T. GAARDER Prepared for · gain (ak is the gain and 0 k is the phase shift) of' the kth path. It is...

Technical Report 2, Part III

oAN EXAMINATIONU0 OF SELECTED HF PHASE MODULATION TECHNIQUES

By: N. T. GAARDER

Prepared for:

U.S. ARMY RADIO PROPAGATION AGENCYFORT MONMOUTH, NEW JERSEY CONTRACT DA 36-039 SC-90859

S S E I N S

M E ~ N L -PRCAF N1

--- 00 AV--- N

-Z -- MEi 4 4~--

February 1)65

Technical Report 2, Part III

AN EXAMINATIONOF SELECTED HF PHASE MODULATION TECHNIQUES

Prepared for:

U.S. ARMY RADIO PROPAGATION AGENCYFORT MONMOUTH, NEW JERSEY CONTRACT DA 36-039 SC-90859

By: N.T.GAARDER

SRI Project 4172

Approved: W. R. VINCENT, MANAGERCOMMUNICATION LABORATORY

D. R. SCHEUCH, EXECUTIVE DIRECTORELECTRONICS AND RADIO SCIENCES

Copy No. ..

CONTENTS

LIST OF ILLUSTRATIONS .. .. ............... .............. v

LIST OF TABLES. .. ................ ................. v

I INTRODUCT ION .. .. ................ ............... 1

II THIE PROPAGATION- MEDIUMI MODEL. .. ........ ................ 3

III BAPSK SYSTEM PROBABILITY-OF-ERROR EXPRESSION. .. ......... ...... 13

IV THE PROBABILITY-OF-ERROR EXPRESSION FOR MODIFIEDBAPSK SYSTEMS .. ........ ................. ...... 29

V CURVES OF THiE PROBABILITY OF BINARY ERROR .. ........ ......... 35

VI CONCLUSIONS .. ........ ................. ....... 53

APPENDIX A .. .......... ................. ........ 57

APPENDIX B .. ......... ................. ......... 67

REFERENCES .. ......... ................. ......... 73

~~j.

IL!,USTRATIONS

Fig. 1 Block Diagram of Propagation-Medium Model ...... ................. 3

Fig. 2 Typical Fade Coordinates ...... ...... ......................... 9

Fig. 3 PJ(D, ) as a Function of P (. .............. . ..... 32

Fig. 4 Curves Showing Probability of Errorthrouigh as a Function of Signal-to-NoiseFig. 13 Ratio ....... ....... ................................... ... 36-45

Fig. 14 Determination of the Probability -of-Error Curve by the ResidualError and the Flat-Flat Probabili ty-of-Error Curve ... ............ ... 49

Fig. 15 Curves Showing P (D) as a Function of Differential Time Delayand DifferentialeFrequency Shift .. .... . ........... ....... 52

TABLES

Table I key to Figs. 4 through 13 ...... ...... .......................... 48

V

I INTRODUCTION

This report develops a probability-of-binary-error expression for a

basic binary-adaptive-phase-shift-keyed (BAPSK) system. The error-ratc

expression uses the inathematical BAPSK system model developed in Technical

Report 2, Part I on this contractit and amathematical propagation-medium

model of the type discussed in Technical Report 2, Part 11.2 After the

probability-of-error expression for a basic BAPSK system has been devel-

oped, this expression is modified to include various BAPSK modifications

(time guard band, diversity, delay compensation, and others).

The propagation-medium model used in the error-rate expression, which

specifically models the high-frequency (HF) propagaticn medium, is dis-

cussed and related to the HF propagation medium. The pzopagation-med.;um

model includes additive noise, and thus it models the additive as well as

the dispersive corruption of the HF propagation med-,m

Because the error-rate expression developed in this report is rela-

tively simple, the probability of error for a basic BAPSK system is

plotted as a farction of the signal-to-noise (S/N) for many types of dis-

persive channels. The most significant feature of the error-rate curves

is that the probability of error approaches an asymptotic nonzero value

for high S/N. This asymptotic value is determined by the time-delay and

frequency-shift structure of the propagation medium. By using this

asymptotic value of the probability of error, the sensitivity of the basic

BAPSKI system's error rate to time delays and frequency shifts is analyzed.

By using the results of Technical Report 3,3 a basic BAPSK system is

compared with a basic binary-differential-phase-shift-keyed (BDPSK) and

quaternary-differential-phase-shift-keyed (QDPSK) systems. The error-rate

performance of a BAPSK system is found to be degraded by frequency shifts

much more than that of either a BDPSK system or a QDPSK system; however,

at low S/N, a BAPSK system outperforms BDPSK and QDPSK systems. The per-

formance of a BAPSK system and that of a BDPSI( system are equally sensitivr

to time-delay effects.

Rleferences are given at the end f the report.

1

I I THE PROPAGATION- MEDIUM MODEL

This report uses a propagation-medium model of the type discussed

by Stein,4 Bello,5 and Daly 2 to model the dispersive corruption of the

propagation medium. However, the model used in the error-rate analysis

is specifically related to the IF propagation medium. This model also

includes additive, white, Gaussian noise to model the additive corrup-

tion of the propagation medium.

The output of the propagation-medium model (see Fig. 1), y(t) [y(t)

is the complex representation of the real-valued outputt y(t)], is assumed

to be the sum of an additive noise, n(t), and the output of a random time-

varying linear filter, z(t):

y(t) = z(t) + n(t) . (1)

X~t) H 0,tf

nt(t

0-4172-173

FIG. 1 BLOCK DIAGRAM OF PROPAGATION-MEDIUM MODEL

The additive noise, n(t), is assumed to be a zero-mean, stationary, white

complex§ Gaussian random process:

S[n(t)] 0 (2)

Throughout this report the complex representation for real processes discussed in Ref. 1 is used.

§ In this report all complex random fields, processes, and variables are implicitly assumed to haveidentically distributed (but not necessarily statistically independent) real and imaginary parts.All random variables are implicitly assumed tu have statistically independent real and imaginaryparts.

3

S(f) f [n(t)n*(t + At)] exp {-i27TfAt} dAt

fN o, f >

(3)0 , f<0

where Sn(f) is the power spectral density of the noise, No is the noise

power per unit bandwidth, n* is the complex conjugate of n, and 8[] isthe usual expectation operator. The use of a zero-mea, stationary, white,

complex Gaussian r process o model the additi'.e noise has been well

justified.6

The output of the time-varying linear filter

z(t) = fJ I(t,f)X(f) exp (i27Tft) df , (4)

where the Fourier transform of the channel input,

X(f) = fx(t) exp (-i27Tft) dt , (5)--CD

and the time-varying transfer function of the filter, ll(t,f), accounts forthe time and freq.uency scattering of the input energy. The time-varying

transfer function, H(t,f), is assumed to be a zero-mean, homogeneous, com-

plex Gaussian random field. Because the transfer function is a complex

zero-mean, homogeneous, Gaussian random field, its statistics, and the

statistics of its output for a known input are completely determined by

the covariance function of the time-varying transfer function,

R(At,Af) = 8[lI(t,f)lt*(t + At,f + Af)] . (6)

This model for a time-varying radio channel has been justified by Stein,4

Bello,5 and Daly.2 To model the HF propagation medium, it is necessary

only to select an appropriate covariance function for the homogeneous

time-va-rying transfer function or, equivalently, an appropriate scattering

func t ion,

S(Xir) = r JRH(At,Af) exp (i27T(-XAt + 7Af]) dAtdAf (7)HX -- -00

for the time-varying filter.

.{4

It is well -known that the [IF propagation medium scatters energy dis-

cretely in time. This IiF phenomenon, commonly referred to as multipath,

is readily evidenced by the records of short-pulse oblique-incidence

ionosphere sounder links. 7 The receivers on such links commonly detect

several short pulses for each short pulse transmitted. These detected

pulses are of different strengths and arrive at different times. The

strengths and particularly the time displacements can be shown to correspond

to various paths or modes of propagation. A suitable model to explain this

multipath phenomenon only is a medium consisting of several paths each with

a distinct time delay, gain, and phase shift. The output of such a medium,

Ky(t) = ak exp (k x t - r,)

(8)K

= X ax(t - rk=1

where x(t) is the input to the medium, K is the number of paths, Tk is

the time delay of the kth path,

a,: = ak exp ( ik )

is the complex gain of the kth path, ak is the gain of the kth path, and

0k is the phase shift of the kth path. If it, is assumed that for each

path a,, the complex gain, is a zero-mean, complex Gaussian random

variablet with variance

S[Iakl 2 ] = ' (10)

the strength of the kth path, and that the complex gains of all the

paths are statistically independent, then the propagation medium is

equivalent to a zero-mean, homogeneous, (time-invariant) Gaussian ran-

dom filter with covariance function,

KRH (4tf) 2- exp (iinAfrk) (11)

k_1 k

The assumption that the complex gain of each path is a (complex) zero-mean Gaussian random variable

is equivalent to the assumption that the gain of each path is a Rayleigh random variable with mean,

[k = (V/2)o- and that the phase shift of each path is uniformly distributed on the interval [0,27).

5

K

R(At,Af) = 2 exp (i27iLk) (11)k 1

and thus scattering function

K

k X 2 3(- - 7- (12)

Such a time-invariant model for the HF propagation medium (consist-

ing of energy scattering in time only) is unrealistic, because even though

it accounts for frequency-selective fading and intersymbol interference,

it does not account for time-selective fading, for interchanne! inter-

ference caused by the propagation-medium scattering energy in frequency,

or for the continual change of the gain and phase of the propagation

medium with time. Perhaps the best example of the presence of energy

scattering in frequency occurs when a CW tone transmitted over an HF link

is envelope-detected. The variations (in time) of the level of the

envelope-detector output (a phenomenon referred to as time-selective fading)

can be caused only by the propagation medium scattering the input energy

in frequency. If the propagation medium did not scatter energy in fre-

quency, the level of the envelope detector output would not change with

time. Thus a comprehensive model of the HF propagation medium must

incorporate energy scattering in frequency as well as in time.

The exact nature of energy scattering in frequency does not appear

to be well known; recent work at Stanford Research Institute,8 gseems to

indicate that the HF propagation medium tends to scatter energy in dis-

crete packets in frequency as well as in time. That is, the propagation

medium tends to shift the input energy by several distinct (Doppler)

frequencies.

