159

Multiple Access to a Hard-Limiting

Communication-Satellite Repeater

JOSEPH M. AEIN

Summary-This paper analyzes the communication capability ofa hard-limiting satellite repeater when spread-spectrum signals areused for asynchronous access multiplexing. It derives formal resultswhich indicate the most suitable system bandwidth and the re-sulting maximum number of simultaneous users as a function ofthe ratio of the received signal power (over the entire systembandwidth) from the satellite to the available noise power densityat the ground station receiver. Equally important, the paper setsforth the assumptions and approximations necessary to achieve theformal results and examines their weaknesses.

T?HE BASIC FEASIBILITY of communication-satellite systems has now been demonstrated,and it is likely that one will be placed in operatioil

in the next few years. The usefulness of any such systemdepends heavily upon solving the problem of multipleaccess, i.e., the problem of making the satellite repeatersimultaneously available to more than one pair of geo-graphically separated ground stations whenever the needarises.' Decisions affecting multiple access also affectthe signaling process and are therefore fundamental tothe design of the system.

This paper presents and solves an illustrative multiple-access problem. The treatment is introductory ratherthan comprehensive. It proceeds from several assump-tions regarding the system as a whole to a formal deriva-tion of the most suitable bandwidth for the system andthe corresponding maximum number of simultaneoususers, as a function of the ratio of two quantities: thesignal power received from the satellite over the entiresystem bandwidth, and the noise power density at theground-station receivers. It concludes with a criticalexamination of the assumptions and approximations uponwhich the formal analysis rests.The hypothetical communication-satellite system ana-

lyzed here has uncontrolled random access,2 or simul-taneous asynchronous multiplexing, through a hard-limiting satellite repeater. The multiplexed carriers arebinary-phase-coded, constant-envelope signals; the mes-sage information is modulated into the carriers by binarycomplementation of n-length sub-blocks of the bit stream.The message source is speech digitally converted to q-bitpulse-code modulation (PCM).

Manuscript received July 28, 1964.The author is with the Research and Engineering Support

Division, Institute for Defense Analyses, Arlington, Va.1 In a strict sense, a single duplex communication link routed

through a satellite represents multiple access to the satellite, butas used here the term multiple access denotes simultaneous accessby more than one duplex link.

2 As opposed to controlled synchronous access, e. g., time-division multiplexing.

The object of the following analysis is to determinethe maximum number of simultaneous signals (denotedby M) that the satellite repeater can handle simul-taneously. Consequently, the maximum number of simul-taneous simplex users is 211; the maximum number ofsimultaneous duplex users is M/2.

I. PRELIMINARY PROBLEMBefore considering the complete satellite problem, one

is restricted to a much less ambitious problem whichhas not yet been fully examined: that of detecting a sinewave with binary phase coding buried deeply in Gaus-sian noise after passage through a hard limiter. Fig. 1depicts the situation.

Band- Correlations'(t) C +3-Opass Hard limiter o(t) detector

I filter + zonal filter matched to s'(t)n'(t)

Fig. 1-Hard-limiting correlation detector.

In Fig. 1n 1

s'(t) = a rectAT (t - 1AT) cos (w0t + u1r/2)z =O

n'(t) = white Gaussian noiseo(t) = output of hard limiter

wherea = signal amplitude, the sign of which carries

the message informationrectAT (x)= 1 if 0 < x < AT

= 0 otherwise

1nATT

1/TWo

4-°1, , n- phase coding sequence

= number of phase-coded samples= length of each phase sample= nAT = total signaling time per message

bit= message bit rate= carrier frequency.

This channel structure has received some attention inthe past as it is a form of constant false-alarm rate(CFAR) fix for radar and sonar detection. Doyle andReed [1] recently carried through an elegant analysisof this type of channel in some detail. They analyzed thechannel with the correlation detector of Fig. 1 replacedby a narrow-band filter centered on the frequency wofollowed by an envelope detector. They considered a sine-

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wave signal and obtained the probability density func-tions of the detector output for both noise only andsignal present cases. For the purpose of this paper, co-herent detection will be presumed. Their formulation ofthe problem will be adopted and the central limit theoremwill be applied to avoid the difficult task of evaluatingthe probability density function of the correlator output.One now makes the following three assumptions:

Assumption 1: The band-pass filter (BPF) is an idealfilter of bandwidth W = 1/AT and gain 1/ -,/NVoW whereNo is the noise power density. (It is further assumed thatthe BPF passes s'(t) with negligible distortion.)

