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IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 39, NO. 1, JANUARY 1993 157

Noncoherent Detection in Asynchronous Multiuser Channels

Mahesh K. Varanasi, Member, IEEE

Abstract-The noncoherent demodulation of multiple differentially phase-shift-keyed signals transmitted simultaneously over an asynchronous code-division multiple-access (CDMA) channel is considered under the white Gaussian background noise regime. A class of bilinear detectors is defined with the objective of obtaining the optimal bilinear detector. The asymptotic efficiency of the bilinear detectors is derived by characterizing the dominant behaviour of error probability functionals of general quadratic detection schemes in the high signal-to-noise ratio regions. The optimality criterion considered is near-far resistance that denotes worst-case asymptotic efficiency over the signal energies and phases which are unknown at the receiver. The optimal bilinear detector is therefore obtained by solving a minimax optimization problem. In the finite packet length case, this detector is shown to be a time-varying multiinput multioutput linear decorrelating filter followed by differential decision logic. In the limit as packet lengths go to infinity, the time-varying decorrelating detector is replaced by a time-invariant multiinput, multioutput decorrelating filter. Several properties of the optimally near-far resistant detector are established. Prominent among these properties are its bit-error rate invariance to the signal energies and phases of the interfering users, thereby alleviating the near-far problem, and its optimality in near-far resistance among all possible detectors, bilinear or otherwise.

Index Terms- Code-division multiaccess, multiuser channels, minimax methods, signal detection, spread-spectrum communication.

I. INTRODUCTION

I N A code-division multiple-access (CDMA) system, several users transmit information simultaneously and indepen-

dently over a shared channel. Each transmitted signal is a carrier signal digitally modulated by a sequence of information symbols and a distinct signatures signal assigned to the corresponding user. The received signal is therefore a superposition of several signals in addition to additive channel noise. Multiuser detection is the study of strategies for the demodulation of such simultaneously transmitted information in a CDMA channel.

There has been considerable research on the coherent multiuser detection problem in recent years. The basic assumption in that work is that the receiver has perfect knowledge of the energies and phases of each signal. Optimum coherent multiuser strategies were studied in [15] and [16] and low-

Manuscript received April 16, 1991; revised May 18, 1992. This work was presented in part at the Conference on Information Sciences and Systems, The Johns Hopkins University, Baltimore, MD, 1991, the IEEE International Symposium on Information Theory, Budapest, Hungary, 1991, and the IEEE Global Telecommunications Conference, Phoenix, AZ, 1991.

The author is with the Department of Electrical and Computer Engineering, University of Colorado, Boulder, CO 80309.

IEEE Log Number 9203019.

complexity, high-performance suboptimum strategies for both synchronous and asynchronous coherent CDMA channels in [61, [71, PO], F'l, and P41.

In this paper, we address the problem of noncoherent multiuser detection where the signal energies and phases are unknown at the receiver. Several reasons may contribute to rapid variations in these parameters such as oscillator phase instability, dynamically evolving positions of transmitter and receiver in a mobile environment, etc., thereby rendering intractable their estimation and tracking. In order to en- able noncoherent detection, we assume that the modulation technique employed by each user of the CDMA channel is differential phase shift keying (DPSK). Furthermore, we consider the general asynchronous CDMA channel where the transmitting users do not maintain any co-operation amongst them, as a result of which, the signals of the different users do not arrive in symbol synchronism. The particular case of a synchronous noncoherent CDMA channel was considered by Varanasi and Aazhang in [13].

II. A MINIMAX PROBLEM FORMULATION

Consider the characterization of the received signal where multiple differentially phase-shift-keyed transmissions of digital information are made simultaneously and independently over an additive Gaussian CDMA channel. The complex envelope of this received signal has the form

T(t) = U(&d) + n(t), (1)

where n(t) is the complex envelope of additive white Gaussian noise with power spectral density u2. The signal component is a superposition of K signals which arrive asyn- chronously at the receiver and contains the differentially encoded information sequence d = {d(~‘)}j”,-~ of the corresponding information sequence b = {b(j)}~~-,, with the K-length vectors b(j) = [bl (j), by, . . . , bK (j)]’ and d(j) = [dl (j), dz (j), e s . , d&‘)lT representing the binary data symbols and the differentially encoded symbols of the K users in the jth time interval, with each of these symbols belonging to the set { -1, +l}. The signal component of the received signal can therefore be written as

M K

V&4 = c c dk(j)uk(j)ok(t - jT - Tk). (2) j=-M k=l

The normalized signature signal {ok(t); t E [0, T]} is a real- valued signature waveform assigned to the Icth transmitter and al,(j) = dmeiek(‘) is the complex amplitude of the kth

0018-9448/93$03.00 0 1993 IEEE

158 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 39, NO. 1, JANUARY 1993

signal in the jth symbol interval. Without loss of generality, it is assumed that the signals are numbered in the ascending order of their relative delays so that 0 5 71 5 72 . . . 5 TK < T. In vinary DPSK, information is encoded into phase differences between successive symbol intervals. A “-1” or a “$1” in the ith time interval of the data sequence is transmitted by shifting the phase of the carrier in the ith interval by r or zero radians relative to the carrier in the previous signaling interval, which can be equivalently expressed as &(j) = bk(j)& (j - 1). The complex amplitudes (i.e., energies and phases) of the signals are assumed to remain nearly constant over two successive symbol intervals.

DefinethevectorspaceL= {~=[~~(-M),...,zK(-~), “‘,~l(M),“‘,ZK(~)]T} as the set of all MC-length complex-valued vectors (with N = 2M + 1 denoting the packet length), each element of which can also be regarded as a sequence of N length-K vectors, so that x = [8(-M), . . . , $ (M)IT. We will also have occasion to refer to the subset LR c L, containing the real-valued vectors of L. Note that the set of possible differentially encoded information vectors is a subset D c LR, obtained by restricting each element to the set { -1, +l}. we refer to x(l), the Zth length- K block of the vector z E L as the Zth vector of z for 1 E {-iv,. ” , M} and to xk(Z), the lath element of the Zth vector of x E L as the (/c, l)th element of x for Ic E { 1, + + . , K}.

Define the normalized signal o(t,d) as the signal component of the received waveform corresponding to unit amplitudes. The function i?(t,v) for any w E L is an element of L2, the complex Hilbert space of magnitude-square- integrable functions. In this space, denote the inner product of two functions fl(.) and f2(.) as (fl(.),fi(.)) = 2-l s-‘,” f;(x)f2(xw with induced norm 11 . 11. The validity of this inner product is assured by invoking the linear independence assumption (LIA) that

v w E L, PI # 0 =s Ilqt,w)l) # 0. (3)

This is a mild restriction as has been demonstrated in [6] where it is shown that if the a priori unknown delays are uniformly distributed, the LIA is violated with probability zero.

Notation: Define the K x K normalized signal crosscorrelation matrices R(j) whose elements are obtained as the crosscorrelations of appropriately delayed versions of the normalized signatures signals according to

cc

&l(j) = 2-l I

&(c(t - rk)if&(t + jT - 71) dt. (4) --oo

Since the modulating signals are zero outside [0, T],

R(j) = 0 V Ij 1 > 1 and R(-j) = RT(j). (5)

If the users are numbered according to increasing delays, the matrix R(1) is an upper triangular matrix with zero diagonal. We assume that the receiver has knowledge of the signature signals and the ability to lock on to the respective time delays so that the matrices R(-1), R(0) and R(l), which depend only on these parameters, can be evaluated from the defining equations for the entries of these matrices in (4).

If the delayed versions of the signature signals (corresponding to all symbol intervals) of all the users are arranged as a vector in the time order in which they are received, the crosscorrelation matrix is an NK x NK symmetric, block- Toeplitz matrix which we define as

R(0) R(-1) 0 ...

R(1) R(O) W-1)

R(1) . (6)

. . .

This matrix is independent of the signal energies and phases and is the noncoherent analogue of the signature signal crosscorrelation matrix defined in [6]. In that paper, the signal phases are assumed to be known and time-invariant and are therefore modeled implicitly as being part of the signature signals. We refer to the (Ic, j)th rtiw (column) of a matrix of dimension of R to denote the lath row (column) within the jth block in the vertical (horizontal) direction. Each row (or column) of a matrix of dimension of R can be viewed as being a member of the vector space L. For instance, the ( !c, j)th column of the NK x NK identity matrix is the (5, j)th unit vector denoted &j and its (m, l)th element is given as t@(Z) = S&&l.

Definition 1: We introduce a bilinear detector for the ith bit of the mth user as the ordered pair of real-valued vector sequences (gm,‘, hml") E LR x LR, the Cartesian product of the vector space LR with itself. The decision of this bilinear detector is defined to be

(7)

In dealing with the demodulation of the ith bit of the mth user, we drop the superscript (m, i) for notational clarity whenever there is no ambiguity. The computational considerations in the implementation of a bilinear detector are easily understood by writing the decision statistic in terms of the sampled outputs of a bank of K matched filters which are matched to the normalized signatures signals of each of the K users. In particular, we define Z&(j) as the output of a filter matched to the normalized &h-user signatures signal .that is sampled at the end of the corresponding jth-time interval so that

It should be noted here that these statistics are obtained without the knowledge of the energies and phases of the component signals. The decision of the bilinear detector defined in (7) for the ith bit of the mth user is in general a function of all the matched filter outputs and can be expressed as *

im(i) = sgnRe 5 2 skid j--M k=l

VARANASI: NONCOHERENT DETECTION IN ASYNCHRONOUS MULTIUSER CHANNELS 159

= w Re [ (g,4 (4 4 *] , (9)

where z E L contains the matched filter outputs arranged in the time order in which they are obtained so that z = [21(--M), . .:, .aK(-M), . . . , xl(M), . . 1, zK(M)]~. Further, (x,y) = ~~=~M&~~(l)yk(l) defines the inner product on the vector space L. The detector whose decision is given by (9) can be interpreted as operating on the outputs of two time- varying linear filters by comparing their phase difference since sgnRe[zr$] = sgnRe[rrr2ej(d1-d2)] = sgn cos(& - $2). Furthermore, the detector is noncausal in the sense that the entire received waveform is processed before a decision on any particular bit of any user is made.

Definition 2: The conventional detector for the ith bit of the mth user is defined by its two-shot decision given as

&(i) = sgnRe[x,(i).&(i - l)]. (10)

From (9), it can be seen that the conventional detector is a degenerate bilinear detector with the vectors g = I@ and h = p)i-l where urnlj denotes the (m, j)th unit vector in L. It is seen that such a detector is optimal for the mth user in a single-user system, i.e., when no other user is actively transmitting. However, in a multiuser channel, the conventional detector is not optimal.

