
...
'' , '.,
I
' i
I ! • ~
1 I
for th.is case is given in (4.38) with
p = (4. 63)
~ erfc(x) cenotes the a:nplerrentary error functioo as cefined
earlier, and
A2 .....
is the up-link Gffi Yu = 2Nu
= A2
is the down-link CNR. Ya 2Nd
Also sh~ in Fig. 4.30 are plots of the bit error rate for the
case of a linear channel using a matdled filter rerei ver. The bit error
rate for th.is case is given by
p = } erfc(~) (4.64) e
where YuYa wr
YT = Yu+ Ya
The plots m Fig. 4-30 exhibit an irreducible error rate far
large cb-m-link carrier to noise pa-.ier ratio due to the presence of the
up-link noise cx:::rrponent. It is clear frcm the plots that in the case 6£
a majority logic receiver, the perfonnanoo over a nonlinear channel is
superior to that over the linear channel. Th.is sarre cx:nclusion was
readled by Jain f49J and Lyons· [59] for the case of a hard-limited
channel •
130
,,
.. i ,. t
'
, I
t
..... ,,
\. •" ' ,. . '•, ? ... 'II:.~ ."" ~
,_:· ·~· : ' . '' ..... ,:
!\..,' ,~ • .:,
'-la•: . .-·,- --
CHAPTER 5 .
RECEIVER S.TRUCTURES FOR NONLINEAR WIDEBAND
BINARY CPSK CHANNELS
The discussion of the effect of thermal noise
• disturbances on the performance of binary CPSK systems in
section 4.2 has been confined to the case of single sample
detection followed by a majority logic decision. · In this
chapt'er we shall extend this analysis to cover two other
receiver structures, namely, the correlation receiver and
the maximum likelihood (ML) receiver. The analysis of the
correlation \t"eceiver is of interest because it is the
optimal receiver for the linear channel. This receiver
structure has been analyzed in the context of a bandpass
nonlinear channel by Davisson et al [29], Jain [48] and
Jones et al [50]. However, the method of analysis yields
results only for the case of pure amplitude limiting and is
not readily extendible to the case of a bandpass nonlinearity
exhibiting AM/PM conversion. Our method of analysis will
be somewhat similar to that of Jain [48] with the notion of
equivalent signal and noise at the output of the nonlinearity
as discussed in Chapter 4.
In section 5.2 we discuss the maximum likelihood
receiver for the nonlinear channel. This receiver structure
-·~-131
. , I ' I . !
' i
.... ,:i.·,. ... ':::_~~ ~· -...:~\ .....
. -· ..
! ·, ' ..
. ; ....
/
132
yields an estimate of the' transmitted phase .that most
likely caused a given received signal at the receiver
over one symbol duration. It i~ well known that for
the case of equally likely transmitted symbols and a bin-
ary CPSK system, as of interest here, the maximum likeli-
hood criterion, as used in section 5.2, yields the
receiver with minimum average risk per decision. Further-
more, for the case of linear channel with only additive
Gaussian noise disturbances the maximum likelihood receiver
can be shown to be equivalent to the co~relation receiver
which is optimum in a decision theoretic sense. However,
it is usually difficult to calculate the performance of
this maximum likelihood receiver and we resort to approxi-
mation techniques in order to assess its performance.
5.1. CORRELATION RECEIVER
A block diagram representing a conventional cor-
relation receiver for a binary CPSK system is shown in
Fig. 5-1.a. The receiver cross-correlates the received
signal with the reference carrier. The output of the·
cross-correlator, assuming that a~l the harmonics higher
than the first are suppressed, is then passed through an
integrator. The outpu~ of the integrator is then sampled
at the end of each symbol duration and fed into a thres-A
hold detector_ which yields the detected phase e .. J.
To simplify the analysis, we approximate the inte-
gration operation by sampling qnd weighted summation as
shown in Fig. 5-1.b. The choice of scaling factor, ~1-, IWT
:·"'-s.-:·----'::""""..,._..,,_ ___ :':'l""_...,. _________________ ~----~----------......
~~~~-----------------........ ---.............. -=-=;;;;..;.=---;;;;;;;;;;;;;;;;;;;;;:;;;;:;;;;;:;;:;;~---------~----------~~--=-.~ ,·. •• ~ ..... ,_ #4r • ' .,
1·,~~ .t; :l;.•,r."' : • ~ , • ·,~,~..._-, 'I"':,,,,,.......,,. ' _> ,_ J ) i1'<±UW~
,-"'.............,. • ~,:'·:. ~-~., ... :.,, ... ·~-;,. 1 _;.,_ f [tt4 M!!2!Jl - . ¥@ £!3.M.43:f:" £~ .... £.t,.;_.;ePwa:wJ-,,, .J.JWWW1l!P,? . ..,..o::.-...;,.,-• ........_,,..,.,...,.,. ,..,,(~ : : ··(rt$> >·.···>l .... ~ ~-··r., -· --- ~- ... ~ .... - -
z(t)
.
z(t)
nit>
/ (
nit>
\ ... . 1--;AIA~-~~ - ;
I I UMP= I ! J FlL'IER
I L
I
--' ~
~ ==~J ·~ J wk
-~ .----- -- ---- ,•
cos(w0
t - 8) sattple at
t - kT
Fig. 5-1 a. Actual correlati.m receiver, (k-l)T ~ t" kT
,------ ---, '
I BALANCED I:EM)OOLMORI I I f ~ • H1' ' I .. lt--,,-1 ~~ ~ I I( w j i wr z '.IHl&5HOID e i I""\. A J o- FlL'JER j j hl'l' ~l wj I . IE'.IrolOR : •
I ' I
L --·--- --- -cos(,;,
0t - 8)
I ..,_/
I ! .I -1
sanple at
t = (k-l)T + _j_ w
sanple at
t = kT
Fig. 5-J: b. ApprOX.itrate rorrelati.cn receiver, (k-llT" t ~ kT
I-' w w
"""'
() ·:
}
' }
·'• ,. . . '· .
'
·~
'•
is optional and is used here only for notational con-
veniences in the subsequent .analysis. In Fig. 5-1.b, the .
output of the cross correlator, after the lowpass filter,
is samp~ed every 6T = ~ seconds over the whole bit dur-'
ation. All WT samples are then fed to the thr~shold
detector. The transmitted phase, e, can take values of
O or n, ~ach with equal probability, and the probability
of making an idcorrect decision does not depend on whether
O or n is being transmitted within any particula~ bit
duration; Hence we shall assume throughout our discussion
in this section that e = o· is being transmitted. The out-
• put signal, z, to th~ threshold detector tnen becomes
z = WT 1 l W(jtiT)
IWT j=l
= /wT' A' + l I WT
WT l [ni(j6T) + , 1 (j6T)]
j=l
( 5 .1)
(5.2)
where A' and ni are the effective signai amplitude and the 0
equivalent in-phase noise component at the output of the •
134
bandpass nonlineari~y. The berm ~l represents the addi~ive
down-link Gaussian noise component as defined, in section 4. 2.
We now define
= WT }: · n' (jtiT)
j=l l cs.. 3)
• •. , .,.. ___________ o'.'!'i-____________________ _
' ·,
~- l
~ .. '
l, ~-' .~~, k·\ ·• .. •
,, , . .;. ~· .. ~ ., ... ; ,~
N 2
= l
rm WT I f;l (jt.T)
j==l (5.4)
Q
It is clear that N2 is still a zero-mean Gaussian
random variable of variance Nd watts. The characteristic
function, CN (v) of N2 can be written as 2
~ (v) 2
(-5. 5)
Since the bandpass nonlinearity of interest is
assumed ta be mern~ryless and each input sample is statis
tically. independent of all other samples, it then follows I
that t~e output samples n1 must also be statistically
independent. If C(V) and CN (v) represent the character-
135
! I .
istic functions of the samplesiW¥ and N1 respectively, then
based on the·"statistical independence of ni and equation
(5.3) we can write
(5. 6)
\ From the Fourier transform relationship between the
characteristic function and the corresponding probability
density function [ 27 J, we can expand the .. characteristic
function C(v) in terms of ·th~ central moments of n1 as
)
C(v) = l p=o
~[ iv] P p.· lwT (5.7)
l ~
. l " l
l l r
~ 1
...
. ·: 1, .
...
where mp = E I(np P)
and E[niJ = 0
136
Taking the logarithm in equatio~ ( 5. 6)
.tn [CN (v) J = WT in [C(v)J l co
~ ~EJ1 = WT in [1 + pk1 p. vWI'
(5.8)
Expanding m (1 + x) into a power series in x as [l,pp. 68] ... . l i
in (1 + x) = I (-l)l.- ~ -1 < x < 1 i=l l.
(5.9)
equation (5.8) becomes
(5.10)
where µr is the rth cumulant of ni and is related to the
moment mr by the following recursive relationship [14]
r l r ! ·
llr+l = mr+l - i.:li ! (r-i) 1 µr-t-i-i I\ (5.11)
Exponentiating (5.10) and expandi~g the exponential term
as an infinite series we obtain
µ2v2/2]. [
...
Jo (WT)m [ ~ r1 ~ (v) = exp [iv'wT µ1 - l µr iv mT r=3 rr ~] 1
(5.12) ,,
The exponential factor is the contribution of a Gaussian
random variable with mean µ1 and variance µ 2• By expand
ing equation (5.12) and collec~ing terms of the same power
of WT, we obtain
,: ~,., ... """""~-~,--~111111111 ..... ~.----------------------------------------
~ l
l t t [ i
1 I i
; ' '
,' ~
.. ,'
.. '
,. .
i \.13 3 rv /wr
137
\J~ 12]+ ......... } 31104 v
(5.13)
The probability of error can now be calculated from
equations (5.2; 5.6) and (5.13) as follows
P • Pr[Z<O] e
• i .. -ir1 j Im(c (v) c..~ (v) eiA' fflTJ 2Y
<- o Nl ~N2 . V (5.14)
where Im(.). denotes that uhe imaginary part must be taken.
