kinds of feedback information on the performance of the iterative receiver. II. SYSTEM ..... Init II: The initial channel estimate is the least squares estimate of a ...
Iterative Joint Channel Estimation and Successive Interference Cancellation Using a SISO-SAGE Algorithm for Coded CDMA †
B. Hu † , A. Kocian † , R. Piton † , A. Hviid † , B. H. Fleury † , and L. K. Rasmussen ‡ Department of Communication Technology ‡ Institute for Telecommunications Research Aalborg University University of South Australia 9220 Aalborg, Denmark Mawson Lakes SA 5095, Australia
Abstract— In this paper we propose an extension of the space alternating generalized expectation maximization (SAGE) algorithm that accepts as input and provides as output soft information. This so-called soft-input soft-output (SISO) SAGE algorithm is applied to joint successive interference cancellation and channel estimation and combined with SISO single-user decoders in an iterative CDMA receiver. Monte Carlo simulations in flat Rayleigh fading channel show that this receiver is robust against channel estimation errors and can support a high system load.
I. I NTRODUCTION The optimal multiuser maximum a posteriori (MAP) sequence estimator exhibits a computational complexity exponential in the product of number of users K and coding constraint length ν and hence, becomes prohibitive for even small K and ν. Iterative processing, in contrast, combines computationally low-complex blocks that exchange soft information in a very efficient way. Lively examples in communications are iterative multiuser detection, iterative (Turbo)-decoding, and iterative (Turbo)-equalization. Alexander et al. propose in their seminal work [1] an iterative receiver structure exchanging soft information between a SISO interference cancellation (IC) device and a bank of SISO single-user (SU) decoders. In [2], Kobayashi et al. extend this approach to include channel estimation, using the expectation maximization (EM) algorithm. In a similar scheme, Chiavaccini and Vietta [3] approximate the MAP symbol vector estimate in a Bayesian EM framework. This approach, however, has exponential complexity in the number of users, preventing practical implementation. A crucial issue for the iterative receiver to work efficiently is the kind of soft information exchanged between the SISOIC device and the SISO-SU decoder. Clearly, the SISO-SU decoder requires extrinsic (EXT) values on the code symbols [4] and so does the SISO-IC device in the large-system limit when the channel is known to the receiver [5]. In case of unknown channel, a theoretical justification for the proper feedback information to the SISO-IC device and channel estimator is still lacking. Experimental investigations of a This work was supported by RTX Telecom A/S, Denmark. R. Piton is now with Siemens A/S, Mobile Phones Development, Denmark. A. Hviid is now with RTX Telecom A/S, Denmark.
CDMA system using convolutional codes (CC) in a frequencyselective fading channel, however, indicate that the channel estimator and the IC device require different kinds of soft information [6]. In particular, it is shown in [6] that when the number of iterations are limited to 4 or 5 and the iterative receiver consists of a MMSE channel estimator combined with a parallel interference cancellation (PIC) device, best biterror-rate (BER) performance is achieved when a posteriori probabilities (APPs) and EXT values on the symbols are fed back to the former and the latter device respectively. In this paper we extend the SAGE algorithm [7] to provide and read soft information on the code symbols. This so-called SISO-SAGE device is combined with a SISO-SU decoder in an iterative receiver architecture performing successive interference cancellation (SIC). In addition, we investigate by means of Monte Carlo simulations the impact of different kinds of feedback information on the performance of the iterative receiver. II. S YSTEM M ODEL We consider a synchronous CDMA system using parallel concatenated CC (PCCC) with K active users and processing gain Nc . For user k, the information bit stream {bk [n] ∈ −1 {0, 1}}N n=0 is encoded by a user-independent encoder of rate R, fed into individual random channel interleaver Πk , mapped M−1 into a frame of BPSK symbols {dk [m] ∈ {−1, 1}}m=0 , where M = N/R, and partitioned into Nb blocks of Ld code symbols (M = Nb Ld ). Each block is then multiplexed with Lp random midamble symbols. Hence, each block and frame has total length of respectively Ld + Lp and L = (Ld + Lp )Nb code symbols. The k th user’s code symbols are then spread by random signature waveform sk (t) with support confined on the R Ts interval2 (0, Ts ]. Furthermore sk (t) has unit energy, i.e. 0 |sk (t)| dt = 1. Here, Ts is the time duration of one BPSK symbol. In the channel each transmitted block of user k experiences quasi-static flat Rayleigh fading. The received signal consisting of the sum of all users’ signals plus additive white Gaussian noise (AWGN) is decoded frame by frame. The column vector z[l] , col{z1 [l], . . . , zK [l]} ∈ CK containing the output samples of a bank of filters matched to the signature waveforms of the K
users in the lth signaling interval is given by z[l] = R[l]D[l]a[l] + n[l],
l = 0, . . . , L − 1.