The HF propagation-medium model used in the error-rate analysis

consists of several paths, each path having a distinct complex gain

(gain and phase shift), time delay, and frequency shift. The output of

such a medium,

K

y(t) = a k exp (i 27TXkt) x (t - k ) (13)k=1

where x(t) is the input to the medium, K is the number of paths, Xk is

the fiequency shift, T k is the time delay, and ak = ake is the complex

6

gain (ak is the gain and 0 k is the phase shift) of' the kth path. It is

again assumed that for each path the complex gain, ak, is a zero-mean,

compiex, Gaussian random variable with variance

&IIaA k (14)

equal to the strength of the kth path, that the time delay, -,k is known,

that the frequency shift, X,, i3 known, and that the complex gains of' all

the individual paths are statistically independent of each other. Under

these assumptions, the propagation-medium model is equivalent to a zero-

mean, homogeneous, Gaussian randcm filter with covaciance function

KRH(At,Af) = 2 exp (-i27r[Xk t - TAAf]) (15)

and thus scattering function

KSH (X,i) - o- 28(X - X ) (-r - -) (16)

k1l

Note that the propagation-medium model can be made frequency-invariant

(i.e., purely time-selective) by setting all the TA to zero, time-

invariant (i.e., purely frequency-selective) by setting all the X to

zero, and time- and frequency-invariant (i.e., flat-flat fading) by

setting all the Tk and Xk to zero. In addition, by making K very large,

a continuous scattering function can be closely approximated. In

a~ialyzing the sensitivity of various systems to time delays and frequency

shifts, it will be convenient to consider time-invariant, frequency-

invariant, and time-and-frequency-invariant propagation mediums. Time-

and-frequency-invariant propagation mediums have been extensively studied

by many authors.-''12 In the literature, the time-and-frequency-invariant

propagation medium model is referred to as the flat-flat Iayleigh rading

channel model.

For a better understanding of the effects of time delays and fre-

quency shifts upon transmitted signals, a particular propagation-medium

model is now discussed in more detail. This particular model, which is

typical of many 11F channels for reasonable time intervals, consists of

two paths each with a distinct and constant time delay, frequency shift, and

complex gain. Let the input to the propagation medium be a CW tone at

an arbitrary frequency, f. If this signal is denoted by its complex

representation,

x(t,f) = exp (i2lTft) , (17)

the output of the propagation medium (neglecting additive noise),

y(t,f) = a1 exp (i2Tr[f - [t - -I)'. a, exp (i27Tn[f - 2] [t -T 2 ]) , (18)

where ak is the complex gain, T-. is tLe time delay, and k is the frequency

shift of the kth path. If the output of the propagation medium is

envelope-detected, the output of the envelope detector,

Iy(t,/)I = 1a1!2 + ja212 + 21a 1lla 2 1 cos (2-7[fAT + t6X] + 4)

(19)

where the differential frequency shift, A = 2 - X 1 , the diffeicntial

time delay, AT = T2 - T 1, and the differential phase shift,

'T1 - 2 T 2 - cos - (, aa 1? )

Thus the output of the envelope detector is at a relative minimum (i.e.,

in a fade) when cos [27r(fAr + tAk) + ] = -1 or, equivalently,

t + 6TfF = ±n + 2 (20)

for some integer n; the subscript F denotes that a fade occurs at that

particular time and frequency.

In a time-frequency space, the set of r.oints (fade coordinates)

satisfying Eq. (20) is a series of parallel straight lines, with each

integer n determining one straight line of the series. Figure 2 is a

plot of a typical set of fade coordinates.

At any fixed time, t., the fades are spaced in frequency by l/AT,

the reciprocal of the differential time delay. At any fixed frequency, f0 ,

the fades are spaced in time by 1/Lk, the reciprocal of the differential

frequency shift. In addition, at any particular time, to, the output

Wp

0tr

=AX

F

I."

t o 0TIME-tD-4172-172

FIG. 2 TYPICAL FADE COORDINATES

of the envelope detector,

Iy(t0,f)12 = la,1 + la21 + 21aIVIa 2 1 cos (27TfLAr + 0 , (21)

wher 00 = L4 + 27rLt o. At any particular frequency, fo the output of the

envelope detector is again of the same form as Eq. (19), since

ly(t,fo)12' = Ja1 2 + la 2 12 + 21a 1l"1a 2 1 cos (27TXt + k0 )

where (op + 2irfozAr. IThe results of this digression can be extended to a more general

propagation medium; however, the spacing of fades in frequency at a

fixed time is still determined primarily by the time delays of the paths,

and the spacing of fades in time at a fixed frequency is still determined

by the frequency shifts of the paths.

9I

This discussion has been partially verified experimentally by

Ames.D The experiment consisted of transmitting a frequency-modulated

signal over an HF link and simultaneously determining the time-delay

structure of the link by using an oblique-incidence ionosphere sounder.

The received signal was envelope-detected and spectrum-analyzed, and the

output of the spectrum analyzer was plotted as intensity in a time-

frequency space. When the propagation medium consisted of two paths,

the points of minimum intensity (i.e,,fades) formed straight lines similar

to those in Fig. 2. In addition, at any fixed time, the frequency spacing

between (frequency-) selective fades was 1/ir, the reciprocal of the dif-

ferential time delay. It was also found that at a fixed frequency the

spacing in time between (time-) selective fades was constant, but not

correlated with 11r. Because no measurement of frequency shifts was

made in the experiment, the time spacing between (time-) selective fades

could not be correlated with the differential frequency shift.

In an extension of his work, Ames considered fade patterns in a

position and time space.14 The fade patterns in the position-and-time

experiment were similar to the fade patterns in the time-and-frequency

experiment. The latter experiment indicated that, in general, most space-

diversity antennas are not uncorrelated. Because of the increased com-

plexity in the mathematical models necessary to model these effects (a

scattering function of many more dimensions and the position and polariza-

tion of the diversity antennas), the error-rate analysis considers only

(space-) diversity antenna outputs that are either completely correlated

(identical) or completely uncorrelated (independent). It was felt that

with this approach the performance of systems using (space-) diversity

outputs that were neither completely correlated nor completely uncorrelated

could be approximated. In addition, the error rate is bounded above by

the error rate for identical space-diversity antenna outputs.

To summarize the propagation-medium model: the output of the model

y(t) = H(t,f)X(f) exp (i277ft) dt + n(t) (23)

where X(f) is the Fourier transform of the input, n(t), is the additive

noise which accounts for additive corruption effects, and H(t,f) is the

time-varying transfer function of the channel which accounts for the

ionospheric dispersive effects. The additive noise, n(t), is assumed to

10

be a stationary, zero-mean, complex, Gaussian random process with

spectrum,

N0 , f > 0'n(f) = (3)

0, f<0

where N o is the noise power per unit bandwidth. The time-varying trans-

fer function,

KH(t,f) = Z ah exp (i27r[fT k - Xkt] ) , (24)

k1l

where K is the number of paths, ak is the complex gain, 'k is the time

delay, and Xk is the frequency shift of the kth path, is a homogeneous,

zero-mean, complex Gaussian random field with power spectral density

(i.e., scattering function),

K

SH(X,T) = o Is(X - Xk) 8 (-r - T) , (16)k=1

where o- 2 is the strength of the kth path. The complex gains, ak, are

statistically independent, zero-mean, complex, Gaussian random variables

with variance 0*2; the time delays, Tk, frequency shifts, X k, and strengths,2'k , are known, as well as the number of paths, K. Because of the special

form of the time-varying transfer function, the output of the propagation

medium can also be expressed as:

K

y(t) ak exp (i 2 7rTkt) x (t - r-) + a(t) (25)

The random filter and the additive noise are statistically independent.

11

III BAPSK SYSTEM PROBABILITY-OF-ERROR EXPRESSION

This section develops an expression for the probability of a binary

error for a basic BAPSK system operating in the non-redundant mode.

The output of a BAPSK system transmitter x(t), which is described in

Sec. IIIof Ref. 1, is the input to the propagation-medium model discussed

in the preceding tection. The output of the propagation-medium model,

K

y(t) F ak exp (i27TX t) x (t - 'r,) + n(t)*k~l

is the input to a BAPSK system receiver, which is described in Sec. IV of

Ref. .t Since the actual probability-of-error expression is complex and

difficult to calculate (even with a. digital computer), a convenient and

accurate approximation to the probability of error is found. This approxi-

mation is valid for a wide range of systems other than a BAPSK system, and

it greatly simplifies the computation of the probability of error. The

pro-bability-of-error expression considers errors caused by dispersive cor-

ruption (the variations in time and frequency of the complex gain of the

propagation medium), additive corruption (the additive noise of the propa-

gation medium), and inherent self-interference (the nonzero information-

tone component of the long-term-average filter output). It does not

consider other errors caused by implementative corruption (the malfunc-

tioning of equipment).

From the results of Ref. 1, it is readily apparent that the nth sub-

system of the BAPSK system will err in the transmission of the kth binary

digit of the nth information sequence In, if, and only if, the kth binary

digit of the nth estimated information sequence I" is not equal to In.

Because both and n[ take only the values +1 and -1, the probability

of error in the transmission cf I

In Sec. IV of this report, modifications [Sec. VI of Rf. 13 to a BAPSK system and operation in 'theredundant mode are incorporated into the probahility-of-error expression.

Preceding Page Blank13

pn = Pr {error in the transmission of iR) = Pr {fln - (ek k k k

(26)

From Eq. (93) of Ref. 1, one finds that the kth digit of the nth

estimated information sequence (for a basic BAPSK system operating in

the nonredundant mode),

i = sgn (Cn Re {iyn(k)yn*(k)}) (27)

where Cn is the kth element of the nth binary code-tone sequence, the

long-term-average filter output,

k-I

y"l(k) = E Py(j) exp (aT[j - k]) (28)

the matched filter output,

O

yn(k) = f y,(t) exp (-i27Tr[fn -At)p (t - t o -kT) dt , (29)

+1 x > 0

sgn(x) j ' (30)

and Re {x} denotes the real part of x. In Eq. (28), pn is the jth binaryj

element of the nth pilot-tone sequence, a is the time constant of the

long-term-average filter,t and 1/T is the signaling rate of the system.

In Eq. (29),

yA(t) = exp (-i27rAt) y (t) (31)

is the output of the propagation medium y(t), hetrodyned to a frequency

A cps less than the transmitter output; fn is the center frequency of thenth subsystem;

I , t E [-T/2 , T/2)p(t) = ; (32)

0 f t [-T"/2 ,T7/2)

T/2)t If the long-term-average filter is implemented digitally in the BAPSK system, e Is constrained to take

one of the values (1 - 2 - , where n is an integur.

14

and t. is the BAPSI( system synchronization time. The probability of error

can thus be expressed as:

P' Pr {IC' Re {iyn(k)yn*(k)] < O} (33)

or equivalently as:

P n Pr {I C ny*Qy < O} (34)

where the column vectort

Y = (35)Ly(k)J

the row vector Y is the transpose and complex conjugate of Y, and the

Hermitian matrix

= K ](36)i 0

Thc probability-of-error expression can also be expressed in the form:

P Pr {IC Pr '*{nZ P i n ny QY < 0 1 j Pr { = 1} (37)

where the set - consists of all possible system transmitter states

Each system transmitter state, J, consists of one possible set of values

for each element of every pilot-tone, code-tone, and information sequence.

