A comparison of trellis coded versus convolutionally coded spread-spectrum multiple-access systems

62 8 l t E E J O U R N A L Oh SFLECTEI) A R k A S I N COMMUNICATIONS. VOI- X. NO 1. MAY 1040

A Comparison of Trellis Coded Versus Convolutionally Coded Spread-Spectrum

Multiple-Access Systems

Abstract-This paper investigates the performance of trellis coding when it is applied to a spread-spectrum multiple-access system. A system model is proposed that allows one to apply both trellis coding a n d a PN spreading sequence to the da ta symbols to be transmitted. Rate t i / n + 1, trellis codes employing 2'" point MPSK signal constellations a r e investigated when Gold sequences a r e employed for purposes of PN spreading. Performance in a n additive white Gaussian noise (AWGN) channel is investigated, with 5-20 users transmitt ing simultaneously. Using criteria of equal complexity a n d throughput, the performance of the trellis codes in a SSMA environment is compared t o that of medium to low ra te convolutional codes through the use of a generalized transfer function bound. The average degradation due to the interuser interference is determined by employing the method of moments. The validity of approximating the interuser interference a s a Gaussian random variable is also investigated. T h e numerical results illustrate that for a given complexity, chip ra te and throughput, that low ra te convolutional codes provide the best performance in a SSMA system. As lower ra te convolutional codes a r e employed, there is a n increase in the effective interuser interference due to the grea te r cross- correlation effects f rom using shor te r PN sequences, or alternatively from the effects of partial cross-correlation. However, this increased degradation is more than overcome by the increased distance properties of the low ra te codes. The coding gains for the codes that were considered range f rom 1-6 d B (depending on the type and complexity of the code) over the corresponding uncoded performances of SSMA in a n AWGN channel.

I. INTRODUCTION HERE has been recent interest in the use of direct T sequence (DS) spread spectrum multiple access

(SSMA) for use in an indoor wireless local area network, in the 800 MHz-1200 MHz frequency range. Some of the pulses of a DS SSMA system over a TDMA architecture are that [I]: it is asynchronous and possesses inherent diversity to multipath and it is robust to time varying channels. Furthermore, SSMA allows frequency reuse if a star local area network is employed, and in addition it provides built in addressing and security.

Disadvantages of employing a DS spread-spectrum system include: the near/far problem; interference to existing

Manuscript received February 2 I , 1989; revised October 30, 1989. This work was supported by funding from the Telecommunications Research Institute of Ontario (TRIO) and a Natural Sciences and Engineering Re- search Council (NSERC) Postgraduate Scholarship.

The authors are with the Department of Systems and Computer Engi- neering, Carleton University. Ottawa. Ont. KIS 5B6. Canada.

IEEE Log Number 9034788.

systems (if frequency overlays are employed), and the in- ferior probability of error performance possible for a given number of users.

Kavehrad in conjunction with other authors [ 11-[SI has analyzed DS spread-spectrum multiple-access systems (i.e., such as BPSK and DPSK) for various types of diversity and coding in an indoor wireless Rayleigh fading multipath environment [I] , [2], [3]. This analysis employed Gold and Kasami codes for DS spreading of the transmitted waveforms. The results of these analyses con- clude that traditional DS SSMA schemes provide poor performance in a Rayleigh fading environment unless some form of diversity is provided.

Recent research into trellis codes has demonstrated that coding gains of 3-6 dB are possible in an additive white Gaussian noise (AWGN) channel with no bandwidth ex- pansion, making them ideally suited to bandlimited channels [6], [7]. It is proposed that the use of trellis codes in a DS-SSMA system be investigated to ascertain if similar improvements in the performance of a SSMA system can be obtained. In the system to be studied, the data sequence is trellis coded before being spread by a pseudo- random sequence. Such an approach allows one to obtain coding gain without any resulting reduction in throughput. For purposes of comparison, the performance of convolutionally coded SSMA systems are also investigated. The comparison is performed on a basis of equal throughput and complexity. In order to maintain the same throughput as an uncoded system, the convolutionally coded system must either use shorter complete PN sequences per coded symbol, or partial PN sequences per coded symbol.

11. SYSTEM MODEL The general architecture of the system that will be ana-

lyzed is illustrated in Fig. l , which has been used frequently in previous studies of asynchronous SSMA systems [2], [8], [9]. The system consists of K users transmitting asynchronously over an AWGN channel. Each user transmits using a different spreading code and the signal transmitted by the kth user s k ( r ) , is assumed to be delayed randomly by a delay of 7 ' . Thus if the receiver of the ith user is attempting to receive the signal trans-

0733-87 16/90/0500-0628$01 .OO @ 1990 IEEE

BOUDREAU er ol . : TRELLIS C O D E D VERSUS CONVOLUTIONALLY C O D E D SSiMA SYSTEMS

0 * c Receiver

r - I r j user #K s K

Fig. I . General system architecture.

mitted by the j th user, the demodulated signal will consist of the desired signal and interference due to a combina- tion of the AWGN and the cross-correlation from the signals transmitted by the other users on the system. The notation required to analyze the performance of such a system will be developed in Section 11-C.

A . Trellis-Coded SSMA The transmitter system model for trellis-coded SSMA

is illustrated in Fig. 2(a). This architecture is based on the coset structure of trellis codes as defined by Calderbank and Sloane [7]. For k input bits from the data source, k , of these bits select a coset of the signal constellation through the convolutional coder, while the remaining k z select a signal point within the given coset. The architecture of Fig. 2(a) is analogous to the above description, with the additional feature that the phase of the signal point chosen within the given coset is multiplied by the PN spreading sequence to be transmitted in the given symbol period (it is assumed that the data symbol transitions are aligned with chip transitions, so that there is no spectral spreading due to the multiplication process). Thus in effect, what is achieved is that in each symbol period, a binary spread-spectrum (SS) signal is transmitted, the absolute phase of which is determined by the trellis code. The two antipodal signal points (denoted as the antipodal coset) employed by the PN sequence as well as the modulation on the PN sequence is selected by the trellis coder-see Section 11-C for further details. For example, if a 8-PSK rate 2 / 3 trellis code is employed, in any particular symbol interval an antipodal PN sequence is transmitted, the absolute phase of which is one of + ~ / 8 , f 3 ~ / 8 , + _ 5 ~ / 8 , or +7n/8 . Fig. 3 illustrates the antipodal cosets for 4PSK and 8PSK signal constellations.