Equation (5.14) follpws from the statistical inde
.Pendenee between N1 and N2 and from the Fourier inversion
rell:ltionship bet;ween the cumulative dist·ribution function
and the characteristic function of a random variable (27) •
. l .. I ·,
. ,,
,'.
•i . \ ·.·
. ',.•• ".
138
Substitution of equations (5.6) and (5.13) into (5.14)
yields an explicit expression for Pe with the use of the <
following integration formulae [611~
f 0
"" J v 2P cos [/WT A'v] exp [- y2v2/2] dv = ~ (-l)WT /2
-p e
2p+I . y 0
2p+l v
(5.15)
sin [r'WT A'v] exp [-y 2v 2/2] dv = (-l)WT jf e-p • y2p+2
• H2p+l ( t,p") (5.16)
where Hp(.) is ~he Hermite polynomial as defined in
Chapter 4 and p =· (A') 2WT/2·t2 •
After some manipulations the bit error rate can be
expressed as follows
Pe = } erfc[ Ip]
exp [-p] [" 4
H3(/2p) .; l + ~ H5(/2p) + WT y4/2,f 24 72y
_.ex;e C-eJ [ µ5 µ3µ4
(/w¥)3y5 12,r ffi' H4 ( {2j;") + 2 H6 (ffp°)
l44y ·
3 Ila cm>] +
\13
1296y4
+( 2
µ3 µ5 } exp ~ .. e] [ µ6 µ4 .+ ffl H~ (ffp) ll52y 2 +
720y 2 (WT) y /2u •
I 1 ~
! f
I J
', -'.~ '
> ,\
!, .
f ,,•,
+
where
power
total
2 y
and p
noise
139
4 \.I 3
H9
( l2p) + 31104y 6 Hll ( /fp) l -....... (5.17)
= \.I 2 + Nd denotes the total received noise
= WT (A') 2 is the total received bit energy to 2
y power spectral density ratio.
The first term in equation (5.17) is similar to the
bit error rate expression for a linear additive Gaussian noise
channel. The other terms represent correction terms due to
the channel nonlinearity. Equation (5.17) and the knowledge
of the moments of n1 are sufficient to calculate the bit
error performance of any nonlinear channel with a correlat1on
receiver. Of computational significance is the presence of
the WT factors in the denominator. · For large values of WT,
our approximate_correlation receiver approaches the.con
ventional correlation receiver and the computational effort
required in the computation of.P0
as shown in equation (5.17)
is greatly reduced.· Equation (S.17) r~sembles equation (4.63),
so the absolute c_onvergence property can -be readily estab-
lished as in . ltppendix E. For WT = i' the correlation
receiver is nothing but. a single sample detection as discussed
in Chapter 4, and equation (5.17) reduces to exactly the s_ame ~
expression as equation (4.63).
The probability of error for the case of a hard-
• •
...
I t. .
\ ' ' j ' 'i j t
' f ' !
'. ,0 ,, . . '
' ',
I I \ l
....
\
140
limited repeater was computed from equation (5.17) for
2 different values of up-link, (CNR)u = A /2Nu' and down-
link carrier to noise power ratio, (CNR)d = (A 1)
2/2Nd' as
defined in section 4.2, and results of the computation
are depicted in Fig. 5-2 for the case of WT= 100. In the
case of the hard-limiter~ different moments 6f ni can be
directly evaluated in closed form as follows
But
and [16]
whe:r:e
= (5.18)
l(j-2) 2 l j l COS [ (j-2i) E] + _j._! __ ....,,._
i=o (j-i) ! i !2j-l 23[(j/2) ! ] 2
j ! (,j-i)
3 l cos [(j-2i)c} i !2 ,..
j even
j odd (5.19)
(5,20)
A2 ~ the up-link carrier to noise power ratio
..
I I I
}
t i ' I i { l l
i
' J t l I :,
i !
l ' .
·,
1
5 - - - LINFJ\R OU\~
I-U\ro-LP1I'IBJ) 0 IANNF:L
WI'"' 100
(~R) d = -13 dB
5
5
10-4
5
crnR> d' dB
10-S
(om}d • -6 dB • -3 dB
5
• O dB
• 3 dB
10-6
-17 -13 -9 -s -1 3 7 11
<om>u' dB
Fig, 5..-2, Probability of error vs up-link. am, 2-phase a>sK where
YoJa i=i (OOU.cl + 10 10910Wr, and YoJu = C~R}u + 10 log1owr.
141
j ~ I l
J
I I I
l
•,. '
. , . ' ,,
' '',
1,,,
'' i ' ,•,
·: ;
and 1
F1
(a;b; -x) is the confluenthypergeometric function
[3].
It is evident from Fig. 5-2 that the error rate
for a hard-llitlted chann~l is higher than for the linear
channel at low values of (CNR)u'but that it gets smaller
as (CNR)u increases. If up-link carrier to noise power
ratio becomes very large the error rate of a hard-limited
142
channel and that of a linear channel will approach the same
value which is dictated by the down-link car~ier to noise
ratio, (CNR)d. The improvement in the error rate for large
(CNR)u is mainly due to the improvement in the output CNR
compared to that at the output of the linear channel (see
Fig. 4-29). Furthermore, in this region of high (CNR)u' /
the major contributien.t0 the error rate is provided by the
leading Gaussian term in equation ~-1~. The successive
higher order terms tend to increase the error rate, but
their contribution is so small that Pe still remains lower
than that of the linear channel •
5.2. MAXIMUM LIRELIHOOD RECEIVER
In the case of wideband binary CPSR signal transmission
tlu;ough ·a linear additive white Gaussian noise channel, it
is well-known that the oombination of a cross-correlator
followed by ~n integra~e and dump circuit and a threshold
detector as described in Fig. 5-1.a., provides optimal
performance. In fact, it is the maximum likelihood receiver
..
.~. ,,
'' ,,,
,, ,,,
'
143
for the linear channeL However, this is not
so for the case of a bandpass nonlinear channel with
Gaussian noise being added to the signal both before and
after the nonlinearity. This particular nonlinear
channel may be regarded (as shown in section 4.2) as a
linear channel with additive non-Gaussian disturbances.
For this type of channel, the maximum likelihood receiver
must be redefined, and should offer an improvement in
system performi,lJlce. Such a receiver is well-known to yield
the minimum probability of error when the transmitted
information digits are equally likely. ~
Following the same analysis as detailed in section
4.2, the output of the balancea demodulator, as shown in
Fig. 5-1.b may be expressed as
w(t) = aA' + an' (t) + t(t) (k-l)T < t < kT
(S.21)
where a = ± 1 depending on whether e = O orn is transmitted
and n'(t) represents the eq~ivalent in-phase noise at the
output of the nonline~r device as defined by equation (4.54).
t(t) is the in-phase component of the Gaussian,
down-link noise which is zero mean with variance Nd watts. /
If we consider WT samples of w(t) within one bit '
duration, we can rewrite equation (5.21) at sarnp~ing instant
t • k llT as
(5.22}
' •'
l '
!' I
where flT = ~ We shall assume in the subsequent analysis that
the probability density function of nk and ,k are known
either in closed form as in section 4.2.1 or in a series
expansion form as in section 4.2.2 .
and
with
and
•
. Define the new random variables
=
:c: -n' + t k k
the following probability density
Pz+ (z) = Pn, ( z) * p ( z) k k tk
' Pz-<z> = Pn,(-z) * p (z) k k ck
functions
where * denotes the convolutional integral.
(5.23)
(5.24)
(5.25)
(5.26)
144
The following conditional probability density function
of wk, conditioned on a, can be obtained from the equations
(5.22), (5.25 - 5.26) as
p(wkla=l) = p (w -A') zt k .
and
(S.27)
(5.28)
Based on the statistical independence of all WT
samples of wk' wit~in one bit duration, the cond!tional
likelihood ratio expression can then be written as.
-----~--""""" .,__...., .. -------·-----------------------------
'
I j
j i
l i l
t f
! I t f: }.
I !
I I )
l
• J I { )
t ,. ' .{
I !
1 i
~-\t , I
' t, 1•, '· ', •,
;
WT p ~f wk -A' J 1T L{ w.k : k=l, ... •.•,WI'} = k=l
p 71<[wk. +A:]
Taking the logarithm in equation (5.29)
.e. [ wk : k = 1 , • • • , WT J t £ n { ~ [ wk : k = 1 , • • •• , WT ] }
wr = l A(wk)
k=l
where h(Wk) = ln ~:::~::::]
145
(5.29)
(5.30)
(5.31)
The nonlinear function A(wK) depends on the statistics
of n' and e ~s shown in equations (5.25 - 5.26) and (5.31).
The decision statistics t [ ~ : k=i';-••• , WT J must now be fed
into the threshold logic and the decision on the transmitted
a is based on
a~ 1(0=0) if t[ wk: k=l, •.. WT] >O
Acoept
a = -1(0=,r) if t[ wk: k=l, •••• WT]< 0 (5.32)
For the case of a purely amplitude limiting channel
as discussed in section 4.2, it can be shown that
1 1 [ ( Wk-U) 2] P (wk-A') = f exp - 2N Pn, (u-A') du zt { 2irNd -1 d k
(5.33)
~nd
exp[- (~+u)2 ] 1 l p ( wk+A') • f 2Nd
p (u-A') du ZJt l2nNd -1 11k
(5.34)
j
l '• i .. ~
/
j i ! ; l
I \ l
t
!
