An Extrinsic Kalman Filter for Iterative Multiuser ... - Semantic Scholar
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An Extrinsic Kalman Filter for Iterative Multiuser Decoding Lars K. Rasmussen, Alex J. Grant, and Paul D. Alexander
Abstract One powerful approach for multiuser decoding is to iterate between a linear multiuser filter (which ignores coding constraints) and individual decoders (which ignore multiple-access interference). Subject to clearly formulated statistical assumptions and the history of input signals provided by the outer decoders over all previous iterations, an extrinsic Kalman filter is suggested. This approach is motivated by the recent observation that decoder outputs are loosely correlated during initial iterations. Numerical results show that iterative decoding using this filter provides better performance in terms of the supportable load and convergence speed as compared to previously suggested linear-filter-based iterative decoders. Index Terms Multiple-access, multiuser detection, iterative decoding, recursive filters
I. I NTRODUCTION A multiple-access channel is a communications channel in which several independent users transmit information to a common receiver. In certain cases it is not possible, or undesirable to provide orthogonality between the users. This causes multiple-access interference (MAI), which limits the performance of any receiver which does not take it into account. The optimal (error probability minimizing) multiple-access receiver must make joint decisions, rather than treating the multiple-access interference as uncontrollable noise. The multiple-user detection problem is difficult. In general, optimal detection [1] is NP-complete [2]. To make things worse, information theory tells us that multiple-user coding strategies should in fact be used [3, Ch. 14]. Once again, complexity is the limiting factor. Brute force joint decoding of a random K user code is exponentially complex in both the codeword length and the number of users. Optimal decoding for convolutionally encoded users This work was supported in part by the Swedish Research Council for Engineering Sciences under grants no. 217-1997-538, 621-2001-2976, 621-2002-4533 and the Australian Government under ARC Grant DP0344856. L. K. Rasmussen has dual appointments with the Institute for Telecommunications Research, University of South Australia, Mawson Lakes, SA 5095, Australia (email: [email protected], FAX:+61 8 8302 3873) and with the Department of Computer Engineering, Chalmers University of Technology, SE-412 96 G¨oteborg, Sweden (email: [email protected], FAX: +46 31 772 3663). A. J. Grant and P. D. Alexander are with the Institute for Telecommunications Research, University of South Australia, Mawson Lakes, SA 5095, Australia (email: [email protected], [email protected], FAX:+61 8 8302 3873).
Submitted to IEEE Trans. Inform. Theory 30 December 2002. Revised 29 May 2003 and 12 September 2003.
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was considered in [4], resulting in a trellis with complexity growing exponentially with both the number of users and the constraint length of the code. One approach to the problem of coding for the multiple-access channel is the use of single-user codes together with sub-optimal, reduced complexity joint decoding methods which attempt to approximate the effect of optimal joint decoding. Of particular interest is iterative multiuser decoding. The basic principle behind iterative decoding is to decode independently with respect to the various constraints imposed on the received signal, rather than considering them jointly. The overall constraint is accommodated by iteratively passing extrinsic information between the individual decoders. For multiuser decoding, there are constraints due to the multiple-access channel and due to the individual users’ encoders. Iterative multiuser decoding therefore separates multiuser channel decoding and error control code decoding. Multiuser channel decoding refers to an estimator that performs inference on the users’ signals without regard to the time domain structure imposed by forward error correction coding. The term “multiuser detection” is avoided in this context, since that term is more properly associated with hypothesis testing, in which one of a number of decisions must be made. The information usually exchanged is a posteriori probability (APP) on bits or symbols, determined by separate APP decoders. Computationally efficient algorithms exist for APP decoding of good codes with suitable trellis structure, but the complexity of the multiuser channel APP decoder remains exponential in the number of users. From a practical point of view, decoders, or estimators with linear (or at worst polynomial) growth in complexity are preferred. A variety of reduced-complexity iterative decoders exist in the literature, mostly based on different types of multiuser channel decoding. Of particular interest is the use of linear filters and cancellation strategies. The linear soft parallel interference cancellation (PIC) strategy suggested in [5] was further developed in [6]. Cancellation followed by instantaneous linear minimum mean squared error (LMMSE) filtering was proposed in [7]. This structure is identical to the feed-forward feed-back LMMSE filter in [8]. An attempt to create a unifying framework for describing these approaches has been presented in [9]. An interesting improvement to the PIC is given in [10, 11] where partial cancellation as proposed in [12] for uncoded transmission is introduced. The main result of this paper, is an extrinsic Kalman filter with potentially superior characteristics. This recursive filter yields estimates based on the received signal and all the successive outputs provided by the error control decoders over all previous iterations. The approach is motivated by the recent observation that these estimates are loosely correlated during initial iterations [10, 11]. Notations: x ∈ S n shall be a column n-vector with elements xi = (x)i , chosen from the set S, commonly the reals R. The space of probability n-vectors (length n non-negative vectors that sum to 1) is denoted Pn . Similarly, X ∈ S m×n is a m by n matrix with elements xij ∈ S. The superscript
t
denotes the transpose operator. For
random vectors x and y, E [x] is the expectation, var x = E [xt x] the variance, and cov x = hx, xi = E [xxt ] the covariance, respectively. Likewise cov(x, y) = hx, yi = E [xyt ].