Spread-spectrum multiple-access systems employing DPSK modulation and the conventional detector have been exten- sively analyzed in the literature. Several authors have dealt with this single user detection scheme for a variety of channel models and multiaccess techniques such as in [l]-[5], and [8]. The primary emphasis in those papers is the evaluation of the multiple-access interference and multipath rejection capabilities of the spread-spectrum signalling schemes when used in conjunction with the conventional detector.

In was shown in [13] for the synchronous CDMA channel that the conventional detector exhibits a dismal overall performance. Operating conditions where its performance is acceptable are of little practical or theoretical significance requiring low bandwidth efficiency and similar signal strengths. Even over such operating points, this detector is very sensitive, degrading significantly with relatively small changes in the Ievel of the multiple-access interference which may be caused by changes in the signal energy levels (the near-far problem) and/or by an increase in the bandwidth efficiency due to the addition of more users.

Definition 3: The asymptotic efficiency of user m whose bit-error rate is P,(a), is the limit of the ratio of the effective energy (that required by the mth user to achieve error-rate P,(g) in the same AWGN channel but without interfering users) to the actual energy of that user, as (T goes to zero, and is formally defined as

m=sup O<r<l;lim

1

Pm(ff> rl -

u-+o exp(-r]a,]2/202) <+m ,

1 (11)

Consequently, in the high signal-to-noise ratio (SNR) regions, the multiuser error probability P,(g) exhibits with decreasing noise variance, a decay with an exponential rate that is equal to that of a single user with energy v,Jum12. This asymptotic efficiency measure was adopted in [13] for the synchronous CDMA problem and is the noncoherent analogue of the asymptotic efficiency introduced in [16] for coherent multiuser detection.

Problem Formulation: Denote the asymptotic efficiency of the bilinear detector (g, h) for the demodulation of the ith bit of the mth user as qm,i(g, h, a) where a E L and denotes the NK-length vector of complex signal amplitudes arranged in the time order in which they appear. Since these signal amplitudes are unknown at the receiver, the problem here is to obtain a bilinear detector whose worst-case performance over all operating points determined by admissible values of the vector a, is highest among the class of bilinear detectors. In other words, we would like to obtain the minimax robust bilinear detector

(12)

where the set of admissible operating points A,(i) = {a E hIam = a,(i - 1) = const.}.

Definition 4: The worst-case multiuser asymptotic efficiency over all admissible operating points corresponding to the unknown time-varying energies and phases of the component signals given by

is defined as near-far resistance. The minimax problem statement can therefore also be

interpreted as one of finding an optimally near-far resistant bilinear detector. The significance of near-far resistance is that if the near-far resistance of some detector of the mth user is nonzero; the error probability decays exponentially with an increase in the mth user’s signal-to-noise ratio at a guaranteed exponential rate, irrespective of what the interfering signal energies and phases may be. Further, note that we consider the general case where these quantities are allowed to vary with time with the mild restriction that they do so slowly enough to be regarded as nearly constant over successive symbol intervals. Near-far resistance as defined above for noncoherent CDMA channels is an analogue of near-far resistance defined in [7] for coherent CDMA channels.

III. PERFORMANCE CHARACTERIZATION OF BILINEAR DETECTORS

In this section, we characterize the performance of bilinear detectors in terms of bit-error probability as well as asymptotic efficiency and near-far resistance. Such an analysis is necessary to address the minimax problem posed in the previous section. Consider the following proposition that contains both the error-rate formula and the asymptotic efficiency functional.

Prtiposition 1: The asymptotic efficiency of the bilinear detector (g, h) for the ith bit of the mth user is given as the


solution of the combinatorial minimization problem given by

qm,i(g, h, a) = min ma? &D,(i) (13)

where Dm(i) = {d E D; d,(i - 1) = fl} and the functions fr and fs are defined as

Proof: Consider the decision of the bilinear detector (g, h) give by (7). Using the alternative representation of the received signal,

r(t) = l?(t,Ad) + n(t),

where A = diag{a} is a diagonal matrix of dimension NK of the time-varying complex amplitudes of the signal components, we have

b,(i) = sgn Re ((fiCWhA4) +A,)

. ((~(0),&4d)) +h)*. (15)

The random variables X, and Xh are the noise components at the output of the cascade of the bank of matched filters, samplers and linear combiners g and h, respectively, i.e., the outputs of the two branches of the bilinear detector. They are defined by the inner products

and Xh = (o(t, h), n(t)).

Let D&(i) = {d E D; d,(i)d,(i - 1) = +l} and D;(i) = {d E D; d,(i)d,(i - 1) = -1) denote the events corresponding to the transmitted symbol of the mth user in the ith time interval being +l and -1, respectively. The bit;error probability of the bilinear detector for the ith bit of user m can be written as

= 2-lPr [6,(i) = -1,d E D;(i)]

+ 2-1%‘r[&,Ji) = fljd E D,(i)]

= z-l33 D$+)Pr be( (o(t1g)! @ t, 4) + 47) . ((@,h),o(t:Ad)) +A,,)* < 0]

+ 2-l EDi (;)Pr R [’ e((~(t,gL~Ct,A4) +A,)

. ((@,h),t$,Ad)) +A,>* > 01,

where Es denotes expectation over d E S(G D) and simply involves averaging over equiprobable interfering bit- combinations in the set 5’. Next, we note that the inner product (I?(t, z), o(t, 31)) defined on & induces an inner product’ on

‘The validity of the inner product on CZ is a consequence of the linear independence assumption, and implies the validity of the inner product on L, which in turn implies that R is positive definite.

the vector space L by observing that it has a representation in terms of the signature signal crosscorrelation matrix R as

(w, 21, w, 9,) = (T Y)R = z*c*Ry. (16)

The expression for error probability can now be written as

Pm,i(a) = 2-1EDA(i)Pr[Re((g, Ad), 4- A,)

+ ((Wd), + h)* < 01 + 2-lE D&(;)Pr[Re((g,Ad)R + ‘~7)

. ((h-M), + Ah)* < 01. (17)

The noise variables X, and Xh are zero-mean Gaussian random variables since they are the outputs of linear filters whose input is the received zero-mean, white Gaussian process n(t). The variances of these noise variables are given as

2-‘E[/Xgj2] = 2-1E[~(~(t,g),n(t))~2]

II II

2 =0 2 W,d

= ~2h7)R, (18)

and similarly,

2-1E[IXb/2] = p2/ln(t,h)112 = ‘~~(h,h),. (19)

The crosscorrelation between these random variables is given bY

2-1E[X,X;I] = g2( i?(t,g), o(t, h))

= a2(!7, NR. (20)

Conditioning on the differentially encoded information symbols and the complex amplitudes of the component signals, the ~ decision statistic in (15) has the form of a statistic which is a quadratic function of two nonzero mean, complex Gaussian random variables. Explicit formulas for the error probabilities in (17) are available (cf. [9, Appendix 4B]) in terms of the Marcum Q-function’ and the zeroth-order modified Bessel function of the first kind. Using that result in our problem, we have

Pm,&) =2-NK

.exp(- w)}

+2-NKd gI){Q( $; 9 -do( 9) E m2

.exp(- v)}, (21)

*The Marcum Q-function has the integral representation &(a, b) = sbm x exp(-(x2 + a2)/2)In(ax)dx.


where the functions fi and f2 are defined in the statement of the proposition and functions ~1 and ~2 are defined according to

(9, hhz ’ (9, &Ah> hhx 1 (24

Substituting the expression for error probability of (21) in the definition of asymptotic efficiency of (ll), we have the equation at the bottom of the page.

Using the critical result shown in Appendix A that establishes the equality

Q($, $) -do(~)

exp(-K!$!Z) < CC

we have

= maxi{ 0, e}, (23)

rlm ,i(g, ha) =

min max2{0,g}} da,(i)

= min max2 da,,(i)

where the first equality follows because in the low background noise region, the summation of error probabilities in P,,i(a) is dominated by the term corresponding to the transmission of the least-favorable bits of the interfering users, and the second equality from the observation that the functions fl and fs remain unchanged when d is replaced by -d. This completes the proof of the proposition. 0

The bit-error probability and the asymptotic efficiency functionals of an arbitrary bilinear detector (g, h) exhibit a dependence on the functions fi and f2 (and CQ, ~2 as well in the case of error probability) which in turn depend on (g, h) only through the normalized vectors

Therefore, without the loss of generality, we can confine attention to the normalized bilinear detectors belonging to the set

R x R = ((9, h) E LR x LRI(g& = 1, W4, = 1).

The asymptotic efficiency of a normalized bilinear detector is given by (13) of Proposition 1 with the functions f! and f2 admitting the simpler expressions

f1,2 = +wW, F VwW,l. (24)

As an application of Proposition 1, consider the asymptotic efficiency of the conventional detector that is a degenerate bilinear detector. In order to demodulate the ith bit of the mth user, the conventional detector corresponds to the two vectors g and h being equal to the unit vectors 2z”li and u?-‘, respectively. ‘Using the fact that (h, h)R = (Cpl, umli-1 )R =

yw~mm = 1 and also that (g,g)R = (u”,~, pL”‘i)R = = 1, the functions fl and f2 defined in (24) simplify

for themcmonventional detector to the form

fl”,2 = +P*i,Ad)R q= (um+l,Ad),l]. (25)

In order to evaluate these functions, we need the ith and the (i - 1)th vector of RAd. Using the structure of the crosscorrelation matrix R, the Ith vector of RAd can be written as

PAdI = e R(l- +WW j=-M

1+1 = c W - ~bW4.9.

j=l-1

Although in addressing the design problem of finding a good bilinear detector, we assume the constancy of the complex amplitudes only ‘over two successive symbol intervals, for the purpose of computing the asymptotic efficiency of the conventional detector, it is convenient to let A(i - 2) = A(i - 1) = A(i) = A(i + 1) = A and define the unnormal- ized crosscorrelation matrices H(j) = R(j)A for Ijj .< 1. Furthermore, since (t~“>~,Ad)~ is the (m,i)th element of RAd, the expressions in (25) can be simplified as

f;,2 = ; 1

2 H(j) [d(i - j) F d(i - j - l)] j--l 1

, (26) m

where {z(l)}m denotes the mth element of the K-length vector ~(1). The functions ft and 1.j depend on the differentially encoded symbols d only through d(i - 2), d(i 1 l), d(i), and d(i + 1) and f ur th er, since the matrices H( 1) and H( -1) are upper and lower triangular matrices with zero diagonal, they are independent of the m symbols {dl (i - 2), + . . , d,(i - 2)) and the K - m + 1 symbols {d,(i + l), ... ,dK(i + 1)) as


well. Consequently, in order to evaluate the asymptotic efficiency of the conventional detector, the discrete minimization of equation (13), viz.,

min max2 de&,(i) C

0, d,(i)w }, (27) am a

has to be carried out over the reduced set B,,(i), which consists only of the 3K - 1 differentially encoded symbols on which ff and fg depend, with the further restriction that d,(i - 1) = + . N 1 umerical computations of the asymptotic efficiency of the conventional detector will be presented in the penultimate section on numerical results.