Since the set , has an infinite number of possible transmitter states, the

probability of error, as expressed in Eq. (37), is impossible to compute

without further assumptions. This difficulty will be ignored for the

moment, while the conditional probability of error for a given input state

= Pr {IkCkYQy < oI1} (38)

k k

t The superscript n and the argument k of the vector Y have been suppressed for simplicity.

15

4

is considered in more detail. This detailed consideration of the con-

ditional probability of error eventually leads to an accurate approximation

of the probability-of-error expression that essentially eliminates the

infinite summation of Eq. (37).

When the transmitter state is known, the vector Y is a zero-mean,

complex,t Gaussian random vector, because each of the components of the

vector Y is the result of linear operations on the propagation-medium

output y(t), which is the sum of two zero-mean, complex, Gaussian random

processes. The statistics of the vector Y, given the transmitter state,

are thus completely determined by the Ilermitian conditional covariance

matrix;

K(e) 2 £YY*l

Skll(e) kl12( )

(39)

where

()= 8[Ij(k)l 2 y-M , (40)

k1 2(e) £ ti(kyn*k)IS] , (41)

and

k22(5) = £[Iyk)I2 Is] (42)

It can be readily verified that there is a matrix [K(5f)]2 such that

K(e) = [K(e)1 [K()]*2 (43)

where [K(e)] */2 is the transpose and complex conjugate of [K(f)]2. Now

by assuming that the matrix K(6) is nonsingular§ one finds that the vector

Z = [g(Y)-Y , (44)

All zero-mean, complex, random vectors considered in this report are implicitly assumed to have compontntswith statistically independent, as well as identically distributed, real and imaginary parts. This as-suniption is consistent with the complex repcesentation 1 used in this report. An immediate result of thisassumption is that for any vector X,IXXT] "- , the zero matrix (XT is the transpose of X); thus thesecond-order statistics of X are comp fetely determined by &XAI1, the covariance matrix of the vector.

The probability-of-error expression developed here is applicable even when K(g) is singular. The de-velopment of the probability-of-error expression is conceptually the same, but much longer, when K(15) issingular; therefore this development is not included.

16

where [K()] -I denotes the inverse of [K(f)]1, is a zero-mean, complex,

Gaussian random vector with covariance matrix,

&[zZ*I ]. [K(f)]

- [~ 1] =I ,(45)

equal to the identity matrix. The conditional probability of error can

now be expressed in terms of the vector Z:

n Pr {ZQ(f)Z < 0} (46)

where the Hermitian matrix

Qn ( )= InCn [K( ) Q[K()] (47)

It is well known,15 that there are matrices A(f) and P() such that:

Q ()f P(e)*A( )P( ) , (48)

P(O) = f' (49)

and A(e) is a diagonal matrix with the real-valued diagonal elements being

the eigenvalues of the matrix Q'(f) [i.e., the roots of the equation

IjX - Qn(e)I = 0 , (50)

where 1Al denotes the deterw..nant of A]. Therefore the conditional proba-

bility of error,

P" = Pr {W*A(6)W < 0} (51)C k

where the vector

W = P( )Z (52)

is a zero-mean, complex, Gaussian random vector with a covariance matrix

equal to the identity matrix, I.

17

By expressing the conditional probability-of-error expression in

terms of the components of the vector W, W1 , and w2, and the eigenvalues

of the matrix Q'(f), X1 (e), and X 2(e), one finds that the conditional

probability of error,

P" () r {k ( )ll2 2

n,()=Pr .{1()w1I + X()II < 01

= {W112 X 2(e)

= Pr {f2 < Y) 1 ) (53)

where the random variable f2 = 1 1 2/1W 21 is "F-distributed" with pa-

rameters m = n = 2.16 The probability density of f2'

(1 + f2 ) - 2 , f2 > 0SPC/2 ) =(54)

1f 0 , f2 < 0

and thus the conditional probability of error,

= (1 + f 2)2df2 = )(55)

0

The conditional probability-of-error expression implicitly assumes

that X,(e) and -X2 (f)/l(e) are positive; to verify these assumptions one

must compute the eigenvalues of Qn(e). The eigenvalues of Q"(f) satisfy

Eq. (50); however,

IXI - Q'-(e)I = Il- p C [K]*/2Q[K()l A

- I - I.Cn K(e)Ql

X kI tr [CnK(e)Q] + IK(e)Ql (56)

where tr [A) denotes the trace of the matrix A. Noting that

IK(6)QI = IK(e)IIQI = -IK(6)1, one finds that

18

Ink tr [C"n,(6) Q]

X = {i + [1 + 4J-1()]2} (58)

2

tr (C)K( 2 Q] 1 - [I. + 4J-1( )]A} ,(58)

and the conditional probability of error,

Pn, = - [ + 4R-(6).]-} (59)

where the conditional hard-limiter input S/N

tr2 [K(()Q]

( = IK()I -(60)

The assumption that -X2(e)/Xl(',) is nonn.:gative is equivalent to the

assumption that K(f) I > 0; however, K(e) is a valid covariance matrix,

and therefore IK()I > 0. Hence the assumption that -X 2 ()/X 1 (f) is non-

negative is valid.

The assumption that X,(f) is positive is equivalent to the assumption

that I' tr [C'K( )Q] is positive, but the trace of CK(e)Q is the expectedvalue (given the transmitter state) of the quantity C"Y QYwhich is com-

pared with the zero threshold (i.e., the input to the hard limiter) in the

final stage of a BAPSK system receiver:

tr {CnK(,-)Q} &[C YQY IQ . (61)

The kth element of the nth estimated information sequence, In, is +1, if

Cny *Qy is greater than zero; if CY*QY is less than zero, then I is -1.

Obviously, if the system is to perform satisfactorily,

I£[CnY* Ftr [CnK(f)QJ must be nonnegative. If In tr [CnK()Q]kc kY k¥ k kk t

is negative, the conditional probability of error is greater than 1/2:

= {] 4 [] ,R-I'()] - } , for I tr [C'K(f)Q] <0 (62)ko 2k

19

At ' -* .

A completely general expression for Lte conditional probability of error,

which ':t. ;or the unlikely event that P tr (C'K(6)Q] is negative, is

P2 sgn Q tr [CK(f)Q])[1 + 4R-'()] - } (63)

For Pn (6) < 1/4, the conditional probability of error can be con-

veniently and accurately approximated by the inverse of the conditional

output S/N:

,,= (64)tr2 {K(f)Q}

Because large conditional probabilities of error result in a high unconditional

system error rate, only values of Pn ( ) < 1/4 are of interest for most prac-

tical applications. Thus, for practical applications, R-I'() can be used as

an accurate approximation to the conditional probability oferror. It can

be readily verified by inspection that R-'(b)is an upper bound on the con-

ditional probability of error for pn (6)< 3/4t; thus, with almost certainty,elk

one can use R-( )as an upper bound on the conditional probability of error.

This approximation to (or bound on) the conditional probability of error and

the observation that /[C Y*QYI ] =~ Itr {C K(:)Q} are employed in the

mathematical justification o: the approximation that alleviates the diffi-

culties in the infinite summation of Eq. (37); an intuitive justification

of the approximation is presented initially, however.

The (unconditional) probability-of-error expression for the basic

BAPSK system as given by Eq. (37), can also be interpreted as an expecta-

tion over the transmitter state of the conditional probability of error:

p" = F1Iek()] (65)C ,k e, k

where the transmitter state (i.e., the values of all the binary, pilot-

tone, code-tone, and information-sequence elements) is assumed to be

random. The natural distribution to ssume for the transmitter state is

a Bernouli distribution--al] the elements of all the sequences are sta-

tistically independent, and Pr {I = 1} =Pr {Pn = 11 = Pr {C" = 1) = 1/2.

t Unfortunately, for Pn () < 1/4, ir( ) is a poor approximation to the conditional probability of

error.

20

This assumption, which is a standard assumption in almost all error-rate

analyses, is made in this analysis also. An immediate result of this

assumption is that the probability of error and conditional probabilityof' error are independent of k; thus pn (fl . pn ( )= pn(f).

C k C,O 0 !

Because the expectation expressed in Eq. (65) is tremendously diffi-

cult to evaluate, it will be approximated. The probability-of-error

approximation, Pn, which is essentially the approximation used in Ref. 3,

is to replace RI'(e) by S[IK(J)I]/2[tr2 [K(s)Q]] in the conditional

probability-of-error expression

P" - [ - 1 _ (66)C 2 L[tr2 [K( )Q] ]

This is an engineering approximation. It can also be intuitively justified

for phase-shift-keyed systems: there exists an actual transmitter state,such that [jK(f)li = K(f0), S[tr 2 [K( )Q]] = tr2 [K(0)Q], and thus P" is

the actual probability of error for a particular transmitter state.

In this report, the approximation is mathematically, as well as in-

tuitively, justified for error probabilities of practical interest. For

these values of the probability of error, the conditional probability of

error is closely approximated by (and bounded above by, for

Cn(k) <3/4) R-'(f); thus the probability of error is closely approximated

(and almost certainly bounded above) by

[R-'( = - P' . (67)

This expression for the probability of error is simpler to compute than

Eq. (65); however, the computation is still difficult. The probability-

of-error approximation used in this report (and in Ref. 3), Pn, can also

be closely approximated for P' < 1/4 (and bounded above for Pn < 3/4), by

R_1 = (68)i) 8S[tr 2 [WflQ] ]

Thus for error rates of practical interest, the expected value of R-(f)

is being approximated by 8[IK(f)I]/&[tr 2 [K(9)Q]]; the ratio of the

expected values of IK(,)l and tr2 [K()Q]

21

.R-1 = IK(e) ]

fri

I= - (69)L[tr2

The quantity tr2 [K(6)Q] k is the expected value (giventhe input state) squared of the quantity compared with a zero threshold

in the receiver; it is readily apparent that, if the system is performing

satisfactorily, this quantity changes very little as the transmitter

state changes:

«[(fl) << 2[tr 2 (K(6)Q)] , (70)

where the variation in tr2 [K()Q] ,

cr(f) : tr 2 [K(,)Q] - S[tr 2 [K(l)Q]] (71)

The variations in tr2 [K(&)Q] will be primarily caused by the variations

in the self-interference (dispersive interchannel and intersymbol inter-ference and inherent self-interference). These changes resulting from

the self-interference must obviously be much smaller than the "signalportion" of tr2 [K( )Q] if the system is performing satisfactorily. The

signal portion of ti2 [K()Q] does not change significantly, because theexpectation over the propagation medium has removed all instantaneous

fading and noise effects, and because the system is balanced-if self-

interference effects are ignored, tr2 {K(9)Q) is essentially independentt

of whether In = +1 or in= -100

Because tr2 {K(9)Q} changes very little and tr2 [K(e)Q] >> 0S[IK(f)I]/&[tr2 [K()Q]1 is a reasonable approximation to[IK( )I/tr - [K(2)Q]]. For all values of practical interest, the prob-

ability of error is equal to

The scattering function of che propagation medium must be symmetric about the origin for this to bestrictly true (see Ref. 17).