. : .... '., ·,'
" ·'~' , ... · .. '. ·r ..
< ',• ,,· !'
146
where Nd= down-link noise power.
A'= equivalent in-phase signal amplitude and p , (.) nk
is the probability density function of the equivalent in-
phase noise component at the output of the limiter. /
From equations (4.25), (4.28) and (5.33 - 5.34) we
can determine A(wk), the desired nonlinear device that is
needed at the receiver. Figures 5-3 to 5-10 describe' the
characteristics of A(wk) for different values of normalized
* limiter softness factor,>.. = >./A, and for various values of
up-link and down-link bit energy to noise,pow~r spectral den
sity ratio. lt is evident in Figures 5-3 to 5-6 that A(wk) is
a bounded nonlinear function of wk which becomes more and
• more linear as A or up-link bit energy to noise spectral
density ratio, [Eb/N0
]u, increases. Heuristically, the inci::aiase
in [Eb/No]u essentially reduces the effect of the limiter and
in the asymptotic case when no up-link noise is present this '
maximum likelihood receiver should be the same as the cor-
relation receiver in Fig. 5-1.b. However, for fixed channel
and [Eb] , an ·increase in [Eb] implies that the contribution No u Nod
due to down-link Gaussian noise becomes less significant than
that due to the equivalent, up~link, non-Gaussian noise com-.. ponent. ;tn such a case the large component of the signal that
appears at the input of the receiver would likely be due to
the up-link noise component and should be severely compressed,
as shown in Figures s-7 to 5-10,in order to reduce this up-...
link noise effect. On the other hand, if the down-link noise
compo?)ent is larger the re~eiver characteristics should again
approximate the car.relation receiver as ~a,..,n in Figure S·l a.
' 1 \ . \ ~
147 ' 'J
I 0 ...,
i N
' ....._
I II
"' ,<
1 O'I ID M 11"1
rl . II II II
0 0 ~ I
I
I "' ... 11 II II LO
I ,< ,< ,< "' fj
) s I~ ,,.l ..-i
f ~ ~
ii:
,•
~ ' . ·~
• ,,
C> ·~ ~ .§ )
...... :.· §
r..> ~ (ll
~ . I ~
.t • LO I ~ 1
• ~ i8 .....
M
·~ ~ 'O;I' ~ • II II II
t ::t ~
~' d,;,,
•rl
:0:0 ~ .... fl i
1
~} I
,•,
\
.~. '
.,. !,
. .'·t • t
0 (· '>' -.. ,
..., i. -· I
·~\ • ~ • • I.I)
? Q ,. • ,',
..... I
-Cl
• ,e-
'\"-""--~ .................... ~----------------
l
~ 148 ;
• l
. ., 0 . -· ~ -l .,
r~ rl t .. f'- ..l It) . 0 ..
\D M N r-l 0
II II II II II .. .. .. .. .. ..... -< ,< ,< ,< ,< • ~ LO
-fj
] ·I ,s
] ~
i, ~· ~ ~
~o ,I
• ~-0 'd
'F' 1 ' 'i I
! •. ;
\
~ 'i ri
~ • ~ '' ) '
~ ' ~ 'j I
18 ~ -c,. I·. ~
~ I':
~ s . .q<
i1~ u II II Ji ,w'
t .,
s:s re, • ~ ~
$ • "o
~ .. :
~ ri. \
~J
L::.t ~ ·, .l ·., '
lo • ~;l \
0 3 .. ~ I -, ! \
1'1·
• ' ~ • • • 1 \' ) t1
(\.I -l 0 1 (\.I I ,
..i;. I I i ~ ~ I
( ' I \
I ~ i
) I
' ' \ \0 M
II II «
-<
"
• ,"(
' . ·:\., '. .. ~·-'
~ ... .._
' $8 18 ...... 0 ~ ~ N u u II
",·"ii
~ ::s re:, .,_ .. r-o '"o > ... • ~ ~l ',1'
'',\t
'. .'t' • • • I 00 U) ....
""' ~ f'"
~
•',
--~
N
II «
-<
II
I'
.. "'<{
.. 0
ti
• ,<
• • • • 0 (\J ~ U)
' I I I
• 0 ~
• lO
• 0
• ?
• ~
-· I
149
-r
.-i
~ 'U
1 ~ I ., ~ M
~c:: ii ·~ ~ .§ s ~ -:§
1 ~ ' ~ • .s-
&'.>, •
j I f J .. j I
i ! ~ r ~
i f } ( ~ t
-~~--=··,.·. .·;;-:~.::;;·:,?:.~.-. ~--------------, ' ~; .. ~,~ ~~·~ .. ~i}',. ~ _,,.,,-- .. , ./' ' .... ...(, ~~~"
- .... ~ - .. --~~.....- --·~~·-....... , ...... t .... ~s~ ..... - - ....... 2W!£ .. t a ....... ~ .. -.·--. ~- -
1.7- ........ ~~~~~~~~~~~~~~~~~~~~r-~~~~~~~~~~~~~~~~~~----,i
s.s·
o.
-9.5
*
WT= 201
[11/NoJu = 22 ea
[y.1oJd = 10 &J
l. = 0,
* l
1
-v. ~~~~~~~~.--~~~~~.~~~~~,.--~~~~~.~~~~~~ . .--~~~~~,,--~~~~~,~~~~~, -10. . --P· o. 5. io.
wk
Fig... 5-6. 91ari.1"ttD-liJr..elihood rntl.i.!Y-....aritv for soft-1..il!li.te<l C!larJJ"lCl
..... V1 0
\ l ' t ! t 151
• \ 0
-I
;a 18 M
I ~ II II 'O 'O '"o
'o ~ } i.;..;;.i H
• I LO
] ·g .... i-l !,
~ 1-1 '8
', h' I:' ~r • I,'~ 0 ·~ ' .
-14 ', .. ,.
I I,:,•
''
1 I ~ IS ,a • ~ ~ M ~
18 s £ 1 ~
M 'i • I ..._ . '. c r--" ',.,.: • u II \h Ii.,•,, .. i:, rt, -, .... .. l"""o '""o • \. ~ .~. ~
l . ~~ .• i.' .. .. .
1·.-. ,, ' ~ l..:J ,., IV
i. I
• '' I
~ I I
-· l
• v: ' • • ~
i4 ,• ,· ~ • 0 1 v .....
' '.-;· /. -< '•
.. ' ; ., .' .;
·:j .. ·-, I
,,.:=-.... : .... , .. "-.·. .•. ... ~ ~ , =·~. ~s>>;: ;-. , . _, ., . . . . =- . -........ ,-o ---"---· ~_:: · :::::::: e J ~ -~w - ,I t > Y5!2Z - .. __ --· · _.., '~
,,.
2---~~~~~~~~~~~~~~~~~~~~--~~~~~~~~~~~~~~~~~~~--,
1..
* l.. - 3
[1i{'liol u = lD d3
[;,,ttola = - 3 to 15 dB
• [fi/Nol d = l5 d3
Ar. I : =~ [V,,o]d ~ -3 <tl I
-2---,-----------,~--~-----,-----------,---..-------.!'----------,------------,-----------1-----------'1 -1.0. --5. · o. 5. 1.0.
wk
I • \., • • 'Pig. 5-8. ~lfr.eliix,od rx:nlinearity for soft-limited d>.n:nnel
.. -.- _ ... .,.-.. '_._.. _____ ¢ .. ~' "~ ..... ,,,"l. ... 'k:'>_..,. __ ~ ..... ---~- - -
1--J U1 ,:,..)
' I
~'
• • (\f . , . '
I ....
~·
• 1
• c:> ~
r • lO
• c:>
I
~
153
•
l
' ; I
l .. ; l
: I l J
i f i I'
I : I
} 1
l ' ' ' l
r l ~ ; i i 1
l
t I~ l t I~ I~
I
_,
z. I I
.1.
A("Jt1
• l = 9
[Yolu = 10 dB
[yoJd = - 3 to 15 dB
------ -
[~oJd = -3 d3
O•-;---:~~~~~~~~~~~~~~~~--:~~~~~~~~~~~~~~~~~~~.....I
·O
-1. t I l~o1 d = l5 &l3
-2.J-- 1 ___________ -----,1
I I ! . I .1.0. -10. -5. o. 5.
"k
Fig. 5-1'>. Maxilr..mt-lil:eli'iood rniline;rritv for soft-li.Jriitea dlannel
.... VJ .r:.
\l :! I
I
~ I , I
,. ' '\
.l . ,,', ' ,, '/ ·. ' ·~· :
. ' •'• ,,
:•
Figure 5-11 depicts a block diagram rep.resenting the proposed lMX.im..un
likelihood rc0:3iver.
155
Hnving selected the test; statistics, t [ wk : k ""l, ... , WT],
the error performnnce of this maximum likelihood receiver
is determined by its distribution function, which is generally
difficult to find without resorting to extensive numericnl
computation. However, for sufficiently large WT, t as defined
in equation (5. 3~ is a sum of a large number of statistically
· independent, bounded variates. Furthermore, since the received
signal wk is a samplQd vorsion of the received signal that is
continuous within the bit duration, then t can be shown to . have an unbounded variance for asymptotically large WT (66],
Consequently, we can invoke tho Central Ll.mit 'l'heorem [37] to
ensure that the random variable t is asynptotically normal
for largo WT. It is then roaeonable to speak of the ratio of
statistical mean oft to its standard doviation as a measuro
of system porformnnce.