Let us turn to the evaluation of near-far resistance of a bilinear detector. Near-far resistance is the worst-case asymptotic efficiency over admissible values of the energies and phases of all component signals. Consider the following corollary which specifies the near-far resistance of any normalized bilinear detector in terms of a solution to a continuous minimization problem.

Corollary 1: The near-far resistance of an arbitrary bilinear detector (g, h) E R x R for the ith bit of the mth user is given as

v,,i(g, h) = m&x2 O,wEi$~cij $(W,!J, h) >

) (28)

where Wm(i) = {‘WE LIwm(i) ~{+l,-l}, wm(i - 1) =+l} and

$(w,g,h) = + [lb + b4d - lb - Oh,ll~ (29)

Proof: Substituting (13) into the definition of near-far resistance, we have

= inf min a&4,, (4 de&,(i)

0 d,(i) ’ 2lam(i)l

[Ib+h,Ad),l - lb-W&l] ’ (30)

where the second equality is obtained by using (24). Next, we note that the objective function depends on A and d only through Ad, and combine the two minimizations problems over a E A,(i) and d E Dm(i) into one continuous minimization problem in w = [am(i)]-l Ad. With this definition, and since the signal amplitudes a,(i) and a, (i - 1) are equal, the admissible sets d,(i) of a and Dm(i) of d induce the admissible set Wm(i) of w that is defined in the statement of the corollary. Finally, making the substitution dm(i) = wm(i), we have

77,hh) = wEi;L(,) max2 (0, $(W,g, h)l

=ma,x2 ‘0, {

wEkL(i) #(WI 9, h) I

’

where the definition in (29) of ,$(w, x, y) is invoked to obtain the first equality. cl

IV. BILINEAR DECORRELATING DETECTORS

Definition 5: A bilinear decorrelating detector for the ith bit of the mth user is a bilinear detector (g, h) E fPi x fPi where Om>i is a subset of 0 given as

R m,i = {x E a]2 E span{s”+l,sm>i}},,

where s”‘j, E L is the (Ic,j)th column of the inverse of R, so that R&j = &.

The existence of d>j for each k and each j is guaranteed by the positive definiteness of R (see footnote 1). The vectors (g, h) can be written as

g = z-&+1 + ~osm,i (31)

h = y-lsm+l + yoP+, (32)

and the corresponding bilinear detector can be equivalently represented by the pair of two-dimensional vectors 2 = [~-r,xe]~ and y = [Y-~, yulT. Since the vectors g and h are elements of R, we have the corresponding constraints on the vectors x and y given by

(TM&! = (XAW = 1 = (h, h)R = (Y, y&v > (33)

where the 2 x 2 matrix Sm,i is defined as

Therefore, without loss of generality, we can denote an arbitrary bilinear decorrelating detector for the ith symbol of user m as (z,y) E A x A where A = {x E R2](x,x)sm,i = l}.

In the sequel, we will show that restricting attention to the class of bilinear decorrelating detectors entails no loss of generality as far as obtaining an optimally near-far resistant bilinear detector is concerned. Consider the next proposition.

Proposition 2: Any bilinear detector which is not a decorrelating detector is not near-far resistant, i.e., it has a near-far resistance which is identically equal to zero.

Proof: Consider the expression for near-far resistance from Corollary 2. We note that if g does not belong to span{p>i-l, p>i}, then there will be at least one element of ‘Rg, which is neither the (m, i - 1)th nor the (m, i)th element, that will be nonzero. Since any w E Y&(i) is restricted only in the (m, i - 1)th and the (m, i)th elements, there exists a m E Wm(i) such that (~,g)~ = 0 and therefore,

which in turn implies (cf. (28)) that the detector is not near-far resistant. Detectors for which h does not lie in span{ Qm+l, s+} can be similarly shown to have zero ’ near-far resistance. 0

As a consequence of Proposition 3, we have the following corollary.

Corollary 2: The conventional detector of the mth user has zero near-far resistance unless the (m,i)th column of R is equal to the (m, i)th unit vector u”>~ for all i = -M, . . + , M.


Proof: In general, ~(~1~) and ~(~,‘-l) do not belong to span{ 0-l) s”>i} and it follows from Proposition 2 that the conventional detector has zero near-far resistance. If rkm,i) E span{s”-li-‘, .?@} however, then Ru(~I~), the (m, i)th column of R, has to be a linear combination of r&m,i) and ~(~,~-l). Further, the coefficient of u((m,i-l) must be zero since &,( -1) = 0 so that the (m, i)th column of R is equal to the (m, i)th unit vector. Owing to the time invariance of the signature signals, this condition must hold for all i = -M, . . . , M. 0

We note that the condition for nonzero near-far resistance of Corollary 2 almost never holds since it requires that the signal Um(t - iT - TV) be orthogonal to each one of the 2(K - 1) signals Uk(t - jT - ok) for k = 1, . + . , m - 1 and j = i,i + 1 and Uk(t--jT-~k) for 5 = m + l,...,K and j = i - 1, i. The situation is hopeless when a nonzero near-far resistance has to be assured for any realization of the relative delays. Therefore, no matter how carefully the signature signals are designed, there will be admissible interfering signal, energies and phases which will cause the conventional detector to be severely near-far limited.

The second significance of Proposition 2 is that it reduces the problem of finding the minimax bilinear detector to that of finding the minimax decorrelating detector. Therefore, there is no loss of generality if we restrict ‘ourselves to the class of decorrelating detectors in finding the bilinear detector which is optimally near-far resistant. If the pair (go, ho) E R x R solves the minimax optimization problem

(9’) ho) E arg g,~~~na~~!(~).rlm.i(g,h,a),

then the optimally near-far resistant bilinear detector (go, ho) is the optimally near-far resistant bilinear decorrelating detector with the equivalent representation (x”,yo) E A x A and solves the minimax problem

(x0, y”) E arg max x,yt*x*,&i) hd? YT a>*

In the sequel, we see that the significance of the bilinear decorrelating detector stems not only from Proposition 2, but also from the result of the next proposition.

Proposition 3: The bit-error probability of an arbitrary bilinear decorrelating detector (2, y) E A x A is independent of the interfering signal energies and phases and so is. its asymptotic efficiency that is given by

min 1 max2{0,L[l(x+y,cl)l - 1(x-Y,cI)II}, max2{0,~[l(x-y,c2)I - I(~+Y,c~)~]} 1 ’

(35)

where the vectors cl = [l llT and c2 = [l - llT. Furthermore, the near-far resistance of a bilinear decorrelating detector is equal to its asymptotic efficiency.

Proof: The bit-error probability of an arbitrary bilinear detector was obtained in (21) of Proposition 1. It is entirely determined by the functions aI,2 and fr,2 of (22) and (14). If these functions are invariant to the interfering signal energies

and phases, so will the bit-error probability. Substituting the expressions of (31) and (32) for the bilinear decorrelating detector into (22) and (14) and using the constraints of (33), we have

and

fl,2 = y I(xFy,c)),

where we defined the 2-length vector c = [d,(i - l), d,(i)] and made use of the assumption a,(i) = a,(i - 1). The asymptotic efficiency of the bilinear decorrelating detector can now be obtained by substituting (36) into (13). We note that the functions fr and f2 depend on the differentially encoded symbols only through c. The discrete minimization over the set Dm(i) in the asymptotic efficiency expression of (13) is therefore replaced by a minimization over the set (I,(i) E { $1, -1) corresponding to the two values of c = cl = [l llT and c = cz = [l llT, respectively. Using these facts, we have (35).

Finally, the invariance of the asymptotic efficiency of a bilinear decorrelating detector to the interfering signal energies and phases together with the definition of near-far resistance as being the infimum of the asymptotic efficiency over those parameters establishes the equality of the two measures. 0

Proposition 3 gives a complete performance characterization of any bilinear decorrelating detector. The invariance of asymptotic efficiency to interfering signal strengths and phases is a desirable property, but we have to ensure that this asymptotic efficiency is the highest that can be achieved within the class of bilinear decorrelating detectors. This is the primary objective in the next section.

V. OPTIMUM NEAR-FAR RESISTANCE

In this section, we address the problem that we originally formulated in Section II of finding the bilinear detector that optimizes near-far resistance. Equivalently, such a detector optimizes near-far resistance within the class of bilinear detectors and both near-far resistance and asymptotic efficiency within the class of bilinear decorrelating detectors, as a consequence of Propositions 2 and 3. From Proposition 3, we know that the highest near-far resistance achievable by any bilinear detector is given by

-0 rl m,i =

sup min { i

max2 O,+-[l(~+~,s)l - I(~--~,c1)11 X,YEA max2 O,,[I(X-Y,CZ)I - I(x+Y,c~)II 11 ’

(37)

Next, we will obtain an explicit expression for ?jk,; by solving this optimization problem under the assumption that the vectors cr and cz are orthogonal and have equal norms so that ]]cl]] = ]]c2]] = ]]c]]. Consider the next theorem.


Theorem I: If cl and ca are orthogonal and are of the same length, the optimization problem in (37) has the following solution:

-0 rl = /1cl14 min m,a {(IlCl +c2ll;J> (IICI --c211q1}.

Furthermore, the pair (x0, yo) = (S(cr + cz), Y(CI - cz)),

with 6 and y determined by the constraint conditions ($0, xo)‘p,* = 1 and (yo,yo)s-,X = 1, solves the maximization problem in (37).

Proof: We begin the proof of this proposition with a view to simplifying the objective function. We first observe that

l(x+Y,C)I-I(“-Y,c)I>O~(x,c)(Y,c)L~, (38)

Therefore, the optimization problem of (37) is transformed to an optimization problem with a simpler objective function and additional constraints given by

--O .= rl m>a

sup min2 -t

L [I(XfY>Cl)I - lb-Y,Cl)ll,

2, YEA ~[I(~-zr:C2)l - I(XfY,Cz)ll 1 .