Submitted to IEEE Trans. Inform. Theory 30 December 2002. Revised 29 May 2003 and 12 September 2003.
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II. S YSTEM M ODEL AND I TERATIVE D ECODER Consider the K-user linear multiple-access system of Figure 1. Let user k = 1, 2, . . . , K encode their binary information sequence bk [l] using a rate R code C, to produce the coded binary sequence dk [i]. Transmission occurs st1
s1
b1
- Encoder C
d1
- π1
.. .
bK
Fig. 1.
- Encoder C
u1
-
.. .
dK
- πK uK -
Mapper .. .
Mapper
? - × J J .. JJ ^
? - × - y1
x1
sK
.
+
6
? xK n
- ×
•
stK
.. .
? - × - yK
The transmission system model.
in blocks of L code bits per user, corresponding to KLR information bits. Each user independently permutes their encoded sequence with an interleaver πk . The sequence output from the interleaver of user k is uk [i], where i = 1, 2, ..., L is the symbol time index. The interleaved code bits uk [i] are then memorylessly mapped onto a binary phase-shift keyed (BPSK) constellation, B = {−1, 1}, giving sequences of modulated code symbols xk [i]. Extension to higher order signalling constellations is straightforward, BPSK is chosen only for simplicity. At symbol time i, each user transmits sk [i]xk [i], the multiplication of xk [i] with the real N -chip spreading N
sequence1 , sk [i] ∈ {−1, 1} . Spreading sequences with period much longer than the data symbol duration are modelled by letting each element of sk [i] be i.i.d. over users and time. For conceptual ease only, users are symbol synchronized, transmit over an additive white Gaussian noise (AWGN) channel, and are received at the same power level. These assumptions however are not required for the following development. Given symbol-synchronicity, the chip-match filtered received vector r[i] ∈ RN at symbol time i = 1, 2, . . . , L is r[i] = S[i]x[i] + n[i],
(1)
where S[i] = (s1 [i], s2 [i], . . . , sK [i]), is a N × K matrix with the spreading sequence for user k as column k. The vector x[i] ∈ B K has elements xk [i] and the vector n[i] ∈ RN is a sampled i.i.d. Gaussian noise process, with cov n[i] = σ 2 I. It will not be required to identify specific symbol intervals, so symbol indices will be omitted. For later use, t
define the following notation, xk¯ = (x1 , x2 , ...xk−1 , xk+1 , ..., xK ) to indicate deletion of user k from x. Likewise, Ik¯ denotes a K − 1 × K matrix formed by deleting row k from the K × K identity matrix. 1 Although
this model is cast in terms of direct-sequence code-division multiple-access, the sequences sk [i] may be used to represent any
type of complex vector modulation, thus admitting a general linear multiple-access system model [13].
Submitted to IEEE Trans. Inform. Theory 30 December 2002. Revised 29 May 2003 and 12 September 2003.
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Application of the “turbo-principle” to the coded multiple-access system just described, where for each user, the error control code is considered as one constraint and the multiuser channel (1) as the other constraint, results in the canonical iterative multiuser of Fig. 2 [14–17]. This is a user-by-user structure where each user has a separate decoder branch. The outputs following each full iteration are however shared among all the decoder branches. r
?
(n)
p1
-
(n−1) q1
-
Multiuser APP
π1−1
-
.. .
-
APP C
.. .
- q(n) 1
π1
.. .