Assumption 2: The zonal filter in the hard limiterpasses only those harmonic components centered on thecarrier frequency coo.Assumption 3: The noise samples are uncorrelated

over AT intervals.

It follows that n = TW. By Assumption 1, the noisepower at the hard limiter is unity. The signal portionof the hard-limiter input, denoted by s(t), is given by

s(t) = s'(t) - E a rectAT (t - 1AT)VNO-0W 1 0

-COS (Coot + ,U I r/2) (1)

where

a

VNoWThus

7' n-1

TjJ" 82(t) dt = 1 E a2T0 1=0

rATf Cos2 (coot + tir/2) dt

aA2 a2 a2/2T 2 2 NoW

But (2/2 is the input power in s'(t) and NoW is thenoise power in the band W, so that a2/2 is the input signal-to-noise ratio at the hard limiter.By Assumption 2, one is only interested in the com-

ponents of the limiter output at the frequency wo. Thus,o-(t) is given by

n-1

a(t) = sgn a E rectA T(t lAT)1=0

*cos (wo t + u17r/2) (2

where

sgn a = 1 ifa > 0= -1 if a < 0

,;(t) = tan-' y(ta+x(t)

The functions x(t) and y(t) are the quadrature com-ponents of the Gaussian noise (centered on wo) at the

input to the hard limiter. The correlator output V isgiven by

snan-1 J(k+1)AT2= Cos (wo t ± I.kWr/2)

-\/c k=O tA T /T

*COS [9ot + JUk7/2 + (p(t)] dt

sgn a n-1 I (k+1) AT

\/H k I\TO A T(3)

where 2/ ( -/ AT) = the correlator gain normalizingfactor.

This motivates our next assumption. Since by As-sumption 3 the noise (x, y) is uncorrelated over AT inter-vals and is hence statistically independent, one expectssp(t) to be statistically independent over AT intervalsalso. In order to avoid the mathematical difficulties in-herent in the stochastic integral of (3), one assumes3 thefollowing:Assunption 4: The correlator output V may be ap-

proximated by

sgn a n-1 1 ((k+l)ATV , E COs (k AT dt (4)

k=0 AT AT

n-i 1V 3 sgn a E /n COS (PA (5)

where 'pk are identically distributed statistically inde-pendent random variables.

It is well known [2] that if a = 0, the 'p, are distrib-uted uniformly over the interval (0, 2ir). Thus,

V2 ~~1n-1 2pE(V la = 0) = - EE COS _k

n k=O (6)

where E is the expectation operator. Since the noisehas a mean value of zero, and if

Pr(a > 0) = Pr(a < 0) - 2

one has a symmetric binary channel and the error prob-ability Pe is given by

P. - Pr(V < Ola > 0)

Thus, if a > 0, V is given byn-1 1

V = Ek0 cos (Pk

and (7)(p, = tan-1 x+

where x and y are zero-mean, unit-variance, uncorrelatedGaussian random variables. It can be shown [3] thatp(sok) = p((p) can be approximated (for small values of a2)by

p = -(11 + <- a cs s) 0 < so < 2wr, (8)

3 A similar technique is used in [2].

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to within terms of order a2. Note that p(qp) > 0 impliesthat

a22 7r

Note further that

E(V) = G cos + a cos2k=O ~~~~~~~~~~(9)

a fliT

= 2 -2Dividing the square of (9) by (6) yields

[E(V)]2 2tE(V2 a = 0) = 4

Now let

d2 A [E(V)]2E(V2 a = 0)

If no hard limiter is present,

d2 = 2 SNo

(10)

where S = the average power of the signal s'(t). Observethat

d = 2 = 2TW (N)

But

which is the inputli'miter. Furthermore,

S a2NOW 2

signal-to-noise ratio at the hard

TW = n

and therefore

d2 = na2 (11)

From a comparison of (10) and (11), it is evident thatthe hard limiter degrades the output detection thresh-old by r/4 or 1.0 db.