22

{[tr2 [K(Z)Q]] -- )

{IK(e)I} _ [R- (_)(]

8{tr2 [K(e)Q] } 8{tr2 [K(G)Q]}

S{tr2 [K( )Q] (72)

or, equivalently, the error in approximation

- - (73)eftr' [K(!f)Q] )

but by using the Schwartz inequality, one finds that

R 12 < [R [ (74)e{tr 2 [Ke)Q] I

Thus the relative approximation error

In-' - R-1' R-2( ) __72_ _

R- 1 < R- 1 {tr 2 [K(e)Q](

t~weer 2 -R 2 n /o-&h[tr2 (K(S)Q1] « 1. Thus the relative

approximation error

<1 (7b)R-1

for all values of interest. Because P ' R-1' the probability-of-errore

approximation is mathematically, as well as intuitively, justified for

all practical values.

At this point, infinite summations are still present in the approxi-

mation; however, these summations can now be evaluated in closed form

because only products, rather than products and ratios, of random vari-

ables are involved. Because the summations are relatively easy to

23

evaluate, the calculation of the probability of error is greatly

simplified. This simplification is nontrivial, for as evidenced by

the work of Bello,18 the summations of Eq. (7) are quite difficult and

tedious to evaluate.

Before proceeding with the evaluation of 2[]K(f)I] and 2[tr 2 [K( )]],one should note that, although the probability-of-error expression has been

developed for a basic BAPSK system, the expression is valid for many other

binary communication systems as well. All binary communications systemst

that compare a quadratic form Y*QY, in (two complex) Gaussian random

variables with a zero threshold to estimate each transmitted information

digit, i.e., I = sgn {Y*QY}, have a conditional probability of error

= {I- sgn [In tr [K(s)Q]] [1 + 4R-}

a2 k

B -R( ), for PC(f) < , (77)4

where the conditional covariance matrix K(e) = [YY*I ], and e is the trans-

mitter input state. The probability of error for such a system,

Pe =i[P,(e)], can be closely approximated by

-P i {1 - sgn (2[Il tr [K(f)Q]])(I + 4R-') , (78)2k

for the values of the probability of error that are generally of interest

(PC < 1/4), the probability of error can be closely approximated by R-1,where the S/N of the hard-limiter input

2 [tr 2 {K()Q}](R 2[IK(f)I] (

The mathematical justification of this approximation for any system of

this type is essentially identical to the preceding justification for

the BAPSK system.

Standard differential phase-shift-keyed, adaptive-differential phase-shift-keyed, and frequency-shift-keyed systems, as well as adaptive phase-shift-keyed systems, are all of this type.

24

The quantity R is referred to as the hard-limiter-input S/N through-

out this section without any justification being given; this nomenclature

can be justified, however. Because the hard-limiter input, Y QY, is the

sum of the transmitted information digit times a random signal voltage

and a random noise voltage,

y*Qy = Ins + n , (80)k

the average hard-limiter-input signal strength is

2[IYnQY] = s] + 8[In

F[s] (81)

since [Inn] is zero. It has already been shown thatk

2[s] = [tr {I"K(f)Q}] (82)

for a BAPSK system; however, Eq. (82) is true for most systems having a

hard-limiter input .hat is a quadratic form in two Gaussian random

variables.

In most systems of this type, the noise portion of the hard-limiter

input is zero-mean; thus one must have 2[n 2] to measure the noise power

in the hard-limiter input. The average signal-plus-noise power in the

hard-limiter input is the sum of the average signal power and the average

noise power because

8[(Is+ )2) = S[s2] + &[n2] + 2£[I'sn]

[ [s] + [2] (83)

In most systems of this type for such a Gaussian channel as that described

in this report, s is a Rayleigh random variable, and thus

S[s2 ] = 3{e[s]} . (84)

25

Hence the average noise power iai the hard-limiter input,

[n2 = 2[(s + )2 - 3O2[s + n] = p[(y*Qy)2] - 3Q2[InY*QA (85)

it can be readily verified that

[(Y*QY)2] - 3&2 [IyyQy] = 4[IK(f)l] = 8[n2] (86)

Thus R is indeed the hard-limiter-input S/N.

Returning to the evaluation of a BAPSK system probability of error,

one can easily show that

IK(f)l = k1 (6)k22( - Ik12( 2 (87)

and

tr2 [K(f)Q] = 4 Im2 {k12( )} , (88)

where Im {k12( )} is the imaginary part of k12( ). After defining the

random "correlation field"

r . - P. Pn [y-j *(-n (j 2 )i ] (89)P I. 2 - 1 -J 2

one can easily show that (see Appendix A)

CO

K(f)I = 7- exp (-a~Ti +j 2)) (rn,0 r" - r0 (90)i1" 2= 1 10 i1P.i2 j 1,r2, ) (0

and

cO

tr2 [K()Q] = 4 exp (-aT[j, + j2] ) Im {r" }Im {r 2}+ j2] 0 2

(91)

the correlation field r . , a function of the (random) transmitter state

is defined for integer values of n between one and N and for nonnegativeinteger values of j, and j2" Thus to evaluate the probability-of-error

approximation, Eq. (78), it is necessary to compute the second moments

of the correlation field.

26

In Appendix A, it is shown that for the propagation-medium model of*

Sec. II,

Kn" 2

r = IT + EG2T Z Z cr; exp (i2lrk[jl -2 27)1 j1j2 k=1 1 1. 12"k1 2

pn pn expI~~/ - I2 I + jfi Wm2,0 1

j ~1 22 2

exp (-i27[m1 -m 2]-k) b(rk + I, /k + MI) * (rk + 12' 'k + M2)

(92)

where the channel gain,

K

G = X - (93)k~k1 I

the normalized strength of the kth path

,,o2

k02 - (94)

the normalized time delay of the kth path

Tk--tok -

the normalized frequency shift of the kth path

, kk

S iT (96)

and

e x p - I sin (1 - 11 < 1

0 1 X~ - ~(97)

27

In appendix A, it is, also shown that (after neglecting terms smallerthan the intersymbol and interchannel interference)

T K E8[IK(l)I] E[k IV + EG2

exp (2aT) - II k=

I~(3k+,2j (98)and (iM)#O,O)

'K8[tr2 {K(j)Q}] E2GVT2 2 AIb~ ,A)12

k-1

exp (UT) cos (2 - k81

-~ (99)exp (2ciT) - 2 exp (rfT) cos (2rtk, + I

where

K

S[ 2K( ))}] Eh2,T I(, X + NOTk

1

K

+ EG2 T . 0" k ) (T k T kk (100)k1 /, ) (0,0)

Thus the probability-o[-error (approximation)

PC - 1 + 4 (101)

where FIK(e)I and Ftr 2 {K(5)} are given by Eqs. (98), (99), and (100).

28

IV THE PROBABILITY-OF-ERROR EXPRESSION

FOR MODIFIED BAPSK SYSTEMS

This section of the report develops a probability-of-error expression

(actually an approximation to the probability of error) for modified BAPSK

systems. These modifications are the modifications discussed in Sec. V1

of Ref. 1. As in Ref. 1, two classes of modifications are considered;

the first class consists of modifications to a BAPSK system that can also

be employed in systems similar to a BAPSK system, the second class consists

of modifications to a BAPSK system that can be employed only in a BAPSK

system or a system of comparable sophistication.

A very simple modification, belonging to the first class of modifica-

tions, is the adjustment of the subsystem signaling rate. Because the sub-

system signaling rate is not a fixed parameter in the error-rate expression,

Eq. (101), this BAPSK system modification is already included in the error-

rate expression.

Two other simple modifications, which are included in most systems,

are time synchronization and frequency synchronization. The synchroniza-

tion time shift, to, of a BAPSK system is very slowly varying, and thus

it is essentially constant. Time synchronization is already included in

the BAPSK system error-raLe expression; as can be readily seen from this

expression, the synchronization time shift, to, is merely added to the

time delays of the individual paths. Frequency synchronization or auto-

matic frequency control (AFC) can be readily included in the BAPSK system

error-rate expression. The synchronization frequency shift is so slowly

varying that it is essentially constant. Because the synchronization

frequency sift is essentially constant, it can be included in the scat-

tering function of the channel by merely adding the synchronization

frequency shift to the frequency shifts of the individual paths. (If the

noise is not wbhte, the spectrum of the noise must be shifted by the

synchronization frequency shift also.) If the time and frequency syn-

ohronization of the system are operating perfectly, the center time delay

and frequency shift of the propagation medium are zero:

KI a2X = 0 (102)

k=1 k A

29

KI 02T = 0 (103)

k1 k k

Because the time delays and frequency shifts of the individual paths of

the propagation medium are not fixed in the BAPSK system error-rate ex-

pression, time and frequency synchronization are already included in the

error-rate expression.

For alleviating dispersive self-interference effects caused by time

delays, time guard band is one of the most beneficial modifications that

can be incorporated into a BAPSK system or similar systems. For a system

transmitting signaling elements of length T + A with a time guard band

equal to A, it can be readily verified that one merely uses in Eq. (101)

the values of &[IK(e)]] and S[tr 2 [K(f)Q]] given by Eqs. (B.5), (B.6), and

(B.7) of Appendix B. From these expressions it is readily apparent that

for time-delay spreads less than A, there are no self-interference effects

caused by time delays, if the system is properly synchronized.

The addition of diversity receivers is another frequently employed

modification in systems like a BAPSK system. The only type of diversity

considered in this report is post-detection combining of statistically

independent receiver outputs. Only statistically independent receiver out-

puts are considered because the analysis of statistically dependent diversity

receiver outputs is complex and because the probability of error for depen-

dent diversity operation can be bounded by the probability of error for

independent diversity operation. The diversity receivers can be displaced

in space, frequency, polarization, or time, as long as they are statistically

independent.