(n)
Fig. 2.
pK
-
(n−1) qK
-
−1 πK
-
-
APP C
- q(n) K
πK
Canonical iterative multiuser decoder.
At each iteration, the multiuser APP decoder takes as input the received signal r and the set of extrinsic (n−1)
from user k = 1, 2, . . . , K calculated in the previous iteration. The extrinsic probability
probabilities qk distribution
(n−1) qk [i]
∈ P|B| is on the transmitted symbols xk [i] ∈ B of user k. With a small abuse of the (n)
(n)
vector subscripting notation, (qk [i])xk is the element of the vector qk [i] corresponding to user k transmitting the symbol xk ∈ B at time i (technically these subscripts should be integers corresponding to an index mapping between B and 0, 1). (n)
The multiuser APP calculates the updated extrinsic probability vector pk [i] for user k and after appropriate (n)
de-interleaving, the extrinsics pk
are used as priors for independent APP decoding of the code C by each user.
These decoders are standard implementations of the forward-backward algorithm [18] on the trellis associated with (n)
C. These decoders produce (after interleaving) the extrinsics qk , which serve as priors for the subsequent iteration. (n)
Calculation of pk [i] requires summation over |B|K−1 terms, which limits the practical application of this receiver. Because of this prohibitive complexity, many lower-complexity alternatives have been proposed while retaining the same basic architecture. Of particular interest are structures that replace the multiuser APP block (n)
with a bank of linear filters as shown in Fig. 3. In this structure, there is a bank of filters Λk , one for each user. The coefficients of these filters may be re-computed every iteration. For the first iteration, n = 1, the input (1)
to each filter Λk
is just the received signal r. For subsequent iterations n = 2, 3, ..., the input to the filter
for user k is the received signal r and a set of signal estimates for all the other users from previous iterations, (m)
{ˆ xk′
: k ′ 6= k, m ∈ M}, where M ⊆ {1, 2, . . . , n − 1} is a set defining the memory order of the iteration.
Submitted to IEEE Trans. Inform. Theory 30 December 2002. Revised 29 May 2003 and 12 September 2003.
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(m)
{ˆ xk
: k 6= 1, m ∈ M}
-
(n)
(m)
{ˆ xk
(n)
- T
.. .
r
Fig. 3.
x ˜1
(n) Λ1
p1
.. . (n)
-
.. .
- π1
APP C
.. .
- T
q1
- U
.. .
(n)
x ˜K
(n) ΛK
(n)
- π1−1
- xˆ(n) 1
.. . (n)
pK
−1 - πK
-
- πK
APP C
qK
- U
- xˆ(n) K
: k 6= K, m ∈ M}
Iterative multiuser decoder with linear multiuser estimation.
Typically in the literature, M = {n − 1}, although recently M = {n − 1, n − 2} has been considered [10, 11]. (m)
These input signals can in some sense be thought of as extrinsic information since the estimates x ˆk
for user k,
have been excluded. (n)
The output of the filter Λk
(n)
is an updated sequence of estimates x ˜k
of the corresponding code symbol for user
k. These estimates are then mapped from the signal space onto the probability vector space using a symbol-wise (n)
mapping T : R 7→ P|B| . The resulting sequence of probability vectors pk
are used as priors for individual APP (n)
decoding of the code C. These decoders can output either posterior or extrinsic probabilities qk . The sequence of (n)
probability vectors qk
is in turn mapped back onto the signal space by a symbol-wise function U : P|B| 7→ R. (n)
(n)
Typically, T calculates the vectors pk assuming that x ˜k is Gaussian distributed with known mean and variance, (n) ˜(n) (n) x ˜k ∼ N µ ˜k , ζk . Likewise, a common choice for U is the conditional mean. Design of improved maps T
and U is an interesting topic, but is outside the scope of this paper, where the focus is on the exploitation of all
the successive outputs from the decoders, rather than just the most recent. (n)
The use of posterior and extrinsic probabilities qk as feedback information has been investigated in the literature, e.g. [9, 11]. When sub-optimal multiuser detection strategies are used in place of multiuser APP detection, it is however still an open problem whether posterior or extrinsic probabilities should be used. For the numerical examples presented here, posterior probabilities have been observed to provide better performance in conjunction with linear filters. The conclusive solution to this problem is however, outside the scope of this paper. III. E XTRINSIC K ALMAN F ILTER F OR M ULTIUSER E STIMATION In the literature, work on linear filters for iterative decoding has focussed on LMMSE filtering [7, 8, 19, 20], linear interference cancellation [5, 6, 9, 21–23], and linear weighted cancellation [10, 11]. Furthermore, these filters have been designed based on the received signal r and the most current code symbol estimates of the interfering users (n)
(1)
(2)
(n)
ˆ k¯ . After n iterations, there are however a sequence of such estimates available, namely {ˆ ˆ k¯ , ..., x ˆ k¯ } x xk¯ , x together with r. It has been observed that the estimates are not strongly correlated during the initial iterations [10,
Submitted to IEEE Trans. Inform. Theory 30 December 2002. Revised 29 May 2003 and 12 September 2003.
