Geometric interpretation of multiaccess joint detection and the alternating projection algorithm

waveforms span a signal space, span{sk(t)} C L (I), of dimension N. Choosing an orthonormal basis of span{sk(t)) The joint detection of all users in a multiple access (MA) The joint detection of all users in a multiple access (MA) allows for us to represent each user signature as sk, the Ncommunication system in which user transmissions are cordimensional column vector of coefficients (with respect to related has been shown in recent literature to enhance the their basis) corresponding to sk(t). Note that span{sk} = system performance relative to that achieved without joint RN. Let the kth user's bit be denoted by bk E {+1, -1}, detection. Over the past several years the area of low comwhere each value occurs with equal probability. The replexity joint detectors has received much attention. This ceived signal is then represented as a coefficient vector, r, paper explains the problem of multiple access joint detection in geometrical terms. Geometric interpretation leads K to the proposal of an alternating projection joint detection r = bksk + n = STb + n (1) algorithm (APJD). Due to some similarities between our k=l APJD and the multistage joint detector (MJD) of Varansi and Aazhangd[5] the MJDtis e 1jisn diecus e1d. T ArDsis * where K is the total number of users, n is the coefficient and Aazhang [5], the MJD is also discussed. The APJD is S guaranteed to converge and a proof is given. The geometric vector of noise', b-[bl b2... bK] and ST-[s . s2 SK]. interpretation of the MA joint detection problem allows for The conventional approach to receiver design is to use the exploration of determining, a priori, the error probabila matched filter and slicer, ity of a joint detector and user waveform set in the absence of noise. Simulations offer empirical characterization of the error behavior of both detectors. where sgn represents the signum function and bk denotes the estimated/detected bit of the kth user. The conventional matched filter (2) represents the optimal receiver in the 1. MULTIPLE ACCESS COMMUNICATION case where the MAI is assumed normal and white. Under AND DETECTION this assumption, as users are added to the system the MAI raises the noise level; this, in turn, limits the matched filter A multiple access (MA) communication system will typiperformance. cally support a large number of users over a given channel. The MAI, however, is not an additive white Gaussian Many users are allowed to transmit simultaneously in the noise process; it possesses a great degree of structure which same frequency band; each user transmits a single bit by can be exploited in building receivers which have better modulating a pre-assigned signature waveform by a +1 or performance than the conventional detector. The optimal -1. The common MA scenario used for this paper is that of joint detector (which accounts for the MAI) maximizes the synchronous signaling through the additive white Gaussian log-likelihood function [7] and results in noise channel with no intersymbol interference. With no loss of generality, we may focus our attention to one bit dub = arg [ max 2rTSTb (STb)TSTb ]. (3) ration, i.e. within a single block of time all users transmit br{1,-1}K a single bit. h This maximization is over 2K possible b vectors (a compuAssign the kcth user a signature waveform, sk (t), which tational complexity that is exponential in K, the number is zero outside of an interval I. Assume that the set of user of users). of users). This optimal method offers substantially higher The work of S. Mallat was supported by the AFOSR grant performance than the conventional demodulator (2), but is F49620-93-1-0102, ONR grant N00014-91-J-1967. The work of of little practical use on account of its computational burthe other authors has been supported in part by the National den [4]. Science Foundation under grant number MIP-9015281 and by the Air Force Office of Scientific Research under grant number 1The irrelevance theorem allows for the portion of the noise F49620-92-J-0002. which lies outside of span{sk(t)} to be ignored [9] Recent communication literature addresses the general our understanding of the problem we examine its fundanotion of suboptimal joint detection which offers compumental structure in the absence of noise. For the remainder tational improvement relative to the optimal method while of this paper, noise is omitted. achieving a significant improvement in performance over the Define the set of bit vectors conventional detector. The state of the art is reviewed in the paper by Verdu [8]. In particular, one approach that has r [bi ... bK]T I bi E {+1, -1} Vi = 1, ,.. K}. been shown to offer good performance compared to other methods is the multistage joint