B. Convolutionally Coded SSMA Fig. 2(b) illustrates the baseband transmitter system

model employed to analyze the performance of convolutionally coded SSMA systems. In this model, the data source produces k data bits per baud, which enter a rate k / n convolutional coder. Each of the n coded symbols output by the convolutional coder is then modulated di-

signal point x(t)

coset coder select

source

PN sequence generator

__

629

(a)

7 coded

data convolutional source coder

signal

PN sequence generator

(b)

Fig. 2. (a) Trellis-coded baseband system. (b) Convolutionally coded baseband system.

a) 4-PSK

Coset 1 * t P

b) 8-PSK

, Coset 3

* \ \ 1 //

d , Coset 0 t

Fig. 3. Antipodal coset structure of 4-PSK and 8-PSK signal constellations.

rectly unto the PN spreading sequence using a BPSK sig- naling format. Each coded symbol can either span one complete PN sequence period, resulting in the necessity to use shorter (by a factor of k / n ) PN sequences in order to maintain the same throughput, or alternatively, each coded symbol can modulate some fraction of the complete PN sequence. In general, the latter approach will result in additional interuser interference due to the greater sever- ity of partial cross-correlation effects.

C. Dejinition of Notation The following notation in complex form is based on

that developed by Pursley [8], [9] and utilized by Kaveh-

630 I E E E JOURNAL ON SELECTED A R E A S I N COMMUNICATIONS. VO1 X. NO 1. MAY 1490

rad in his analysis of SSMA for indoor wireless radio [ I ] , [2]. This notation is based on the architecture that was briefly described in Section 11-A, and illustrated in Fig. 1. Assume a SSMA system consisting of K users each

one data symbol. Define the kth user's complex baseband

Assuming an AWGN channel, the received signal may be expressed as

1 Re ( ' ( r - 7') exp [ j ( w , r + ph)] utilizing a PN sequence of length N, chips, which spans r ( r ) =

information signal as + ( 7 ) in which r h is the random delay of each user's signal ar-

! . = I

m

x k ( ( t ) = C x;P,(r - PT) ( 1 ) riving at the receiver, and n = -m

in which T is the baud period and Pr( e ) is a rectangular pulse of T seconds duration. The term x i is the complex baseband symbol of the kth user during the pth symbol period, defined by

xh = bi exp [ je;] ( 2 ) in which b; and 0; are the amplitude and phase of the complex baseband output, a n d j = G. Similar to the definition of x k ( r ) , the DS spreading chip waveform is defined as

ph = Cp!. - w , r h ( 8 ) is the random phase of the kth user's carrier at the receiver. Furthermore, n ( t ) is a sample of AWGN with variance N0/2.

For the purposes of analysis the receiver for the ith user can be implemented as two correlators matched to the or- thonormal carriers [9], [ 101

(9)

chip waveform, and T,. is the duration of the chip pulse with N,. = T/T,. In the above and subsequent definitions, the superscripts refer to the user under consideration, whereas the subscripts denote the time interval under consideration (i.e., a particular symbol or chip). The sequence a k ( r ) is multiplied with x k ( r ) [as was mentioned previously the data and chip symbol transitions are assumed to be aligned], and the resulting sequence is used to biphase modulate a carrier of frequency fi.. The resulting transmitted signal of the kth user is

s k ( ( t > = Re { t k ( ( t ) exp [ j (w,. t + C p k ) ] ] (4)

in which E k ( t ) is the complex baseband signal of the kth user, defined by

T

( ' ( r ) = a k ( t ) x k ( t ) ( 5 )

and Cpk is the random phase of the kth carrier. A point of note is that this architecture in effect uses the same PN sequence for both the in-phase and quadrature channels, which differs from Pursley 's [9] technique of employing a different PN sequence on each of the I and Q channels. Examining (2) and (5) it can be seen that the trellis-coded SSMA system transmits a biphase modulated PN sequence, the phase 0; and amplitude bi of which are determined by the signal point selected by the trellis code. The binary PN sequence is transmitted using the signal point x : and the signal point antipodal to x i . Furthermore, E, is t'le symbol energy defined by

E, = j ' [ s k ( t ) ] ' 0 dr. (6)

which has the form of a conventional I and Q type de- modulator. Thus, the receiver consists of a correlator matched to the ith user's PN sequence for both the in- phase (I), and quadrature (Q) channels (see Fig. 4) fol- lowed by a coset correlator which provides inputs to the Viterbi decoder. This receiver is optimum in an AWGN environment, but not in a SSMA environment due to the presence of the interuser interference. In complex notation, the receiver performs an operation equivalent to correlating r ( r ) of (7) with m a ' ( r ) exp ( - j w , r ) . De- fining Y ( r ) to be the equivalent complex low-pass output of the correlation process gives

Assuming without loss of generality that r i = 0 and 0' = 0, the sampled complex input to the coset correlator at some arbitrary baud interval p becomes

BOUDREAU e/ U / . : TRELLIS C O D E D VERSLIS CONVOLLITIONALLY CODED SSIMA SYSTFMS

t = T

r

Fig. 4. Receiver model

The result of (12) is valid if one assumes that U,. >> T-I , in which case the higher order frequency terms are negligible. The even and odd continuyus-time partial cross- correlation functions R k , ; ( 7') and Rk,; ( 7 ' ) are defined by [SI, [91

7 i

R k , i ( ~ k ) = io a k ( r - # ) a ' ( r ) dr

= C',;(1 - N c ) R + ( T ~ - /T . )

+ Ck.;( l + 1 - N , , ) R+(T' - 1T,.) (13)

= i' a'(r - # ) a i ( r ) dt TI

= Ck, f ( l ) &(T' - lT,.)

+ Ck.;(1 + 1 ) R + ( T ~ - lTc) (14)

in which the partial autocorrelation fyctions of the chip waveform are defined by R, (s) and R, ( s ) 191

R,(s) = j'' $ ( r ) $ ( r + T,. - s) dr (15)

T,

= s $ ( f ) $ ( r - s ) dr (16)

and the aperiodic cross-correlation function Ck.; ( * ) is defined by [9]

For the rectangular chip waveforms employed in the analysis (1 3) and (14) take on the form

R k , ( ~ ) o = C k f ( l - N O T , + (T!' - IT,)

* [C' , ( l + 1 - N O - Ck f ( l - N , ) ] (18)

* [ C d + 1 ) - C ' f ( U ] . (19)

R ' , ( T ' ) = C k f ( 1 ) T c + (d' - IT,)

63 I

The coset correlator correlates the complex signal YI, with each of the possible signal points of the partitioned signal constellation, as was explained in Section 11-A. This can be accomplished by correlating YI, with only one signal point from each antipodal coset, since the value of the correlation for any two signal points in the same antipodal coset, will have the same magnitudes but opposite polarities. Thus, the number of computations can be reduced by half. In general, for each decoded signal point, the estimate of the data bits transmitted by the decoded symbol is stored, as well as the distance of the received signal from the decoded symbol. The Viterbi algorithm is used to identify the signal path (i.e., the sequence of symbols) through the code trellis which has minimum Euclid- ean distance from the received signal sequence.