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11], so in principle, using all the previous estimates in the filter design could improve performance. The main result (n)
of this paper is the derivation of an extrinsic Kalman filter Λk
for use in the iterative decoder of Fig. 3 that
outputs the LMMSE estimate for xk , given r and the output of all the APP decoders for every previous iteration. We call this filter an extrinsic Kalman filter, since it does not use previous estimates of xk in its update procedure. (n)
Consider the following recursively defined vector of observations as input to the filter Λk , r n=1 (n) (n−1) (2) ck = ck n = 2, 3, . . . (n−1) x ˆ k¯ D ED E−1 (n) (n) (n) (n) Direct application of the LMMSE design criterion results in Λk = xk , ck ck , ck . It is clear however (n)
that Λk
grows in dimension with n, which is impractical. Application of the principles of Kalman filtering [24,
25], or recursive Bayesian LMMSE estimation [26], solves this dimensionality problem by giving a recursive form (n)
for Λk , subject to the following assumptions. The received signal is r = Sx + n, according to (1) where n is Gaussian with cov n = σ 2 I, and σ 2 and
A1
S are known. (n)
ˆ k¯ The interleaved code symbol estimates of the interfering users x
A2
coming out of the single user APP
decoders can be written (n)
x ˆk (n)
where it is assumed that vˆk
(n)
= xk + vˆk
(3)
is uncorrelated with x and also uncorrelated over time and iterations, but
not over users at a given iteration,
(n)
The matrix Qk
A3
(n)
Let ck
(n)
defined as Qk
E D (n) =0 x, vˆk E 0 n 6= m D (n) (m) = vˆk , vˆj q n=m kj D E (n) (n) ˆ k¯ , v ˆ k¯ = v with elements determined by (5), is known.
(4) (5)
be according to (2). Then the system has trivial state evolution x(n) = x and observations Sx + n n=1 (n) (n) (n) y = Hk x + z = (n−1) I¯ x + I¯ v n≥2 k kˆ (n)
The usual Kalman equations, under A1-A3, yield the LMMSE estimate x ˜k
(n)
of xk given ck
as the k-th element
of (n)
˜ (n) = x ˜ (n−1) + Mk x
(n)
˜ (n−1) y(n) − Hk x
Submitted to IEEE Trans. Inform. Theory 30 December 2002. Revised 29 May 2003 and 12 September 2003.
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where (n) Mk (n)
Wk
(n)
Rk (0)
=
(n−1) Wk
(n) Hk
t
(n) (n−1) Hk Wk
(n) Hk
t
+
(n) Rk
−1
(n−1) (n) (n) Wk = I − Mk Hk σ 2 I n = 1, = cov z(n) = Q(n) n > 1. k
˜ (0) = 0. with initial conditions Wk = I and x
Figure 4 shows how the filter exploits feedback to avoid growing dimensionality. Observe that the filter for each user is required to estimate the code symbols for the interfering users based on the user-specific input signals to the user-specific filter. These estimates are for internal filtering use only and are not available in the external iterative structure.
(n = 1)
r ˆ (n−1) x ¯ k
H H•- + • − 6 (n > 1) •H
(n)
Mk
- x˜(n) k
de - + - mux 6
˜n x
? z
(n)
Hk
Fig. 4.
−1
˜ n−1 x
(n)
(n−1)
ˆ k¯ The extrinsic Kalman filter Λk . For n = 1 the input signal is r while for n ≥ 2 the input signal is x
.