detector (MJD) developed Geometrically, r comprises the vertices of a hypercube of by Varanasi and Aazhang in [5] and [6]. The MJD corredimension K. For K > N the N-dimensional signature vecsponding to the received signal vector of Equation (1) is tors, {Sk } , are linearly dependent. This means that the solution, x, to b(m + 1) = sgn[Sr + (E SST)b(m)], (4) r = STx (7) nA is not unique. By definition of linear dependence, we have where the energy matrix E = diag (< Sk,Sk >)k=E1 The MJD is motivated by separating Equation (1) in the noiseSTc -O (8) less case (for each user k) into two parts, the MAI and the signal of interest, and then applying the appropriate for any a E AJr(ST), the nullspace of ST. We may, then, matched filter for user k to yield express the solution of Equation (7) as T STTi T x + a, (9) s~ r = bisk si + bksk sk. (5) x=p+c>, (9) i~k where : is the solution of Equation (7) for which P 1 a. The only solutions of interest to the MA problem stated in After rearranging and rewriting Equation (5) for all users this paper are contained in the set r. For every 3 1 Af(ST) in vector form, we obtain in vector form, we obtain which solves Equation (7) we are interested only in the soEb = Sr + (E SST)b (6) lutions for which (3 + c) E F, where ca E g(ST). A geometric interpretation of the above discussion folthe impetus for the MJD. The MJD estimates the MAI lows. We have our set of possible solutions, r, the verand subtracts it from the output of the bank of matched tices of a K-dimensional hypercube. We separate our sofilters to obtain an estimate of the desired bit. This process lution, x, into two parts, ce and 3. This corresponds to is iterated to obtain "better" estimates of the MAI in the viewing our vector space, 1RK, as the Cartesian product hope of improving the estimate of the desired bit. of two subspaces, N(ST) and the space which is orthogThe problem of finding the correct bit vector from the onal to A(ST). Given the uniquely determined solution, aggregate can be shown to be N-P complete. Primarily, 1 MJ(ST), the general solution must lie in the affine space there is the problem of heuristic approaches converging to WV = A((ST) + 3. The MA joint detection problem correlocal minima. MA joint detection, therefore, is not going sponds, geometrically, to finding the point, x, which lies in to be solved by a simple trick. the intersection of the set r and the affine space W. This paper looks at the problem in geometrical terms in The definition of 3 may be specified further. The set of Section 2, revealing its structure. The structure is reminisuser signature waveform vectors, {Sk}K, comprise a frame cent of other problems which are solved via alternating profor the space span{sk}. Let {Sk}K be the corresponding jections. An appropriate alternating projection joint detecdual frame, defined via the dual frame operator tor (APJD) is deduced and shown to converge in Section 3. The MJD, viewed as a sequence of operators within our S [1 S... SK] = S(STS) 1 geometry and can easily be shown (in some cases) to loop between two incorrect bit vector estimates. The differences We may decompose r using the dual frame operator between the MJD the APJD are discussed in Section 4. The geometric framework allows for us to begin the character< sk,r >= Sr, (10) ization of errors for suboptimal joint detectors. This idea is briefly discussed for the APJD and MJD and an empirand we may reconstruct using the frame reconstruction forical examination of the errors is done via simulations for mula K both the APJD and the MJD in Section 5. The paper is > S = ST concluded in Section 6. k=l 2. GEOMETRICAL PRESENTATION OF THE Note the similarities between Equation (11) and the MA MA DETECTION PROBLEM aggregate signal of Equation (1). Since Sr has the same 3 This is the case in which the number of users is greater than In order to examine the MJD and develop an approprithe dimension of the span of their signature waveforms. ate low complexity joint detection algorithm the detection For our purposes it is sufficient to note that a subset of problem is described in a geometrical framework. To begin the signature waveforms {Sk} constitute a basis of the considered space and that no sk is identically zero. See the text by 2 This issue is examined by Verdu in [4]. Daubechies for a tutorial treatment of frame theory [1]. properties that were required of /3: Sr E IZ(S) = RT(S) Unlike alternating projections between two intersecting and R(S) I j\/(ST) 6, we see that the frame reconstruction convex sets, the APJD is an alternating projection between Equation (11) corresponds to the unique portion of the soa convex set, W, and a non-convex set, r. In such a sitlution of our MA joint detection problem, thus, P = Sr. uation, the alternating projection procedure may