When the order of diversity is D, it can readily be verified that the

conditional probability of error

DP = Pr Z * QYd < 0

id1 k d d J(104)

Pr{ < 1~12 -

30

where the subscript d denotes the appropriate diversity receiver, the

eigenvalues, XI and X 2' are given by Eqs. (57) and (58), and the random

variables, xk and y,, are 6tat.istically independeit, complex, Gaussian

random variables. However, it is well known that 16

1D

(Aixd2)( ~ Yd 1) f f2D

has an F distribution with parameters m =n =2D). Thus the conditional

probability of error

P(DX2()/(e) (2D) fD-1 dfV I-2. + r%2D

•'2D D (l\ 1] (Dj .1* ~~~~-2 F(D) i ^-) TVL (1 _

(105)

but P ,) Pe() = ( - X' 2 ()) -1, and hence (see Appendix B] the

conditiqnal probability of error for Dth-order diversity

-'(2D)p (()]D { + I [-P()]J (106)e['2 (D) j=1 Dj

where the constants

D (-1)1 + - I+1 D-jD. j 1 D + Ic+-jD1 +

Figure 3 is a plot of P(D,6) as a function of P

The unconditional probability of error for a BAPSK system with Dth-

order diversity,

['(D) (2) {[(())D] + 7- a &8[(-PC(6))1+.] (108)

-['2() Da j e f

31

10-5~

0-6

10- 7

10-8

10'9

10- 10

105 0- 0- 1-0. (c D -4127

FIG.3 PD,6 AS FUNTIO OF o~e

320

For relatively low orders of diversity and practical values of PC(D),

such that U D,1 Pe < <

P(2D)P (D) =_ - [((, ))] 1(i09)

-2 (D)

however, by the argunients used in the previous section,

P (D) r(D [JK(f)' ) J (110)eF 2(D) £D/2{tr2[K(C)Q]}

By using the Schwartz inequality, it can easily be verified that[(2D)/r 2 (D) peD is a lower bound on P (D). Because P (D, ) is a deter-

ministic function of Pe(f) for any system, this report concentrates

primarily on the evaluation of the (unconditional) probability of errorfor a system without diversity. The benefits accrued by adding diversityare essentially the same for any system.

The transmission of M-ary signals is not included in the error-rate

expression. The approach used in this report could be extended to in-

corporate M-ary signals; this extension would be similar to that used

in the analysis of a QDPSK system in Ref. 3.

The second class of modifications are those that can be made only

to a BAPSK system. This class of modifications includes alterations to

the long-term-average filter characteristic, the division of power between

the information-tone and the pilot-tone portions of the transmitted

signals, and decision feedback. Other modifications are essentially

complete system changes.

The simplest BAPSK system modification (of the second class) is the

adjustment of the tim. constant of the long-term-average filter. This

modification is actually only a system adjustment, and since a is noL[: fixed in the probability-of-error expression, it is already included in

that expression. Another alteration to the long-term-average filter

characteristic is delay compensation (see Sec. VI of Ref. 1). One can

include delay compensation in the probability-of-error expression by

substitutingcn

8[IK(1)d] = exp (-aT[j 1 + £ [r r" -rn r * ] (111)

1 1 , 2 = 1 1 J 1 J 1 2 1 1 ,1 J1 2,3

33

and

2[tr 2 [K()Q]] 4 7 exp (-oT[j 1 + j2]))[Im{ra }Im{r .}] (112)j 1 ,j 2 =1

in Eq. (101). Delay compensation and a double-pole long-term-average

filter can be included by substituting

2[IK([)I] = 1 J112 exp (-aT[j 1 + j2 ] ) J[r JrI 2 J r ) * 2 1J 1,]2= 1

(113)

and

S[tr2 [K( )Q]] = 4 1 J1 J 2 exp (-aT[j 1 +j2 ]) &[Im{r *}Im{r2,.1] (114)jl'j2=1 112" ])SI ' ,j 11

in Eq. (101). Similarly, the combination of long-term-average filter out-

puts of adjacent subsystems and the division of pilot-tone and information-

tone power can also be included.

Decision feedback is the only modification that cannot be included

easily. The inclusion of decision feedback in the probability-of-error

expression would necessitate major theoretical modifications that do not

seem justified, because of the unknown value of the modification itself.

34

V CURVES OF THE PROBABILITY OF BINARY ERROR

The probability of a binary error for basic BAPSK and BDPSK systems

(as well as several modified APSK and DPSK systems) is plotted as a func-

tion of the S/N in Figs. 4 through 13. In each figure, the curves repre-

sent different systems; however, all the curves in each figure are for the

same (propagation-medium) scattering function. Different figures display

probability-of-error curves for different scattering functions. The ex-

pression used in computing the probability of error for various systems

is actually the approximation expressed by Eq. (101) of this report.

Time and frequency synchronization, second-order post-detection

diversity, and the transmission of quaternary signals by DPSK systems are

the only modifications explicitly included in the probability-of-error

curves. By proper interpretation of the curves, however one can approxi-

mate (or bound) the effects of several other modifications to basic BAPSK

and BDPSK systems.

In each figure, the probability-of-error curves labeled K are for a

basic BAPSK system, as described in Ref. 1; the curves labeled B are for

a basic BDPSK system, as described in Ref. 3. These systems have the same

signaling period T, and frequency separation between subsystems 1/T. The

time constant of the BAPSK system long-term-average filter is aT = 0.1.

This value was chosen, even though it cannot be easily implemented

digitally, because it offers a reasonable balance between inherent self-

interference and frequency-dispersive effects. A more thorough discussion

of the effect of the time constant appears in Sec. VI.

Because the center time shift and frequency shift of each of the scat-

tering functions considered are zero, time and frequency synchronization

are essentially included in all systems. The curves labeled K2 are for a

BAPSK system with second-order, post-detection diversity with statistically

independentt diversity inputs. The curves labeled B2 are for a BDPSK

system with second-order, post-detection diversity with statistically in-

dependent diversity inputs. Thus the K2 and B2 curves are lower bounds on

the probability of error of the appropriate system with second-ordei,

t All diversity inputs are implicitly assumed to be identically distributed in this report.

35

12

0

Mi K2

F

r-T

02

B2

16 CRSE 31____ ____

0 10 20 30 40Q s0SIGNRL TO NOISE ARTIG -- 06D-27-I'

FIG. 4 CURVES SHOWING PROBABILITY OF ERROR AS A FUNCTION OF SIGNAL--TO-NOISE RATIO

Note: All BAPSK system curves are for a fixed time constant (see paragraph 3, p. 35);for a discussion of the effect of changing the time constant, refer toparagraph 2, p. 55.

36

iEH_-

__J

CD ! K

cc

>_ -FCc

0:

0 10 20 30 4 50SIGNAL TO NOISE RRTIO - DB -,4172.181

FIG. 5 CURVES SHOWING PROBABILITY OF ERROR AS A FUNCTION OF SIGNAL-TO-NOISE RATIO


37

-210

CEr'm

cc

C

a-LO

S TF1 :CE2.

1OD "~m ~ J

kCF~I_5_ _ _

TO-NOISE AT ...O

F , Note: All BAPSK system curves are for a fixed time constant (see paragraph 3, p. 35),for a discussion of the effect of changing the time constant, refer toparagraph 2, p. 55.

38

i

M -1

10

r-

r-CD

'- -2 _ _ __ _ _Q'10a:

FcrCI

i6 Ix I I/\-j

CSE 13

0 10 20 30 u4o 50SIGNAL TO NOISE IRTIO - DB o-4,72-,70



39

1

M10

clOcMCDccCL

ILl

cE

2

B2

CASE__46 ____ ________

010 20 30 410 50SIGNAL TO NOISE RATIO - EB D-4172-182


Note: All BAPSK system curves are for a fixed'time constant (see paragraph 3, p. 35);for a discussion of the effect of changing the time constant, refer toparagraph 2, p. 55.

40

-*1I

I i

CD

CDa:Cc

wii

Fa:CE

Z

r r 1 I ___ ___ __ _ _ _ I _ _ _ _

CASE 44

0 10 20 310 40 50SIGNAL TO NOISE -RTIO - D8


Note: All BAPSK system curves are for a fixed time constant (see paragraph 3, p. 35),for a discussion of the effect of changing the time constant, refer toparagraph 2, p. 55.

14

10

-l2

1 'A

Cc

0

CD

CC -3

>_ FcC~

X

-s CFISE 17 _____

010 10 20 30 4I0 50

SIGNRL TO NOISE RRTIO - DB D-472-184


Note: All BAPSK system curves are for a fixed time constant (see paragraph 3, p. 35);

for a discussion of the effect of changing the time constant, refer toparagraph 2, p. 55.

I42

1 I-. .- -I '' ' ' 'J T ' - -::' '- - - -, -

. . .. -' "

..

2'

0

'-' -3

CE

10 4 102T0'05

SIGNAIL TO NOISE RRTIO - B047-8


Note: All BAPSK system curves are for a fixed time constant (see paragraph 3, p. 35);I for a discussion of the effect of changing the time constant, refer to

paragraph 2, p. 55.I4

-14

CDr-

D-

r-

02

cEcc -

0~ ___

1c !__RSE 33__

0 10 20 30 40 50S I NAL TO NO ISE RATIO - r',rB D-,41,-18,0

FIG. 12 CURVES SHOWING PROBABILITY OF ERROR AS A\ FUNCTION OF SIGNAL-

TO-NOISE R ATIO

Note: Ali' BAPSK system curves are for a fixed time constre ,",ee ," paragraph 3, p. 35':

for a discussion of the effect of changing the time constant, refer to

paragraph 2, p. 55.

44

12

0

C-

n

0-'1 5

F O-OSE ATI

Note: ~ ~ ~ ~ ~ ~ ~ ~ ~ Al--SK s se-cr e-ae f r a-ie-tm-ost n-se -a 'gr , . 3 )

d~ ~ ~ ~ ~~ ~aarp 2,te p.l 55. yimcre r orafxdtm osat(sep,'g h3 .3)

45

post-detection diversity; the curves labeled K and B are ,pper boundIs on

the probability of error of the appropriate .vstem itih second-order,

post-detection diversity.

The curves labeled Q are for a DPSK system that transmits two linary

digits per signaling element (i.e., a QDPSK systc.nt); the curves labeled

Q2 are for a QDPSK system with second-order, post-detection diverbitt and

statistically independent diversity inputs. The curves labeled Q and Q2

can be interpreted as bounds on the probability of error of a QDPSK s-.stm

with second-order, post-detection diversity.

Comparison of the B and K curves seems to be the best method oft con-

trasting APSK and DPSK systems, even though commercially produced systems

operate at various signaling rates and incorporate various modifications.

If a comparison of the B and K curves is to be a valid comparison of the

basic concepts of the two systems, the systems must have the identical

bandwidth, signaling rate, number of subsystems, transmitted signal pover,

and modifications. For the curves of Figs. 4 through 13 these proper-

ties are identical. Neither system employs such modifications as time

guard band or M-ary signals; modifications of this type should not enhance

the performance of one type of system any more than the performance of the

other type of system. Perhaps the major inequality in the comparison of

the two systems is that various modifications to the long-term-average

filter of a BAPSK system are not included. In particular, delay compen-

sation should improve the performance of a BAPSK system significantly.

The curves labeled Q2 are the probability-of-error curves for a DPSK

system transmitting quaternary signaling elements and employing (statis-

tically independent) second-order post-detection diversity. When the

outputs of different subreceivers (rather than those of space- or

polarization-diversity receivers) are diversity combined, the curves

labeled Q2 are for a system that has the same bandwidth, signaling rate,

data transmission rate,, and transmitted signal power as the basic BAPSK

and BDPSK systems, which have probability-of-error curves labeled K and B.