In order to obtain the probability density functionfor V as given by (7), the central limit theorem is in-voked in a form [3] which states that if all quantitiesXk (k = 0, 1, 2, ...) are statistically independent and iffor some a > 0 they satisfy the relation

n-1

E2 E IXk Xk 12 + 8rT = liMk=O =0 (12)z -n-1 1 + 8/2

n(o E E Xk Xk | )k =O

where Xk = EX,, then the random variable() n-1

y(n)= Jl-E):kVnk=o

this case, Xk = cos Sp, and X = b/2 = a/2 xV71. Fora = 2,

IX XkI 4os2

d~oE lXk -X = A (cosp - b/2)4(1 + b cos () 2d-

< (cosp - b/2)4 d < 25; for small a2/2

E Xk - XkI = (1 - b2/2)

Whence the test r becomes

25r = lim n 2 - =

Consequently, one concludes that for sufficiently large n(i.e., for sufficiently large time-bandwidth product), Vis Gaussian with mean v\¶ b/2 and variance I (1 -b2/2).Thus

XT 2

(EV)2 n8a(d)= 2 = 2

0u (1 _ 12)

T2 (1 +1r 2\ nr 2/t-- n-a 1+a J2~,-a4 \4/4 (13)

for small a2/2. Therefore (see (10)),

(d') = d

It is the above argument which leads to the statementthat the hard limiter is a quasi-linear device that de-grades the signal-to-noise ratio by 1.0 db. Of course it isimportant to know the relation between the size of n andthe non-Gaussianness of V, especially when dealing withvery low error rates.

II. HARD-LIMITING SATELLITE LINK

In this section the satellite down link is incorporatedinto the framework of Section I. Fig. 2 is the correspond-ing block diagram. In Fig. 2, nl(t) and n2(t) are statis-tically independent zero-mean Gaussian random variables.

If p2 is the space loss (power attenuation), normalizedto a hard-limiter output power of 2 watt (see (2)), ablock diagram equivalent to Fig. 2 is Fig. 3.

HardBand- limiter Space Correlation

s (t) f ilrpass oe+ zonal loss + detectorfilter filter

nl(t) nc(t)

Fig. 2-Hard-limiting, communication channel.

HardBand- limiter Correlation

s'(t)- pass + zonal detector+ filter filter

n,'(t) p-Ln2 (t)

approaches a Gaussian random variable as n - o . For Fig. 3-Equivalent hard-limiting satellite communication channel.

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Since the system following the hard limiter is linear,one may use superposition and apply the results ofSection I. Namely, the correlator output V can be con-sidered as the sum of two random variables V, and V2which, due to the independence of n'(t) and n2(t), arealso statistically independent. Furthermore, V2 will beGaussian with zero mean and a variance which one nowcalculates.The function n2(t) will have an available noise power

density of KTr watts/cps, where K is Boltzmann's con-stant and Tr is the effective receiver noise temperature.Consequently, the noise power density at the input tothe correlator of Fig. 3 is pV2KTr. Thus,

1 n-I 2 (k+1)ATV2 = EV\n kO= AT kA T Cos ot + k72)p-n2(t) dt

and

V2 4 n-1 (k+ 1) A T -2

E 2-n= ,T) EO tT P KTc COS2 (t+L2)dt

p-2Krr -- p *~Tr = p~2KTrW (14)

If V1 is regarded as Gaussian with mean (a/2) Vnir/2and variance 1(I- ira2/4) - 2 (see (13), then V = VI+V2is also Gaussian with mean (a/2) )n/r2 and variance2 + p-2KT,W, and the average error rate P. (assuminga binary symmetric channel) is given by

Pe = J exp[ 2] - (15)

where27ra

= --_2 mv 2 2-a 1 + 2p 2KTrW

The quantity a2/2, as before, is the input signal-to-noise ratio at the limiter (assumed to be small).