(TCl)(Y,Cl)to (X,C2)(Y,C2)50

(39)

Next, we invoke the two identities:

2-l(lt + u( - It - ~1) = min{ltl, IuI}, when sgn(tu) > 0

and

2-l(lt-ul - It+ul) = min{ltl,lul}, when sgn(tu) 5 0,

to obtain a further simplification of the objective function so that

--O .zz r! m>a sup min2{l(x,cdl, I(y,cdl, l(x,c2)l, I(~,ca)l).

2, YEA (~,Cl)(Y,Cl)>~ (2, CZ)(Y, c2)53

(40)

If cl and cz are aligned with the abscissa and the ordi- nate of a Cartesian coordinate system, then the constraints (x,cI)(Y,c~) 2 0 and ( x, ca) (y, cz) 5 0 together imply that if x lies in the first quadrant then y lies in the fourth quadrant and vice versa, and if x lies in the second quadrant then y lies in the third quadrant and vice versa. Note that the function f(x, y) = min{ I( X>Cl)l, I(Y>“l)l, Ihc2)l, l(Y,C2)1> is even and symmetric with respect to x and y, i.e., f(x, y) = f(-x, -y) and f(x, y) = f(y,x). Furthermore, if (x, y) satisfies the constraints of the problem then so do (-5, -y) and (y, x) and (-y, -2). Therefore, the inequality constraints on the vectors x and y can be made more restrictive by confining x to lie in the first quadrant and y to lie in the fourth

quadrant with respect to the (cl, ca) axes. Therefore, a compact expression for the highest achievable near-far resistance in (40) can be obtained as

-* .zz 77 m,z xcy;EY~~lin2 {I(x,cdI, l(y,cdl, lbc2)l, Iiy,cz)lI,

(41)

where we define the set of admissible values of the vectors x and y as

X = {x E A: (X,CI) > O,(x,c2) L 0}

and Y = {Y E A : (Y,CI) 2 0, (Y,CZ) IO).

The constraint equations (2,~)s = 1 and (y,~)~ = 1 (we suppress the superscript (m, i) for clarity) require that the vectors x and y lie on an ellipse since S is positive definite (this follows from S being a 2 x 2 submatrix of the positive definite matrix R-l (cf. (34)). The major axis of this ellipse has to either pass between the cl and cz vectors, a condition we denote as Al, or between the cl and the -ca vectors, a condition we denote as A2. Let us consider condition Al that is depicted in Fig. l(a). This constitutes our first assumption. We denote the line segment of the major axis passing between cl and cp as OM. The second assumption Bl (B2) is that in order to reach the vector cl from the line segment OM via the smaller angle, one would have to proceed in the counterclockwise (clockwise) direction. For the moment assume Bl is valid. Rotate the ellipse so that the line segment OM of the major axis coincides with the new abscissa as in Fig. l(b). Under the two assumptions Al -Bl stated, we would always have the scenario depicted in this figure, i.e., cl would lie in the first quadrant and ca in the fourth quadrant. Let cl make an angle 6’ with the abscissa as shown in Fig. l(b). The constraints in the optimization problem require that x must lie on the ellipse in the quadrant formed by cl and ca and y must lie on the same ellipse between the quadrant formed by cl and --~a. In Fig. l(b), consider the point on the ’ ellipse given by y, = y(cl - ca), which is the intersection of the ellipse and the bisector of cl and -ca (since cl and c2 have the same length), where y is the constant determined by requiring y. E y, or equivalently, the positive-valued constant determined by the condition (yo, yo)s = 1.

We first show that

I(Yo,c~I = I(Yo~c~)I > min{l(y,cdl, I(Y,cz)II ~YEY. (42)

From the Al-B1 assumptions,on the orientation of the ellipse and the relative position of cl with respect to the segment OM of the major axis of the ellipse, we have 0 5 0 5 7r/2. Consider the point y that is the intersection of the line with slope tan 4 and the ellipse for some 4 in the interval [8,8 + r/2]. E ac such y satisfies the constraints imposed h by the problem. The proof consists of showing that ((y, cl) I is monotonically decreasing and I (y, cz) I is monotonically increasing as 4 increases from 8 to 6’ f ~12. The point where these two projections are equal, ‘i.e., for y = y. will therefore satisfy the claim in (42).


(4 (b) Fig. 1. Illustration for the proof of Proposition 5 under the Al-B1 assumption.

The squared length of the vector y is

lIYl12 = a2b2 (If tan2 4)

b2+a2tan2$ ’

where a and b denote the lengths of the major and minor axes of the ellipse, respectively. Therefore, we have

I( = lhll 2 a2b2 (1 + tan2 4)

b2+a2tan2$ cos2 ((Lb - 0)

and

lb1412 7 llcall

It is readily verified that (see equation at bottom of page) and

w $ I(Y, 41” = -1, v I9 5 $5 r/2.

Consider the next case where 1r/2 5 4 < 8 + n/2, and denote c) = 7r/2 + cp. We have

b2 ___ a2 - b2

+ cos2 p 1 - sin 29 sin2 (0 - ‘p) 1

= -1, VJo<cpl~, where the two equalities use the fact that a2 > b2 and for the second equation, it is sufficient that sin 2(0 - cp) cos2 cp - sin 29 sin2(6 - ‘p) > 0, which is indeed the case, since for the range of cp and 0 under consideration, sin (6 - cp) 2. 0 and cos ‘p > 0 and cos 8 2 0. Therefore, we have established that I (y, cr >1” is a monotonically decreasing function of $ as 4 increases from 6’ to 0+~/2 for any fixed 8 with 0 5 0 < 7r/2.

By similar argument it can be shown that I(y, ca)p is a monotonically increasing function of 4 as I$ increases from 19 to 6’ + x/2 for any fixed 0 such that 0 5 0 5 n/2.

At the point y = yo, which lies on the bisector of cl and ~2, its projections on both cl and ca are equal. Furthermore, as a result of the previous argument, these projections are greater than or equal to the smaller of the two projections from any other y satisfying the constraints in (42), thereby establishing the claim of (42).

We now bound from above the highest achievable near-far resistance, under the Al-B1 assumptions, as follows:

I(Y~,cI)I~ 2 min2{l(y,cl)l, I(Y,cz)~> VYEY

2 min2{l(x,cdl, l(x,c2)l, I(y,cdI, I(y,cz)l> VXEX, VyEY

L sup XEX,YEY

min2 {l(x,cl’l;~~,“;i~ I(x,c2)l, ,c2

oje<x/z -0 =vmi* (43)

Finally, if it can be established that the function f(x, y) can actually achieve the value I (yo, cl) I for some (x0, yo) E X x Y, the statement of the proposition would hold under two assumptions stated. We will show that this is indeed the case for the choice of x = x0 = S(c1 + ca), where 6 is the positive- valued constant obtained by letting (XO,XO)~ = 1. Suppose that the slope of the vector x0 is m, then it follows that the slope of y. is -l/m. Since x0 is a bisector of cl and ~2, both projections 1 (x0, cl) I and I (x0, ca) I are equal and are not smaller than I(~~,s)l(= I(yo,c2)l) provided l lx011~ L 11~~11~. It can be readily verified that

ll~ol12 = a2b2(1 + m2)

b2 + a2m2

and that

llYol12 = a2b2(1 + m2)

a2 + b2m2 ’

$ l(Y,412 = - llc~l12a2b2[sin2(q5 - O)[b2 cos2$ + a2sin2$] + ( a2 - b2) sin 24 cos2(# - O)]

(b2 cos2 4 + a2 sin2 $)2


I 4-=2

(4 (b) Fig. 2. Illustration for the proof of Proposition 5 under the A2-C2 assumption

and therefore that ]]~u]]~ > ]]y0]]2, provided that a2(1- m2) > b2(1 - m2), which is the case since a2 2 b2, and since ]m] 5 1 (because 0 5 0 < 7r/2). Therefore, we have ](ya,cr)] = f(zu,~,,) and (43) holds with equality. This establishes the proposition under the Al -Bl assumptions.

Consider the second case where assumption Al-B2 hold. In this case the scenario in Fig. l(b) has to be modified by interchanging the vectors cl and c2 so that the new vectors cl and c2 lie in the fourth and first quadrant, respectively. This case can be treated in a manner identical to that of the Al-B1 case with the only exception of measuring angles in the clockwise sense. Therefore, 0 denotes the angle between the abscissa and cl and 4 spans the interval (in the clockwise direction) [0,B + 7r/2] to denote each y E y as before. The proof thereafter, and the result remain unchanged.

Next, let us consider the case with assumption A2 where the major axis of the ellipse lies between cl and -cz. We denote the line segment of the major axis passing between cl and -cz as OM. The second assumption Cl (C2) is that in order the reach the vector cl from the line segment OM via the smaller angle, one would have to proceed in the counterclockwise (clockwise) direction. Let us consider the case where assumptions A2 and C2 hold. This is depicted in Fig. 2(a). Rotate the ellipse so that the major axis between cl and -cs coincides with the new abscissa as depicted in Fig. 2(b).

Here again, the angles are measured in the clockwise direction with 0 as depicted in the figure and the range of values of $ (measured in the clockwise direction) over the interval [0,8 + r/2] spanning the set of admissible values of z E X. From the obvious symmetry of the problem, we have

SUP XEX, YEY

ss.t.o<s<7r/z (measured clockwise)

.min2{Ihcd, I(~,cdl, lhc2)l, I(Y,cz)I = lhcd12, (44)

and that the maximum is again achieved at (~0, yu). The last case with assumptions A2 and Cl yields a result

identical to the one obtained in (44) by considering the measurement of angles 19 and 4 in the counterclockwise direction.

Combining (43) and (44), and using the established result that equality holds in (43), and that (~0, ye) = (S(cr + cz), y(cl - c2)) solves the maximization problem in both cases, we obtain

--o rlm,i = SUP

XEX,YEY

-nin2{I(vl)l> I(Y,Cl)l> l(x,c2)l, l(Y,C2)1>

= min( Ih, c1)12, I(YO> c1,12}

= 11414 min{ (lh + c2ll~)-l, (llc1 - c211~)-1},

where the last equality is obtained by using the relations S = (/ICI +c211s)-‘,7 = (11~1 -czII~)-~, and (~1 +CZ,Q) = ]]c]]~. The proof of the theorem is complete. 0

It is worth mentioning here that the pair (x0, yu) = (S(cr + cz), y(cr -~a)), is not the only one that solves the optimization problem in (37). Owing to the property of even symmetry of the objective function f(x, y), the pair (ye, xu), (-20, -ya) and (-~a, --5e) also solve the same problem. Furthermore, we leave it to the reader to show that no other solutions exist because of the strict monotonicity of the functions ](y, cr) I2 and ](y, c2)12 under the Al-B1 (B2) assumptions and that of I(x,cd12 and lh412 under the A2-Cl (C2) assumptions.