Thus, contrasting the B, K, and Q2 curves offers a valid comparison of

these basic systems (actually, the curves labeled Q2 are for a modified

DPSK system). The curves labeled Q are for a QDPSK system without di-

versity; thus they are a bound on the performance of a QDPSK system with

second-order post-detection diversity when the diversity inputs are not

statistically independent.

t The QDPSK system is described in detail in Ref. 3.

46

Li

The scattcring functions used in computing II,, I I es iII 1.he. %ar'ils

figures have been normalized. The time delays are nlormal Ized 1,v IIe

signaling period, T, of the systems; the frequency -AiI 't.s ale normal Il,(-(

by the frequency separation between adj acent subsystems, I /7', of the

systems; and the path screngths are iormatized by the invver.se of* the

channel power gain,

1G-2 = (115)

KE 2

k1lk

(G2 is the channel powei gain). Thus the probability-of-error cr~es for

a given normalized scattering function,

SH(X,-) 28(X- X, )(r - Tk) rk=1 k

are applicable for systems using a signaling period T and a channel with

power gain G2 and an (unnormalized) scattering function

KSH(XL, -r ) = Z 0_2 G2 (116)

SH~,T ,.,~G 3(X. - L./T)3(-r -rkT) . (116)k=l

For systems with a signaling period T and a channel power gain G2 , the

actual s n of 2 = Gk, is G times the normaltzed

strength of the kth path '2 the actual frequency ,hi ft of the kth path,

Xk = k/T, is 1/T times the normalized frequency shift of the kth path,

X and the actual time delay of the kth path, Trk =kT, is T times the

normalized time delay of the kth path, Tk"

The normalized scattering function is depicted by the plot in the

lower-left-hand corner of each figure. For each path, an X is plotted on

the T-F plane; the size of the X is proportional to the normalized

strength of the path, '; and the center of the X is located at the point

(T,F) = (Ork,k ) (117)

To accomodate a wide range of ' and X values, the 7' and F axes are log-

arithnic. In addition to the plot of the normalized scattering function,

9 the parameters of the normalized scattering function, {0,k,kk for

k = 1, ..., K) are tabulated in Table I.

47

Table I The probability of binary -rror

KEY TO FIGS. 4 THIROUGH 13 for each system is plotted as a !funct, 011

_ _ _ _ _of the receiver input S/N (ito dec ibels),

FIGURE SCATTERING FUNCTION PARAMETERS

NUMBER 2 rkIT AkT G 2 Ek k (S/N)db 10 Iog,0 / -- (118)

*1 4 1 0 o 0 No )

1/2 -2.5 x 102 01/2 -2.5 x 10-2 0 where the power gain of the channel iS.

1/2 5 x 10_ 0 G Z 0- the channel noise power per6 1/2 -5 x I0-2 0 k =1 I

unit bandwidth is N0 , and the trans-7 1/2 -10-1 0 mitted system energy per signaling

1/2 0 2.L x 10 -3 element is E.1 1/2 0 -2.5 x 10-

9 1/2 0 2.5 xThe significant feature of the9 1/2 0 -2.5 x 10-m s in fcn

22 probability-of-error curves is that the1/2 10 ~ .10.2o-ro osnt~1/2 -10-2 -10- probability of error does not approach

1/2 2.5 x 10-2 2.5 x 10-3 zero as the S/N increases (unless there1/2 -2.5 x 102 -2:5 x 10-12 2 is only one path with no time delay or1/2 2.5 x 10 5 x 10-

12 1/2 -2.5 x 10-2 -5 x 10-2 frequency shift, t. e., a time- and

13 1/3 2.5 x 102 5 x 10-2 frequency-invariant channel). Careful

13 1/3 0 -5 x 10-1/3 -2.5 x 10-2 0 consideration of the probability-of-

error expressions shows that this as-

ymptotic value of the probability of error (as the S/N increases) or residual

error Pe (0), results from the dispersive self-interference (intersymbol and

interchannel interference), the changing (with time) phase shift of the

propagation medium, and, in aBAPSK system, the inherent self-interference.

To study the sensitivity of the various systems to the dispersive effects

that cause this nonzero residual error, it is desirable to have a parameter

that is indicative of the performance of the systems and essentially inde-

pendent of the S/N.

Careful inspection of the probability-of-error curves indicates that

the curves are essentially determined by the time- and frequency-invariant

probability-of-error curve and the residual error P,(co). Figure 14 illus-

trates the determination of a probability-of-error curve by the residual

error and the time- and frequency-invariant curve. At large values of the

S/N, the probability of error is essentially constant and equal to the

residual error, because the operation of the system is essentially deter-

mined by the energy-scattering effects of the propagation medium. For

S/N such that the time- and frequency-invariant probability of error is

48

III

i0 - a

>--

_ 0-21

m,0

,- ACTUAL PROBABILITYOF ERROR CURVE

n.-

0cra:

iO-3

-Pe ()z

RESIDUAL ERROR PROBABILITY OF ERRORFOR A TIME-AND- FREQUENCY-FLAT CHANNEL

O-4

1O5 I I I

0 10 20 30 40 50SIGNAL-TO-NOISE RATIO db D-4172-174

FIG. 14 DETERMINATION OF THE PROBABILITY-OF-ERROR CURVE BY THERESIDUAL ERROR AND THE FLAT-FLAT PROBABILITY-OF-ERROR CURVE

49

greater than the residual error, the probability of error is essentially

equal to the time- and frequency-invariant probability of error, because

the performance of the system is essentially determined by the additive

aoise (rather than by the dispersive effects). Because the time- and

frequency-invariant probability-of-error curve is independent of the

residual error, the probability of error for various scattering functions

is essentially determined by the residual error. Thus tile residual error

is a useful parameter for study of the sensitivity of various systems to

scattering-function parameters.

Careful analysis of P,(o) for a variety of scattering functions in-

dicates that it is relatively insensitive to the normlized strengths of

the paths and the number of paths. The predominant factors determining

P,(o0) are the maximum differential (normalized) time delay,

-- max {I7 - (119)j,k=l, ... , K

and che maximum differential (normalized) frequency shift,

= J max K - . (120)j ,k=l . .... k )

When the maximum differential time delay and frequency shift were main-

tained, P,(co) decreased somewhat as a single path became weaker (i.e.,

as the normalized path strength decreased) and as tile number of paths in-

creased. In addition, for a fixed scattering function with a maximumdifferential time delay, /A0' and maximum differential frequency shift,

, X0, P,(co) is essentially equal to the sum of P,(W) for a scatteringfunction with no time delays and maximum differential frequency shift /AX0 ,

and P (CO) for a scattering function with no frequency shifts and maximum

differential time delay / 0 " In other words, for the systems considered,

the residual error, P,(00), is essentially determined by the maximum differ-

ential time delay or the maximum differential frequency shift.

The residual error, P,(°), was calculated for various scattering

functions consisting of two equal-strength paths with either zero differ-ential time delay or zero differential frequency shift, so that the

sensitivity of the various systems to time delays and frequency shifts

could be evaluated. Figure 15 is a plot of P,(0o), for scattering functions

of this type, as a function of the normalized differential time dela),

50

A, and the normalized differential frequency shift, AX. By using the

lower horizontal axis, Fig. 15 is a plot of P,(00) as a function of A' X

for various systems: K(X) denotes a basic BAPSK system: K2(1) denotes

a BAPSK system with perfect,t second-order, post-detection diversity. By

using the upper horizontal axis, Fig. 15 is a plot of P (cO) as a function

of A for various systems: K(Tr) denotes a basic BAPSK system; K2(') de-

notes a BAPSK system with perfect, second-order, post-detection diversity.

The curves for BDPSK and QDPSK systems are the same for both the lower

and upper horizontal axes (AX and AT, respectively). The curve labeled

B[Q] is for a B[Q] DPSK system with no diversity; the curve labeled

B2[Q2] is for a B[Q] DPSK system with perfect, second-order, post-

detection diversity. The curves for BDPSK and QDPSK systems are applicable

for use with either horizontal axis; however, the curves for a BAPSK

system are applicable for use with only the appropriate horizontal axis.

All the BAPSK curves are for a long-term-average filter time constant,

aT = 0.1.

For a valid comparison of the basic concepts of the systems, the

curves labeled B,K(X) or K(r), and Q2 should be contrasted.

Ii

SIn this report, perfect diversity implies diversity with statistically independent inputs.

51

IO **------ II i I I

r Ar;

K W

B2 B K21)

0

w

0

z0

< 10-3z

LL / 0 I K()z

0 02 XB2 aK2( -

t: 10-74-7 1

0

0.

0

z

10-61_ _ _ _1 _ _

2x103' 5X10-3 10-2 2xI102 5x10-2 I0- 2X1011TIME DELAY SPREAD-A-rT

2xI103 5xA1 3 i0-3 2x10-2 5xI1O 2 I0- 2x10-DOPPLER FREQUENCY SPREAD -AXT

Ra-4172-2R

FIG. 15 CURVES SHOWING Pe ( 1° ) AS A FUNCTION OF DIFFERENTIAL TIMEDELAY AND DIFFERENTIAL FREQUENCY SHIFT

Note: All EAPSK system curves are for a fixed time constant (see paragraph 3, p. 35);for a discussion of ihe effect of changing the time constant, refer toparagraph 2, p. 55.

52

VI CONCLUSIONS

The relatively simple probability-of-error expression, Eqs. (98)

through (101), and the relationship between the probability of error and

the S/N at the hard-limiter input, Eqs. (82) and (85), are the most general

contributions of this report. Since the probability of error is a monotonic

function of the S/N of the hard-limiter input and since it is proportional

to the inverse of this S/N for practical values of the probability of error,

the relation is intuitively satisfying. Although the probability-of-error

expression is developed specifically for a BAPSK system, the techniques

employed in developing it are applicable to a large class of digital commu-

nications systems. The probability-of-error expression developed in this

report is of value, because it is significantly easier to compute than the

expressions of other authors. It is interesting to note that, the BAPSK

system probability-of-error expression is essentially equal to the sum of

three other error probabilities (See Fig. 14): the probability of error

for a time-and-frequency-invariant propagation medium, the probability of

error for a nonadditive (i.e., the additive noise is identically zero) and

time-invariant propagation medium, and the probability of error for a non-

additive and frequency-invariant propagation medium.

Another significant, but less general, contribution of this report

is the comparison between a basic APSK system and a basic DPSI( system pre-

sented in the probability-of-error curves of Figs. 4 through 13. As can

be readily seen from the curves of Fig. 15, the performance of a basic

APSK system (with signaling rate lT) and the performance of a basic DPSK

system (with the same signaling rate, l/T) are equally sensitivet to time

delays; this is not surprising because the time-delay sensitivity results

from dispersive intersymbol and interchannel interference primarily, and

because both systems are equally prone to dispersive intersymbol and inter-

channel interference. For a time-and-frequency-invariant propagation

medium (or a propagation medium that is dominated by the additive noise)

the performance of an APSK system is better than that of a DPSK system.