III. MULTIPLE AcCESS UTILIZINGSPREAD-SPECTRUM SIGNALS

Since the necessary mathematical framework is con-tained in (15), this section is devoted to the applicationof (15) to the problem of multiple access to a hard-limiting communications satellite with spread-spectrumsignals. Fig. 4 depicts the situation.

It is assumed that there are m unsynchronized spread-spectrum signals of equal power at the input to the satellite.A generic signal is singled out and denoted by s'(t). Theother (m - 1) signals are added together; their suIml isdenoted by n' (t). It is our aim to endow n'(t) with thestatistical properties of Gaussian noise, white over thefrequencies passed by the BPF and of a power in - Itimes that of s'(t). Of course, the correlationi detector ofFig. 4 is that receiver which is tuned to s'(t). If m isreasonably large (?), if the binary phase codings appearwell distributed, if the constant RF referenice phase ofeach signal is independent of all others, and if the signalsare unsynchronized, then it is customary to bless n (t)with Gaussian noise properties and assert that nl(t) isstatistically independent of S'(t).4 This is justifiable bythe central limit theorem. If these assumptions are ac-cepted and if the thermal noise at the satellite input isneglected, (15) is valid for an input signal-to-noise ratioof 1/m - 1. It should be borne in minid, however, thatas the system bandwidth increases, the satellite thermalnoise may no longer be neglected. Appendix I discussesthe influence of this noise source on the system, but forthe moment one shall continue to neglect it even to thepoint of letting the system bandwidth go to infinity, be-cause by doing so one obtains mieaningful results whichmay then be reinterpreted in the light of Appendix I.The derivation of (15) presupposed a smnall input signal-to-noise ratio, which corresponds to the necessity of alarge m.

One's purpose now is to obtain the niumber of siniul-taneous users m as a function of the RF bandwidth,given the quality of message transmission q and the re-ceiver signal-to-noise ratio. From [3] and [4] one makesuse of some well-known formulas relating PCM to out-put signal-to-noise ratios. Let q be the number of bitsused to encode speech samples at a rate of f, cps. Thatis to say, speech is sampled every 1/f, seconds andquantized into 2' levels. There are (nominally) twosources of noise for PCM: quantization noise and channelnoise. The quantization noise is independent of thechannel and results from the quantization process. It is afunction of the choice of quantization levels and the input-signal probability distribution. However, if the quanti-zation levels are chosen to match the input signal statistics

4This assumption is discussed further in Section IV.

thermalnoise

Fig. 4-Hard-limiting satellite communication channel.

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Aein: Multiple Access to Satellite Repeater

(the process known as companding), then the signal-to-noise ratio S/N of the process is given [4] to a closeapproximation by

S/N = 4a (16)

The other source of noise is channel dependent, i.e.,it arises from the errors introduced by the channel indecoding the PCM word. If P. is the channel error prob-ability and if P. < 10-2, then the output signal-to-noiseratio due to channel errors is closely approximated [5] by

SIN = p1 (17)

Equating the right-hand members of (16) and (17)defines a very sharp threshold on P. so that for small P,the output signal-to-noise ratio is quickly dominated bythe quantizing noise. This follows from the fact that forreasonably small P. (i.e., P. < 10-2), a 1-db increasein channel noise decreases P. by an order of magnitude.

Thus, P, at threshold denoted by P. t, is given by

Pe, t = 4-( + 1) (18)

The system parameters are uniquely related by intro-ducing (15) as follows:

Pe,t = 4- + = exp [- ] d (19a)

- TWd2 2 (M+p-1) (19b)t 1 + 2p-2KT,W

1 (M _ 1)-l Prf (W)d= w

t1 +wWo

(21)

where WO = P,/KT,.Let d, (q) be the solution to (19a) for a given value of q.

Then

(22)[qd2(q)]X- W

1+ WO

where

Pp \p total received satellite power-to-noise(p,) = KTr = ratio referred to the sampling band-

KTf, width.Eq. (22) yields the maximum number of simultaneousspread-spectrum users5 of a hard-limiting satellite re-peater as a function of the RF bandwidth for a givenoutput signal-to-noise quality 4' and total receivedpower-to-noise ratio (referred to the sampling bandwidth).If one assumes that the actual number of simultaneoususers m exceeds M, the system as a whole will breakthreshold. If m is less than M, the output signal-to-noise quality of the users will increase to the value 4'.If m equals M, the system is at threshold and the result-ing output quality is 3 db down from 4' (i.e., 24a). Thus,one sees that this system has a user loading character-istic which saturates sharply, an effect which is attribut-able to the PCM mode of message modulation.

