ITERATIVE JOINT DETECTION USING RECURSIVE SIGNAL CANCELLATION

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1 ITERATIVE JOINT DETECTION USING RECURSIVE SIGNAL CANCELLATION Christian Schlegel Department of Electrical Engineering University of Alberta Edmonton, Alberta, CANADA Fax: Abstract It is argued that iterative signal cancelation receivers with or without using error control coding are the perferred practically viable multiuser detection methods. Achievable spectral efficiencies are calculated for both cases and arguments are given on how to achieve maximum spectral efficiencies by rate or power layering of the different users in a multiple access environment. The achievable spectral efficiencies of iterative cancellation receivers which include the error control codes are compared to those which do not. The latter are found to be essentially viable only in power controlled environments. 0 Capacity bits/dimension AWGN Capacity Optimal Processing MMSE Filter Preprocessing I. INTRODUCTION Ever since the initial application of direct-sequence spreadspectrum (DSSS) signaling to multiple access, in what is now known as code-division multiple access (CDMA), it was evident that jointly detecting the different data streams held the potential to vastly increase spectral efficiency of CDMA. Optimal detection [27] being too complex motivated the application of simple linear filtering or separation methods such as zeroforcing (accomplished by the decorrelator) [2], and minimum mean-square error fitlering (MMSE) [3], a transplant from estimation theory to signal detection. Figure shows the spectral efficiencies per signaling dimension achievable with different processing filters for a system using rate R = /3 error control codes with random signature sequences at equal received powers of all users [26]. There clearly is a substantial gap between these filtered systems and optimal processing for this case. If the users are received with unequal power, the situation worsens for the filtered systems. It is well known that unequal power causes the near-far problem for matched filter receivers, and it has been shown [24] that equal power also maximizes the spectral efficiency of the MMSE filter. It can be shown that the spectral efficiency of the decorrelator [2] is also maximized by the equal power distribution. Equal received power levels is therefore the most advantageous operating condition, which implies some for of power control in a network as exercised in commercial CDMA networks [29], [6] With the advent of turbo coding and iterative processing [4], [5], [9], new decoder structures employing iterative decoding have emerged and proven to be superior to alternative receiver structures. Iterative decoding [], [2], [5], [30] breaks the daunting task of a joint decoder into two operations, viz. (i) 0. Matched Filter Preprocessing Fig.. Information theoretic capacities of various preprocessing filters. soft estimation of the coded symbols, and (ii), parallel singleuser FEC soft-output decoding. The processing cores for these two operations form a closed loop around which soft information on coded symbols is exchanged in the form of extrinsic. Moher [5] used this method employing a CDMA channel detector for a small number of users, while Alexander et al. [] used a simple interference cancellation (IC) operation and achieved surprisingly good results. Wang and Poor [30] extended the canceller by appending per-user MMSE filters to improve interference suppression. Figure 2 shows the basic transmission system model of coded CDMA with K transmitters that generate independent binary information symbols which are encoded by K parallel error control encoders. Random interleavers may separate the error control encoders from the spreading operation as is customary in serial concatenation, but for many codes, such as LDPC codes, this is not necessary. The outputs of the encoders are mapped into BPSK, and modulated by K DS spreaders.