result in a "locally best" solution. By this we mean that the APJD 3. THE ALTERNATING PROJECTION JOINT will converge to a point, b ¢ r n W, where at each step of DETECTOR the iteration the distance between b(m) and WV decreased, and where b is a fixed point of Equation (12). As discussed in Section 2 the MA joint detection problem Our problem of finding the intersection between W and reduces to finding the point r can be shown to be N-P complete. No solution which is polynomial in complexity is known to solve the N-P comb E n wF . plete problem. Moreover, any approach which is polynomial in complexity will suffer from possible convergence to The problem of finding the intersection between two conlocal minima. With this in mind, we know that there is no vex sets is known to be solved iteratively by alternating low complexity suboptimal joint detection procedure which projections between the two sets. Our problem differs from converges to the solution. Instead, we strive to understand this in that one of our sets, r, is not convex. Noting the the problem so that joint detection algorithms can be desimilarities between the two problems, we propose the alveloped in order to minimize the probability of converging ternating projection approach for the MA joint detection to a local minima. problem and prove convergence. We wish to derive the operators, Pr and Pw. It is easy to see that Pr is the sgn function. To find Pw we begin Theorem: We define two projection operators; Pr maps with the definition of WV a vector in IRK to the closest vector in r, and Pw maps a vector in RK to the closest vector in W. By closest, W ' Sr +Af(ST). we mean shortest Euclidean distance. Thus, the following alternating projection joint detector (APJD) is guaranteed The projection onto W is, therefore, the projection onto to converge in a finite number of steps. NA(ST) translated by Sr, b(m + 1) = PrPwbl(m) (12) Pwx = Plx + Sr, (13) Proof: Let d(x, y) denote the Euclidean distance between where P± is the projector onto .A(ST). Using the identity the two vectors x and y. For guaranteed convergence we Pi = (I-P) and the projector P = S(STS)-1ST we find need to show that the APJD in terms of the frame and dual frame operators which define the user signature waveforms d(f(m + 1), Pwb(m + 1)) < d(f)(m), Pwlb(m)), b(m + 1) = sgn[ Sr + (I S(S T S)S )b(m)]. (14) where the equality holds only for b(m + 1) = b(m). In words, we wish to show that with every iteration of the If we initialize the iteration with b(O) = 0 then b(1) = APJD, the estimate gets closer to the affine space W. PrSr. We assert the following: d(b(m±1),Pwb(m±1)) • d(;(m~+),Pwvv(m)) 4. COMPARISON OF APJD AND MJD < d(bf(m),Pw[b(m)). A brief comparison of the two detectors is offered. Note the similarities between the APJD of Equation (14) and The validity of the above equation is explained. The left the MJD of Equation (4). Below are the corresponding comparison: by definition of Pw, we know that b(m + 1) components of each detector. is closer to Pwb(m+l) than it is to Pwlb(m) with equality only in the case Pwlb(m + 1) = Pwb(m). The right comparison: by the construction of t(m+l1) from Equation (12) decorrelator matchedfilter and from definition of Pr, we know that Pwb(m) is closer S = S(STS)-i ST to /(m+ 1) than it is to [b(m) with equality only in the case when the points are the same, b(m + 1) = b(m). We have orthogonal projector o linear operator equality in both comparisons if and only if Pwb(m + 1) = (I S(STS)-1ST) (E SST) Pwbl(m) and the points are the same. Since there are a finite number of points in r, the algorithm is guaranteed to The dual frame operator, S, applied to r gives S(STS) 1STb, converge in a finite number of steps. [ the orthogonal projection of b onto R(S) while the matched filter, S, applied to r gives SSTb E R(S). The orthogonal 5It is easy to show that 1(S) = 14(S) but is not proved here. 62(S) denotes the range of S. For more details on the rela8The orthogonal projection operator, P which maps a vector tions between vector spaces, see the text by Strang [3]. onto the closest point in T(A) is P = A(ATA)-1AT. These projectors are not required to be linear, i.e. P(a+b) : 9 This b(1) is the decorrelating linear detector of Lupas and Pa + Pb. Verdu [2]. projector (I S(STS)-lST) applied to 1(m) gives an "es07 Dimensionof signalspace: N=4 timate", & E Af(ST), of the true cz while the linear operator (E SST) applied to b(m) gives an estimate of the MAI which lies in the space AJ(ST ) GR(S). Note that the APJD consists of orthogonal projectors while the MJD does not. The MJD has been found to have limit cycle behavior, 05 i.e. for some correlated waveform sets, the bit vector estimate loops between two incorrect elements of r. For lack ~ 0.4 of space, we leave this topic for another paper.