Notice that (22) is of the form

detectability at PCM thresholdmaximum number of simultaneous users, m

l/qf. = sampling interval1/AT = RF bandwidthnormalized space lossBoltzmann's constanteffective system noise temperature.

Eq. (19a) can be solved for d, as a function of q, thePCM bit size. Then from (19b), threshold values of theindicated parameters may be computed.The quantity 2p 2KT, has an interesting interpreta-

tion. The numerator is the space loss normalized to a

power output of 2 watt. If the satellite total outputpower is P, watts and the actual space loss is L2, then thetotal received power at any one station Pr is equal toL2Pf. Writing PT as A., one sees that P, =por p2 = 2Pr.

Consequently,

2KT P,

2KT, KT, (20)

is the total received satellite power-to-thermal-noisedensity ratio. From (19b) and (20),

(M - 1) = (M;. - 1)W/Wo1 + W/W0 (22a)

which resembles the transfer function of a simple high-pass RC filter with break frequency WV0 and gain (M. -1).Thus the value that M approaches asymptotically as Wapproaches infinity, denoted by M., is given by

M. -1 = T(2 )[qd(q)d

and the critical RF bandwidth W0 is given by

W - P'KTr

(23)

(24)

where

q = PCM bit sized2(q) = the function of q given by the solution to (19a)Pr = total received satellite powerK = Boltzmann's constantTr = effective noise temperature of the ground

stationN = KTrfa = noise power referred to the sampling

bandwidth.

6 As noted in the introduction, one duplex communication linkcomprises two simultaneous users.

where2

T =

W =

2pK =

Tr=

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_x

w/W

Fig. 5-Normalized plot of M -1 vs W.

TABLE Idt SOLUTIONS OF (19a)

q dt [q d12]- 1 (7/2)[q d2]- 1 = (1Am - 1)(P/IN)-4 3.094 0.0262 0.0415 3.489 0.0164 0.0266 3.843 0.0114 0.0187 4.169 0.00823 0.013

Fig. 5 is a normalized plot of (3ll - 1)/Me. - 1) as afunction of W/Wo. Table I presents d, solutions of (19a)together with values of (7r/2)[q d2] 1One takes as an example the parameters of a medium-

altitude communications satellite wherein a Pr/KTr of67 db is available (see Appendix II) which implies thatthe critical RF bandwdith W0 is 5 Mcps. This in turnimplies that any RF bandwidth greater than 5 Mcps in-creases M marginally. Let us arbitrarily choose W = 10Mcps. At least a 3-db margin must be provided for un-controllable losses (e.g., rain); consequently, if f, = 8kcps, the available Pr/N is given by

Pr/N = (64 - 39) db = 25 db

and substitution of values from Table I for 5-bit PCMand from Fig. 5 for W/Wo = 4, yields6

11-1- = (0.8)(0.026)(315) = 6.5

Therefore,

31 = 7

Thus, there is seen in this example a set of param-eterswhich seriously strains the assumption used to modelthe access problem; namely, that Pr/KTefs is not suf-ficiently large to produce 3/1-values greater than 10.Under this condition, the above 111-value of 7 can onlybe taken as a qualitative indication. This example waschosen because the characteristics of a conmmunications-

6 Note than when Pr fades 3 db, Wo is halved.

satellite repeater with an isotropic antenna are of currenitinterest. If gravity stabilization of the satellite provesfeasible, then it may well turn out that a 10-db increasein Pr/KTr is possible, in which case 1V0 becomes 50Mcps and 111 about 70.

It is worthwhile to compare these results to sonmeideal systems. The first is one where it is assumed

1) that the ground station-to-repeater input links arefree from interference and noise;

2) that the repeater output is multiplexed aiviong Mlorthogonal subehannels;

3) that the modulation is q-bit, binary PCM with asampling frequency of fe cps; and

4) that the ground-station receivers are perturbed bywhite Gaussian noise of available power densityKTr watts.

