1
ITERATIVE JOINT DETECTION USING RECURSIVE SIGNAL CANCELLATION Christian Schlegel Department of Electrical Engineering University of Alberta Edmonton, Alberta, CANADA Email:
[email protected] Fax: 1-780-492-1811
Capacity bits/dimension Abstract—It is argued that iterative signal cancelation receivers with or without using error control coding are the perferred prac- 10 tically 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. 1
AWGN Capacity Optimal Processing MMSE Filter Preprocessing
I. I NTRODUCTION Ever since the initial application of direct-sequence spreadspectrum (DSSS) signaling to multiple access, in what is now Matched Filter Preprocessing 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 0.1 of simple linear filtering or separation methods such as zeroforcing (accomplished by the decorrelator) [12], and minimum Eb mean-square error fitlering (MMSE) [31], a transplant from es-2 0 2 4 6 8 10 12 14 16 N0 timation theory to signal detection. Fig. 1. Information theoretic capacities of various preprocessing filters. Figure 1 shows the spectral efficiencies per signaling dimension achievable with different processing filters for a system using rate R = 1/3 error control codes with random signature sequences at equal received powers of all users [26]. There “soft estimation” of the coded symbols, and (ii), parallel singleclearly is a substantial gap between these filtered systems and user FEC soft-output decoding. The processing cores for these optimal processing for this case. If the users are received with two operations form a closed loop around which soft informaunequal power, the situation worsens for the filtered systems. tion on coded symbols is exchanged in the form of extrinsic It is well known that unequal power causes the near-far prob- APP. Moher [15] used this method employing a CDMA chanlem for matched filter receivers, and it has been shown [24] nel APP detector for a small number of users, while Alexander that equal power also maximizes the spectral efficiency of the et al. [1] used a simple interference cancellation (IC) operation MMSE filter. It can be shown that the spectral efficiency of and achieved surprisingly good results. Wang and Poor [30] the decorrelator [12] is also maximized by the equal power dis- extended the canceller by appending per-user MMSE filters to tribution. Equal received power levels is therefore the most improve interference suppression. Figure 2 shows the basic transmission system model of advantageous operating condition, which implies some for of power control in a network as exercised in commercial CDMA coded CDMA with K transmitters that generate independent binary information symbols which are encoded by K parallel networks [29], [16] With the advent of turbo coding and iterative processing [4], error control encoders. Random interleavers may separate the [5], [19], new decoder structures employing iterative decod- error control encoders from the spreading operation as is cusing have emerged and proven to be superior to alternative re- tomary in serial concatenation, but for many codes, such as ceiver structures. Iterative decoding [1], [2], [15], [30] breaks LDPC codes, this is not necessary. The outputs of the encoders the daunting task of a joint decoder into two operations, viz. (i) are mapped into BPSK, and modulated by K DS spreaders.
FEC Soft-Decoder Πk
Spreading Function
Inner
r1
a1 d1
x1 π1
FEC Encoder 1
×
τ1
r
a2 u2
FEC Encoder 2
d2
π2
×
τ2
x2 x +
CDMA
u1
Soft-Decision Decoder
Outer
r2
rK
w1
−1 Π1
APP
Π1
w2
−1 Π2
APP
Π2
wK
−1 Π K