into [8] (1)
K×K
The entries of the matrix R[l] ∈ R are the crosscorrelations between the signature waveforms of any two users in the lth signaling interval i.e., ρk,j [l] , [R[l]]k,j = R (l+1)Ts sk (t − lTs )sj (t − lTs ) dt. The column vector a[l] ∈ lTs RK contains the received complex amplitudes of the K users in the lth signaling interval, i.e. a[l] , col{a1 [l], . . . , aK [l]}. The symbol matrix D[l] ∈ RK×K reads D[l] = diag{d[l]} with the column vector d[l] , col{d1 [l], . . . , dK [l]} ∈ RK . Finally, n[l] ∈ CK is a circularly-symmetric complex zero mean Gaussian random vector with covariance matrix N0 R[l]. Under the assumption of quasi-static flat Rayleigh fading on each block, the k th user’s received amplitude ak [l] is modeled as circularly symmetric Gaussian random variable with zero-mean and variance σk2 . This amplitude is assumed to be constant within matrix of a[l] one block. The covariance 2 reads Σa[l] = diag σ12 , σ22 , . . . , σK ∈ RK×K . We also need to define the column vectors d , col{d[0], . . . , d[L − 1]} ∈ RKL , a , col{a[0], . . . , a[L − 1]} ∈ CKL and z , col{z[0], . . . , z[L − 1]} ∈ CKL for later use. For the problem at hand, we further define the column ¯ k [l] ∈ RKL−1 that are obtained vectors dk¯ [l] ∈ RK−1 and d th by deleting the k user’s lth code symbol in d[l] and d respectively . III. T HE SISO-SAGE A LGORITHM In their recently proposed applications to uncoded DSCDMA [8], the EM and SAGE algorithms are used to compute an approximation of the maximum likelihood (ML) estimate of the uncoded symbol vector d based on the observation z when the channel is unknown. These algorithms can also be applied to approximate the MAP estimate of the code symbol vector d [9]. We focus on the SAGE algorithm subsequently. A. The SAGE-JDE Scheme [8] Let us briefly review the application of the SAGE algorithm to joint (uncoded) data detection and channel estimation (JDE) [8]. At iteration i, the data symbols dk [l], l = 0, . . . , L − 1, of user k = k[i] are updated based on the observation vector z, while keeping the estimated code symbols of the other users [i] dk¯ fixed. In the SAGE framework z is referred to as the incomplete data. The so-called admissible hidden data for dk is selected in [8] to be the observation vector z together with the nuisance parameter vector a, i.e. X k = {z, a}. Notice that X k can only be partially observed. 1) E-step: The SAGE-JDE scheme computes in the expectation (E)-step the following estimate of the log-likelihood function of dk for the hypothetical observation X k o n [i] Qk dk |d[i] , E log p X k , dk , dk¯ |z, d[i] . (2) If the channel is memoryless, the objective function [i] computed in the E-step (2) can be decomposed Qk dk |d
L−1 X Qk dk [l]|d[i] . Qk dk |d[i] =
(3)
l=0