2 FEC Soft-Decoder Outer Π k Inner Spreading Function r w Π Π u u 2 FEC Encoder FEC Encoder 2 d d 2 π π 2 a a 2 τ τ 2 x x 2 + x r CDMA Soft-Decision Decoder r 2 r K w 2 w K Π 2 Π K Π 2 Π K u K FEC Encoder K a K d K π K τ K x K Fig. 2. Asynchronous CDMA with error control system model. Fig. 3. d d 2 d K Block diagram of the joint iterative decoder Figure 3 on the other hand shows an iterative receiver based on iterative decoding of serially concatenated error control codes [4]. The CDMA interference receiver is used to generate soft outputs of the encoded symbols given a received frame r, and separate these into K parallel signal streams suitable for the K parallel FEC decoders. Initally, complete a posteriori probability CDMA decoders [5] were proposed which generate log-likelihood ratios (LLR) of the encoded symbols d of Pr(d k(i)= r) Pr(d k (i)= r) the form λ(d k (i)) =, but complexity prohibits this even for moderate numbers of users K. A more practicable way is to use linear filters to suppress the residual interference. The received filtered signal samples z k,l = w T k,lr k,l () where r k,l and w k,l are the outputs of the CDMA decoder and the filter for the k-th user for symbol l, respectively, are fed into the FEC decoders which treat them as noisy signals, each decoding its specific data streams. Simple interference cancellation in the CDMA detector as proposed in [] uses matched filters, w T k,l = a k,l, to generate the input signal to the individual FEC decoders. The more sophisticated and complex approach to use per-user MMSE [30], has been shown [22] to lead only to a small increase in spectral efficiency of the order of + /α, which is significant only for small system loads α = K/N <. The soft-outputs of the various FEC decoders are combined to generate an estimate of the transmitted signal which is cancelled from r to obtain an improved received signal for each of the users, given by r k = Pk d k,l a k,l + l= l= m= Pm (d m,l d m,l )a m,l + n (2) where d m,l is a soft estimate of the coded symbol of user m at time l. The signal r k now passes through the same decoding and cacellation process again, whereby the residual noise and interference remaining at the j-th iteration is η k,l (j) = l= m= Pm (d m,l d m,l (j)) ( wk,la T ) m,l + nk,l (3) Under some quite general conditions, η k,l (j) is well approximated by and approaches in the limit an independent Gaussian random variable. Assuming a matched filter w k,l = a k,l with w T k,l a m,l = R k,m,l and unbiased estimates of the i.i.d. distributed coded symbols d k,l allows us to calculate the variance of the zeromean random variable η k,l (j) as E [ ηk,l(j) 2 ] [ = P m E (d m,l d m,l (j)) 2] E [R k,m,l ] + σ 2 l= m= (4) Furthermore, for i.i.d. random spreading sequences with P (a k,l,n = / N) = P (a k,l,n = / [( )] N) = /2 it follows that E a T k,l a m,l = /N and σ 2 k(j) = E [ η 2 k,l(j) ] = N m= [ P m E (d m d m (j)) 2] + σ 2 (5) which is independent of the time index l, which we have dropped in equation (5). For equal powers P k = P, the normalized effective variance is independent of k also and assumes the particularly simple form [2] of σeff 2 = σ2 P + K [ N σ2 d; σd 2 = E (d k d k ) 2] (6) Note that σ eff is the effective signal-to-interference ratio of each user s channel after cancellation.