If a•l, wao actually transmitttod, then fran tha asynp
totically normal distribution oft it can bo shown thnt ~
io aloo aoymptotically normally distributed with moan
µ • (5. 35)
. ---- ... : - ~ . . .....
,· ' :- : -;. <~/' .;. 1 .,, •• ,\ • .,.
,,': ·~ .. :·,'-; , .. _...,._~_ ,,_saz_ ~---- ~-- .. ·--it M ~-tL* ~-·'--- .h..f:-?..--~
z(t:)
~-nd(t)
~ A(.)
IEMU1U{.[OX
sa:i:ple at
t = (k-1)'1' + _i_ w
1 WT -v- IA(w.) j=l J
i'-ii
Semple at
t = kT
Fig. S-ll. .Ka:ximm-likelihcx)d xeoeiver, Ck-l)T < t ~ kT
~
IE'lE:'lOR.
EL l.
t-J VI
"'
l I
\
I j
...
• A'
,, ,; . '
.. ': ,•
157
and variance
1 "" -w2
(5.36)
We now define y as the ratio of the statistical
mean oft to its standard deviation as
y - ·O (5.37)
t and since W is asymptotically normal distributed, the
asymptotic performance of the system ise-completely determined
by this parameter. Since the transmitted phase o can take ...
the value of O or n with equal probability and the cost of
making an incorrect decision is the same whether O or n is
being transmitted, then the single threshold to.which the
log-likolihood ratio,~, must be compared can be shown to be
zero. Baaed on the binary docision that is mado at the
receiver according to equation (5.32), the asymptotic error
probability ia than given by
P0
• -Pr{t s Ola•l}
a ! orfc (y//'2) (5.38)
• whoro orfo ( ·.) donoteo tho oomplomontary orror function.
Figura s-12 illuotrat:ao tho aoymptotio orror porformanco of
tho maximum likolihood roooivor for a hard-limited ohannol
with difforont valuoa of up-link and down-link bit onorgy to
noioa epootral donoity ratio. For omall [Eb] tho parform~nco ~d
-~------------------------------------------~
l "i
j
I I ~ }
J
' I.
I t l
I
\ l '\ I
\ :
'.
. '
' ' ...
'.' ,·. \ .. ,
. ',l •' ·~: : ; • l ~ •• ' ' .. . \ ,,__ ..
t ••• ~' . ' '.'
, 1'.
I
: I. i
'' .' , . :,: ,·:,/•: I
•' .". •, •. I
''• ,: ;'~ ,'
p e
158
wr • 201
u •• M1\JORI'f'{ IOOIC ~.IVER
MZ\XI~ LIKELIHOOD nEO'.IWR
HM'CIIEO FILTER ~IVER (LINF..AR}
... __ . ·--. ·-·· - .. ··-··-
-- - - - --------
.. ---- ..
.. I -.... _ ___ -.;;;;.J ................... .._
>
s 10 · 1s 20 2s dO, [\.Mold Fig. s-12. l'orfor:m"lnco of binary CI'SI< trMcmiooim tht'O\XJh hard·limitoc\ channol
of the maximum likelihood receivor is slightly inferior
to that of the matched filter receiver for the linear I
channel. The explanation for this is evidont in Figure 5-7
159
which indicntes that the maximum likelihood rocoivor chnrnc-
teristic is more linear for small _1?. and tho stutistiC9 i · [E l NO d
of the received noise approach the Gnussinn normal distribu-.. -tion as assumed earlier. For large [Eb] th~ performance of
Nod
the maximum likelihood receiver yields an improv~mont ovor
that of the majority logic receiver, however, it is poorer
than that of the matched filter receiver for the case of tho
linear channel. Figuroa 5-13 to 5-15 illustrate tho perforrn
anoa of the maximum likelihood roceivor for tho piocowise
tinaar limiting channel. Of importance is the fact that the
porformance of tho maximum likolihood receiver approaches that
* of the linonr m~1tchod fil tor caao as X incroaeoe. Intu-
itivoly this ohould bo so no tho matched filter io nothing
but tho maximum likolihood rocoivor for tho linoar channel
and the oamplod output of tho matchod filtor io a normally
diotributod random vari~blo.
To conclude thio eoction wo omphaoizo tho point that ,•
our analyois of maximum likolihood rocoivor io not r~otrictod
to tho caco of a purely anplitudo limiting channel but a.loo is
oxtondiblo to covor tho oaoo of nn actual trWT ohnnnol •. In
ouch a oaoo tho Gram•Charlicr norioo oxpanoion muot again bo
uood, ao in ooction 4.2.2,in ordor to oharactorizo .tho proba-
I , l
l\ ''i I: ! !
; i
'. l • l , I ' !
\ f
'
. , .
I, ? f ~\ ~
. ':
-3 10 r.:>
10·4
MAJORIT'i IOOIC ~IVER
--.:::--..:::...:.-==... ---
--- ·--' ..
M1tl'OmD FILTER ~IVER __ ___,
LINFJ\R a 11\NNF.I.,
--- ). ... 0
-·- >.*. 1
............. >.*. 3 ,
........ - . >.* • 6
160
-- ·- - ·--
10·_5--"-------------------------r----------,---------.,_~-----r"'---
I • !
i
't
' . '
,, \ ,.,
" .
JI
•
., '
. -
·--· -u-
• A • 0,5
A. •• 3 ~· ~' A•• t
,
_.__ L.'CWAR Q~ . ,
10.
[rb/No1 u • 10 cm WI' .. ~ 201
. . .. '-'
• 161
;,
?
Fi9. 1•14, Port~~ of binM,y . .t:rm{ tl\inoroir1nicn tlu:o~ r.:,r~·limi.tcd Ql - ·1 "
I ' 1 ~ \
t \ ! \
1' '"l
i
l . ! .. r I I
' i •
ti
l
·f '.'
~
5
l ' 1
! I l
l 10·6 1
,, ,i s . II ~
1-' ',: i
Po t ... . ~ ".: "'""-•
10·1 ..
5
10·0
s .· .
r. c:• ... .. " .. 1it9
I .l ., I
10· 10 'I ; '
l2 I
!
1.H"
• A•• l A • 2
~room numR m:iUW'J\ .
t.nmAna~
• A •·6
" A*• 3
---- . IU\m-tJMt~ CRANNm.
\
. 14 . 16 · 18 .
(F0,'N0] u • 13 '19
wr • 2()1
,I
IB
. ao ta, ~A'fot,
162
i I I l
l j 1
. ! l {
l. -
. ,.
bility donoity function of n1. Onco thio io known, a
proo~duro, aimilnr to that davolopod in thio oaction,
may bo uood to dorivo tho maximum likolihood roc6ivor and
to aotJ(H.lD tllo por!oxm:u1<Xt of ouch n roooivor, 110'.-.\)VC\r,
bocnuoo olouod form oxpr~ooionc nro not nvailablo for tho <,
probnbility donnity function of n1 tho nooooumont of tho
porformnnoo ot thio rccoiv~r io a vory diftioult taok.
•'
• r
. ;,"
,, .I
( '•
163
I •
l J j
I \ I '
CHN'TER 6
6 , l) ctffl'R!llt1l'IOOS OJ? '.l'HE 'l'IW:SIS
In thin thooio an attonpt hM bocn mr.\dl to invootigo.to tho.
ofi'oct of bMdpMa nofilinonritioo on tho porfoxm.,nco of ooheront phaoo
' Dhift koying oyo~, A nood for ouch invoati9ntion.a.riooo in tho caou
of catallito cammi<lltiain whore tho trM!lmittod oignal (CPSI<) and
up-link noico aro pMeod through a 'lW1' on bcx\rd tho catollito prior to
rot.rnnomioaia-a to a rocoiving outh otntion, ~ ll\:\jor cootribu.tiono
of tho t:hooio moy bo o\llll\lrizod M toll<Maa
(l) I\ novol ~dmtuxo m:xbl of 11wr hM boon d::lvol.q?Qd, 1VJ
c:onplrod t-9 tho ot:hor oxioting m:xbln our propoooc:1 nonlin~ qu~c\raturo
~l ottom tho tol.lcwing <1dwn~at
(4) it. a\J.y nce4a tnJ ohoio:2 of tour ~taro to ~ivo o.
UOCld fit to tho natUQl tubo. na1Unoarity up to Md boycnd · ontumtion,
(b) · it ia . wo~; l>olll.\Vtld fot' 411 inp\l~ lovolo and pomu. ta
G~ghtfoxw~ OV3l.Uilti.Ctl of ~ noico stat,iotico . at ~ output ol tho
tub;l.
· (3) Fer ncnlinoor <.twi~o with AWPM oonvc:oicn, t:ho . ,,,,,-·
chomatoriotic:, of tho cpt.i.n\.'\l bM~o nonlinoarit:t the1t yiQld •
~ oicanal. to intorfor<.mco ~ m.tio ~vo .~ ~ivcd, . . '
(3) ~cd ex\ (l) M4 l2) t a po:Joibl.O in\)lem\~ti.on of ,ll
..
165
oignC\l prodiotortion ca1\xmootor in CMCncti with tho nct~l 'M hruJ boon
ou99ooteid •. 'l'ho carp<montor, conoit1ti~ of a oitq:>lo nrrMgontmt of
opproprinto a.twnua.toro Md (XMOr law ~viooo, hao tx,",n ohcwn, by
, oarputor oim.llat.ion, to yield m.x)Ut l dl3 il19i:ovorront in oyotcJn pt'r!'o.r
m..,n~ for tho c.,oo of n ningle c..u-rior por channol rrocb of oporation.