APP
ΠK
aK dK
uK
πK
FEC Encoder K
×
τK
˜1 d ˜ d2 ˜K d
xK
tanh() tanh() tanh()
Fig. 2. Asynchronous CDMA with error control system model. Fig. 3. 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 [15] were proposed which generate log-likelihood ratios (LLR) of the encoded symbols d of Pr(dk (i)=1|r) the form λ(dk (i)) = Pr(d , but complexity prohibits k (i)=−1|r) 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 zk,l
T = wk,l rk,l
(1)
where rk,l and wk,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 [1] uses matched T filters, wk,l = ak,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 1 + 1/α, which is significant only for small system loads α = K/N < 1. 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 rk =
L p L X K p X X Pk dk,l ak,l + Pm (dm,l − d˜m,l )am,l + n l=1
l=1
m=1 (m6=k)
(2) where d˜m,l is a soft estimate of the coded symbol of user m at time l. The signal rk now passes through the same decoding and cacellation process again, whereby the residual noise and
interference remaining at the j-th iteration is ηk,l (j) =
L X K p X T Pm (dm,l − d˜m,l (j)) wk,l am,l + nk,l l=1
m=1 (m6=k)
(3) Under some quite general conditions, ηk,l (j) is well approximated by — and approaches in the limit — an independent Gaussian random variable. T Assuming a matched filter wk,l = ak,l with wk,l am,l = Rk,m,l and unbiased estimates of the i.i.d. distributed coded symbols dk,l allows us to calculate the variance of the zeromean random variable ηk,l (j) as K L X h i 2 X Pm E (dm,l − d˜m,l (j))2 E [Rk,m,l ] + σ 2 E ηk,l (j) = l=1 m=1
(m6=k)
(4) Furthermore, for i.i.d. random spreading sequences with √ √ P (ak,l,n = h 1/ N ) =iP (ak,l,n = −1/ N ) = 1/2 it fol-
lows that E
aTk,l am,l
= 1/N and
K h i 2 1 X Pm E (dm − d˜m (j))2 + σ 2 σk2 (j) = E ηk,l (j) = N m=1 (m6=k)
(5) which is independent of the time index l, which we have dropped in equation (5). For equal powers Pk = P , the normalized effective variance is independent of k also and assumes the particularly simple form [2] of 2 σeff =
σ2 K −1 2 + σd ; P N
h i σd2 = E (dk − d˜k )2
(6)
−1 Note that σeff is the effective signal-to-interference ratio of each user’s channel after cancellation.
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 Eb /N0 ≈ 4.5dB the interference-free performance of the convolutional code is achieved. Bit Error Rate 1 K = 45 N = 15 dual-stage filter
1 iteration 5 iterations
−1
10
10 iterations
α as the number of users Kj that share the common rate rj , Schlegel et. al. [23] have shown that the rate constraints ! 1 1 ; b = σ 2 /P (7) rj ≤ log 1 + Pj 2 α + b m m=1 can achieve the capacity of the CDMA channel as the number of users grows large under certain mild restrictions on the partial system loads αj 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.
10−2
10 le U
20 iterations
10−3
30 iterations
10−4
System Load K/N = 3 −5
10
2.5
3
3.5
4
α = 50 α = 20 α = 10
15 iterations
ser
4.5
5
Eb N0
Fig. 4. Illustration of the turbo effect of iterative joint CDMA detection.
Sum Capacity [bits/dimension]
Sing
α=5 α=1 1
0.1 II. U NEQUAL R ATES AND U NEQUAL P OWER D ISTRIBUTIONS A problem that shows up in attempts to achieve high spectral efficiencies with symmetrical systems of equal powers, Pk ≡ P, ∀k, and equal rates Rk ≡ 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 P can also be used. Defining the partial loads αj = Kj /N, αj =
−2
0
2
4
6
8
10
12
14
Eb 16 N 0
Fig. 5. CDMA spectral efficiencies achievable with iterative decoding with equal power groups assuming ideal FEC coding with optimized rates according to (7).