With the above assumptions, one may use (19a) anidmodify (19b). Our measure of performance is the sameas that used for the spread-spectrum analysis. Thus d'is the same as before, being uniquely specified once q isselected. It is well known that d2 is given by

d2 = 2E/KTr

where E is the energy available for transmissioni of oniebit of each PCM sample word. Thus

qf,, AIl,lwhere Pr = total average received signal power from therepeater. Whence

Ml = 2 [qdt] (N) (25)

or

Me.-1 4rM11 4 (26)

Consequently, the channel output obtainable with thespread-spectrum technique can be as high as 7r/4 timesthe output of the ideal orthogonally multiplexed channel.Next one compares the spread-spectrum technique to

an ideal linear, average-power-limited Shannon channel.The information rate on the Shannon channel with infi-nite bandwidth Re. is given by

TPr (27)

where

Pr = average received powerKTr = available noise power density.

Each of the Me., - 1 communication links of thespread-spectrum system produces qf, loge 2 natural bits persecond. Thus the spread-spectrum multiplex efficiency rcan be defined as

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A (Me. - 1)qf. loge 2Pr/KTr (28)

- 2 log, 2d7-2

- 1.1d,-2where d, as a function of q is given in Table I.Note that q is a function only of the quality of the

communication system. Table II presents -q as a functionof q.

TABLE IIUPPER BOUNDS ON MULTIPLEX EFFICIENCY

Quality of Multiplex System OutputTransmission, q Efficiency, v Quality, S/NDimensionless per cent db

4 11 245 9 306 7 367 6 42

IV. CRITICAL REVIEW

The following paragraphs summarize the critical de-ficiencies in the foregoing analysis. These deficiencies are

attributable to sins of omission and sins of commission;they will be reviewed in that order.

A. Undiscussed Problems

Two problems have been overlooked: the choice of theactual bit-stream codes and bit-stream synchronization.Directly related to the coding problem is the micro-scopic interference between bit streams which can cause

loss of synchronization between the transmitting and re-

ceiving stations of a particular link. This, in effect, placesanother threshold (loss of synchronization) into thesystem aside from that induced by signaling errors P,.Given that a system is coded reasonably well, this thresh-old should be below the signaling threshold and theanalysis of Section III then applies. However, the codingmust be chosen somewhat intelligently. In this context,it will be important to establish, for a given codingmethod, the relationships between the bit-stream length,the correlation length n = TW, the totality of the ad-dresses (potential ground stations), and the maximumnumber of simultaneous users, so as not to lose bit-stream synchronization.

Also of importance is the actual acquisition and main-tenance of bit-stream synchronization by the receivingstations. Doppler tracking at the ground stations may bean important consideration for medium-altitude (6000n.mi.) satellite systems.

B. Analytical DeficienciesPerhaps the greatest weakness of the analysis pre-

sented is that for a q as small as 4, the channel Pe is of

the order of 103. Consequently, one must work withthe tail of the distribution of the correlator outputvoltage V. Thus, all expedient assumptions regardingthe Gaussian nature of a sum ofM -1 binary-phase-codedsinusoids are critical. Even granting this assumption, itis clear that this sum does not have a white spectrum.Moreover, the assumptions leading to the technique ofobtaining the correlator output in Section I (especiallyAssumption 4) need a closer examination.

Further justification is needed for the use of the CentralLimit Theorem in Section III in summing the undesiredsignals to Gaussian noise which is then also assumedto be independent of the desired signal. The difficultyis that some small but nonzero correlation between thephase-coded bit-stream generators does exist. However,an argument [6] for justifying the statistical independenceof these bit streams can be based upon the fact thateach transmitter adds an arbitrary, completely inde-pendent, and random RF reference phase to the trans-mitted bit stream.A further serious deficiency is the assumption that all

the signals arriving at the satellite input are of equalpower. This point must be examined much more closely. Ifone signal has an average power slightly greater thanthe sum of the average powers of the other signals, thereare indications that the strong signal will capture thelimiter. In fact, the analysis of Section I specificallyassumes that such is not the case; for if one consideredthe strong signal as s(t), the signal-to-noise ratio wouldnot be sufficiently small for the analysis to hold and (8)would be quite erroneous. Thus future work must devoteconsiderable attention to the strong-signal case.