The first term on the right-hand side of (12) is the transmitted symbol in the pth baud interval, whereas the second term under the summation is the interuser interference at the output of the correlators. Define the interuser interference seen by the ith user in the p th baud interval as

26 = 6 - c [ x ; , - ~ R ~ . ~ ( T ' ) + x j R , , , ( ~ ' ) ] i' T I = I ':'

in which

k # f

Combining (12), and (20) one obtains

(22) Y; = x;, + ZI, + TI,

in which X h = &xh. In the above and subsequent analysis, lower case has been employed to denote symbols or random variables that have been normalized with respect to 6, whereas upper case has been reserved for symbols or random variables that implicitly contain the square root of the symbol energy E, .

If a convolutionally coded system employs PN sequences that span n coded symbols, it is necessary to modify (17), (1 S), and (19) appropriately. Define the rth subsequence of a ' ( r ) of length L as

a i r ) = ( a ; [ ? a;L+I? * * . 1 a ; r + l ) L - l ) . (23)

Generalizing (17) to include subsequences give [ 1 I ]

C a,b,+, 0 I 1 I L - 1 / = o

c a,-,b, 1 - L I 1 < 0 (24)

111 1 L

632 lb.Ek J O U K N A L O N SEL.F.C'lF.I> AREAS I N COMMUNICATIONS. VOI. X. NO 1. M A Y IY90

in which a = ( u o , al, . . b12). This allows one to rewrite (1 8) and (19) as

, u L ) and b = (bo, b , , . . , can be calculated, the general form of which is (using the super state notation of [ 131)

m

T ( D , I ) = P ( S O ) /d ( ' s 'u ' q ( U ) P ( X -+ X). N = I S ( 0 . N )

(33 ) \ I

In (33) S is a sequence of super states and U is a sequence of "super input signals." These sequences are defined in an analogous manner to (30) and (31), with the individual elements being defined as

S,, = (6,,, 6,)) and Up = (U,,, z i p ) (34)

. [ C [ a ; r - , ) , + 1 - L )

- C[a; , - l , , 4 \ l l ( / - L ) ] (25)

ff [a: , ) , a ; 5 ) I ( 7 ' )

= C[a: , , , a : \ , ] ( q T + (7' - U,) * [ c [ a ; r ) , a; \ ) I ( I + 1 ) - c[a;r ) * a ; , , ] ( / ) ] .

(26)

In order to obtain the required values of (21) one must average (25) and (26) over all possible subsequences utr) and ai7) .

111. PERFORMANCE ANALYSIS The performance of trellis coded SSMA can be evalu-

ated using transfer function bound techniques that have been developed for evaluating the performance of convolutional codes and have recently been applied to evaluating the performance of trellis codes [ 121, [ 131, [ 141, [ 161. The generalized transfer function bound involves calculating the painvise errors using the Chernoff bound and summing over all possible error patterns using the union bound. The following notation used in the devel- opment of the Chernoff bound parallels that of Divsalar and Simon [15]. Define a complex coded symbol sequence of length N by

x = ( X I , x,, . * * 7 X,) (27)

in which X,, is the complex symbol transmitted in the pth baud interval. The value of the symbol X,, is determined by the state of the trellis encoder 6, and the previous n input bits (for a rate n / n + 1 code) defined by the vector

up = (up , U , , - l , . . * , U p - r , + l ) . (28)

x p = f ( 6 , , , U,,) (29)

Thus,

in which " f ( - ) " is a nonlinear function. Corresponding to the channel input sequence X define a complex channel output sequence

Y = ( Y , , Yz, * * * 9 Y N ) (30)

z = (Z,, z,, * . , Z N ) . (31)

(32)

and a complex sequence of interuser interference as

From (22) the channel output signal at time p is

q, = x,, + z,, + v,,. The random variable vn is a sample of complex Gaussian noise (with zero mean and variance of a 2 ) .

From the state transiton diagrams, a transfer function

in which (A,,, U,,) is the correct input, state pair, whereas ( S I , , z i p ) is an incorrect input, state pair. S ( 0 , N ) is a pair of correct and incorrect sequences, that diverge at time 0, and remerge at time N, and P ( S o ) is the probability that the sequence started in state zero. Furthermore, d ( S, U ) is the distortion measure (in this case the number of bit errors), q( U ) is the probability of super input U ( 1 /2'lN for the codes under consideration), and finally P ( X -+

X ) is the probability of a pairwise error in which the incorrect sequence X is decoded in place of the correct sequence x . Frequently, P ( X + 8 ) can be replaced by the Bhattacharyya bound D , ( S , U ) [ 121, [ 131, however, for the present analysis the Bhattacharyya bound is not nec- essarily tight. In order to obtain the tightest bound on P ( X -+ X ) it is necessary to optimize the more general Cher- noff bound.

In terms of the above definitions, the probability of bit error can be bounded by [ 131

(35)

in which the transfer function bound has been averaged over the interuser interference as is indicated by the over- bar. The probability of bit error as expressed in (35) can be evaluated in matrix form for any general code (see [ 131 or [ 161 for details).

By using the method of moments [2], [ 171, [ 181 one can account for the interuser interference due to other users in the SSMA system and thus evaluate the Chernoff bound for any painvise error. Conditioning on the random vector 2, and subsequently taking the expectation over Z in order to obtain the average probability of a painvise error, gives

P , = P ( X -+ X ) = E [ P ( X -+ XlZ) ] . (36)

If it is assumed that the interuser interference is independent from one symbol to the next, then employing the Chernoff bound (see Section 111-A) allows one to upper bound PE of (36) as a product of painvise errors over individual symbols. The method of moments can subsequently be employed to evaluate the painvise error over a single symbol period as a weighted summation. The resulting expression for PE takes on the form

N N,,,

PE 5 rI c WP,P(X,, -+ 51ZP)lz,,=,, (37) 1 1 = ~ I = n

BOUDREAU PI al.. TRELLIS CODED VERSUS CONVOLUTIONALLY CODED SSiMA SYSTEMS 633

in which W,,, and r,,, are thejth weight and node (i.e., the abscissa at which the weighted function is to be evaluated), respectively, in the pth symbol period, calculated from the moments of the interuser interference, and N,,, is the number of moments utilized to calculate the weights and nodes. The details of the calculation of the moments is provided in Appendix A.