Consider the particular case which is relevant to the problem we have, i.e., cl = [l, llT and cs = [l, -llT. The following proposition can be deduced from Theorem 1.

Proposition 4: The highest near-far resistance achievable by a bilinear detector is given by

and the bilinear decorrelating .detector

hY0) = ( 11,,‘1-‘11 [ JT7 (0) &JT] (46)

achieves optimum near-far resistance for the ith bit of the mth user. Furthermore, the asymptotic efficiency of this detector is equal to its near-far resistance and is highest among the bilinear decorrelating detectors.

Following the statement after the proof of Theorem 6, the bilinear decorrelating detectors (yo, x0), (-x0, -yo) and ( -yo, -x0) have the same optimal performance as well.


We know from Proposition 2 that all bilinear decorrelating detectors have asymptotic efficiencies (and bit-error rates) that are invariant to the interfering signal energies and phases. However, this property alone is not sufficient since some of these detectors could have zero asymptotic efficiency. In fact, we saw in the course of the proof of Theorem 1 that among such detectors, only a subset of them satisfy the necessary and sufficient conditions for nonzero asymptotic efficiency which were added as constraints in (39). The significance of these detectors is that they alleviate the near-far problem in that, in addition to their performance being invariant to the interfering signal energies and phases, they exhibit an exponential decay in error rates with an increase in the signal-to-noise raiio of the desired user. Within this subset of near-far resistant detectors, Proposition 4 characterizes the detector which achieves the highest asymptotic efficiency. This detector is also the optimally near-far resistant detector among all bilinear detectors. The next proposition gives a finer performance analysis of this detector in terms of bit-error probability.

Proposition 5: The bit-error rate of the optimally near-far resistant bilinear decorrelating detector for the mth user’s ith bit is independent of the interfering signal energies and phases and is given as

p&) =.Q ($,2) - ;Io(z) exp(-&s),

where the functions fl and f2 are define‘d as

7l 2 = la&>l 1 1 2 p+llln + JlP~iIIR .

Proof: For any arbitrary decorrelating detector (!?,h) E fpi x flwi, the functions fl and f:! are given by (24).

(47)

Substituting the optimal decorrelating detector (g, h) =

into that equation, we have

1 f1,2 = - 2 (1

a,(i - l)d,(i - 1) ~ am(i)d,(i)

Il~YR Ilp>i II?2 1) . Substituting these functions in the expressions for error probability in (21), and noting that the functions fl and f2 for the optimal decorrelating detector depend on d only through the symbols d,(i - 1) and d,(i), we have

P&$(a)= +[Q(p) -do($

’ exp(-w)] ldm(i)=dm(i_,,.+l

+;[Q($,k) -do(~)

The functions fl and f2 given in (48) when evaluated in the two cases corresponding to the values of the symbols d,(i - 1) and d,(i) are obtained as

fl / d,(i)=dm(i-l)=+l

= f2 1 = 71 d,(i)=-d,(i-1)=-l

and

fl =f2 1 =72, d,(i)=-d,(i-1)=-l d,(i)=-d,(i-l)=+l

where ~1 and T2 are defined in the statement of this proposition. Finally, making use of these relations and the fact that a1 + al = 1, we deduce the result of this proposition from (49 0

VI. IMPLEMENTATION

In this section, we discuss the issues relating to the implementation of the optimally near-far resistant bilinear decorrelating detector. We will consider the finite sequence length case in Section VI-A and then pass to the limit as the sequence length N + cc in Section VI-B. The limiting case is shown to result in a noncoherent multiuser detector that is easily implementable in the sense that in addition to the bank of matched filters, it consists of a single K-input K-output linear time-invariant decorrelating filter followed by a bank of identical differential decision logic circuits each being identical to that used in a single-user DPSK demodulator.

A. The Finite N Case

Even though the optimal decorrelating detector is bilinear, the specific relation between the two linear transformations that describe it enables its implementation by a single time- varying K-input K-output decorrelating filter in conjunction with K delay operations. As a starting point, consider the decision statistic of an arbitrary bilinear detector given in (9), which we repeat here for convenience,

im(i) = sgnRe[(g,z)(h,z)*].

A more detailed look at the decision statistic is afforded by a closer examination of the matched filter output statistics. Substituting the expression for the received signal in the expression for the matched filter output of (8), we have

(j+l)T+Tk

Zk(j) = 2-l s

U(t, d)&(t - jT - 71~) dt + ~/c(j)

and (j+l)T+Tk

-yrc(j) = 2-l s

n(t)ak(t - iT - TV) dt

jT+Tk

is the noise component of the &c(j) statistic. Denoting the outputs of the K matched filters at time 1 by the vector z(Z) = [Zl (l), . . . , zK(Z)]~ for each 1 E {-M, . . . , M}, and substituting the expression U(t, d) from (2) in the previous


expression, and invoking the definition of the partial crosscorrelation between signature signals from (4) and (5), it is readily verified that

z(Z) = R(-l)A(1+ l)d(Z + 1) + R(O)A(Z)d(Z)

+ R(l)& - l)d(l - 1) + r(l), (50)

where we adopt the convention that d(-M - 1) = 0 and d(A4 + 1) = 0. The noise sequence y(Z) is the matched filter output noise vector [n(Z), . . . , ~K(Z)]~ for the Zth time duration. Equation (50) can be used to write a convenient expression for the matched filter output vector z given by

z=RAd+y, (51)

where y = [r’(-M), . . . , rT(M)IT. Substituting the expression for z into the decision of the bilinear detector yields

&(i) = sgnRe[gT(RAd + y)(RAd+ r)*h]. (52) G(z) = [R(-1)~ + R(0) + R(l)&] -’ (56)

Consider this decision statistic for the optimal bilinear decorrelating detector g = ,,a~~~~;;, and h =

gm,i m. UPon

substituting these equations in 02) and noting that the nor- malizing constants can be factored out without affecting the decision, we have

&(i) = sgnRe(+(i - l)&(i - 1) + vm(i - 1))

(Gn(+&n(4 + &n(q)*, (53).

where we made use of the fact that srn,i is the (m, j)th column of the inverse of R. Alternatively, all the decision statistics represented by equation (53) can be obtained by operating on the matched filter output vector z by the inverse of the matrix R to obtain the intermediate statistic

2 = R-~,z = R-‘@Ad + y), (54)

followed by the differential decision

!.&(i) = sgnRe[&(i - l)&(i)]. (55)

The operation of (54) can be thought of as being performed by a single K-input, K-output linear, time-varying decorrelating filter with input sequence {z(n)} and output sequence 1 >

a(n) which, in the absence of noise coincides with the sequence {A(n)d(n)}. I n o th er words, the decorrelating filter eliminates the multiple-access interference from the matched filter output statistics. This linear filter is similar to the decorrelating filter obtained for the coherent multiuser detection problem in [6], but it should be noted that the filter obtained in that paper depends on the phases of the received signals which are locked on prior to the demodulation process and assumed time- invariant thereafter. In the noncoherent problem considered in this vlork, no such information is needed since the decorrelating filter can be implemented without the knowledge of the signal phases.

B. The Limiting Case N + cc

The decorrelating operation involves inverting the NK x iW matrix Y?, that is acceptable only for very small data sequence lengths but can otherwise be computationally too intensive to be of practical use. In this section, we will consider the limiting case where the data sequence length goes to infinity. It results in replacing the linear time-varying decorrelating filter of the previous subsection by a linear time-invariant decorrelating filter. The following proposition provides a succinct descrip- tion of the limiting optimal decorrelating multiuser detector and essentially follows from the discussion in Section VI-A of the finite sequence length implementation of the optimal decorrelating detector.

Proposition 6: In the limit as the transmitted sequence length increases (N + co), the optimally near-far resistant decorrelating detector for the simultaneous demodulation of all K users approaches a K-input, K-output linear time-invariant filter with transfer function matrix

followed by a differential decision circuit, the decision on the symbols of the K users in the ith time interval of which is

i(i) = sgnRe p(i - 1) 8 i*(i)], (57)

where {2((i)} is the output vector of the K-input, K-output filter with transfer function G(x). (The symbol @ denotes component-wise multiplication of two vectors, and Sgn and Re with vector-valued arguments denote component-wise Sgn and Re operations, respectively).

Proof: The argument used in [6, Proposition 21 is applicable in asserting the first part of the stated proposition. Evaluating the z-transforms on both sides of (50) and letting N -+ 00, we have

Z(x) = S(x)b(z) + N(z), (58)

where

S(z) = R(-1)z + R(0) + R(l)+

and Z(z),Djz) and N( z are the vector-valued z-transforms ) of the matched-filter output sequence, the sequence {i(Z) = A(Z)d(Z)} and the noise sequence {y(Z)} at the output of the matched filters. Next, we find the limiting form of the optimally near-far resistant decorrelating detector. In the finite sequence length case, we know that this detector consists of a time-varying decorrelating filter as front-end followed by the differential decision logic. The decorrelating filter eliminates the multiple-access interference so that its output in the absence of noise is the sequence {k(Z) = A(Z)d(Z)}. Therefore, as N + W, this filter is replaced by its time- invariant limiting form whose transfer function matrix G(z) is such that when the input to this filter is the vector sequence z(Z), the output in the absence of noise is the sequence {ii(Z) = A(Z)d(Z)}. F rom (58) therefore, the limiting filter G(z) has the transfer function matrix given as

G(z) = [S(z)]-1 = a.

VARANASI:NONCOHEREN?:DETECTIONINASYNCHRONOUSMULTIUSERCHANNELS 169

C n (1 )I i DPSK decorreiating detector :--_--_,--_---------------

Fig. 3. Equivalent communication system.

The rest of the proposition is established by noting that in the finite sequence length case, the optimal decorrelator makes a decision on the data symbol b,(i) by comparing the phase difference between the consecutive outputs of the decorrelator d,(i - 1) and &(i) as in (55). This implies that in the limiting case, the optimal decision is obtained by comparing the phase of the corresponding consecutive outputs of the limiting decorrelating filter. Writing these decisions in vector form, we have the result of this proposition. 0

The matrix S(Z) can be viewed as the transfer function matrix of an equivalent multiuser communication system between the differential encoder and transmitter and the DPSK demodulator as is depicted in Fig. 3. It is not surprising that this system is similar to that of its coherent counterpart in [6]. A major difference is the absence of the differential encoder and decoder operations in the coherent system. Moreover, the phases of the component signals in the coherent case are assumed to be known and time-invariant and part of that transfer function matrix. In contrast, in the equivalent noncoherent system, the signal energies and phases are part of the transmitter with S(Z) being independent of the unknown, time-varying energies and phase of the component signals.