( 4) Anw.yticru. oxproaoiono for tho probwili ty dQnoi ty
funot:icna of tho cquiv(\lont in-phaoo Md qundrnturo noioo CQ'l{)OOonto at
tho output of a pioccwiao-linoar onvolopo limiting c.\:,viQ.) havo boon
c.bri vod, '111000 ~ooiono allCM otrniijhtfoxw(U'd ovolu~tioo o! tho
pxebobill ty of orror tor. M~ . a'SI< oignalo tranomi ttod thrQ.\,t\lh auoh a
limi.tor with both up- and cbm·link noioo prooont.
(5) In tho °'-"WO of an natual, 1.Wl', tho CkC\m-ChClrlior ca.rioo ox-
panoion h:la bee.fl tlR)liod to a.,~to tho pdf ot tho OCJui vulont in-ph"eo
noica OOITf?Otlent dt t:ho output ot 'lWl', Thia pdf ~ ~ uaod· to oval~to
too porto:rn.,nal ot binary G'SK oi9t1als trancmittod throucah. tho o.ot\14l
'lWl' channol with a oinglo DM\l~ d::ltoatic:n and m..,jority locaio deoioion
d:lviexl (lt tho xoooivor ao in eoyo (4). 'rhio onalyoio io ro~d.Uy oxt:on• . . . d1bl.o to tho \70,llOl"Ol co.co of M-m.y C,SK oyotom pravitbd {\ oiroi.l~
~w,icn io ~lco u:ad to ~to tho pdf of tho oqu.lva.lo.nt. ~atu.m : . . '
noico ·'tcnpcneht at thO output. .of 'lWl'".
.. (G) · ~r oxploro.tory rooulto hn~ alto· boon obtained for
tho =oo of binmy trDI< oigMlo tmtom1ttod tlu.'o~h a puroly enpli.tu=
limiting ncnUno~ chonnol with tho CC)t'J:Olat:I.CI\ tocaivor or too. m."Ud.nun
]4kolihood tctcivor, ln tho o.:\Oo o! ei ~lo.~c:n roQ:\ivor tho bit . . ~ ~ . .
..
166
t1ooowhnt oimilar to tho aloe of a m.,jority log;o roooivor. 'l'ha oxact ox
proooion t:OJ: tho porfol.il'Mco of a n\:Udmun, likelihood roC\.iivor io gonorally
di1'1'icult to do.riw. HO't«}Wr, M npproxim.,te oxproanioo hao ~.n dori-. .
wd baood on too ooourrption that tho otntiotioo of tho output of tho
m:ud,.nl\un likolihood roooi vor, prior to tho throohold cbtootor, r:xxJY tho
nOl.'I\Ul difltributioo lciw.
6 • 2) SUOJESTICNS I1'0R IlUl\IRE S'J.WY
'l'horo <U"O m.umrouo di.rooUono in which tho procont otudy «Juld
bo oxt.ontbd, A few ot thoco aro liatod bolc.wa .
( l) 'IM Qm)l:' .r~to Molyoio of ~ ~Sl< O~il\al.O tronoroi ttod
tfm:>ugh tl na\linOM' ontcllita c:honnol with cooi)inQd Offoottl of int.el"llym-' .
bol intortorcnooa Md ndditivo, thoxm.'\l, ~~ and dcMn·link noiooo ohould
bo invootig~tod, In ~r to colw cuch. a o::rtplox pxoblom, tho choired
oignol and intorfet'OOQ) tol\'ro at. tho O\\ti,Jt of tho ncnlJ.nO(lt' d:lvico
nuot co cpocificall.y (bfinod in ouch a way that m cvtlluation of tho
ot:L\tiotica of tho intortorcnco ~ Ci.'1\ bo .reM.11.f nQQQ\\)liohod, ,\ . . .
. . . . (2) ~ o\l'OrQll t>:onofor ~O.d~iot.:Lcn ot a bM~o ncn•
U.n~i ~ ~at m:ud..l\\tCQO 1 to O\,l~\\t. .=ioi to intorfoJ."llOJ. pQ\'Ol' r..,tJ .. . ' ' .~ th:l =co of l'l\\ltti~iar ~t o1~l oow bcon ~oct\\l'Cd 151,
, • I '
92) to bo thot. of n piOa:'R,i'co-linotj: cnvcl~ limitor with com cnvolo-. . ' . .
pp tbpcn~t phaco ohift at tho· qutp\l~, Ucr..iovo;o, thol."Cl ia no oxiat.tn1 . . -. . .. . (
cn~it1 tho.t j\\Otiftoa th.to ~oot~: ~ tho coco o!.- l\\\3lticmio.r ' ..
' ' ~. ,•
' I
\
'J l I
l I
I l l \
167
input. ·Ot equal inportnnw in tho ~oator onalyoio io tho har<,\,/arc
in{,lannn tntictl ct tho maul ting COlll,)('l10ntion notwo:'k and tho oe;,noi ti vi ty
in tho portotn\,nco ot tho ovor~ll C(ll\Xmontod 1Wl' an oelch catp00ont in
tho ~oation not.work varioo.
(3) '11\o h(U"('woro inl>lom:mtatiM and tho IX)rfol.'n'(U'\CO analyoio • of tho nwc:1.nun likolihood (ML) tocoivor ohould be caltii&:n.\1d, 'l'ho cxnot
<»q>rooDial for tho porfo::m.inco of ouch a roooivor m."\y not bo poaoiblc
and a\O ll'AY havo to rocort·to a conput02: oimulation tochniq\x> in order
. to accooo tha ~t in dctootability of ouch <l ~ooivor ovoi: m:,ro
eawontiori~l, oxioting roooivom, . . .
(4) . Tho porfoxm.,noo malyois of tho corrolation i:eooivor and
tho Mt. rocclvor in Ch~to,: s Dhoul.d bQ omon~d to tho ~o of M""M'Y
.. . '
(S) 'Ibo ov~luat:J.on of tho pcwor opoatr\Ul\ l.lproad of CPSK
ai~al.o ®u.cqd by 'lWl' anplifioz:o nonUnom:itica io ~o Qll ~t
prcblcm, . Exc:oooivo cp~c.d pl'Odutco <lajoeont tronopm.tbr · (or <'ldj~cx,.nt . ' . . .
ch.OMQl.) . .intol'tol'QnQO which m.,y bo unflcccpt.abl.o. ElhM"PQr filto.ro at tho ' .
aato1U to ~pontor oU.~\\t.. Md/or Clt. tho oN.'th ota.tion input:. · c.,o\\l<l bo . . '
u:o4 to.keep .thi.o un~tod intorfoJ:cnOl wi~ ~=pt.eiJ)io lovo~a. now-. . aver, it io WJl.1 knam 1-.Mt DllQ.l'Par filtom will introduea ndditicnal
:in~ intorZarcnQQ. 'lbuo, a. ~co io no~ootu.'Y, M4 in oMr . . . . to ~\b ~J\ ~t.int\1 ~"Off ~t\>.iQen int.oroynto~ intorf~rcncc o.n4
r~ao:nt eQ\~1 intori'.~al_, nn naaumta o\r\\l~tiQ\ of tho ~ror • ' • • l ... • .
QpOCtr\Ul\ cprce.d of ~ oi,~"l io jWl~ CO ~Qnt. ® M t\OtOClTOllt qf ') . .. , ,
·.~ oi~tcm'o porform.-.n~ i,n.~ p::cccn~ ~ ~tcl'Q~1 int.crfcnmo:. '
' ' '
..
& '
168
•
and ttdd.itivo up- Md cbwn-link ~uoaiem.noiooo.
(6) 'lho ontiro analyoin pre,oemtcd in thitJ thaoio io OOtJ{Xl on
tho MOUlll.)tiCtl of par.foot oynchralizatioo in both C&Tior and bit timing
:rocovory, Such M Mowrptian, though groatly oirrplitying our Malyoio,
io not readily juotifiod without oncrifioo in harcworo ca,plQXi.ty {lt
tho rocx.ii vor. carrier phoco on:or and bit timing error al\l woll kna.ffl
to rooult in furthor ~rnda.Uan in tho porlol.'ffl.'lnco of CPSK trMomiooion
oyotorn thro\lgh n linoo.r chMriol. Tho invootigat:J.on at Guch oync:hra'li·
iation problem, tor tho nCtlli.noor channol Dhould aloo bo ~iod out •
\
'•
, I I
1 J
J\PPENOIX 1\
Tho bMio input-output rolatic:ru:ihip of tho 'lWl' io no~lly
ob~od f-rom output pc:MQr (dBW) Md output LW'CO (dogrooa) iroaourom.mtn
f1>r VL\l'iouo VL\lu:ia ot input pcMOr (dBm) [l3, 31], Tho output daU\ l1ro
• • tnon tranafOlm!id into inotMtMOoUa onvolcpo to:- tho caoo of oinglo
cru:rior rrocwurom:int by WJin; tho woll·knc.wn rol11tionohip
2
p ·+ whore I> io tho inotMtanoour.i oignal pcwor
Md' A io tho.oi~al Ml?lit\\Cb,
'Ibo inotontMO.ow.t cnvolopo t:.rMtlfor funotic:n Z (2') and pho.co
chift, +he) arc thon aal'binod to yiold tho I\l.')O.OUl'Od qu~..dratw:o irocbl "'}
of tho ~ubo to which 4 ow.~ly ~ eot of ip{x) and Zq(x) c:M bo
dltonni.nod, Uoing oqunti?'a (2,18) and (2.U) WJ m.,y· n'"""' fOlTt\ on
\namotmincd c::bjcati~ fUnctj.on
. M . .