A . Pairwise Error Probability As was outlined above, the pairwise error probability

bound is required in the transfer function bound in order to evaluate the performance of the coded SSMA system. It is well known that the pairwise error between any two events can be bounded using the Chernoff bound which is defined by [ 191

Prob ( X > 0) < E[exp ( A X ) ] ( 3 8 )

in which X is a random vector. Assuming the optimum metric for the AWGN channel,

conditioning on the sequence of interuser interference Z , and applying the Chernoff bound, one obtains [ 151

P ( X + X(Z) I II E(exp [ x ( I Y , , - x,,12 - IY,, - X , , 1 2 ) ] }

P S V

( 3 9 )

in which v is the set o f p such that X, # X,,. Simplifying (39) gives

~ ( ~ 4 x 1 ~ ) 5 II E { e x p [ h ( ( x P l 2 - I x , , ~ * P E ”

- 2Re{ qx,, - X J * } ) ] } . ( 4 0 )

Substituting (32) into (40) gives

P ( X + X ( Z )

5 II E (exp [ A ( ~ X , \ ~ - I X , , ~ ~ P E ”

- 2Re{ (Xp + 771, + Zp>(X,, - %)*})I} P ( x + a l z )

I II exp ( - x E , I ~ , , - x,, 1’) P E ”

x E(exp [ - 2 h J F ~ e { q , , ( x , , - i,,)*)])

X E(exp [-2AE,Re{z,(x,, - i f )*} ] } ( 4 1 )

in which z, = Zp/&. The above and subsequent analysis assumes that zP is independent from one symbol to the next. The first expectation of (41) involving the complex Gaussian variable q,, can be evaluated by expressing q, in terms of its’ real and imaginary components. Denote the real (i.e., in-phase) component of q,, by q1 and the imaginary (i.e., quadrature) component by qQ, in which the subscript p has been dropped without loss of generality. Furthermore define a vector d,, = x,, - X,, (the norm of which is the distance between x,, and X,,), and as in the case of q,, define the real and imaginary components of d,,

as dl and d,. Noting that TJ! and qQ are uncorrelated, zero mean Gaussian random variables, each with variance a 2 / 2 = N 0 / 2 , one obtains

+P ( - 2 M & { 1 7 , ( x p - .,,*})) = +XP [ - 2 h J F ( r l / d , + 7QdQ)Ij

= exp (N,E,x~(~ , , - .t,,I2)

= exp (N,E,h2d:) * exp ( N , E , h 2 d i )

(42) which follows from the definition of the characteristic function of a Gaussian random variable. The second expectation of (41) can be written as

* P(Z/, ZQ) dZ/ dZQ. ( 4 3 ) In (43) the in-phase and quadrature components of z, (i.e., z I and z Q ) are in general dependent and as such the moments must be evaluated jointly. Defining U = zldl t z p d Q , (43) can be evaluated using the method of moments, as follows [see (37)]

+P (-2hE,Re(z,(x,, - f / J * } ) )

= j exp ( - 2 h E , v ) p ( v ) dzi

= c w,,, exp ( - 2 W ! & , ) .

m

- m

NO,

( 4 4 ) I = I

Substituting the results of (44) and (42) into (41) gives a pairwise error of

~ ( x + a ) I II exp [ - x E , ( I - XN,)I~ , , - i/> l ’ ) ] NI,,

* 1 = I w,,, exp ( - 2 W l,,!). ( 4 5 )

In order to obtain the tightest possible upper bound on the probability of error, (45) must be optimized with respect to the Chernoff parameter A . This can more easily be achieved if h is redefined as h’ / N o . Dropping the primes, gives

Expressing the pairwise error bound as in (46) enables h to be optimized in terms of E,s/No, a system input parameter. However, it should be noted that X cannot be optimized independent of the sequence length N . Thus, in practice one must evaluate the transfer function bound using some fixed value of A, which will be optimum for

634 1t.t.E JOLIRNAL O N SELEC'l't.I) A R E A S IN COMMUNlCArlONS. VOI X . NO -1. MAY I Y Y O

only one particular path length. Even so, a tight bound can still be obtained. Furthermore, the X of (46) is in fact the same h as in the original Chernoff bound, allowing easy reference of the deviation of the optimum value of A from the Bhattacharyya value of 1 /2. Also note that for the case of a convolutionally coded SSMA the value of I xfJ - f,, 1' is 4.0 for all x,, # 5. B . Gaussian Approximation

A common assumption that is used to simplify the analysis of the interuser interference in a SSMA system is to assume that such interference is Gaussian. If this assumption is used to evaluate the expectation with respect to z in (41) one obtains the approximation

E(exp (-2hE\Re{z/,(x,, - f/J*})} = exp ( 2 ~ * ~ f a f l x , , - f/,l') (47)

in which U; = $ E [ 1zP1*] (i.e., the variance of the interference in the in-phase or quadrature channels). In arriv- ing at the result of (47) it was assumed that z I and z Q were independent, which is a valid assumption since if z I and zp are jointly Gaussian and uncorrelated (by definition) then it follows that they are independent. Substituting the results of (47) and (42) into (41) one obtains the following approximate painvise error bound:

P ( x + 2) 5 n exp {-A( 1 - 2X(N0/2 + a ; E , ) ) P E V

* E,Ix, - . , , I 2 ) . ( 48 )

Optimizing (48) with respect to the Chernoff parameter X gives

(49) 1

4(N0/2 + a ; E , ) h =

Substituting (49) into (48) gives a resulting bound of

A point to note is that the bound obtained in (50) is in fact that Bhattacharyya bound for the Gaussian approximation. This result is comparable to one derived by Pursley [8] for uncoded binary SSMA.

IV. SYSTEM PERFORMANCE The performance of a spread-spectrum multiple-access

system utilizing either various trellis or convolutional codes has been calculated and is presented below. In order to facilitate a comparison between the performance of the trellis codes and the convolutional codes it has been assumed that the minimum chip period T:. is fixed ( i . e . , TI. must be chosen such that TI. > T:.). Furthermore, it is

assumed that the data rate is constant, requiring compar- isons on the basis of equal throughput or equal energy per bit to noise spectral density ( & / N o ) . Rate k / n trellis codes transmit k bits per coded symbol for which it is assumed that each coded symbol spans one complete PN sequence. Bearing in mind the assumptions of fixed chip periods and constant throughput this results in two possible methods of choosing an equivalent convolutionally coded system. A rate k / n convolutional code transmits k / n bits per coded symbol, and if one wishes to have each coded symbol span a complete PN sequence, the period of the PN sequence must be shorter by a factor of k / n than the PN sequence used for the trellis-coded system. This results in increased interuser interference due to the poorer cross-correlation properties of the shorter PN sequences. Alternatively, one may choose a PN sequence that spans n coded symbols, which would be the same length as the PN sequence employed in the corresponding trellis code. However, this approach also results in increased interuser interference, since partial cross-correlation effects are more severe than cross-correlation over complete sequences [ 1 I ] .