2) M-step: The value of dk [l] that maximizes the corresponding summand on the r.h.s. of (3) is given by [i+1] dk [l] = arg max Qk dk [l]|d[i] dk [l] o(4) n = sgn Qk dk [l] = +1|d[i] − Qk dk [l] = −1|d[i] , l = 0, . . . , L − 1. In (4) sgn{·} denotes the signum function. These two steps are repeated in subsequent iterations until convergence is achieved. Notice that the SAGE-JDE scheme reads and returns hard decisions on the symbols. B. A Modified SAGE Algorithm for Joint Successive Interference Cancellation and Channel Estimation (JCE) The iterative structure of the SAGE algorithm makes this scheme a potential candidate as a constituent element of an iterative receiver architecture with sub-components capable of exchanging soft information. To obtain soft estimates on the symbols, Chiavaccini et al. propose in [3] to reverse the role of the parameters in the SAGE framework, i.e. to assume that the code symbols are the nuisance parameters, while the amplitudes are the parameters of interest. This approach leads, however, to an E-step that returns soft estimates of the code symbols at exponential complexity in the number of users. We propose in the sequel a modification of the SAGEJDE scheme that accepts as input and provides as output soft information. This approach still conceptually keeps the code symbols as parameters of interest and the amplitudes as nuisance parameters. The proposed algorithm performs the following two steps: 1) Modified E-Step: n o ˜¯ [l][i] |z, d ˜ [i] , E log p X k , dk [l], d ˜ [i] Qk dk [l]|d k o n ˜¯ [l][i] . ˜¯ [l][i] |z, d ˜ [i] + log p dk [l]|d ∝ E log p X k |dk [l], d k k (5) 2) Modified M-Step: ΛSAGE (dk [l])[i+1] ˜ [i] ˜ [i] − Qk dk [l] = −1|d , Qk dk [l] = +1|d p X k |dk [l] = +1, ˜¯dk [l][i] [i] ˜ |z, d = E log [i] p X k |dk [l] = −1, ˜¯dk [l] {z } | λeSAGE (dk [l])[i+1]
(6) ˜¯ [l][i] p dk [l] = +1|d k . + log ˜¯ [l][i] p dk [l] = −1|d k | {z } λaSAGE (dk [l])[i+1]
The difference between (5) and (2) is twofold: (i) the ˜ as input modified E-step accepts soft symbols denoted by (·)
and (ii) it includes an additional term that accounts for the statistical dependency between the code symbols. Step (6) results by dropping the hard decision operation sgn{·} in (4). Clearly, this step computes an a posteriori log-likelihood ratio (LLR) of the symbol instead of a hard-decision. We coin this modified version of the SAGE-JDE algorithm SISO-SAGE. IV. I TERATIVE R ECEIVER U SING A LGORITHM
THE
SISO-SAGE
In this section we consider an iterative receiver architecture where a successive JCE scheme using the SISO-SAGE algorithm is combined with a SISO-SU decoder implementing the BCJR algorithm. A. SISO-SAGE Based JCE This sub-section is devoted to the derivation of a JCE receiver architecture within the SISO-SAGE framework. We start with the log-likelihood function of the admissible hidden ¯ dk [l][i] ). Discarding terms independent of data log p(z, a|dk [l], ˜ dk [l], it follows that ˜ ¯ k [l][i] ) ∝ log p(z[l]|a[l], dk [l], d ˜ ¯ [l][i] ). log p(z, a|dk [l], d k