The result of the last proposition gives a particularly simple form for the limiting noncoherent decorrelating detector as shown in Fig. 4. The decorrelating filter admits the same interpretation as its counterpart in the coherent CDMA channel [6] in that, it can be viewed as a cascade of an FIR filter with transfer function matrix adj S(Z), followed by a bank of K identical filters with transfer function [det S(z)]-r. The filter with transfer function matrix adj S(Z) eliminates the multiple-access interference from the matched filter outputs but introduces intersymbol interference of maximum length 2K - 1 including at most the past K - 1 and future K - 1 differentially encoded symbols of the same user that were previously noninterfering. To see this, one need only observe that in the absence of noise, the z-transform of the output vector of the filter adj S(Z) is of the form det So and that det S(Z) is a polynomial in z with positive and negative powers of degree at most K - 1 each. The bank of K filters, each with transfer function l/det S(Z) represents a bank of K identical IIR linear equalizers corresponding to each of the K users. They eliminate the IS1 introduced by the multiple-access interference rejection filter adj S(Z).

For a detailed discussion about the issues of stability, causality and implementation of the decorrelating filter G(z), the reader is referred to [6]. However, for the sake of com- pleteness, we will enumerate the main points found in that paper.

1) The stable version of the decorrelating filter is noncausal and exists, if and only if the,signal crosscorrelations are

Fig. 4. Optimal limiting noncoherent decorrelating detector.

such that

2)

3)

detS(ej“‘) = det [R(-l)ej“’ + R(0) + R(l)e-j”] # 0, v’w E [0,27T], (59)

a condition that is equivalent to the LIA as N -+ 00. An exact implementation of the decorrelating filter ne- cessitates the processing of the entire received waveform in order to make optimal decisions on any symbol. In practice however, the more recent symbols will count less heavily. Therefore, by truncating the noncausal part of the decorrelating filter after a sufficiently long delay, the near-far resistance of the truncated filter can be maintained to within an arbitrarily small deviation from that of optimal performance. In particular, for a specified suboptimal performance, the delay depends on the rate of convergence of the IIR part of the impulse response. In the two user case, the impulse response of the IIR part of the decorrelating filter is of the form ,!$“I and decays faster with lower crosscorrelations and consequently, a given suboptimal performance can be achieved with smaller delays.

In

VII. PERFORMANCE OF THE OPTIMALLY

NEAR-FAR RESISTANT DETECTOR

the sequel, it is shown that the bit-error rate of the optimal limiting decorrelating detector behaves identically to that of the optimal single-user detector in a single-user channel with an effective energy that is equal to that of the desired user in the multiuser channel scaled down by a factor that is equal to the desired user’s multiuser asymptotic efficiency. The implication of this result is that the bit-error rate performance (and hence asymptotic efficiency and near-far resistance) of the optimal decorrelating detector is invariant to the arbitrarily time-varying interfering signal energies and phases.

Let us denote the transfer function matrix of the decorrelating filter G(x) as

G(z) = [S(z)]-1 = g D(n).P, (60) j=-cc

and consider the following proposition which is a limiting analogue of Proposition 5.

Proposition 7: The bit-error rate of the optimal limiting decorrelating detector for the mth user’s ith bit is given as


where {Dmm(j)} is the mth d iagonal element of the inverse z-transform of the decorrelating filter transfer function matrix G(x).

Proof: Consider the decisions made by the limiting optimal decorrelating detector that is given in the statement of Proposition 6 in terms of {d(j)}, the output vector sequence.of the decorrelating filter when it is driven by the matched filter output vector sequence {z(j)}. This output sequence can be written as

&I = 4Md + 4j), (62)

where {v(j)} represents the output noise process due to the input noise process {r(j)}. The input noise process is a zero-mean complex Gaussian vector sequence associated with {z(j)} and can be shown to have an autocorrelation function

2-L?+&*(j)] = a2R(i - j),

implying that its power spectral density is a?S(z). Therefore, the power spectral density of the output noise process {v(j)} of the filter G(z) is g2G(z), whence it follows that the autocorrelation matrix of the output noise process is specified by the equation

2-1E[v(i)v*(j)] = o?D(i - j). (63)

The decision made by the limiting noncoherent decorrelator for the ith bit of user m is given in (57): and can be written by using (62) as

im(i) = sgnRe[(a,(i - l)d,(i - 1) + vm(i - 1))

* (%n(wn(q + &n(i)>*] = sgnRe[(a,(i - 1) + vm(i - 1)) ’

. (%n(~YJm(~) + bn(~>>*], (64)

where we made use of the relation d,(i)d,(i - 1) = b,(i) and the assumption that a,(i - 1) = am(i). Using the expression for the autocorrelation matrix of the output noise process of (63), it is seen that

2-1E[lvm(i - l)[“] = 2-1+&)/s] = 0%,,(O)

and

2-!lqv,(i - 1)2&(i)] = a2Dmm(-1) = 02&&,(-l),

where we made use of the property of the matrices D(j) wherein D(j) = D’(-j).

The decision statistic in (64) is a quadratic function of the two complex-valued Gaussian random variables with means am(i) and a,(i)&(i) and equal variances a2Dmm(0) and cross-covariance a2 D mm(l) The problem of determining the error probability can be regarded’ as a special case of the problem encountered in the proof of Proposition 1. In particular, if we replace the corresponding characteristics of the Gaussian variates in that case with their limiting decorrelator

counterparts in this proposition, it can be shown that

where

and

Substituting these expressions in (65), we have

where the last equation follows from the Marcum Q-function wherein Q(O,z) = exp( of the Bessel function wherein Ia(O) = 1.

property of the -x2/2) and that

q

As a consequence of this proposition, we have the following corollary.

Corollary 3: The near-far resistance (and asymptotic efficiency) of the mth user optimally near-far resistant limiting decorrelating detector is strictly positive and is given as

--0 - 1 17

nz- D,,(o) (66)

[R(-l)ejw +R(O) + R(l)e-j”]-l dw]-l. mm

(67)

Proof: The first equation is obtained by substituting the expression for error probability derived in the previous proposition into the definition of asymptotic efficiency. The second equality is shown in [6, Proposition 51 where it is obtained by integrating the mth diagonal element of both sides of (60), evaluated on the unit circle, with respect to w over the interval [-X, rr]. The strict positivity of asymptotic efficiency follows as a direct consequence of [6, Proposition 61. q

Remarkably, the expression for the asymptotic efficiency of the noncoherent limiting decorrelator has the exact same


form of the asymptotic efficiency of the limiting coherent decorrelator -found in [6]. However, it is important to note that the number l/Dmm(0) has a different meaning in the coherent and the noncoherent problems because of the different interpretations of the signature signals in the two cases. In the coherent formulation, the phases are implicitly part of the signature signals, whereas in the noncoherent formulation, the signature signals are independent of the phases.

The asymptotic efficiency of the optimal limiting decorrelating detector for the two-user case can be shown as in [6] or by evaluating the definite integral in (67) to be

777 = r/g = J (1 - d2 - dd2 - 4d2& , (68)

where PIZ = [R(O)],, and PZI = [R(l)],,. When Ip121 % lpzll = 1, the asymptotic efficiency is equal to 0. In fact, this condition violates the condition for stability of the decorrelating filter and hence the linear independence assumption as well.

Throughout this paper, we have restricted attention to the class of bilinear detectors. A natural question is whether there exist other forms of detectors that outperform the optimal decorrelating detector. If the performance measure is near-far resistance, we prove that there are no such detectors, thereby establishing that the choice of bilinear detectors was not a restrictive one.

Proposit ion 8: The highest near-far resistance achievable by any detector, bilinear or otherwise, for the demodulation of the ith bit of the mth user, is bounded from above by the reciprocal of the (m, i)th diagonal element of the inverse of the crosscorrelation matrix ‘FL, i.e., ,

which in the limiting case can be expressed as

The significance of Proposition 8, is that the optimal decorrelating detector in the infinite sequence length case, whose near-far resistance for the ith bit of the mth user is’ given in Theorem 1, achieves at least the minimum of the upper bounds on achievable near-far resistances for the ith bit and the (i.-- l)th bit, among all detectors, bilinear, or otherwise. In the limiting case as N + co, it is seen from Corollary 3 that the optimal limiting decorrelating detector has a near-far resistance which is equal to the upper bound on achievable near-far resistance. Hence, the limiting case result of this proposition together with the result of Corollary 3 provide a proof of the optimality in near-far resistance of the limiting optimal bilinear decorrelating detector among all detectors, bilinear or otherwise.

Proof: The idea behind the proof of this proposition is .identical to the proof of the corresponding result for the synchronous DPSK-CDMA channel considered in [13]. We find the upper bound on the highest achievable near-far resistance by analyzing the performance of an optimum receiver, that ob- serves the received signal and has additional side information

as well. In particular, suppose that the receiver has a perfect knowledge of the signal energies and phases. The minimum error probability that can be achieved is equal to the error probability of the coherent minimum error probability detector. Therefore, the asymptotic efficiency of this detector constitutes an upper bound on the highest achievable asymptotic efficiency in the noncoherent CDMA channel. This quantity was shown in [15] to be equal to the normalized Euclidean distance between signals corresponding to the two closest hypotheses that differ in the ith bit of the mth user.3 Therefore,

where Em(i) is the set of error sequences E = {e(i) E {-LO, 1> K, i = -M, . , M, em(i) = l} that affect the ith bit of the mth user and W = u;l(i)A so that wm(i) = 1. Substituting the previous expression for asymptotic efficiency in the definition of near-far resistance, we have

17+ , ,i<min 1, C

inf min FW*RWe Wmv,(i) E%(i) I

= min {

1, inf WEW,(i)

w*lzw ) I

where Wm(i) = {w E L, lwrn(i) = l} and where the last equality follows by noting that the dependence of the objective function on w and E is only through WE, the admissible values of which E W ,(i). The minimum norm optimization problem above has the same form as that obtained in [6] except that here we have a complex Hilbert space. It is easily established that this problem admits a closed-form solution and is given as the reciprocal of the (m, i)th diagonal element of the inverse of the matrix R and the first part of the proposition is proved. In the limit as N -+ 00, the (m, i)th diagonal element of the inverse of R approaches emm(0) whence the result of the second part of the proposition follows. 0

VIII. NUMERICAL RESULTS

In this section, we undertake a comparative study of the asymptotic efficiency of the conventional detector and the limiting optimal decorrelating detectors via numerical computations. The asymptotic efficiency of the conventional detector was obtained by particularizing the result of Proposition 1 and is expressed in (25) and (27). The asymptotic efficiency of the optimal decorrelating detector on the other hand, involves the numerical computation of the definite integral of Corollary 3 and admits a closed-from expression for the two-user case given in (68).