It may be objected that the satellite BPF before thehard limiter has a gain normalized to 1/ VNOW, andthat in the multiple-access case NoW is directly pro-portional to in, the number of users. However, this gainwas chosen to set the average power in the output of theBPF equal to the input signal-to-noise ratio. Any fixedvalue of gain for the BPF will provide analyticallyidentical results (to within the dynamic range of theequipment used).

V. CONCLUSIONSThe foregoing analysis permits the following con-

clusions:1) Spread-spectrum signals with PCM digitized speech

message sources produce a system loading character-istic exhibiting sharp saturation with an increase in thenumber of users. This effect is due to the performance ofPCM encoded speech. Graceful degradation (soft satura-tion) is not an inherent attribute of the spread-spectrummultiplexing modulation, although systems can be de-signed which do have this property (e.g., spread-spectrumcarriers with the message information contained in anarrow-band FM subcarrier). This suggests that the totalmodulating process be analytically factored into the

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component modulations of carrier multiplexing andmessage encoding so that generalized statements re-garding system behavior can be more easily understood.

2) The selection of the system RF bandwidth is ofgreat importance. The foregoing analysis suggests thatthis bandwidth is intimately related to the received power-to-noise density ratio P,/KT, at a typical ground station.That Pr/KT, is a critical bandwidth is gratifying sinceit is the Shannon limit for an average, power-limitedchannel. An increase in this ratio is necessary to obtaindesired increases of multiple-access capability. This, inturn, requires an increase in satellite bandwidth. Everincreasing bandwidth will eventually force the designerto be concerned with thermal noise at the satellite input.

APPENDIX I

EFFECTS OF SATELLITE THERMAL NOISE

This Appendix accounts for the effects of the satellitethermal noise terms of the hard-limiting satellite com-munication channel shown in Fig. 4. It is assumed thatthis noise is white, Gaussian, and statistically inde-pendent of n2(t), and that it has an available single-sided power density of KT. watts/cps. Thus, the inputof the satellite contains a signal s'(t) of power PTL andGaussian noise of power (m - 1) PTL + KT,W, whereit is assumed that all ground stations transmit PT wattswith the same ground-to-satellite power loss factor L.Thus (S/N)i8, the input signal-to-noise ratio at thelimiter (see (15)), is given by

(S) PTLN/ in PTL(m-1) + KT8W

1(n - 1) + KT,W/PTL

Normalizing W to Wo produces

N j. (m- 1) +aw

(29)

(30)

=Pr' °-KTr

KT. Pra PTL KTr

If the down-link losses are assumed equal to the up-link losses, P, is equal to P8L, where P8 is the satellitetransmitter power in watts, and a is then the power-temperature product ratio

P8T,PTT,

Conisequently, the expression for d} becomes

Pr. 12 2q N W,,X ~ ~ ~ O_

Eq. (31) is easily put into the following form:

(AlI- 1)

~ ~ ~ P P L

where

W=Pr P',L- KTr K7Tr

M"k - 1 = '[qd(' )

Thus, it is seen that (22a) is modified by the term

[1 + 1Wo]

(31)

(32)

To calculate the effects of this term one considers as anexample the following system parameters:

PS = 4 watts; T, = 20000K

PT = 1 kw;w

To

T, = 2000K

= 4.

Whence

a = 4 X 10'2

a W= 0.16

and since (M - 1) must be no less than six

0.16 0] .1- 601(M -1)j-'_ (M -1)

which is a negligible correction even for an M as smallas six. Note, however, that if W/TW0 is made sufficientlylarge, this correction is not negligible. In fact, if Wbecomes too large, the maximum number of simul-taneous users decreases. This follows from the fact thatthe satellite input power contributed by the spread-spec-trum signals remains fixed, whereas the satellite noisepower increases linearly with W.The foregoing suggests the existence of an optimum

system bandwidth W*. Solving (32) for (M - 1) yields

(33)