This is to be expected, however, since an APSK system makes its decision

(as to whether the transmitted binary information digit was plus or minus

In this Sec. the performance sensitivity of a system refers to the sensitivity of the probability of binary

error for the system. One system is said to be more sensitive than another if the probability of a binaryerror for the system is larger than the probability of a binary error for the other system.

53

IA

one) by comparing a noisy matched filter output with a relatively noise-

free long-term-average filter output, while a DPSK system makes its

decision by comparing two noisy matched-filter outputs. For a time-and-

frequency-invariant propagation medium, the performance of an APSK system

approaches the performance of a coherent phase-shift-keyed (CPSK) system

as the time constant of the long-term-average filter becomes larger; a

CPSK system is well known9 to be the optimum system to combat additive Gaus-

sian noise. Unfortunately, the long-term averaging that makes an APSK

system performance approach the performance of a CPSK system for a noise-

dominated propagation medium also makes the performance of an APSK system

extremely sensitive to variations (with time) of the complex gain of the

piopagation medium. This performance sensitivity is most readily evidenced

by the residual probability of error, Fig. 15, for frequency-invariant

propagation mediums. It can readily be seen that the APSK system perfor-

mance is much more sensitive to frequency shifts than the DPSK system

performance. This is reasonable because a DPSK system is prone to complex

gain variations during only two signaling intervals, while an APSK is

prone to complex gain variations during 1 + rc signaling intervals, where

,r is the time constant (in signaling intervals, T) of the long-term-

average filter.

The discussion of the preceding paragraph has been concerned with the

comparison of a basic APSK system and a basic DPSK system when both systems

have an identical subsystem signaling rate and no modifications (other than

time and frequency synchronization.), and an APSK system has a long-term-

average filter time constant of aT = 0.1. Existing APSK and DPSK systems

have different subsystem signaling rates and modifications; in addition, the

time constant (of the long-term-average filter) of an APSK system is never

equal to 0.1. One can glean a great deal of information from the comparison

of the preceding paragraph, however.

When an APSK system and a DPSK system have the same signaling rate, the

performance of a basic APSK system and the performance of a basic DPSK system

are equally sensitive to time-delay effects, while the APSI( system perform-

ance is more sensitive to frequency shifts than that of a DPSK system; thus

if one were to pick aa optimum sybsystem signaling rate for an APSK system,

one could always find a DPSK system (i.e., a DPSK system having a subsystem

signaling rate equal tc that of an APSI( system) that would perform at least

as well as an APSI( system for dispersively dominated propagation mediums.

54

Hence comparing an APSK system and a DPSK system with the same subsystem

signaling rate is not really a constraint. Because most propagation mediums

are time-varying, the optimum subsystem signaling rate of the DPSK system

would be less than that of the APSK system (it is assumed that the optimum

subsystem signaling rate is the rate that makes the system equally sensitive

to time-delay and frequency-shift effects). Note that, as shown in Ref. 1,

changing the subsystem signaling rate does not change the data transmission

rate of the system.

As discussed earlier in this section, the performance of a DPSK system

is more sensitive to additive noise than the performance of an APSK system;

however, this gain is only for large values of the time constant of the long-

term-average filter. Computer calculations of the probability of error for

other values of the time constant indicated that, as the time constant be-

comes small, the noise dominated behavior of an APS.K system quickly approaches

that of a DPSK system; in addition, increases in e- above 10 produce little

change in the noise-dominated performance of an APSK system. The same com-

puter calculations also indicated that the time-delay sensitivity of an APSK

system is essentially independent of the time constant, and that the frequency-

shift sensitivity of an APSK system approaches that of the DPSK system as the

time constant is decreased.

The modifications that can be made to both an APSK system and a DPSK

system have essentially the same effect. The effect of the addition of

perfect dual diversity is indicated by the probability-of-error curves in

Figs. 4 through 13. It can readily be seen from these curves that the ad-

dition of diversity does not enhance the performance of one system more than

the other; however, diversity does enhance the differences in the (nondi-

versity) performance between the two systems. It can be readily seen from

the curves that diversity is a very beneficial system modification.

As noted in Ref. 1, a time guard band is one of the most beneficial

system modifications for the purpose of alleviating the effects of disper-

sive intersymbol and interchannel interference resulting from time delays.

In Ref. 3, it is shown for a DPSI( system that if the time guard band is A,

then the effect of all time delays ol magnitude less than A is essentially

negated. From Eqs. (B-5), (B-6), and (B-7) of Appendix B, it is readily

apparent that if the time guard band of an APS( system is A, then the effect

of all time delays of magnitude less than A is negated also. This negation

of the time-delay effects results in a slight degradation in the S/N and

55

in the data transmission rate of both a DPSK system and an APSK system.

Thus the time guard band modification does not benefit or degrade the

performance of the APSK system any more than it does the performance of

a DPSK system.

The effect upon the performance of an ADPSK or a DPSK system resulting

from the transmission of M-ary signals can be readily inferred by comparing

the probability-of-error curves for a QDPSK system and a BDPSK system of

Figs. 4 through 13. These curves are for a modified DPSK system; however,

the effect of transmitting quaternary information signals in an APSK system

should be essentially the same. As indicated by Pierce, 20 diversity

combining the independent M-ary subsystems does enhance the system perform-

ance, this can be readily seen by comparing the Q2 and B probability oferror

curves in Figs. 4 through 13. A QAPSK system with diversity combining of

the outputs of independent subsystems should show the same performance

improvement.

Perhaps the major inequality of the system comparison is that several

modifi-ations that can be incorporated only into an APSK system are not

included. These modifications include alterations to the long-term-average

filter [delay compensation, double-pole filter characteristic, and aver-

aging of the outputs of adjacent (in frequency) long-term-average filters],

decision feedback, and the division of power between information and pilot

tones. Delay compensation is probably the most beneficial of those modifi-

cations, and its omission is unfortunate; however, most of these modifica-

tions tend only to make the performance of an APSK system approach that of

a DPSK system.

[

56

APPENDIX A

57 0

APPENDIX A

The probability-of-error expression is expressed as a function of

E[[K(,)1] ana S[tr 2{K(f)Q}] in Eq. (78); thus to evaluate this expression

F[IK()I1 and F[tr 2{K(f)Q}] must be computed. By using the definitions

of determinant and trace and the fact that P" is rcal and (Pno) 2 - 1, it

can readily be verified that

8[IK( )I] l[k k 2 2 )- 4P k 2 ( )123 (A.1)

and

S[tr2 {K()Q} 4S[Im2 {Pnk 12( )}] (A.2)

where [see Eqs. (28), (40), (41), and (42)]

k, : = [1-n(0)12 j

Y exp (-ctT[j 1 + j 2] )P.j P.j 8[yn(-jl)yn*(-j2)lf]

(A. 3)

-) 0 Po n[ (O)y* (0)! ]

- ~ exp (-aTj)P_ PS[y (-j)y (0)I] (A.4)j=1

a nd

k2 2 ( ) ; PoP~S[yn(o)yn (0) X1 (A.5)

Aft.er introducing the random correlation field

pn pf E(yn(-j,)yn*(-J 2 )I ] (A.6)JIJ 2 iI -J2

one Cinds that

FX Ixp (-aT[j + , 2] [r 0 3 I , j 2 j r, J2

J P Pj 2 Ba

59 Preeding Page Blank

and

2Ltr [K(6)Q]] = 4 exp (-aT[j + ' 2 J) & [Im{rI. }Im{r'2}]

jlJ'j 2 =1 j 12'

(A. 8)

Thus to evaluate the probability-of-error expression, one must com-

pute the autocorrelation function of the random correlation field r . .

Since

y( = f y(t) exp (-i27f. t)p(t + to + jT)dt-a0

f y(t)hP(tdt (A. 9)-O

where the channel output

y(t) = f H(t,f)X(f) exp (i27ft)df + n(t) (A.1O)

the random correlation field

P'J P- af.f ha (t 1 )X(fj)h" (t 2 )X ]2)Jl'J 2 -J 1 2 - CO 1

* RH(t2 - t1,f2 -fl) exp (i217[ft 1 - f2 t2]) dt I dt2dftdf2

+ ff Rt ( 2 - t1)h (t )b2*(t)dt dt2O 1 2J

P n. pn1 fI S(-rT)qIJn ('r,X) ', (-r, X drdl\

+ Sn (f)H. (f)H , (A. 11)

where the scattering function of the channel is

K* SH(X,-) = - - -rk) (A.12)

k-1

the spectrum of the additive noise is

N0, f > 0Sa(f) (A.13)

t0, f<O

60

the cross-ambiguity function of x(t) and y )

Py ('rX) = (t-1) y*(t + 1) exp (-L2nX) 1t , (A. 14)

and the Fourier transform of h(t) is H(f). Because of the special naturej j

of the channel scattering function and the spectrum of the additive noise,

the correlation field

KP.P + (Tk) ,r') C (A.15)

{0 P (0,0) A.. (-r'~x 0 ~X~kP ,J2 -] I -j2 h I hi k I h h 2

I 2 j I )2

Because the matched filters of the receiver are orthogonal, i.e.,

JT, 11 = 12

h n (0,0) = = (A.16)

1 J2 0, / J2

the correlation field

K

rn No TS.I + Z. o 2pn pn. qp (Tkk) • rT x. ) (A.17)11 2 2 k -2 h 2 x

The essential problem remaining in the computation of the autocorrela-

tion function of r' is the evaluation of

1 2)I T,) exp (i27T[fo +~* [t 1

--(t + t+ jT) (t + exp (-i2vit) dt (A. 18)9 .p 2

however, by using the definition of x(t) [Eq. (11) of Ref. 1], one finds

that

P'jq (-rX*@) E Y, P!_ exp ti i + Mi

e(ia,[jf0 +m T) f (t 0: + jT)p (ti + - -1 1T

" exp (i2nL + i (A.19

61

and by making the substitutions

to tL - jt = t + O + - T, 1 = -j -l

2 2

and

M = n -m,

one finds that

h'.0 OrPX) +xp+i - T + 2XTj +

exp(i277--t o] + iT[l- - + lT[+T])

Piz -+ ) exp -2 ( t it.