The modified M-step of the SISO-SAGE algorithm calculates the a posteriori LLR ΛSAGE (dk [l])[i+1] in (6) that can be decomposed into the EXT LLR λeSAGE (dk [l])[i+1] and the a priori LLR λaSAGE (dk [l])[i+1] . Inserting (8) in the EXT LLR and evaluating for dk [l] = ±1, we readily obtain after few straightforward algebraic manipulations λeSAGE (dk [l])[i+1] = X 4 ∗ [i] ∗ ℜ (ak [l] )[i] zk [l] − (ak [l] ak′ [l])[i] d˜k′ [l] ρk′ k [l] . N0 ′ k 6=k
(10) Notice that in a communication system with memory, the derivation of the a priori LLR λaSAGE (dk [l])[i+1] in (6) is cumbersome. Hence, we resort to suboptimal methods. By neglect˜¯ [l][i] in ing the coding constraints the prior p dk [l] = −1|d k ˜¯ [l][i] . In addiλaSAGE (dk [l])[i+1] becomes independent of d k a [i+1] tion, λSAGE (dk [l]) is approximated by the soft information returned from a constituent SISO-SU decoder discussed next. B. SISO-SU Decoder
From (1) the right-hand expression can be rewritten as ˜ ¯ [l][i] ) ∝ 1 2ℜ (D[l]a[l])H z[l] log p(z[l]|a[l], dk [l], d k N0 − (D[l]a[l])H R[l]D[l]a[l] . (7) [i] ˜ ¯ [l] dk ¯ [l]=d k
From the EXT LLR λeSAGE (dk [l])[i+1] provided by the SISO-SAGE scheme a constituent SISO-SU decoder iteratively approximates the MAP estimate of the data symbols. Various decoding algorithms were proposed in the past. We employ a modified version of the BCJR algorithm [11] that computes soft information on both the systematic and parity symbols as requested by the SISO-SAGE scheme. After it has converged, the SISO-SU decoder returns the a posterior LLRs
The modified E-Step of the SISO-SAGE algorithm computes ˜ [i] }: the expectation of (7) with respect to {z, d ˜ ¯ k [l][i] + ˜ [i] = log p dk [l]|d Qk dk [l]|d Λdec (dk [l])[i+1] = λedec (dk [l])[i+1] + λadec (dk [l])[i+1] , (11) X 2dk [l] p [i] [i] ∗ ℜ (ak [l] )∗ zk [l] − (ak [l] ak′ [l])[i] d˜k′ [l] ρk′ k [l] , where dk [l] ∈ {dsk [l], dk [l]} and λadec (dk [l])[i+1] = N0 e [i+1] λSAGE (dk [l]) . The superscripts s and p refer to systemk′ 6=k atic and parity symbols respectively. The EXT LLRs of the (8) systematic and the parity symbols in (11) are given by [11] X [i+1] [i+1] where λedec (dsk [l]) = λedec,p (dsk [l]) o o n n [i] [i] [i] p ˜ ˜ = E a[l1 ]|z, d , ak [l] = E ak [l]|z, d k and [i] (ak [l]∗ ak′ [l])[i] = (ak [l][i] )∗ ak′ [l][i] + Σa[l1 ] . [i+1] [i+1] ′ k,k λedec (dpk [l]) = λedec,p (dpk [l]) Without loss of generality we have considered the l1 = respectively. The subscript dec, p indicates that the corre⌊l/(Ld + Lp )⌋-th signaling interval of each block in a. The sponding EXT values are computed by the pth decoder. Eisymbol ⌊·⌋ denotes the largest integer not larger than the ther the a posterior LLR Λdec (dk [l])[i+1] or the EXT LLR argument. The conditional distribution of a[l1 ] given z and [i+1] λedec (dk [l]) in (11) is then forwarded to the SISO-SAGE ˜ [i] is Gaussian with expectation [10] d scheme. n o ˜ [i] = 1/N0 Σ[i] E a[l1 ]|z, d a[l1 ]
(l1 +1)(Ld +Lp )
X
˜ [i] z[l] D[l]
(9)
l=l1 (Ld +Lp )+1
and covariance matrix [i] Σa[l1 ]
h = N0 N0 Σ−1 a[l1 ] +
(l1 +1)(Ld +Lp )
X
l=l1 (Ld +Lp )+1
˜ [i] R[l]D[l] ˜ [i] D[l]
i−1
.