Before discussing the performance results there are two points to be noted. All of the performance curves calculated are based on the assumption that the interferers and the desired signal arrive at the receiver at the same rela- tive power level. Furthermore, the curves are Chernoff upper bounds on the average performance. The true average performance can be from 0.5 to 1 dB superior depending on the E,,/NO at which one is operating (see Fig. 1 I ) .

The performance of 2, 4 and 8 state rate 1 /2 trellis- coded SSMA employing a 4-PSK constellation is illustrated in Fig. 5 (these are the same codes found in [12], but with no asymmetry-see also [ 2 0 ] ) . Gold spreading sequences of length NI = 51 1 are assumed, and K = 20 users are transmitting simultaneously. For benchmark purposes, the probability of error curves for uncoded BPSK and uncoded BPSK with K = 20 users in a SSMA system are also plotted in the figure. It can be seen that at a P , of lo-' there is an improvement of 4 dB of the four- state code over the uncoded BPSK system with SSMA.

Fig. 6 illustrates the performance of 4 , 8 and 16 state trellis codes employing an 8-PSK signal constellation. These are Ungerboeck codes and the encoders and generator polynomials can be found in [6]. Since these codes transmit two bits per symbol, their performance has been benchmarked against uncoded 4-PSK. Gold spreading sequences of length N, = 51 1 were assumed with K = 20 users present in the system. At a P , of there is an improvement of almost 4 dB of the 16 state code over the uncoded 4-PSK with SSMA.

Figs. 7 and 8 illustrate the performance of rate 1 / 2 and 1 /8 convolutional codes on a basis of equal throughput and chip period. The generator polynomials for these codes can be found in [ 2 2 ] . In octal form the generators for the rate 1 /2 code are: 5 and 7 (4-state); 15 and 17 (8-state); 23 and 35 (16-state). The generators for the rate

BOUDREAU et al.: TRELLIS C O D E D VERSUS CONVOLUTION ALLY C O D E D SSiMA SYSTEMS 635

IO'-

CPSK TRELLIS CODES N e 5 1 I K=20 1 00

IO' =!A! wth SSMA NsS11 K=20 -2-state CPSK trek code: K=20 -&state CPSK treW code K=20 -- 8-snte CPSK trehs eocie. K=20 102

8 5 1 0 3

+j 104

IO 5

0

0 c

k

104

0 2 4 6 8 I O 12 14 16 10'

-BPSK t u B P S K with SSMA K=20, N ~ 5 1 1

t -BPSK

Eb/No Eb/NoldBl - .

Fig. 5 . Performance bounds of 2 , 4, and 8 state 4-PSK trellis code with Gold spreading sequences of length N , = 5 I 1 and K = 20 users.

Fig. 7. Performance of 4 , 8, and 16 state rate 1 / 2 convolutional codes with Gold spreading sequences of length N , = 255 and K = 2 0 users.

.II; 8 5 IO-' -

+ 104 -

P c

6

Eb/No Eb/NoldBI . I

Fig. 8. Performance of 4 , 8, and 16 state rate 1 /8 convolutional codes with Gold spreading sequences of length N , = 63 and K = 2 0 users.

Fig. 6 . Performance of 4 , 8, and 16 state 8-PSK trellis codes with Gold spreading sequences of length N , = 5 I 1 and K = 20 users.

1 / 8 code are: 7, 7, 5, 5, 5 , 7, 7, 7 (4-state); 17, 17, 13, 13, 13, 15, 15, 17 (8 state); 37, 33, 25, 25, 35, 33, 27, 37 (16 state). It has been assumed that for these codes that the PN sequences employed span one coded symbol. Using N , = 511 for the trellis codes as a benchmark it follows that the rate 1 /2 code must employ PN sequences of length N , = 255 (i.e., two coded symbols per bit giving 510 chips per bit), whereas the rate 1/8 code must employ PN sequences of length N , = 63 (i .e. , 8 coded symbols per bit giving 504 chips per bit). Though the number of chips per bit do not exactly match those of the trellis codes the difference has negligible effect on the cal- culations. Examining the performance of the 16 state codes one observes an improvement of over 4 dB at a P, of lop5 between the rate 1/2 code and uncoded 2-PSK with SSMA, whereas an improvement of nearly 5 dB is observed for the 16 state rate 1/8 code. In Fig. 9 the

performance of 4 and 16 state rate 2 / 3 ( N , = 127) and 2 /7 ( N , = 63) convolutional codes are plotted. The generator polynomials for the rate 2 / 3 code are: 17, 06, 15 (4 state); 27, 75, 72 (1 state). The generators for the rate 2 / 7 code are: 05, 06, 12, 15, 13, 17 (4 state); 33, 55, 72, 47, 25, 53, 75 (16 state). These codes transmit two information bits per n coded symbols (3 and 7, respectively) and allow a more equitable comparison with the rate 2 / 3 8-PSK trellis codes. It can be seen that the 16-state rate 2 /7 code offers a 1.3 dB gain over the 16-state rate 2 / 3 convolutional code at a probability of bit error of lop5 . Both of these exhibit superior performance to the rate 2 / 3 trellis coded system. Fig. 10 illustrates the performance of a convolutionally coded SSMA system that employs a 4-state rate 1 /7 code (having generator polynomials 7, 7, 7, 7, 5, 5, 5 ) and Gold spreading sequences of length N , = 5 1 1. Each coded symbol used

636 IEEE JOURNAL O N SELECTED AREAS IN COMMUNICATIONS. VOL 8 . NO. 4. MAY 1990

code

RATE 2/3 AND U! CODES: 4 AND 16 STATES 100

rate no. PN sequence Ea/,V, dB for gam over gam over

states length for Pc = lo-' BPSK [dB] QPSK [dB]

L

c o n v o l u t m a l

I" 0 2 4 6 8 10 12 14 16

Eb/No[dBl

Fig. 9 . Performance of 4 and 16 state rate 2 / 3 ( N , = 127) and rate 2 / 7 ( N , = 6 3 ) convolutional codes with K = 20.

118 16 63 5.8 5 0

RATE 111 4-STATE CONV. CODE PARTIAL CORRELATION

loo 1

8-PSK trellis

EbiNoldBI Fig. 10. Performance of 4 state rate 1 / 7 convolutional code with partial

correlation of length 73 over N , = 51 1 .