III. I TEATIVE S YSTEMS W ITHOUT E RROR C ONTROL C ODES If the iterative CDMA decoder is operated without the integrated error control decoder, we obtain a simplified update equation, given by dˆk,l (j +1) =
L p X Pk dk,l l=1
+
L X K p X
Pm (dm,l − d˜m,l (j))Rk,m,l +nk,l (8)
l=1 m=1
(m6=k)
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 = dˆm,l is the estimate from the previous iteration, the update equations are given by
Capapcity bits/dimension 10
AWGN Capacity dk,l (j + 1) = zk,l −
K X
Rk,i,l di (j)
(9) α = 0.5
i=1 i6=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−1 r
MMSE Capacities 1
α=2 BPSK
(10) BPSK
since equation (9) iteratively implements the inversion of R [3]. Decorrelator This application of iterative matrix inversion methods to deα = 0.5 coding CDMA systems was first proposed by [10], but an early form of the cancellation receivers following the format of (9) 0.1 was proposed in [25] without exploring the connection to matrix theory. A thorough analysis of iterative cancellation reEb -2 0 2 4 6 8 10 12 14 ceivers in a random CDMA environment in [13] revealed that N0 the Jacobi decorrelator suffers from a serious drawback. For Fig. 6. Spectral efficiency achievable with the linear filters and low system large systems using random signature sequences, it converges loads, both for unconstrained signaling – solid lines, as well as for binary BPSK √ to the decorrelator if and only if K < N ( 2 − 1)2 , i.e., for a modulation – dashed lines. system load of only 17% or less. There are many other iterative matrix solution methods As in the case of the coded iterative detector, the feedback which more or less can all be used in cancellation receivers non-linearity tanh(?) can be introduced in the loop. However, [3], [13]. Their performance does not typically differ by much, whereas tanh(λk,l /2) = E [dk,l |rk ] is the exact conditional exhowever, arguably the most efficient structure is the Gauss- pectation of a given transmitted symbol derived from the LLR Seidel canceler. It is based on splitting a matrix ratio of a symbol, tanh(dˆk,l ) in the FEC-free loop has no such direct interpretation, and is an ad hoc measure that takes note of M = diag(M) − ωL − (1 − ω)L − L† (11) the fact that d ∈ (±1). The performance of this non-linearity k,l in the feedback loop has not been explored in much detail, but where −L is the lower triangular part of M. This leads to the a certain improvement in performance has been observed [25], general form of over-relaxation iterative inversion. Choosing [13], [9]. the relaxation parameter ω = 1 we obtain the Gauss-Seidel In fact, Buehrer and Nicoloso [6] observe also that directly iteration equation using the matched filter outputs in the feedback entails a loss (diag(M) − L) d(j + 1) = r + L† d(j) (12) in achievable system loads. They modify the feedback signal by introducing weighing factors, after observing that the direct The major consequence for an interference canceler is that feedback in the uncoded system inadvertently reduces the usecomputed values are used as soon as they are available, unlike ful signal power of the symbols. This strategy allows them to the Jacobi canceler, which performs a complete block update achieve notably increased system loads. A coherent theory is before new values are used. The iteration equations to imple- however not presented. ment (10) are therefore given by A further consideration of utmost importance in the applicability of such decoder in real-world systems is the effect of k−1 K X X asynchronicity. The proposed signal cancellation receivers are dˆk,l (j) = zk,l − Rk,i,l dˆi,l (j)− Rk,i,l dˆi,l (j−1) (13) all easily adapted to asynchronous transmission, due to the simi=1 i=k+1 ple subtraction operation at the receiver front-end. Direct inFor lower system loads, remarkably high spectral efficien- version of the correlation matrix (10) [12] is practical only for cies can be achieved with simple linear filter front-ends, imple- synchronous systems. mented by multistage filters. Using (13) also avoids the conIV. D ISCUSSION vergence problem of the Jacobi canceler. Letting the iteration matrix either be R or R + σI in (13) We have argued in this paper that practically viable joint deallows for either the decorrelator or MMSE filter to be imple- tection of CDMA signals is synonymous with signal cancellamented [13] as a multistage filter. tion receivers. The question remaining is merely whether the
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 Eb /N0 , 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 > 1. 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. R EFERENCES [1] P. Alexander, A. Grant and M. Reed, “Iterative Detection in CodeDivision Multiple-Access with Error Control Coding”, European Transaction on Telecommunications, Special Issue on CDMA Techniques for Wireless Commmunications Systems, vol. 9, pp. 419-426, Sep. - Oct. 1998. [2] P. Alexander, M. Reed, J. Asenstorfer and C. Schlegel, “Iterative Multiuser Interference Reduction: Turb CDMA”, IEEE Trans. Commun., vol. 47, no. 7, July 1999. [3] O. Axelsson, Iterative Solution Methods U.K., Cambridge Press, 1994. [4] S. Benedetto, D. Divsalar, G. Montorsi and F. Pollara, “Serial Concatenation of Interleaved Codes: Performance Analysis, Design, and Itera-tive Decoding”, IEEE Trans. Inform. Theory, vol. 44, pp. 909926, May 1998. [5] C. Berrou and A. Glavieux, “Near Optimum error correcting coding and decoding: turbo-codes”, IEEE Trans. Commun., vol. 44, pp. 1261-1271, Oct. 1996.
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