166 December

I I

(M 1) =III",) 1 -(XwI +

w wowo i

Aein: Multiple Access to Satellite Repeater

Differentiating (33) with respect to W/W0 and settingthe result equal to zero, yields the value of W* thatmaximizes M. That is to say,

(34)wo ~~aand

M*-1 { 2

M. I M.( 1))

where M* denotes the maximum value of M. This im-plies that for an optimum bandwidth to exist the follow-ing condition must be satisfied:

aorPTrL 2 2

KTJS > 2 qdt

The quantity PTL/KT,f. is the received power-to-noiseratio at the input to the satellite from one ground stationreferred to the sampling bandwidth. Condition (35) simplystates that the up-link signal-to-noise ratio must exceedthe PCM threshold. If condition (35) is not met, thesystem will break the PCM threshold established byequating channel noise to quantizing noise.For the example cited above, a = 4 X 10-2 so that

Ma-1 = (25) (M0. -1) > 1

If (M,. - 1) is as small as six, then

W*wo

andM*-1 =0.85

In practice a W*/W0 of more than four would seemimplausible. The example considered here seems to em-phasize this even more, because for W/W0 = 4 and(M. - 1) = 6,

M* - t=(1 - 0.16) (0.82) = 0.8M"Ic -16That is to say, with a bandwidth one-third of optimumone has come within 95 per cent of the maximum num-ber of users. If PTL/KT,f. is increased, a W/W0 of fourdecreases the yield of simultaneous users to a floor of82 per cent of the optimum number that could be ob-tained with ever increasing bandwidths.

APPENDIX II

PERTINENT CHARACTERISTICS OF A MEDIUM-ALTITUDECOMMUNICATIONS SATELLITE SYSTEM

This Appendix presents the satellite to ground-stationpower budget associated with a proposed medium-alti-tude (6000 n.mi.) communications satellite system. Thepertinent features of the system are: 1) a 4-watt satelliteoutput into an isotropic antenna, and 2) a 30-foot dishat the ground station feeding a receiver with a noisetemperature of 200°K and an operating frequency of8 kMcps.The up-link budget provides satellite input signal-to-

natural-noise ratios exceeding 20 db; so that in com-parison to the noise of the (m - 1) unwanted signals,the natural noise is neglected (see Appendix I).

P,(= 4 watts)G.(=0 db)lossesG, (30-foot dish)167r2X2= (0.123)2tracking lossesD2 (maximum slant range)PrKTr(Tr = 200°K)Pr/KTr

6 dbw0 db

-3 db+54 db-22 db-18 db-2 db

-154 db-139 dbw-206 dbw

67 db

ACKNOWLEDGMENTThe author wishes to thank R. A. Hoover of the Insti-

tute for Defense Analyses (IDA) for his careful editingof the manuscript, and W. E. Bradley, J. Kaiser, andJ. W. Schwartz of IDA for their stimulating discussionsof the subject of this paper.

REFERENCES[1] W. Doyle and J. S. Reed, "Approximate band pass limiter

envelope distributions," IEEE TRANS. ON INFORMATION THEORY,vol. IT-10, pp. 180-184; July, 1964.

[2] P. Bello and W. Higgins, "Effect of hard limiting on the proba-bilities of incorrect dismissal and false alarm at the output ofan envelope detector," IRE TRANS. ON INFORMATION THEORY,vol. IT-7, pp. 60-66; April, 1961.

[3] W. B. Davenport, Jr., and W. L. Root, "An Introduction to theTheory of Random Signals and Noise," McGraw-Hill Book Co.,Inc., New York, N. Y.; 1958.

[4] B. Smith, "Instantaneous companding of quantized signals,"Bell System Tech. J., vol. 36, pp. 653-708; May, 1957.

[5] H. F. Mayer, "Principles of pulse code modulation," in "Ad-vances in Electronics," Academic Press, Inc., New York, N. Y.,vol. 3; 1951. See p. 251.

[6] J. M. Aein, "Multiple Access Capability of a Hard-LimitingCommunication Satellite Repeater with Spread-Spectrum Sig-nals," IDA/RESD, Washington, D. C., Research Paper P-121;April, 1964.

1964 167