The choice of signature signals is an important problem in CDMA communication. In order to obtain high asymptotic efficiencies, it is necessary that the signature signal crosscorrelation be kept as low as the constraints imposed by the particular application demand. One of the primary constraints is bandwidth. Given a fixed rate of data transmission, and a

3Although the result in [15] was establ ished for the asymptotic efficiency measure for coherent detection, it can be shown as in [13, Proposit ion 51 that it remains valid for the asymptotic efficiency for noncoherent detection considered in this paper as well.


Optimal Decorrelator

-5 -3 -1 1 3 5 E2/E I (in dB)

Fig. 5. Asymptotic eficiency comparison of the conventional and the optimal decorrelating detectors as a function of relative interfering signal strength.

fixed bandwidth, it is desirable to design signature signals that have good crosscorrelation properties for any arbitrary relative delays amongst them. Direct-sequence spread-spectrum signals are examples of such signals. However, it must be noted that these signals were designed to keep the multiple-access interference levels low in the conventional detector decision statistics. There is no reason why these signals would optimize the performance of multiuser detectors such as the optimal decorrelating detector found in this paper.

We suggest FDM-like signature signals here based on the following hueristics: given the overall bandwidth constraint of a K-user CDMA channel, and a fixed data transmission rate for each of the K users (= l/T bits/s), we observe that by choosing the signature signals to occupy frequency bands that are as removed from each other as the constraints would allow, it is possible to achieve low correlations between them; furthermore, the partial crosscorrelations between the delayed version of these signals would be low for all relative delays as well, since the magnitude of the frequency spectra are unaffected by time shifts.

Consider the set of sinusoidal signals with distinct frequencies for each user. The ith signal may be written as

Q(t) = Ai rect - cos (;) (F); (69)

where rect(t) denotes a pulse of unit amplitude over the interval [0, T] and the constant Ai is chosen by requiring that U,(t) be a unit-energy signal. The crosscorrelations between the signals, can be made as low as the bandwidth constraint allows, within this class of signature signals, by proper choice of the parameters { oi, 02~ . . . , oK}.

Let us define the width of the contiguous set of frequencies from 0 to the highest frequency that lies within the main lobe of the .frequency spectrum of at least one of the K signature signals as being the null-to-null bandwidth of the K signature signals, and the spectrum spread-factor N as the ratio of the null-to-null bandwidth occupied by the K signature signals to the null-to-null bandwidth occupied by a single rectangular pulse of duration T. Finally, we define bandwidth efficiency as the ratio K/N, of the number of users to the spectrum spread

k 0.8 0) 2 Conventional Detec (E2/El = -5 dB)

5- 0.6 5 .- .u 5 0.4 .o Conventional Detec (E2/El = 0 dB) cl ‘, E 0.2

6 Q

0.0 0.00 0 33 0.67 1 .oo

Relative Delay

Fig. 6. Asymptotic efficiency comparison of the conventional and the optimal decorrelating detectors as a function of relative delay of interfering signal.

factor. As an example, consider the bandwidth efficiency of the signature signals defined in (69). If the parameters are arranged in increasing order, then the spectrum spread factor is CXK + 1 and the bandwidth efficiency is K/(CXK + 1).

We first consider the two-user case where the signature signals employed by the two users are of the direct-sequence spread-spectrum type and are shown enclosed in Figs. 5 and 6. The spectrum spread-factor for this example is three and the bandwidth efficiency is two-thirds. These signature signals have been considered previously in [15] and [12] in the context of coherent multiuser communication. Fig. 5 depicts the asymptotic efficiency of the conventional and the optimal decorrelating detectors for this signal set as a function of the relative energy of the interfering user with respect to the desired user in dB, when the relative delay of the second user is fixed at r = T/3. From this figure, we observe the strong dependence of the conventional detector performance on the interfering signal strength, deteriorating rapidly as the latter parameter increases. When the relative signal strength reaches 5 dB, the conventional detector becomes multiple- access limited. The multiple-access limitation of the conventional detector is to be expected in any K-user system in general, for it was established in Proposition 2 that its near-far resistance is identically equal to zero. On the other hand, the asymptotic efficiency of the optimal decorrelating detector is invariant to the interfering signal strength (and phase as well). This invariance property of the optimal ‘decorrelating detector holds in any general K-user system as was shown in Corollary 3.

Fig. 6 depicts the asymptotic efficiency of the two detectors as a function of the relative delay of the second user for three different values (-5 dB, 0 dB, 5 dB) of the relative signal strengths of the second user. The rapid degradation of the conventional detector with increasing signal strength is clearly evident from this figure. In fact the conventional detector becomes multiple-access limited for every possible relative delay of the second user when the relative interfering signal strength is 5 dB. The asymptotic efficiency of the: optimal decor-relator on the other hand, remains invariant to


I 0 Noncoherent Decorrelatmg Detector

I

1.0

-5 -3 -1 I 3 5 E2/El (In dB)

0.0 00 0.2 04 0.6 0.8 I .o

Relative Delay

Fig. 7. Asymptotic efficiency comparison of the. conventional and the optimal decorrelating detectors as a fknction of relative interfering signal strength.

Fig. 8. Asymptotic efficiency comparison of the conventional and the optimal decorrelating detectors as a function of relative delay of interfering signal.

this parameter while it rises and falls between 0.89 and 0.94 with a variation in the relative delay of the interfering user, remaining relatively high throughout. The invariance of the asymptotic efficiency of the conventional detector with respect to the relative delay parameter is only incidental and due to the high degree of symmetry that the signature signals exhibit. It is not to be construed as a general property of the conventional detector.

direct-sequence signature signal example that had a bandwidth efficiency of only a two-thirds. This indicates the greater suitability of the sinusoidal signature signals.

In the second set of examples, we choose a pair of bandwidth efficient unit-energy signature signals given by (69) for (or, 02) = (0,0.95) corresponding to a bandwidth efficiency of 2.0/1.95 which is greater than 1. Fig. 7 shows the variation of asymptotic efficiencies of the conventional and the optimal decorrelating detector for this signal set as a function of the relative interfering signal strength. The relative delay of the second user is fixed at T/3. The observations made in Fig. 5 are relevant in this case, the conventional detector becoming multiple-access limited when the relative interfering signal strength is only 3 dB and the optimal decorrelating detector exhibiting a relatively high asymptotic efficiency independent of the interfering signal strength.

In summary, the numerical examples of this section show that the optimal decorrelating detector has a clearly superior performance as compared to the conventional detector, this superiority becoming more significant in near-far scenarios. In fact, it can also be established that even when the signal strengths are weaker than the desired signal strength, as the number of users becomes sufficiently large, the optimal decorrelator begins to perform significantly better than the conventional detector. The only situation where the conventional detector would out-perform the decorrelating detector is when the multiple-access interference plays a subordinate role to the additive noise in contributing to the performance. This rules out all but the uninteresting cases of low bandwidth efficiency and/or when all the interfering signals are sufficiently weak relative to the desired signal (which of course can only be true for the strongest signal).

In Fig. 8, the variation of asymptotic efficiency of the optimal and the conventional detectors is depicted as a function of the relative delay of the second user for three different levels of relative ‘interfering signal strengths (-5 dB, 0 dB, 5 d?). This variation leaves the performance of the optimal detector unchanged while the conventional detector shows a drastic overall degradation. The optimal detector exhibits a uniformly high asymptotic efficiency varying in the range [0.83-0.9951 as the relative delay parameter is varied. On the other hand, the conventional detector performance is largely controlled by the relative delay. Any small decline in the optimal detector performance is accompanied by a magnified degradation of the conventional detector, this magnification being larger with increasing interfering signal strength. When the relative signal strength is 5 dB, the conventional detector is multiple-access limited for nearly half of the range of values of the delay parameter. Furthermore, the performance of the optimal decorrelating detector for the higher bandwidth efficiency example is comparable to its performance for the

As a final note, the second set of examples that we considered in this paper where the bandwidth efficiency was greater than 1 suggests that in uncoded CDMA communication with a given overall bandwidth and a fixed uncoded bit-error rate, there is a continuous trade-off between the rate (per user) at which data can be communicated and the signal energies required. Higher data rates require higher energies and vice versa. Similarly, for a fixed overall bandwidth and fixed data rate (per user) an increase in the number of users can be supported by increasing the signal energies to maintain the same performance. These conclusions should be of particular interest in radio applications where users are plentiful and bandwidth is scarce.

IX. CoNCLUsIoN

In this paper, we consider the problem of noncoherent demodulation of differentially phase-shift keyed signals over an additive Gaussian code-division multiple-access (CDMA) channel. We derive a minimax robust multiuser detection strategy which which does not require a knowledge of the


energies and ‘phases of any of the transmissions for its implementation. It is a minimax detector in the sense that it optimizes the worst-case multiuser asymptotic efficiency over the interfering signal energies and phases. In the limit as the packet length + cc, this detector is shown to be is equivalent to a time-invariant K-input, K-output decorrelating filter followed by a time-invariant differential decision logic. A significant property of this detector is the invariance of its error probability to the interfering signal energies and phases. The long-standing premise that the near-far problem is endemic to the CDMA channel is thereby disproved in the context of noncoherent CDMA communication and shown to be only a characteristic of the conventional spread-spectrum detector. Significant gains in error-rate and hence throughput can be achieved over the conventional strategy by the optimally near-far resistant decorrelating detector. In addition, the optimal decorrelating detector lends itself to a relatively simple practical implementation. It does not need any knowledge of the signal strengths and phases, and requires the use of well- established building blocks such as matched filters and linear time-invariant multiinput multioutput digital filters and time- invariant differential decision logic circuits. In summary, the remarkable performance gains over the conventional detector together with its near-far resistance and ease of implementation render the optimal decorrelating detector a detector of choice in noncoherent CDMA channels.

X. APPENDIX

In the proof of Proposition 1, we made use of the key result in (23) that we repeat here for convenience:

sup O<r<l; lim Q($, 5) - alo < 3.

o-+cc exp(-$#)

= max’{O, e}.