3 ~ Ji [ (II!!\) coo t+(3t)l • ~(~1)2 + (Z(3tl ai.n. l9(3tl I • .
. ' • , . . •q<11~.u· J. . (A.ll
~ M !.11. tho. tot:Ql n~ of. o\lb41,v.l.ai.aio. ot ~ i:onu:i ot ~ UlP\lt
. ' ',
\
170
S~oo tho right hood oido of oqutition (A.l} io a oontinuouo,
diffo
zq ( •), n vontional optimi.z~tioo oubroutino oon then bo uood to mini-
mizo J by p r colootion of thoco ox,ffioio.nto [8].
~·
\,
\
If o nnrrowband oignnl,
u(t) - R(t) ooo(2wf0t + c(t)J (B, l)
with a.mtor trcquonoy f 0, io fod ton rrcncrylooa noolinoar dovioo whono
output v io a tunatic:n of it:o input u at tho owm inatMt, then thio output ,.
o:ul in 90J10rnl bo wri tton .M
V(\l) • V(R (XX) 0) (D.2)
whol'o o • 21rt0t .+ c (t>
Sinoo (Jl,2) io M ovon poriodio tunoticn ot o, it io roprooon
txid 02<Aotly, tor all o, . by tho Fcurior corioo
end
• •
V(R CCO 0) ~ f v0(R) + -L Vm(R) ~ mO m-l
(ll, 3)
ff
i ! v(n coa o) coa mo do (D,4)
r GOO •l(WR) ,cqllQticna (D,3) end (D,"4) OM llo .,
v(u) (D,5)
" ..
'
l I I
\
172
R
vrn(R) • i J v(u) Tm(u/R) [R2-u2J-l/2 du (D,6)
•R
Tm (x) A a,n (ll'OO!:I •1x) , io tho Chobyohov polynanit1l of
~ m {l), Equationo (B,5) and (B.6) nro gt"norally knONn no tha '--- -
Chobyohov t.rm10form pair. Equtltioo (B.G) rolatoo tho mth zono cinvolcpo
ncnllnonrity to tho inotnntanoouo voluigo trcmofor cnnrnowriotiaJ of
tho bandpnoo non.linadr dovioo. ' ' In m..,ny awo9, ~vor, tho problem is not .to tine\ vm(ru tor n
qi~ v(u) but rat:llor to find what .. v(u~ will yiold cam dooircd vm(R).
It io thorofom noCX)ooaxy t:o knew hew to invort:. tho tt-onotornl\ticn in
(B,6) fer v(u). Dlachmln (18J hao pointed out th(lt tho invoroo
Olobyohov tt'Mof'cxm toz: ony m:bitrnry m can lxi o><proacod an tt/2
v(u) • f [ (Vm (u coo ~) om (ooo ~'. +
• l. l(-l)m ,I uv• (u ooa ,~I !i tl d, (B.G) m
V~(x) • .. ~ (Vm (x) J
Do (X) • · 0
n1 (x) • l
~1 (x) • Dm-1 (x) + 2m Tm()(} (D. 7)
Equutia, (B,6) holclo t.rm to within nn arbitrru:y odd (wh~n m in
oven) or own (who.11 m in ocld) function of u nncl/or MY polynani.nl in u
of cb9roo looo thM m. For tho cru.10 whc~ m • l, ~Uoo (0.6) CM bo
einplit'iod to
V(U) • ! r2
[V l (U CX>O +) + UVi (U COO +) CX>U ,j) d + (D, 9)
pluo an ntbitrnry ovon function of u. :_;.
To cxnolooo thio appendix wo ohnll work C\tt. tha oxprooaioo for
tho in-phMo intJtMtMooua voltll(Jo nonlinoari ty NP (u) that will yiold
tho ~oirod firnt-zcno onvolopa na,llnoo.rity ~(R) o,cproocod in te.xtro
ot a polynanial in R M
M c&lit.ivo ca\Ot:lnt 0.0
M'
\ (U) • Ji k°i(U2k-l
-- Uoing tho followinta integration fornuln (39]
. f 2k•l 4 . • _L k;1 (ak•l) 11l&JF'1•t=.ill1 co:i • t ~, ·j --2~ • :1•0 .
' . I ,f
~(U)
(D,10)
(D,11)
173
.,
174
, ..
. (
I • I
' I
.. • 0
•
APP1:ND1X c
THE OOISE STATISTICS NID CAI.aJIATla.l OF A' FOR PIF..O:.'WISE
LrnFAR AMPLITUDE UMI:TING (SOFT-LIMJ:TING) OINMI.S.
C.l) Derivaticn of the noise statistics
The equivalent in-phase and quadrature noise ~ents at the
output of an envelope.soft-limiting repeater can be expressed as;
x - A' 0 ~ l.+y2 ~ 1
' .. n' m .,, 1 2 2 x
- A' Y. +y > 1
I x2+y2 (C.l)
and
2 2 y 0 ~ x +y :£ 1
n' 2 =
i: .;+y2 > 1 I I X2+':f2
(C.2) ('/ •
where x A+~
= ).
,, n2 y = T (C.3)
'!, , ~~
'
;, .. >'•
·'!,
175
,·.
176
Sinre· n1 and n2 are assurred to oo zero rrean, normal random
variables with varianoo Nu' then the probability density functions of x
tmd y, as defined in {C.3) can be written as
Px<x> (C. 4)
Py(y) (C.5)
where NI \.I, r/ J denotes the pdf of normal randan variable with rrean u, and
2 var.i.ance a •
C.cnsider first the' in-phase noise cmponent. The probability
density function (pelf) of n1 is defined as the derivative of its oorrea-• ,.
J?Onding probability distributiai function (PbF).
Pn,(a) A 1
d da {Pr[ni < a]} (C. 6) ,.
Fran equatioo (C.1) , the probability distrirutioo functicn of. ni
can be obtained by integrating the pelf's of x ani yin equatioos (C.4-
C.5) over the appl.'q)rlate regicns in x-y plane. To facilitate the
analysis~ shall sulxli.vide the range of a into.different regions.
For a+A' < -.1, it follows that
2 2 Pr lni < a] = Pr [x < a+A' < -1 ; 0 ! x +y ;: 1] +
Pr( x < et+A' < -1 ; 1 < .,l,+y2
] I >t'-+y2
== 0 cc. 7)
,I < I 'j
,, ) •, 1
11,
...... _ ..... .,...
.....
Similarly for a+A' > l,
2 2 Prfn1 < a] A Prfx < a+A' , 0 ~ x +y ~ 1] +
< crl-A' 1 1 < x +y 2 2]
• 1 (C. 8)
In the range O < crtA' .: 1, the region of integrati~,,.d ", is
given by
x ;S a+A'
,d/ •
Jtl > /l-(r,f-A'~2 x (a+A')
(C. 9)
It can J:e shewn that for -1:; l)+A' ~ O, the regicn of inte
gration, ,,J I is the cx:,nplenent Of ,,J, ~ and iS given by
x < a+A'
d =
:J.tl , /l-(r,f-A'~2
x (a+A')
(C.10)
The regions .eJ and d / are as shcMn in Fig. C-1.
_Too probability distributicn functiai of ni is then 'given by .
177
' i
,1 ~ ,
" ' ' I
,, I i
. '
. ,
~' i
•, . 1 •
" :, I
O c, a + A'
Fig. c-1.
178
y y
' x
d 1 -1 d a+ ,.. , " 11!1. 0
Integration of regioos u:aed in the .in-pha the dete se noise s tati rmi.na tioo sties.
.,
·.'
0 " < -1-A I
1 a > 1-,A I
Pr[n1 < a] • 0 < {a+A') < 1
-1 < (a+A') < 0 -(C.11}
Substitution of equations (C.4-C.10} into (C.11) yields
Pr[ni < a)
where
0 a< -1-A'
1 a< 1-A'
0 < a+A' < l - -
a+A' [ } f ~[~] t(-}..x/ 1-(a+A')i] - i dx
- 1Nu /ca+A')i; u
-1 < et+A' < 0 - -(C.12)
+ {x) 1 =- exp(-x2/2)
x
t(x) = -1:._ J exp(-i /2} da = -21 erfc(-x/lI )
Iii·~'
179
I
I I
r
) I
,i
'
~--
•
.. l :
.. ,;
Tho probability oonsity function of the in-phase noise, n1, ia
cbtru.ned by differentiating the distribution function in (C.12) with
respect to a to get
Pn' (a) = 1
A(a+A')
nr u
t [ A(ctl-A')
l"N ·u
la+A' I ;:,, 1
- ~] } la+A'I < 1
(C.13) ..,
In the case of a hard-limited channel substitution of >.=O into
(C.13) yields
[pn' (a) J >.=O = 1
0'
A(a+A') t[A(a+A')/r' Nu J}
/Nu
I a+A'{ > 1
I tl+A' I < 1 4
(C.14)
The in-phase noise pdf as e,cpressed in (C.14) is identical to
that cerived by Lyons [58, eqn 20] •
Similarly it can be socwn that the probability distribution
180
I 1
' i
1 , j A
' l
' ' {
•,
' • l . ,,.
' function of n2 is given by
•
0
1
l - !.Ji, Px (x) Py Cy) dY.dx
~
i : L : ,tr1l r.1,, {y) dY..dy ~.,- f\ \ l
~-~:/:' I
'
a< -1
B > 1
-1 ~ a :5. o·
(C.15)
where O < 8 < 1 ~d -1 <'f. < O are as shewn in Fig. C-2. After eorre - .. ... -
181
I ~
manipulatia,s , equatic.n (C. :Ls) can be expressed as .