C. Iterative Receiver Structure With the SISO-SAGE based JCE scheme derived in Section IV-A and the SISO-SU decoder presented in Section IV-B we are now ready to present an iterative receiver architecture that works as follows: From iteration i through i+Ld , the code symbols in corresponding signal intervals of user k, k = κ[i], are updated. First, the SISO-SAGE algorithm performs MMSE
z
10 0
˜ ′[i] d
Soft MAI Cancellation for user k
λeSAGE (d′k )[i+1]
λeSAGE (dk )[i+1]
Π−1 k
b bk
SISO SU Decoder for user k
λedec (dk )[i+1]
Init I: Kmax = 34
10−1 K=16
˜¯ ′ [l][i] d k
10−2 Λdec (dk )[i+1]
k = κ[i] mod K + 1
APP or EXT values
ap
˜ )[i+1] (d k
˜ [i] only Update d k i+1→i APP or EXT values
Πk
APP
10−3
SD e
˜ )[i+1] (d k
Πk
EXT values
SD
10−4
d[0]
Fig. 1. Block diagram of the proposed iterative receiver architecture embedding the SISO-SAGE algorithm (SD: Soft Decision; ap: a posteriori).
channel estimation (9) for all the users in parallel and mitigates the k th user’s multiple access interference (MAI) according to (10). The more reliable probabilities are then de-interleaved and fed into a SISO-SU decoder providing soft information on the symbols that are fed back to the SISO-SAGE device, which might be APP or EXT values. Since the JCE scheme requires soft decision (SD) on the code symbols rather than their LLRs, the output LLRs of the SISO-SU decoder are fed into a SD device computing tanh( 12 Λdec(dk [l])) APP d˜k [l] = . tanh( 12 λedec (dk [l])) EXT values ≈ E {dk [l]|z} The result is interleaved and fed back to the SISO-SAGE scheme. The resulting receiver architecture is depicted in Fig. 1. D. Initialization Issues
Init II: Kmax = 31
BER
[i] Channel a Estimation
0
SU, known chan. EXT /EXT (Init I) AP P/AP P (Init I) EXT /AP P (Init I) AP P/EXT (Init I) EXT /EXT (Init II) AP P/AP P (Init II) EXT /AP P (Init II) AP P/EXT (Init II)
1
2
5 8 6 7 3 4 γ¯b,1 = γ¯b,2 =, · · · , = γ¯b,K [dB]
9
10
Fig. 2. Bit-error-performance of the iterative receiver in quasi-static flat Rayleigh fading for different kinds of feedback information and initialization.
length Nc = 8 are assigned to different users. For the sake of simplicity, the received amplitudes are only estimated once after all symbols of one user have been updated. In a socalled stage of the iterative process all symbols of all users are updated once. In Fig. 2 we compare the average BER performance BER of the proposed receiver for different initialization and soft information fed back to the SISO-SAGE scheme. All users are received with the same average signal-to-noise ratio γ¯b = γ¯b,k , k = 1, . . . , K. The labels SIa /SIi indicate that the soft information SIa and SIi are fed back, respectively, to the channel estimator and the soft IC device in the SISOSAGE scheme. The iterative process is terminated when
[i] 2 1 ˜ [i+1] ˜ ≤ 10−5 or 20 stages have been −d either
d LK
The proposed iterative receiver is initialized as follows: [0] • Init I: The initial estimate of the received amplitudes a is the MMSE estimate of a given the observation z and the user specific midamble symbol vector dp . Based on a[0] and z, a linear MMSE multiuser detector computes the initial symbol vector d[0] . • Init II: The initial channel estimate is the least squares estimate of a instead. The output signal of the k th user’s [i] ∗ matched filter output, weighted by ak , k = 1, . . . , K, forms the initial symbol vector estimate d[0] . V. N UMERICAL E XAMPLES A theoretical analysis of the proposed iterative receiver architecture is cumbersome. Hence, we resort to Monte Carlo simulations to evaluate the performance of the receiver. The system parameters are chosen as follows: All users employ the same rate R = 1/3 PCCC with generator matrix G = (1, 58 /78 , 58 /78 ). Each codeword contains M = 3000 symbols. Each block consists of Ld = 150 code symbols and Lp = 6 pilot symbols. Random signature waveforms of