213 4 511 9.6 2.8

a subsequence of length 73 chips, which conveniently di- vides evenly into 51 1 seven times. The curve that is plotted is for K = 10 users, and even though the partial cross- correlation properties tend to be quite poor, it can be seen that the code provides significant coding gain. In fact, the performance is almost identical to the performance of rate 1 / 8 convolutionally coded SSMA with K = 10 users and N , = 63 [21]. Finally, Fig. 11 provides a comparison between the accuracy of the Chernoff bound form of the transfer function bound, the use of the Gaussian approximation for the interuser interference in the transfer function bound, and a Monte Carlo simulation of the code performance. The two codes considered here are a 4 state rate 1 /2 4-PSK trellis code ( N , = 127) and a 4 state rate 1 /2 convolutional code ( N , = 63 ), both in the presence of K = 5 users. It can be seen that the Gaussian approximation underbounds the Chernoff bound by about 0.25

convolutional

IO0

, conv. code simulation - N=63 K=5

217 16 63 6.5 5.9

Eb/No[dB]

Fig. 1 1 . Comparison of Chernoff bound performance against the performance of the Bhattacharya bound with the Gaussian assumption and comparison against Monte Carlo simulations.

TABLE 1 COMPARISON OF TRELLIS A N D CONVOLUTIONALLY CODED SSMA A T P , =

IO..' FOR K = 20 USERS

8-PSK trellis 1 :I; 1 [: 1 ;;; 1 ;,; 1 I ;:; I 8-PSK trellis

convolutional 213 127 9 9 2.5

convoiutianal 213 16 127 7.8 4.6

convolutional 217 63 8.3 4.1

dB, whereas both of these bounds exceed the true performance of the codes by over 1 dB, as indicated by the Monte Carlo simulation curves. The Monte Carlo simulations points are plotted with 95% confidence intervals on the abscissa with a second-order least squares error polynomial fit. The fact that the Gaussian approximation of interuser interference is overly optimistic in a SSMA system is consistent with previous work (i.e., see [2]). As the number of users K increases, this approximation becomes even poorer (see [2 l]).

Table I compares the various trellis and convolutional codes that have been presented on the basis of equal throughput and constant chip rate at a probability of bit error of for K = 20 users. It can be seen that for both the codes with a throughput of 1 bit per baud (i .e. , the rate 1 /2 trellis, rate 1 /2 convolutional and rate 1 /8 convolutional codes ), that the convolutional codes are su-

BOUDREAU r f U / ' TRELLIS C O D E D V E R S U S C O N V O L U T I O N A L L Y C O D E D S S / M A S Y S T E M S 631

perior, with additional gains being obtained by the lower rate codes. The codes with a throughput of 2 bits per baud also exhibit similar trends in performance.

V. CONCLUSIONS The results of this study indicate that if coding is to be

employed in a SSMA system that utilizing a convolutional code provides superior performance than utilizing a trellis code applied to the data symbols. The gain of the convolutional code over the trellis code for a given complexity and throughput, increases as one employs lower rate convolutional codes. In a band-limited environment, the use of a convolutional code would result in a penalty being paid for in bandwidth. However in a SSMA system a convolutional code can be employed with no resulting expan- sion in bandwidth or decrease in processing gain (this is consistent with the analysis of Viterbi [23] who used the Gaussian assumption to model the interuser interference). This allows one to exploit the greater distance properties of the lower rate convolutional codes over those of the corresponding trellis codes. When employing the lower rate convolutional codes there is a tradeoff due to the increased cross-correlation from either the use of shorter PN spreading sequences or partial cross-correlation effects. However, the results of this study demonstrate that this effect is secondary to the additional coding gain due to the increased distance of the lower rate codes.

APPENDIX A INTERUSER INTERFERENCE MOMENTS

In order to evaluate the performance of a SSMA system one can use the method of moments algorithm [ I ] , 1171, [ 181. For a system which employs trellis codes or convolutional codes, one must evaluate the probability of error for path lengths which vary from one symbol to an infinite number of symbols. The method of moments algorithm requires the evaluation of the moments of the random vector v [see (44)] up to an arbitrary order N,,, at which a desired accuracy is obtained. Rewriting the random vector v in complex form as

(51 1 ( d " ) : + (Z* fd

2 v =

will prove expedient in the subsequent analysis. Thus, in general one must calculate

is equal to one. In practice, this can be achieved by a simple interleaver of length two, however, even without an interleaver this assumption does not significantly de- grade the accuracy of the results [21]. Dropping the subscript p without loss of generality and evaluating only even moments (the odd moments are zero since the density function of z is even), gives

( 5 3 ) Substituting the expression for 2 of (20) into (53) gives

(54)

Noting that the second expectation in (54) is nonzero only when i = m enables onelto simplify (54) to

Define the real and imaginary parts of the random variable GP as E P and F". The resulting expression for the expectation on the right-hand side of (55 ) becomes

( 5 6 )

in which j = f i . Due to the assumed randomness of transmissions from

one user to the next, the interuser interference random variables U' are independent (between users), and as such one way calculate the interference random variable U' due to the kth user independently of the other users. Once t ih

is obtained for each of the users, the total interuser inter-

for m = 1, 2 . . . N,,, 2

E[u"'] = E

The random variables z , are not independent from one symbol period to the next and in general the evaluation of (52) is quite cumbersome (see (211 for details). However, if it is assumed that the interuser interference random variable zp is independent from one symbol to the next then the evaluation of (52) is simplified considerably since N

ference can be obtained using the cumulants algorithm based on the fact that [ 11

K

2) = c t J h . (57) h = I k f l

638 I E E E JOURNAL ON SELLCTED .AREAS I N COMMCINICATIONS. V O I ~ . X. K O 1. M A Y I9YO

Expanding (56) for any interferer k , and neglecting the user superscript as well as the subscript p without any loss of generality, one obtains

3 . ( 5 8 ) . ( - j )”’I-’ - / E { El+/F?ff f - - I - /

Thus, the problem reduces to determining the expectation of the expression on the right-hand side of (58). Rein- troducing the subscripts p , the expectation of the function of the random variables E,, and F,, can be evaluated using the definition of (21). Defining $(E,,, F,,) as the final product in (58) gives

E { 5 (E , , , F,,)} f + /

= E { (x;-,R, ,(TA) + x;R‘ ,(r’))

. (x$- IR’ ( r‘) + X P R ‘ I (2)) 2 f f I - l - / ] ~ (59)

In (59) the notation x ; and x $ has been used to denote the in-phase and quadrature components of x j . Defining a and b

a = i + l ( 6 0 )

(61 1 b = 2m - i - 1

(59) can be expressed as

. ( ;; ) - [’ ( xp ) /’ - f

* (Rp.;(r~))~’+s(~,.;(r“)) (1 + h - c -f

In the derivation of (62) the expectation was split into two products, based on the fact that the in-phase and quadrature components of the transmitted symbols of the interferers (i.e., x: and x P ) are independent of the continu- ous-time partial cross-correlation functions R‘,, ( r‘) and

( r’). Define the expectation of the first product as X and the expectation of the second product as @ ( r ‘ ). For a given set of spreading sequences { a ( ’ ) ( t ) }, @ (7‘) can be evaluated using numerical techniques (see [ 11 or [21]). Calculation of X depends on the constellation being employed to transmit the data. Expressions for X are derived in Appendix B for MPSK signal constellations employed in the analysis.