3 The dynamical behavior of this system can be analyzed using density evolution and gives rise to the typical turbo code behavior with its steep threshold turbo cliff and error floor caused by the single-user performance. This error floor is entirely due to the error control code and is a consequence of the fact that the additive noise component cannot be removed with the cancellation operation (3). The effect is clearly visible in Figure 4, which shows a CDMA system using convolutional codes. At / 4.5dB the interference-free performance of the convolutional code is achieved. 0 Bit Error Rate K = 45 N = 5 dual-stage filter iteration 5 iterations 0 iterations α as the number of users K j that share the common rate r j, Schlegel et. al. [23] have shown that the rate constraints ) r j ( 2 log + j m= α ; b = σ 2 /P (7) m + b can achieve the capacity of the CDMA channel as the number of users grows large under certain mild restrictions on the partial system loads α j are observed. Again, the decoder essentially progresses in an successive cancelation operation, stripping lower rate users off before decoding higher rate users. The particular decoder setup, however, neither requires a fixed successive decoding schedule, nor error-free decoding as in the traditional successive cancellation procedure [8]. Achieving capacity under this method is possible, but potentially large system loads α = K/N are necessary as shown in Figure 5 which shows achievable spectral efficiency assuming AWGN Shannon-bound achieving codes. 0 2 Single User 5 iterations 0 α = 20 α = 50 α = iterations 30 iterations 0 5 System Load K/N = Fig. 4. Illustration of the turbo effect of iterative joint CDMA detection. Sum Capacity [bits/dimension] α = α = 5 0. II. UNEQUAL RATES AND UNEQUAL POWER DISTRIBUTIONS A problem that shows up in attempts to achieve high spectral efficiencies with symmetrical systems of equal powers, P k P, k, and equal rates R k R, k, is that there is a limiting maximum spectral efficiency which lies significantly below the optimal value, and what is worse, has an asymptotic slope of zero [22]. Caire et. al. [7] have studied the use of different power levels which can overcome this problem and dramatically improve spectral efficiency. However, the optimal power distribution to achieve capacity is exponential [23], which may be highly impractical. In fact, using an optimal power distribution, in essence, transforms the receiver into a successive cancellation receiver which migrates along the edges of the capacity polytope as explained in [8] and explored for convolutional codes by Viterbi [28]. Apart from different power levels, different coding rates can also be used. Defining the partial loads α j = K j /N, α j = Fig. 5. CDMA spectral efficiencies achievable with iterative decoding with equal power groups assuming ideal FEC coding with optimized rates according to (7). III. ITEATIVE SYSTEMS WITHOUT ERROR CONTROL CODES If the iterative CDMA decoder is operated without the integrated error control decoder, we obtain a simplified update equation, given by ˆd k,l (j+) = + Pk d k,l l= l= m= Pm (d m,l d m,l (j))r k,m,l +n k,l (8)

4 Due to the strong correlation between signals of different iterations, the large-system analysis used for the coded systems does not apply. If d m,l = ˆd m,l is the estimate from the previous iteration, the update equations are given by d k,l (j + ) = z k,l R k,i,l d i (j) (9) i= i k which is known as the Jacobi Canceler. In the limit as the number of iterations becomes large, its outputs are identical to that of the decorrelator d = R r (0) since equation (9) iteratively implements the inversion of R [3]. This application of iterative matrix inversion methods to decoding CDMA systems was first proposed by [0], but an early form of the cancellation receivers following the format of (9) was proposed in [25] without exploring the connection to matrix theory. A thorough analysis of iterative cancellation receivers in a random CDMA environment in [3] revealed that the Jacobi decorrelator suffers from a serious drawback. For large systems using random signature sequences, it converges to the decorrelator if and only if K < N( 2 ) 2, i.e., for a system load of only 7% or less. There are many other iterative matrix solution methods which more or less can all be used in cancellation receivers [3], [3]. Their performance does not typically differ by much, however, arguably the most efficient structure is the Gauss- Seidel canceler. It is based on splitting a matrix M = diag(m) ωl ( ω)l L () where L is the lower triangular part of M. This leads to the general form of over-relaxation iterative inversion. Choosing the relaxation parameter ω = we obtain the Gauss-Seidel iteration equation (diag(m) L) d(j + ) = r + L d(j) (2) The major consequence for an interference canceler is that computed values are used as soon as they are available, unlike the Jacobi canceler, which performs a complete block update before new values are used. The iteration equations to implement (0) are therefore given by k ˆd k,l (j) = z k,l R k,i,l ˆdi,l (j) i= i=k+ R k,i,l ˆdi,l (j ) (3) For lower system loads, remarkably high spectral efficiencies can be achieved with simple linear filter front-ends, implemented by multistage filters. Using (3) also avoids the convergence problem of the Jacobi canceler. Letting the iteration matrix either be R or R + σi in (3) allows for either the decorrelator or MMSE filter to be implemented [3] as a multistage filter. Capapcity bits/dimension 0 0. BPSK AWGN Capacity MMSE Capacities Decorrelator α = 0.5 BPSK α = α = 2 Fig. 6. Spectral efficiency achievable with the linear filters and low system loads, both for unconstrained signaling solid lines, as well as for binary BPSK modulation dashed lines. As in the case of the coded iterative detector, the feedback non-linearity tanh( ) can be introduced in the loop. However, whereas tanh(λ k,l /2) = E [d k,l r k ] is the exact conditional expectation of a given transmitted symbol derived from the LLR ratio of a symbol, tanh( ˆd k,l ) in the FEC-free loop has no such direct interpretation, and is an ad hoc measure that takes note of the fact that d k,l (±). The performance of this non-linearity in the feedback loop has not been explored in much detail, but a certain improvement in performance has been observed [25], [3], [9]. In fact, Buehrer and Nicoloso [6] observe also that directly using the matched filter outputs in the feedback entails a loss in achievable system loads. They modify the feedback signal by introducing weighing factors, after observing that the direct feedback in the uncoded system inadvertently reduces the useful signal power of the symbols. This strategy allows them to achieve notably increased system loads. A coherent theory is however not presented. A further consideration of utmost importance in the applicability of such decoder in real-world systems is the effect of asynchronicity. The proposed signal cancellation receivers are all easily adapted to asynchronous transmission, due to the simple subtraction operation at the receiver front-end. Direct inversion of the correlation matrix (0) [2] is practical only for synchronous systems. IV. DISCUSSION We have argued in this paper that practically viable joint detection of CDMA signals is synonymous with signal cancellation receivers. The question remaining is merely whether the

5 FEC system should be included in the iteration loop or simply concatenated at the output of an iterative symbol-based CDMA receiver. The answer depends on the particular application, in particular on system parameters, such as load, rate and power distributions. We have argued that capacity results for the achievable capacities in each of the subchannels should be used to settle this question. Such calculations reveal that simple symbol-based receivers are quite sufficient in a number of cases, such as for very low values of /, for low system loads α, and in certain power-controlled systems. Note, however, that in low loaded system, achieving high spectral efficiencies may require higher order modulations to be used (see Figure 6). The dashed curves in the figure show the capacity if signaling is restricted to BPSK (or QPSK for complex channels). Coded iterative systems on the other hand can perform close to the theoretical limit of the CDMA channel if optimal power or rate distributions can be applied to the user population. In this case, the high capacity can be achieved with simple binary signaling (QPSK on complex channels) and larger system loads α >. In equal received power, equal rate systems, coded iterative receivers still outperform the simple symbolbased iterative receivers, but can no longer approach the capacity of the channel. Schlegel et. al. [23] have shown that the performance of a coded iterative receiver using very simple error control codes at equal received powers and equal rates is lower-bounded by the performance of an MMSE filter receiver, the best of the linear symbol receivers. All this points to the conclusion that power controlling a CDMA system to equal received power levels as currently practiced in cellular CDMA system [29] may not be advantageous. Finally, we would like to note the remaining problems of synchronization. Each of the different CDMA data streams will arrive at the receiver with a different timing phase and frequency, and a different carrier phase and frequency. While carrier phase can possibly be kept accurate enough such that differentially coherent decoders, or decoders with built-in iterative phase correction [] can be used, the timing signal will have to be derived for each of the users separately. REFERENCES [] P. Alexander, A. Grant and M. Reed, Iterative Detection in Code- Division Multiple-Access with Error Control Coding, European Transaction on Telecommunications, Special Issue on CDMA Techniques for Wireless Commmunications Systems, vol. 9, pp , Sep. - Oct [2] P. Alexander, M. Reed, J. Asenstorfer and C. Schlegel, Iterative Multiuser Interference Reduction: Turb CDMA, IEEE Trans. Commun., vol. 47, no. 7, July 999. [3] O. Axelsson, Iterative Solution Methods U.K., Cambridge Press, 994. [4] S. 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