0 B < -1
l B > 1
-1 ~ B ~ 0
(C.16)
j '!
,)
i I
-1 l,
i ·j
i I ,, ! '
. '
I I " ~ ,•
y
' ' . • l
.,
- \
Fig .. C-2 • ·:'
II y
-1 t'I 8 -' 0
~ the determinatioo ims u:sed,in Integraticn reg statistics. .
ture noioo of too quadra
182
''-- 1
,
' \
183
Differentiation of (C.16) with respect toe yielda the pdf of
' p~ (8) ~ ~B {Pr(ni < 81}
0 I al > 1
I I
). +[ YL] [ 0[ Ji7 - ;-;:;:! J } - A] _ t[ -~ 1 e - A
l"N I Nu I Nu ~ u u
(C.17)
Substitution of A = 0 into (C.17) yields the fdf of n2 for the
. · case of hard-limi~ dlannel as
0 ;;
,-
e,q;,(-A2B2/2NUJ
I 21r c1-e2>
I Bl > 1 }
{2+ [A/ 1.-e,2 I IN:] + AQ { ,· u l1r'
u
(.,.....-,-'
'!
' 'l
l I
1l
1 1 I
,I !
' l,
I . I
I
.,
I ..
! ( I
184
c.2) calculatioo of A'
oignal anplitude A' fa &fined in section 4.2.1 ag
A' • ER (f(R)cx,s c] ,c (C.19}
Pran tie definiticn of f (R) and the well-Jcnc,..m joi'flt probability
dznsity functioo of R and c, equatia1 (C. · 9) can oo written as
A' = E Ja::,a c) -e
( t !HR/A ) J i:ir =[- i? + A2 ~u 2AA coscj d<dl!
(C.20}
Integratia, of equat.f.oo (C.20) with respect to£ yields
A' = E !cost} -&
). ! (1- (11/~J ) :u e -p e><p [-it /2Nu) I l [AIWUJ di!
(C.21)
' Dafining x = R/>. and suootituting the well-k?la,m idantity (17)
Ee(a:::>s e) = ~,,--;;e-p/2c10
(p/2) + r1
{p/2)}
into equatia, (C.21) yieldz the desired result in equation (4~25).
I ...
I ''
\ )
,.,, .. '~··
'
J\PPI:1-IDIX D
. CALO.JI.ATICUS Of' A' , e AND VARIOOS ~f(.)ISE M)MF.2'TI'S FOR
SATELLITE OWlNEIB
D,l) Calculaticn of A"' and e The effective signal anplitu:Je, A', and phage ahift, e, at the
ootput of 'IWr are oofined as
A' • / {ER [Z (R) ccgl] }2 + {ER [Z (R) cogi:_] }
2 ,e p 'f. q
(D. l)
where ER [ • l denot:P..s the eY.pectatioo over R and e • ,e
For the case wh:n the .input signal to 'IWl' is the sum of sine
wave plus narrCY.v'Dand Gaussian noise, the joint probability density
function of t:re envelcpe, R, and phase,c, is
p(R,c)
woo-re 112
R 2 21ra [
A2
- 2AR a:,s e + R2
] e,q;, - ---....,,.....----20
for o < R < ~ and o !£ 1 21r
iB the input noise variance
(D.2)
A is the anplitu:Je of the transmitted sinusoidal signal.
185
I
I
,,
I J
and
x • R c.rx: i:;
y • R sin ,;
P0 • ER {Z (R) 0:XJ ~} ,,.. p
P1 • ER (Z (R) a:,g r.} ,, q
(D, 3)
(D. 4)
(D. 5)
Substitutioo of equationg (D. 2-D. 3) into (D. 4) and (D. S) yielda
--
Since
p ,.. 0 2 2 2
1f C1
p =
11'
f e)'.p { c2 ( x2 +~/ > cos 1i J d~
0
is the input CNR
(D. 6)
(D. 7)
(D. 8)
186
r I
t
•,
,, ..
,,
r +-
2a2
" ,c1 f e,i:p{-cdl-~ ) }di$
O 2M (D. 9)
(D.10)
where n • 1
p • 0
(D.11)
: In oroor to determine P
1, we start by substituting
11'
2 2 1 ( 2 2 Il {S/x +y ) } ~ 7 J e,i;p{S2 (~ +y ) 009 t}C03 t d~
0
into equatioo (D. 7) , and after sore manipulatloos similar to the
previOUB steps, we cbtain
187
'/
• '• '•
' ;
..
' ~-\
L
. '
188
- µ ] {~
(D .12)
where
and
n• = 1
In terms of PO
and P 1
....:e can write
A' =
=
/ (Po>2 + (P1)2
-1 -tan (P 1/P 0)
D.2) Nurrerical evaluation of nrnents of in-phase ooise rorponents.
'Il1e output in-phase noise o::rrponent, ni, is expressed in
teD11S of the input envelope and phase, R and E:, as
-C R2 2 n' 2 oos (E:+B) -= c
1R e r
0 (C
2R )
1
s1
R e -stl" 2 1
1 (S
2R ) sin (E:+B) - A' (D.13)
'Ihe kth rrorrent of the in-phase noise is cbtained by averaging
the kth ~r of ni with respect to R and E:.
--....-.-x,,..·'"'1 -~.;azt---.-• ---
. ,
...
. -,
"" 'IT
= I f 0 -n
= [ -_R_2 _+_A_2..,..2N_-u_2_AR __ cos __ e:_ J dRdt
(D.14)
In order to use nurrerical integration to evaluate (D.14) ,
the infinite limit of integration must be replaced by a finite linn.t.
Using the transformation of variables
R = tan[ ,r~x J tan z
£ = y
and substituting (D.13) into (D.14), the kth m::nent of the in-phase
noise ni can be expressed as
'If 'If
1 f f -,r -,r
-s tai/z cos(y+l3) - s
1tan z e 2 r
1 (S
2tan2z)sin(y+B) - A' }k . {
[
2 2 tan z sec2z exp - tan z t A ~ 2Atan z cos y
u (D.15)
Equaticn (D.15) is cx:nputed nurrerically using the m:xlified
Gauss proouct fonwla [45]. First, the integration regicn in equaticn
(D.15) is subdivided into 22M icentical squares. To eadl square, the
15th degree, 64-p::,int Gauss product fomula [78) is applied to evaluate
the &:ruble integraticn. 'lhese 22M values are then sunned to yield the
kth m::mmt of the in-phase noise. In all the cases a:nsidered 1t was
189
~
' .
'• . 1
' ;
1 t ,, i 1 ' l
,, '
I ;-
found errpirically that the value M = 5 is sufficient to give results
aca.rrate to 6 significant digits. 'Ihe value for ~ can then be
cx:xrputed frcm equation ( 4. 49) once the first k rrarents of the in-phase
noise are kno...m.
The cx:::nvergenoe rates of the final value for different noise
m:::rren ts as a functicn of M are sh:Jwn in figures D-1 to D-7, for
different values of up-link carrier to noise p::wer ratio.
190
l• .,, .. -- ' ' ' ~\--... -~ ... ~~ .,..,._., -----as,:~:S!-n' j21 M 1111 1 & & I I ,.._ ~~.,. .. .._...,.,,..... _ _... •. _....... il .. ..._w_.,~"'-~~~~C'~'?C~~~-·-~1'
100%
......
I ~ -~ ~ 8 ::>
!~ I
90%. I I 0 0 (<NR) = -10 dB u
.s ~ (CNR) = - 5 dB u ~-~
I II ~ (<NR) = ~ ! u O dB
! ij I --- (OlR) = 5 dB u
] a 80% I 0 (:) (CNR) = 10 dB u
~ ~ ij I -
I I
70% I I -- -------- ..... ..., ..... 0 l 2 3 4 M 5
Fig. D-1. Convert;;enCE of the seccnd rronent of in-phase ooise to the final value as a funct.ioo of M.
r ~ . ~ ~ ' .,. . ----·~. +~ _<>;..::::::.__ - __ ......-,.... .. "' ---~ ,._ ~ .... .......,., ) :r , ~~-~~ ...
100%
-3l I ] ~ ! i 9
°'· l 0 0 {am.) = - X) dB u
(CNR) = - 5 dB )( k u .s ~ ~ !
(<NR} = O dB 0 u .µ
1 ij (<NR) = 5 dB --- u
8. 80% j 'E El 0
(<NR} ::: 10 dB RS
u
~ .s
70%
0 l 2 3 4 M
Fig, D-2. a:mvergence of the third nommt of in~ noise to thµ. final value as a function of M. ,:
.,>~~• -"":' ~-~A .. ~~~::..~!~•-:,•,• t;,• .. ~ ... ---·~----n,.,.._ -·--
5
..... \0 N
"> .
I ,
i . I
193 h \'.
Ill
t l i
18 @ @ @ @ IH 0
0 If) 0 If) 0 :,:.:
8 _. _. I I .B II II II II II
::, ::, ::, :::, ::, ] i ~ ! ~ ! g g ..,. nj
:a ~
l I l i
J' ]
I: lH
,.., ~ B i ..... 8
! .. • ' , . .'
.. ·~ -~ l
N < lH .. 0 .µ
! t ¢1
~ ~ !'
' IH 0
,, ~
!° I 0
. ,.., b .l .
dP "" dP "" t1'I ~ ·l 0 0 0 0
~ .,
0 0\ co r-- 1, .... , . ·t
(an.DM tl?U"TJ' an Jo ~uro.iad ~ S!i?) '~ ,j
asTOU asettd--tr1' JO :iuau:lll lf.l.lT\Od 'l
f • 1\ ·--!