All of the expressions required to calculate the moments of (52) have now been defined. Recapitulating, the moments of (52) can be calculated numerically in the fol-

lowing steps. Initially, the functions @ ( r h ) and X can be calculated as defined in (62). These results can subsequently be utilized to evaluate 5 ( E,, F,,) of (59). More- over 5 ( E,,, F,,) can be used to determine E [ I G12ff’] of ( 5 5 ) , which when evaluated provides the desired moments.

From the above analysis of the moments it can be seen that the interuser interference is a function of both the cross-correlation properties of the set of PN sequences employed for spreading the transmitted signals [i.e., through @ (r‘)], and the signal constellation employed by the trellis code (i.e., the factor X ) . Thus, in order to optimize the performance of the trellis code one must optimize @ (7‘) i n conjunction with X

APPENDIX B CORRELATION MOMENTS OF MPSK CONSTELLATIONS The codes considered in the analysis of this paper em-

ploy MPSK signal constellations, the correlation moments of which are derived below. With reference to Ap- pendix A, signal points in X are chosen from the constellation under consideration according to the output of the convolutional coder of the trellis code. Though this produces some correlation between adjacent signal points, due to the symmetry of the constellations and the coder, the in-phase and quadrature components are chosen with equal steady state probabilities in any two symbol intervals. Thus, for the purposes of averaging, the symbols can be treated as being independent from one symbol period to the next. In this case X can be reduced to a calculation of the form E { ( x ~ ) ” ’ ( x ~ > ” } for m and n > 0.

A . 2-PSK Constellation Choose an arbitrary 2-PSK constellation with antipo-

dal signal points at ( 1 /&, 1 /&) and ( - I / & , - 1 /&). It then follows by inspection that

E { (x , : ) ” ’ (x , ” , “ } =

if m + n is even, otherwise the expectation is zero.

B. 4-PSK Constellation Choose a 4-PSK constellation with the in-phase

63 1

and quadrature components of the si nal points taken from the set { ( I / & , I / & ) , ( I / & , - I / & ) , ( - I / & , l / & ) , ( - 1 /&, - l / & ) } . Due to the symmetry of the constellation, if m or n is odd, then

If both m and n are even then the expectation takes on the form

E { (x:,)2ff’(x,”)2”} = - : [(5&)? 2fff 2’1 + . - + ( $ ($1

BOUDREAU c/ < I / . : TRELLIS C0Db.D VERSUS CONVOLU.TIONAL1.Y COI)l<D S S ’ M A SYSTF,MS 639

C. 8-PSK Constellation Choose an 8-PSK constellation with the in-phase and

quadrature components of the signal points taken from the set { ( + I , 0), (0, k l ) , (+l/h, k l / h ) } . As in the case of the 4-PSK constellation, if m or n is odd, then

Moreover if both m and n are even and nonzero, then as in the 4-PSK case

+ . . . + - (-l)?”J ( - l ) ” l ] ~

Furthermore, if n is zero then

L

A similar analysis holds for E { (x;) 211 } if tn = 0

REFERENCES

[ I ] M. Kavehrad and P. J . McLane. “Spread spectrum for indoor digital radio,” IEEE Conimun. Mug.. vol. 25. pp. 32-40. June 1987.

[2] M. Kavehrad, “Performance of nondiversity receivers for spread spectrum in indoor wireless communications.” AT&T Tech. J . . vol. 64, no. 6 , pp. 1181-1210, July-Aug. 1985.

131 M. Kavehrad and P. J . McLane, “Performance of low-complexity channel coding and diversity for spread spectrum in indoor wireless communication.” AT&T Tech. J . . vol. 64. no. 8 . pp. 1927-1965. Oct. 1985.

141 M. Kavehrad and B. Ramamurthi, “Direct sequence spread spectrum with DPSK modulation and diversity for indoor wireless communications.” IEEE Truns. Corntnuri., vol. COM-35. pp. 224-236, Feb. 1987.

[SI M. Kavehrad and G. E. Bodeep, “Design and experimental results for a direct sequence spread spectrum radio using differential phase shift keying modulation for indoor wireless communications,” IEEE J . Select. Areus Commuri.. vol. SAC-5. pp. 815-825. June 1987.

161 G . Ungerboeck, “Channel coding with multilevel phase signals.” 1EEE Truns. Inform. Theory. vol. IT-28, pp. 55-67, Jan. 1982.

171 A. R. Calderbank and N. J . A. Sloane. “New trellis codes based o n lattices and cosets.” IEEE Truns. Inform. Theory, vol. IT-33, pp. 177-195, Mar. 1987.

181 M. B. Pursley. “Performance evaluation for phase-coded spread spectrum multiple access communications-Pan I : System analysis,“ IEEE Trans. Connnutt.. vol. COM-25. pp. 795-799. Aug. 1977.

[9] -, “Spread spectrum multiple-access communications.” in Mulri- User Conimunicurioris, G. Longo. Ed. Vienna and New York: Springer-Verlag. 1981. pp. 139-199.

[ I O ] Viterbi and Omura. Principles ofDiyittrl Co,,ir,iuriic.crtiorr trnd Coding. New York: McGrdw-Hill. 1979.

[I I] D. V. Sarwate. M. B. Pursley. and T . U . Basar. “Partial correlation effects in direct-sequence spread spectrum multiple-access communications systems,” 1EEE Trcrris. Connnun., vol. COM-32. pp, 567- 573, May 1984.

[ 121 D. Divsalar and M. K. Simon. ”Trellis coding asymiiietric modula- tions.” lEEE Turns. Coniniun.. vol. COM-35. pp. 130-141. Feb. 1987.

[ 131 J . K . Oniura and M . K. Simon. “Modulationidemodulation techniques for satellite communications-Part IV: Appendices.’’ J f L Pu/~/ic.rrtiori 8/ -73 . Pasadena. CA. Nov. I . 19x1.