2 2

(A.20)

After performing the integration in Eq. (A.20), one finds that

(Tx)_ vTT eT+Xxp i27 fo + "+(T+to)-

-J

7Te: - x p f + ic -l e x p - T - I T t- )

[M

-t + iT, x + ) (A. 2i)

where

(xpiiXr slT sin(7XT [ T <

7T7- T) =

O, (A.22)

62

U>t

and thus the random correlation field

Krn = TNo + TE Z C2 exp (i27&k T[jl - 12])

1 p. P 2 j 1-i2 m1 + j2)

!ll _2 ~ 1 j_2 x i4_2lt

e x p ( - i2n - -T , k + / + 1

T2 t) (k + ... " +

(A. 23)

For convenience in interpreting later results, it is instructive to

compute the mean of the correlation field

[r " ] ET- (1 +8 2 o- l(k~2 . exp (i2m[] - 1]k' l 2 2-1J 2 k1 9 +

+ 8. {NoT [Tko + 1 TX( 1) -rk + 1T, ?k + I2}

el-p k 0 ( 1 k +0,0

(A.24)

in the above expression, the

ET K

K 2

(1/2) ET (1 + 8 ) "I

11 -= 2 k+

term results form the signal portion of r . ;the No1T 2 term results

form the additive-noise portion of rt h; and theK

(1/2)ET (I2 k j 2k I

term results from the dispersive self-interference portion of r. 2

Using Eq. (A.24), one can also compute

63

Fe[k exp (-aT[j + j])2 )[r n )

P ,J 2 = 1 1 P1 1 2

[a K 12eLxp (aT)E 2ep(2aT) I k--1 c k k

exp (aT) - cos ( 2iiakT)

exp (2aT) - 2 exp (aT) cos (27A, T) + 1

K ~2+ N O + E Z 2 k + T , k +

0t rk (P) (+ o)T,k 1 k 1 , m ~ 0 ++ T

(A.25)

and

22 0,0

K

0-21 -12

k I,

+ ET Z cr m ( "I k + 1T, Nk +kl ((rk(o o

(A.26)

It can also be readily verified that

g = -02exp (UT) cos(27n\kT) - 18[PgUocr{K( )Q} Z ET k r' Xk )12

k=1 exp (2aT) -2 exp (aT) cos (2mXk T) + 1

(A. 27)

By neglecting terms that are smaller than the dispersive self-

interference terms (and assuming e' T ' 1), it can be shown that for

j and j2> 0

, r r o = 8[r, 0] r" ] - 2(1 + 6. )2[r ][rnP[ 2 '1 2" V ' 2 1-J2 Jl' 0 2'

(A.28)

64

and

£[Im{rn 0 }Im{r o} ] 1 + 6 J)[r S ]8[rn* n (A.29)j2 - 2 1 11 0 2

and thus

2[IK(f)I 2 [k22 ~ eaT f + E 7- k lm0(-V( +iX 2)2

(A.30)

and

[tr )Q}] =02[I , tr {K(f)Q}]

E 2V K 2 e r cos 2nXkT- 1

[k= k e - 2 e a T cos 2nXT + I

(A. 3 )

65

APPENDIX B

67

APPENDIX B

For a BAPSK system with time guard band, the transmitted signal is

given by Eq. (94) of Ref. 1. It can be readily verified that Eqs. (A.1)

through (A.18) of Appendix A are valid for a BAPSK system with time guard

band if one uses

A/ (rX T (2 r .~j X

S(T+XylT,X+mT) , (B. 1)

whbere thle constant

and

si [7TXT T 2

______- -<07r7T ' 2T

rn n mkrr, f + Lx (on-X ex i2um- -'1 2T

-- -11T X + a (B. I)

171 2+ (B)

Thus the random correlation field for a BAPSI( system with time guard band

is

69

-- J

A ET KNoTa + 7 cra2 exp (i2.yyXkT[jl -j2')11'j2 ] - 2 "/ k-1

1 2 2-1, i x~27[ii- 2

+ ([ i 2]) -i27 T (m 2T m

* + T l l T , k + T + yl2T;Xk

(B. 4)

It can also be readily verified that Eq. (101) is valid with

2(A T2[IK(f)I] k2 Texp (2yoaT) - 1

+- E o- mEOO + ylT, k 12

2V k1T (B.5)

[tr 2 [K(e)Q]] E 0-2 27 1 (T,xk)(B

exp (aT) cos (27'ykkT) -1 2

exp (2-yaT) - 2 exp (TyaT) cos (27ryXkT) + 11 (B. 6)

and

A ET K A8[k2(2 (6 -)) OI- k Xk) 2 + NoT") k~l

ET K 2IA+M 1+ 2

+a ,0) k + ylT,Xk + 2

(B. 7)

70

In Sec. (IV) of this report, it is shown that the conditional prob-

ability of error for a BAPSK system with perfect Dth -order diversity is

____D D- 1. ID ( 1 2 D+J)

P2D) Do j) (1 L - ( I - ; (B.8)r2 (D) J

0 D+

thus the conditional probability of error for the basic BAPSK system

PC( ) PC(I,{) = 1 - (i -e(

X 2(e) (B.9)

Using (1 - X2 ()/Ll(e))- = 1 -P( and the binomial expansion theorem,

one can easily show that

/ .r D+j (D + j \[ 1 k]

I"(~~ (XD) D-1 k __

(D(D 1)(D + j) (- k0 k )[7eeA]-__ I /['2) D-1 D 1 D + (1 J k +

1F2 (D) j=o k=O k D + j

F'(2D) D- 1 D- 1) D=1 D + j)[ (e)]k+ (Z) + Il ( PL ~ J

r 2 (D) j=O D + L ko

(1. 10)

however, it can readily be verified that the second term is zero, and thus

_ -(2D) D-1 D+j (D -1) (D+j) (-)+k [p(.)]kl

[2 (D) j=0 kl=D k1 1+j

(B3.11)

By letting k1 : k - D and by interchanging the order of summation, one

finds that

71

-DR [pc(!)]D k I0 -P'2 (D) 0

D-1 -k1 (D+ikIZ (-l)D-1- D A 2 (B. 12)

and since

D (-WDl1j( 1V +1 1) (Dj 0 \D +j/ i\ i

P,(D,) ,!f)]D [I + ZD [-p (~()kak] , (l

(D) k =1DW(B13

where

=D-i-k (~)1 D + ]+ k\(D-D, k j~o D + j+k + kk(B+14J

As particular examples,

P (2,e) = 3 2(g) -2p3 (e)

P (, = lOP3( ) -l5p4( ) + 6P5(e)

P(4~ 35p4(e) - 4P(,c) + OP6( )-cO~

P P(5,e) 126P5(e) - 42Op6 (e) + 54Op7 (e) -3l5p8() + 7OP'(fl

and

P (6,e) 462p 6(l) 198OP7(e) +3465P 8(e) -30P'e

+ 1386P 0 ( ) -252P"'(e) (B.15)

72

REFERENCES

1. N. T. Gaarder, "A Mathematical Model for the Kathryn System," Technical Report 2, Part I,Contract DA 36-039 SC-90859, SHI Project 4172, Stanford Research Institute, Menlo Park,California (April (1965).

2. R. F. Daly, "On Medeling the Time-Varying Frequency-Selective Radio Channel," TechnicalReport 2, Part II, Contract DA 36-039 SC-90859, SRI Project 4172, Stanford ResearchInstitute, Menlo Park, California (July 1964).

3. N. T. Gaarder, "An Examination of Selected High-Frequency Phase Modulation Techniques,"Technical Report 3, Contract DA 36-039 SC-90859, SRI Project 4172, Stanford ResearchInstitute, Menlo Park, California (August 1964).

4. Seymour Stein, "Statistical Characterization of Fading Multipath Channels," ResearchReport 321, Applied Research Laboratory, Sylvania Electronics Systems, a Division ofSylvania Electric Products, Inc., Waltham, Massachusetts (2 January 1963).

5. P. A. Bello, "Characterization of Randomly Time-Variant Linear Channels," IEEE Trans.PGCS-11, pp. 360-393 (Decamber 1963).

6. V. B. Davenport, Jr., and W. L. Root, An Introduction to the Theory of Random Signalsand Noise (McGraw-Hill Book Company, Inc., New York, N.Y., 1958).

7. D. L. Nielson and G. Ii. Hlagn, "Frequency Transformation Techniques Applied to Oblique-Incidence Ionograms, "Research Memorandum 13, Contract DA 36-039 SC-87197, DASA Sub-task 938/04-014, SRI Project 3670, Stanford Research Institute, Menlo Park, California(January 1964).

8. R. A. Shepherd, "HF Communication Effects Simulation: Instrumentation and Operatioiof the Field Experiment," Interim Report 3, Contract DA-36-039 SC 87197, DASA Sub-task 938/04-014, SRI Project 3670, Stanf'jrd Research Institute, Menlo Park,California (December 1963).

9. K. D. Felperin, D. L. Nielson, and N. T. Gaarder, "Propagation-Related Outa es on DCACircuits--A Field Experiment," Research Memorandum 2, Contract SD-189, SRI Project 4554,Stanford Research Institute, Menlo Park, California (July 1964).

10. G. L. Turin, "Error Probability for Binary Symmetric Ideal Reception Through NonselectiveSlow Fading and Noise," Proc. IRE, Vol. 46, pp. 1603-1619 (September 1958).

11. B. Elspas, "The Effect of Jamming on Error Probability and Message Intelligibility inFrequency-Shift-Keyed Teletype Communications," Technical Report 4, Contract DA 36-039SC-66381, SRI Project 2124, Stanford Research Institute, Menlo Park, California(December 1959). SECRET

12. J. G. Lawton, "Theoretical Error Rates of Differentially Coherent Binary and KineplexData Transmission Systems," Proc. IRE, Vol. 47, pp. 333-334 (February 1959).

13. V. W. Ames, "The Correlation Between Frequency-Selective Fading and Multipath Propagationover an Ionospheric Path," J. Geoph ys. Res., Vol. 68, No. 3, p. 759 ('February 1963).

14. V. W. Ames, "Spatial Properties of Amplitude Fading of Continuous 17-Mc Radio Waves,"Technical Report 87, Contract NOnr-255(64), NR 088 019, ARPA Order 196-64, RadioScience Laboratory, Stanford University, Stanford, California (March 1964).

15. F. R. Gantmacher, The Theory of Matrices, p. 274 (Chelsea Publishing Company, New York,N.Y., 1959).

16. E. Parzen, Modern Probability Theory and Its Application, p. 326, (John Wiley and Sons,Inc., New York, N.Y., 1960).

17. P. A. Bello and B. D. Nelin, "'lle Effect of Frequency Selective Fading on the BinaryError Probabilities of Incoherent and Differentially Coherent Matched Filter Receivers,"IEEE Trans. PGCS-l1, pp. 170-1.86 (June 1963).

73

-- - - mk.-. A

18. P. A. Bello, "Evaluation o1 Special Modulation Techniques for Improved System Design,"Vols. I and II, Report ICS-64-TR-530, Contract AF 19(628) -3414, ITr CommunicationSystems, Inc., Paramus, New Jersey (30 June 1964).

19. D. Middleton, An Introduction to Statistical Communication Theory, p. 814 (McGraw-!tiliBook Company, Inc., New York, N.Y. 1960).

20. J. N. Pierce, "Approximate Error Probabilities for Optimal Diversity Combining,"IEEE Trans. PGGS-11, pp. 352-354 (September 1963).

74

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T. GAARDER Prepared for · gain (ak is the gain and 0 k is the phase shift) of' the kth path. It is...

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