performed. For medium system load, e.g. KR/Nc = 0.66˙ with K = 16, it can be seen that for a fixed SIi , the iterative receiver with SIa = EXT outperforms the one with SIa = AP P over the entire range of γ¯b . For given SIa , in contrast, the performance of the iterative receiver is roughly independent of SIi in the high γ¯b region. The performance of the iterative receiver with Init I is only slightly better than that with Init II in the medium loaded system. It can also be seen that best performance is achieved with EXT /AP P feedback. The iterative receiver can support a maximum system load of KR/Nc = 1.42 (Kmax = 34) for Init I and KR/Nc = 1.29 (Kmax = 31) for Init II, however, with AP P/AP P feedback only. The behavior of the receiver varies with the system load. The best performance is achieved with EXT /AP P feedback for a medium system load, while the receiver with AP P/AP P feedback performs best and can support more users in a high loaded system. Fig. 3 shows the convergence behavior of the iterative receiver for different kinds of feedback information and initialization in a system with K = 16 users and SNR γ¯b = 6 dB.
APP, which is fed back to the IC device, also incorporates a significant amount of uncancelled interference when the system load is high.
10 0 SU, known chan. EXT /EXT (Init I) AP P/AP P (Init I) EXT /AP P (Init I) AP P/EXT (Init I) EXT /EXT (Init II) AP P/AP P (Init II) EXT /AP P (Init II) AP P/EXT (Init II)
10−1
BER
10−2
VI. S UMMARY
10−3
10−4
10−5 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Number of stages
Fig. 3. Convergence behavior of the iterative receiver for different kinds of feedback information and initialization: K = 16, γ ¯b = 6 dB.
10 0
K K K K K K K K
10−1
EXT /EXT AP P/AP P EXT /AP P AP P/EXT EXT /EXT AP P/AP P EXT /AP P AP P/EXT
10−3
10−4
1
10
20
30 Number of stages
40
C ONCLUSIONS
R EFERENCES
BER
10−2
= 24, = 24, = 24, = 24, = 30, = 30, = 30, = 30,
AND
The SAGE algorithm has been extended to read and provide soft information. This so-called SISO-SAGE algorithm has been applied to joint channel estimation and successive interference cancellation. The resulting receiver exchanges soft information on the code symbols with single-user (SU) decoders. Monte Carlo simulations in quasi-static flat Rayleigh show that the iterative receiver is robust against channel estimation error and depending on the kind of initialization, can support a system load of KR/Nc > 1.3. Simulation results further indicate that soft information fed back to the channel estimator and the interference cancellation (IC) device mainly influences, respectively, the BER performance and the convergence rate of the iterative receiver. It is worth mentioning that the iterative receiver supports the highest system load when APP is fed back to both channel estimator and IC device. Ongoing research focuses on analyzing the properties of the SISO-SAGE algorithm.
50
Fig. 4. Convergence behavior of the iterative scheme for different system loads: Init I, γ ¯b = 7 dB.
It can be seen that for a given initialization scheme and a fixed SIa , the iterative receiver with SIi = AP P converges faster than that with SIi = EXT does. In contrast for fixed SIa and SIi , the receiver with Init I converges faster than that with Init II. Clearly, Init I provides more accurate initial symbol estimates than Init II does. As shown in Fig. 4, the convergence rate decreases as the system load increases. This effect can partially be explained by the increasing amount of MAI in the system. Limiting the number of stages to 50, the iterative receiver can support the loads previously mentioned with AP P/AP P feedback (fast convergence) while that with EXT /AP P (slow convergence), cannot. One possible explanation for this behavior is that
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