141 E. Zehavi and J . K. Wolf. “On the performance evaluation oftrelli\ codes.’‘ IEEE Truns. 1rrfi)rnr. Theory, vol. IT-33. pp. 196-202. Mar. 1987.

IS] D. Divsalar and M . K . Simon. “Trellis coded modulation for 4800- 9600 bit/s transmission over a fading mobile satellite channel.” IELE Trcim. J . Se/ec,r. Arcwx Coniniun., vol. SAC-3s. pp. 162- 175. Feh.

161 E. Biglieri. ”High level modulation and coding for nonlinear satellitc channels.” IEEE Truns. Coinniun.. vol. COM-32. pp. 616-626. May 1984.

171 M. H. Meyers. “Computing the distribution of a random variable via Gaussian quadrature rules,” Bell S?..,/. T d i . J . . vol. 61, n o . 9. pp. 2245-2261. Nov. 1982.

[ 1x1 D. Laforgia. A. Luvison. and V. Zingarelli. ”Bit error rate evaluation for spread spectrum multiple access systems.“ /E€€ Truns. Conin7un.. vol. COM-32. pp. 660-667. June 1984.

[ 191 M. K. Simon, J . K. Omura. R. A. Scholtz. and B . A. Levitt. Sprctrcl Spcc/runi Corrirrrroiictr/io,ls-Volrrr,rc, 1. Rockville, MD: Computer Science. 1985.

1201 G . D.‘Boudreau, D. D. Falconer, and S. A. Mahoud. “The application of trellis coding to spread spectrum multiple access systems.” MILCOM’88 ConJ P r o c . . vol. 3. no. 5. pp. 616-626. Oct. 1988.

1211 G. D. Boudreau. “Analysis of the application of trellis coding to spread spectrum multiple access systems.“ Ph.D. dissertation. Dep. Syst. Comput. Eng.. Carleton Univ., Ottawa, Canada. 1989.

[ 221 Proakis. Digitcl/ Cor,i,rirrriiccrtrorls. New York: McGraw-Hill. 1983. 1231 A. J . Viterbi. ”When not to spread spectrum-A sequel.” IEEE

1987.

Cornrnun. Mrr,q., vol. 23, pp. 12-17. Apr. 1985.

Gary D. Boudreau (M’85) received the B A Sc degree in electricdl engineering from the Univer sity of Ottawa, Ottawa. Ont . Canadd, in 1981 and the M Sc degree i n electricdl engineering from Queen’s University, Kingston, Ont , Can- ada. in 1984 He I\ currently in the hndl \tages of completing his Ph D degree at Carleton Univer- sity, Ottawa. Canada

He joined Canadian Astrondutic\ Limited of Ottawa, Canada, in 1984 where he is currently employed as a communications sy\tem\ engineer

His interest\ include digital communications. bandwidth efficient moduld- tion techniques. coding theory, and digital signal processing

I

David I). Falconer (M‘68-SM‘X3-F‘86) was born in Moosc J a w . Sash. . Cmadu. on August I S . 1940. He received the B.A.Sc. dcgrec in cngi- [leering physic\ from the Uni\,crsrt! o f Toronto. Ont.. Canada. in 1962. and the S.M. and Ph.D. degrcca in clectrical enginccring 1roiii the Mas- sachusetts Institute of Technology . Camhridgc. in 1963 a n d 1967. rcspcctivcly.

Alter a yciir iis ii Postdoctoral Fcllow at the Royal Institute 01 Technology. Stochholm. S N C - den. he was with Bcll Lohoratoric\. Holmdel. N J .

I’roiii 1967 to IY8U. as a iiiciiibcr ol . thc Technical Stall and Iatcr as G r w p Supervisor. During 1976- 1977 he w;i\ ii Vlsiting Prolc\sor ;it LinhOping University. Linh(iping. Sweden. Since 1980 hc ha\ heen at Curleton Uni- . vcrsity. Ottaua. Ont.. Canada. ahc rc hc i s a Profcssor iii thc Dcpartniciit o l System and Computer Engineering. His intcrcsts arc in digital coiiiiiiu- riicationa. signal proccaaing. and coii~~iiunication thcor) .

Dr. Fdconcr was Associatc Editor for Coiiiiiiuniciition\ Thcory ;ind kiter Editor lor Digital Coiiiiiiunication\ of the IEEE TK\\sI(’I i o \ s o\ Cos1 h l i \ i ( ~ \ i i o \ s lroin 1981 to 1987. Hc i s i i l ~ ;I mcn~hci- d t h c As\ociotion o f Prolcs\ional Enginccra o f Ontario. Hc wa\ awardcd the Conimunica- tion\ Socicty Pri/c Pnpcr Award in C~)miiiiinications Circuits ancl Tcch- niques in 1983 and in 1986.

“*

640 IEEh JOURNAL O N SELECTF.1) AREhS I N COMMIINICATIONS. VOI. X. NO. 1. MAY I Y Y O

Samy A. Mahmoud (SM'83) received the M.Eng. and Ph.D. degrees in electrical engineering from Carleton University, Ottawa, Ont., Canada. in 1971 and 1974, respectively.

From 1974 to 1975, he worked as a Consultant to the Canadian Department of Communications where he did basic research in the area of robust data transmission over high-frequency radio net- works. He joined the Faculty of Engineering at Carleton University, Ottawa, Ont., Canada, in 1975, where he is currently Professor and Chair-

man of the Department of Systems and Computer Engineering. He is a cofounder and Vice President of Intellitech Ltd., a telecommunications research firm. He is also the principle investigator of a major research project

in tlic area 01 VLSl applic;itions IO iiiobile and portable radio systems. He \pent the Suniiner 01' 197X as ii Research Fellou in the Nuclear Research Center. Karlsruhc. Fcderal Republic of Germany. working on distributed proccssing probleiii\ o\'er local area nctworhs. He has published on a wide range 01 topics including technical articles on distributed inlormation net- works. His current research and consulting activities include distributed processing systems. mobile and portable radio coiiimunication systems. ap- plications o l VLSl technology in digital coinniunieations. and ISDN sys- leiiis and prolocols.

Dr. Mahmoud i\ ;I member ol the Association o l Prolcssional Engineers olOntario. and he was a Guest Editor lor two IEEE J O L : R ~ A L O ~ ~ S t ~ i ~ i x r i ~ i ~ ARIAS ip\i C u h i h i i \ K A I IOX\ Special Issue on Portable and Mobile C o n - munications. puhlished in J u n e 1987 and January 1989. rcspectively.

A comparison of trellis coded versus convolutionally coded spread-spectrum multiple-access systems

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