(70)

In this appendix, we establish the validity of this result. In the process, we explore the issues involved in the computation of asymptotic efficiency in general terms via error probability bounds.

Let us define the set S(P(a)) corresponding to a positive, real-valued function P(a) as

S(P(a)) =

{

0 < r < 1; )Lo PC,)

exp(-+$)

and consider the following lemma. Lemma A.1 Suppose 3 a 00 such that V 0

positive real-valued functions PI(~) and P2 (0) are such that PI(O) > Pi, we have for any Q E [0, l), the set equality

S(P1(a) - aP2(a)) = S(P1(a)). (71)

Proof: If r E S(PI(a) - crP2(a)), then we have the inequalities

whence T E S(PI (CT)) since 1 - cy > 0. Therefore, we have the set inclusion S(PI(a)) -d(a)) C S(Pl(a)). The proof of the converse statement is straightforward. 0

We use the result of this lemma to establish the next proposition.

Proposition A.1: The following equality holds for ~\i E

<oO .I (73)

Proof: Let

Using the integral definition of the Marcum Q function,

Q(;,;) = cxp(-5) l;,,iexp(-;}r,($) dt,

(74)

integrating by parts, and using the property of the modified Bessel function that d/(dz)le(~) = II(Z) [ll], we have

Q(;>;) =e~p(-~)1~($) +;izc

.exp(-;}Il(Jg) dt

> exp(-~)Io(fg). (75)

With PI(O) = Q(fl/a,f2/a) and h(g) = Io(f1f2/g2) exp( -f,” + fi/a2), the inequality in (75) and the fact that a: E [0, l), Lemma A.1 is applicable and leads to the result of Proposition A. 1. 0

From the definition of al,2 in equation (22) and by the use of the Cauchy-Schwarz inequality we have 0 5 CX~J 5 1 with ~1,s = 1 if and only if g = ch. In this case, we leave it to the reader to verify that the error probability function Pm,i(cJ) = l/2, and therefore the asymptotic efficiency is identically zero. Without loss of generality, we exclude this degenerate case from further consideration.

The next lemma emphasizes the usefulness of upper and lower bounds on error probability in characterizing the asymptotic efficiency.

Lemma A.2: Suppose 3 a go such that V g < 00, the error probability function P(O) is bounded above and below by

VARANASI: NONCOHERENT DETECTION IN ASYNCHRONOUS MULTIUSER CHANNELS

P”(a) and P”(c), then the asymptotic efficiency corresponding to P(g) is bounded above and below by the asymptotic efficiency corresponding to PL(g) and P’(g), respectively.

Proof: Since V CT < (ro, we have P”(a) >_ P(cr), we have

so that we have the set inclusion

s(W 4 ) c S(P(a)), whence it follows that

Tf = supS(P(a)) 6 supS(P(a)) = q.

The upper bound for 7 can be sho,wn by the same argument to be equal to vu = supS(P(a)). This completes the proof of the lemma. cl

We obtain in the following proposition, a closed form expression for the right-hand side term in (73).

Proposition A.2: The following equality holds:

Q($>$) exp(-y) < cc

= max2{0, e}.

(76)

Proof: The main idea here is to establish upper and lower bounds of the Marcum Q function in the low-noise region and apply the result of the preceding lemma.

The Marcum Q function defined in (74) can be shown to be the tail probability of a normalized Rician random variable 2 = (Xl” + Xi) / ( 2a2) where Xl and Xa are normal random variables with a common variance g2 and means ml and ms, respectively (cf. [9]). The noncentrality parameter rn: + rni = ff and the “tail” begins at (fl) / (2~“)) i.e.,

where the integrand is the probability density function of 2. Lower Bound on Asymptotic Eficiency: To upper bound

the Marcum Q function, we will invoke the Chernoff bound (cf. [ll]). For notational convenience, let us introduce the parameters Q! = (ff)/(202) and p = (fz)/(2r~~). Using the parametric form of the Chernoff bound, we have

for any v 2 0 for which P(V) = In E [evZ], the natural logarithm of the moment generating function (MGF) of the random variable 2, exists. It can be shown that the MGF of 2 exists provided v < 1 and is given as

E[evZ] = & exp( E),

175

so that we have the parametric upper bound for any u E [0, 1) given by

Q(- -) fl f2 <

CT’0 - &exp(e -b). (7%

Following the central idea of the Chernoff bounding technique, we have the tightest upper bound when the error exponent P(V) - PV is minimized, a necessary condition for which is

It is well known that the function P(Y) - ,Bv is a convex function so that (SO) yields a minimum of P(V) - PY, provided of course that such a solution exists in theinterval [0, 1). Since /J(Y) is a monotonically increasing function of Y [ll], we require that p be such that

From the expression for the moment generating function in (78) we have

b(v) = & + ~ (1$2’ O<u<l. (81)

Therefore, b(O) = 1 + cr and b(1) is unbounded thereby requiring that ,0 > 1 + CL Since the validity of this inequality is required for any arbitrary c, it is necessary that fz > fi. Substituting the expression for b(y) into the necessary condition of (80), we obtain a quadratic equation for which the roots are

Y,2 = -1+2/3&J-

2P .

It can be shown that only the root ~2 (with the negative sign) lies in the interval [0, 1). The error exponent evaluated at this root yields

4~4) - ~4-4 = - In(l - ~2) - a: - /3 + Ji?Z$

Therefore, the Chernoff bound on Marcum Q-function is given under the condition that fl > fi > 0 by the following set of equalities and inequalities:

Q(- -) fl f2 I

oi a & exp(-a - j3 + JFG3)

< $-p&&+xpi-~-8+~)

< fiexp($==J exp[-(fi- fi)“]

= ~exp(&)exp(-“~“i2), (82)

where the second inequality was obtained by using Taylor’s theorem to bound exp (dm). Finally, in order to ensure that ~2 2 0, we need the mild restriction that D be selected such that c < min(f2, (dm)/2) = 00.

At this stage, we can find a lower bound on the asymptotic efficiency corresponding to the function Q (f 1 /a, f2 /g). Using


Lemma A.2, and the low noise region upper bound in (82) we have

Q($ $) exp(-+$) < O”

2 max2{0, e}> (83)

where we used the fact that [ll]

fl fl Q(- -3 g’ CT ++,xp(-~)lo(~) >$

and that Q(fr/a, fs/a), being a tail probability function, increases monotonically with decreasing f2.

Upper Bound on Asymptotic Ejjiciency: We obtain a lower bound on the Marcum Q function in the low noise region. Starting with the inequality obtained in (75) we seek to lower bound the right-hand side of that inequality by lower bounding the modified Bessel function via Taylor’s theorem as follows:

cl- >- -

rdm exp ‘ifi

( ) 7r .

V- - - Lexp(-?!S&E)] 2 rdm-

vff < co, (84)

where the penultimate inequality follows from using the upper bound for the error complementary function [ll] and the last inequality by choosing au to be sufficiently small for the constant in the square bracket to be greater than c which in turn can be chosen to be arbitrarily close to &$. Substituting in (75), the lower bound for the modified Bessel function obtained in (84) we have

Q($!$) > &exp((f22::)2), vo<go. (85)

Now we are ready to invoke Lemma A.2 again to obtain an upper bound on the asymptotic efficiency function

corresponding to the Marcum Q function so that

5 max2{ 0, e}. (86)

Finally, the statement of this proposition follows from the fact that the upper and lower bounds found in (86) and (83) are identical. q

The results of the two propositions of this appendix together yield the sought-after result of (23) which was the key to the proof of Proposition 1.

PI

PI

[31

[41

[51

[61

[71

ts1

[91

WI

IllI

WI

[131

1141

WI

VI

REFERENCES

G. R. Cooper and R. W. Nettleton, “A spread-spectrum technique for high-capacity mobile communications,” IEEE Trans. Vehicular Technol., vol. VT-27, pp. 264-275, Nov. 1978. E. A. Geraniotis, “Performance of noncoherent direct-sequence spread- spectrum multiple-access communications,” IEEE J. Select. Areas Com- mun., vol. SAC-3, pp. 687-694, Sept. 1985.

“Noncoherent hybrid DSiSFH spread-spectrum multiple-access communications,” IEEE Trans. Commun., vol. COM-34, pp. 862-872, Sept. 1986. M. Kavehrad and B. Ramamurthi, “Direct-sequence spread spectrum with DPSK modulation and diversity for indoor wireless communications,” IEEE Trans. Commun., vol. COM-35, pp. 224-236, Feb. 1987. A. W. Lam and D. V. Sarwate, “Multi-user interference in FHMA-DPSK spread-spectrum communications,” IEEE Trans. Commun., vol. COM- 34, pp. 1-12, Jan. 1986. R. Lupas and S. Verdu, “Near-far resistance of multiuser detectors in asynchronous channels,” IEEE Trans. Commun., vol. 38, pp. 496-508, Apr. 1990. -> “Linear multiuser detectors for synchronous code-division multiple-access channels,” IEEE Trans. Inform. Theory., vol. 35, pp. 123-136, Jan. 1989. R. W. Nettleton and G.R. Cooper, “Performance of frequency-hopped differentially modulated spread-spectrum receiver in Rayleigh fading channel,” IEEE Trans. Vehicular Technol, vol. VT-30, pp. 14-29, Feb. 1981. J. G. Proakis, Digital Communications, seconded. New York: McGraw Hill, 1989. C. Rushforth and Z. Xie, “Multiuser signal detection using sequential decoding,” presented at IEEE Military Commun. Conj, Oct. 1988. H. L. van Trees, Detection, Estima‘tion, and Modulation Theory, Part I. New York: Wiley, 1968. M. K. Varanasi and B. Aazhang, “Multistage detection for asynchronous code-division multiple-access communications,” IEEE Trans. Commun., vol. 38, pp. 509-519, Apr. 1990. -> “Optimally near-far resistant multiuser detection in differentially coherent synchronous channels,” IEEE Trans. Inform. Theory, vol. 37, pp. 1006-1018, July 1991. -, “Near-optimum detection in synchronous code-division multipole- access systems,” IEEE Trans. Commun., vol. 39, pp. 725-736, May 1991. S. Verde, “Minimum probability of error for asynchronous Gaussian multiple-access channels,” IEEE Trans. on Inform. Theory, vol. IT-32, pp. 85-96, Jan. 1986.

“Optimum multiuser asymptotic efficiency,” IEEE Trans. Com- mun.,‘vol. COM-34, pp. 890-897, Sept. 1986.

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