••,;
: ! "I
111
I
100% -
~ I /
i /
-~ 8 g l
11,4
~ 90~ 0 0 (CNRJ = -10 dB u
-~ 11-4 0
~·} (CNR} = - 5 dB u (CNR) :::: O dB
! l u
'~ I --- (CNR) = 5 dB
~ cu u
"&! Yd 80~ 0 a {CNR) • 10 dB u -
70..__ ____ ~-----------,---~----------.------------~-,--------------.-------------~
0 l 2 3 4 M 5
Fig. ~4. Ccnvergence of the fifth rrarent of in-phase noise to the final value as a functicn of M.
~""
:--=-:.-,-.... ~ ... "!.,~ ...... ~ ,,..._<W-;t~-_........_.-----~'J ... 'l!ft.~ .. $~~~~~--,:;..."lt.~--~-:-~~~---'--~· .. ,,,,.i~· • ·! ' • ,( ~ ·- .. ~ •• .,...., >,I. --~..._t~ ..
...... \0 A
.......
] i i! .s 11-4
0 \1.-1-'~ ! ! l ! ~ fJ) ~ .....
, .. ,,.,. ... _ ..
100%
90% ·1 I 0 0 (Qffi) = -10 dB ...... u (rnR} = - 5 dB
)4 ~( u
I {<NR) = u O dB
I --- (<NR) = 5 dB u
80% ~ {.. D G (<NR) -u 10 dB
70%
0 1 2 3 4 M
Fig. D-5. Coo.vergenoe of the sixth m::rrent of in-phase noise to the final value as a functic:n of M.
____ ,_..~ .. ~~-t-""-".,..,,.,.,. ,l"l'kl.'•.,,...,...,._..,,...,,_.,_ ......... ~~ ...... - ... -~.~ .... "'~"'"'-~ '" ,.,. __,.__ - - -
5
...... \0 Vl
i L . ..
........
·I ~ j ] ?' ~ -~ ~ 11-l ~ 0 0
! ! ll ~ I ; -
100%
I /
90\ J 0 0 (CNR} = -10 dB u
)4 ~( (CNR) = - 5 dB u
(rnR) == u O dB
(CNR) .. 5 dB --- u
80% j GJ (:) (CNR) .... u 10 dB
10%
0 l 2 3 4 M
Fig. D-6. Convergence of tha seventh rrarent of in-phase noise to the final value as a function of M.
5
.... \0 O'I
~, ........ ~ > ,l
.' I
1 '
I I
' 'l
_,, -;'- . -~..-- 'i-,--.. - -· ~,, --,·-= .___...--.-. r~-
\ I
\
I t 197 ! •
n ~ ll'l ~ '
: '
.\ 18 @ @ ~ @ i
1 0 Ill 0 Ill 0 ~
.... ...... ...... 0 I I
8 • • 0 • If
I ::::, ::::, ::::, ::::, ::::, -a ~ ~ ~ ~ ~ ~ - - - - - ~ lfj
I ra ! i l
l l I I rl
] ....
("') i _.J
B m
-.-f
8
- J "r..,., ,I
('f !i ~ +J
! fj -§. '@
rl
~ 'M B r~ ! §
0 . r-~ . ', .
d? g dlP dP 0,
0 0 0 ·rl 0 0\ 0:, r,. rs. ....
(.mteA TVUJJ 34l JO agelua:>iad v sv)
~SJOU asvqd-u1 JO lUdUICX!l qlq~l3
t . J .,
I, I,
i,
'i f
APPENDIX E
Calsider too infinite gp_ries e;,:pansioo for p, the prcbability
of cne sanple ~ing in error, as given by J
-o ao n [ N' J n/2
p = _21 erf c ( r"P) + ~ l J.:.!L h ~ Hn-·l ( f2o) & n=J nf n NT
~ (E. l)
where p =
The rand:rn variable ni is ~ by a finite f,OSitive
nuxber K, sud1 that
sup(x) = K > 0 for all x £ Q
l\
Using the well-known inequality for He.rmite polyncmi.als [40]
roe can slx::,w that ~
4N' lhnl u
< (n-1) f!e - for n even (E. 2)
~ ..m•
and lhnl (n-2) ! ! hie u < - for n cx3d (E. 3)
wnere (2n-l)!! = 1·3·5·7·9········(2n-3)·(2n-l)
and K is as def .ired above.
198
t' 't
t ~·· •
J
i. j t ( f
'
FUrtherrrore tre Hermite polynani.al is uppertx:>unded by [38]
<
n-1 ;;, -< 2 > 2 o< 2 {n-1) l e
n-1 < ->!
2
n-1 -y~
...,'here <x> ~tes the integ>..r part of the real nl.lilf:P-r x.
(E. 3)
The error whidl arises as a result of truncating the series
~ioo of equatioo (E.l) at the (L-l)st tenn can b:! written as
..,
!\, = }: n=L
< -p CD I h I ~ I-n.;r; n=L n
n/2
(-1) n [ N~l -- h -n! n NT
(E.4)
n/2
[ :;] n-1 / n-1 -< - > 2 p<-- > 2 2 e 2
(E. 5) n-1
< 2 >l
Without loss of generality we m:i:y assure that L is even in the
subsequent analysis.
Substituticn of (E.2) into (E.5) and rearranging as sums of
even and cdd ter.ns we cbtain
(E.6)
~ the sequenc:ES s2k and s2k+l are defined as follcws
[
N' ]k -(k-1) 21 p (k-1) (2k- l) ! 1 u 2 e 2k N.r --~-(k---1-,1--~- (E. 7)
199
...,
:
l t
l
I f .. 1 J
l
.) 'j I i ' t ' I i
t ,t c .t '•
':\ I
I C~.,;.
.. {' ,, I
. \
l • 1
I i: ' r ' i
r r----"
2k+l
s2k+l (2k-1) ! I [ N~ ]-2-/ 2k+l 2-k e20k (2k+l) NT k!
It is then easy to sho,/ for the even and crld terns
and
s2k+2
s2k = (2k+l) [ N~ J 2/o{ >'k - lk-1}
(2k+2) N.r, e ,
(2k+l) (2k+2)
(E.8)
(E. 9)
(E.10)
Fran D' ~rt' s ratio test of absolute cawergence and I
equatims (E.9) and\{E.10) it then fol.lows that the senes of S2k and
L S2k+ 1 are jointly absolutely a:n~.1ergent for -any k :::. 2 sudl that
L > 2 + ! [ 2 /p 2 ln [N'l"'1'1 ~]
_1n_[N_~_~1-]2 2,tp
(E.11)
The value for L, as indicated in (E.11), depends en the ratio
of total noise p:,v.er at the receiver, N.r, to the equivalent up-link
noise pa,,er, N' and tends to diverge as N' approact-ies zero. Ho...ever, u u
for the trivial case of no up-link noise, N' = 0, the series terms in u
(E. 1) are redured to z.ero and the system perf orm3l'loa, p, is affected
cnly by the da.ln-link Gaussian noise.
l · ..... ...,,.,... . ~~~~-""""= •'"'"*' ~ iE .. •<
200
i '
.,.
J f
·~ .. -,<
::{;; ,,. r~J
, , .
I
'l !
'
j ..
For any values of L k::: 2 and L satisfying (E.11), equations
CE. 9} and (E.10) teoorre
s2k+2 -·- < s2k [ :~ l 2-1'; f r'L/2 - I< L-2 > 12 } e . l Q (L) < l
- e
(E. 12) and similarly
rota.in
and
s2k+3 - < s2k+l
[ Nu' l 2;'o{ l(Lt2) /2 - /(L/2)} e ~J
(E .13)
Using tbe results for the sum of infinite georret.ric series, we
L
... (L-2)
--2- 210 (L-2) /2 (L-1) !! r:~ r 2 e
L
rL s2k+ 1 < '"'"c1r __ 1} __ !_! [ ~ l.J.
~ l(Ltl) --r
L+l 2
[(L-2)/2]! (1-<J (L)] e
2 e (L/2) ! [ 1-Q (L) f
0
(E .14}
(E.15)
Cmbining (E .14) and (E.15) and substituting into (E. 6) we get
£K2 /4N' J
~ e-Pe u
< /Ti
L
(L-1} ! ! [~r -~ 2~
2 2 e
(L-2) 2 lo (L-2) /2 --2-
2 e L [ (L-2} /2 ) ! (1-() (L)]
e
(E.16}
201
,. , .
. •
..
,•
?
I
. i i
202
and since the series s2k, s2k+l are shewn to re absolutely cawergent,
then for a small truncation error bound, S , it is always p:>5s1.l>le to
find a sufficiently large L satisfying equation (E.11) sudl that
< 6 .
(
4 1 r 1 1 iili rtw
j ' ' 1 I
J. t !
i l i t
! """"
I
l 'j
! •
.,
.;,
I i
l r·
I I I.
[1]
[2]
[3]
[4]
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.. ,,
..
I12J
{13)
{14)
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.... za a .... ~ ex; ",:; -
,~
°\..
~ ,. t ij
I l I I I
• I
I f t i
t J
. ,
L~ '
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205
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-
j . . ,,
:.
'
l': ' . '
), . ' f f
.. .
'•.
-f: . ,, }.
"' •• 1
...........
[33]
[34]
J35)
206
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i I , l l t t j ! f. i ·~
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• ,I f
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•• ~· y
~ t h ~ .
!64]
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