Assumptions A1-A3 yield a tractable filter design problem and are the key to obtaining the convenient iterative structure. In practice, these assumptions hold only approximately, and some performance loss can be expected due to this inaccuracy in modelling. The filter is referred to as the extrinsic Kalman filter since the estimate for user k from the error control decoders is excluded by choice from the observation vector input to the filter. This is not a requirement of the Kalman filter derivation, rather it is neccessary for good performance of the iterative decoder. Similar deletion of user k has been widely used in the litertature with cancellation/filtering based receivers. It is envisaged that many other simpler filter structures exploiting all or part of the iteration history of the decoder could be investigated. The Kalman filter just described provides however a useful benchmark, just as the post-cancellation LMMSE based iterative decoder provides a reference for first-order iterative decoders. Submitted to IEEE Trans. Inform. Theory 30 December 2002. Revised 29 May 2003 and 12 September 2003.
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IV. N UMERICAL R ESULTS In this section numerical results are used to illustrate the performance benefits obtained by the proposed filter. For the purposes of simulation, each user applies the maximum free distance four state convolutional code (with generators (5, 7)) mapped onto BPSK. Each user therefore transmits one bit per two channel uses. Binary spreading sequences with length N = 8 were generated i.i.d. at each symbol for each user. Transmission is symbol(n)
synchronous and all users are received at the same power level. The matrix Qk
is determined based on decision-
directed estimation in the simulations. The performance of the new recursive filter (RF) is compared to the parallel interference canceller (PIC) [6], the improved parallel interference canceller (IPIC) [10, 11], and the LMMSE filtered parallel interference canceller (FPIC) [7, 8]. In Figure 5(a), the bit error rate (BER) performances for the four cases are shown as functions of the load in terms of the number of active users. Here, the number of iterations is limited to five and the performance is captured for Eb /N0 = 5 dB. Observe that the PIC can support up to 8 users and still provide close to single user performance. The IPIC can support 9–10 users, while the FPIC and the RF can support 12–13 users. As predicted, the RF provides better performance than other linear estimators, although it is only marginally better than the FPIC after five iterations. In Figure 5(b), the BER performances are shown after ten iterations as functions of the load. The PIC can now support 9 users, the IPIC 12–13 users, while the FPIC and the RF both can support up to 14 users. After ten iterations, the advantage of the RF over the FPIC is still marginal, although becoming more significant. The BER performances after twenty iterations are shown in Figure 5(c) as functions of the load. The PIC can still only support 9 users, however, for more iterations, 10 users may be supported. The IPIC can comfortably support 13 users, while the FPIC has not improved with iterations and thus can still support 14 users. In contrast, the RF has improved with iterations and can now support 14–15 users with close to single user performance. In order to investigate the convergence behavior for each system, the BER performance is now given as a function of the number of iterations at Eb /N0 = 5 dB for a load close to the maximum for the estimator in question. In Figure 6(a), the BER performance for the PIC as a function of the number of iterations is shown. As was the case in the previous figures, the PIC can support 8 users after 5 iterations, 9 users after 7–8 iterations and possibly 10 users for a large number of iterations. However, 11 users cannot be supported for any number of iterations. In Figure 6(b), the corresponding plot for the IPIC is shown. Again, the supported loads observed in previous figures are confirmed. The IPIC can support 9–10 users after 5 iterations, 11 users after 7 iterations, 12 users after 9 iterations and 13 users after 11 iterations. It is also possible that 14 users can be supported for a large number of iterations, however, 15 users cannot be supported even for a large number of iterations. The convergence behavior for the FPIC is shown in Figure 6(c). Here, it is observed that 11 users can be supported after only 4 iterations, 12 users after 5 iterations, 13 users after 6 iterations and 14 users after 9 iterations. It is, however, not likely that the FPIC can support 15 users even for a large number of iterations. Finally, the convergence behavior for the RF is demonstrated in Figure 6(d). As for the FPIC, 12 users can be supported after 5 iterations, 13 users after 6 iterations and 14 users after 9 iterations. Note that 15 users can be Submitted to IEEE Trans. Inform. Theory 30 December 2002. Revised 29 May 2003 and 12 September 2003.
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9
10
BER
10
10
10
10
0
10 PIC IPIC FPIC RBF
−1
10
−2
10
BER
10
−3
10
−4
10
−5
6
7
8
9
10
11
12
13
14
15
10
16
0
PIC IPIC FPIC RBF
−1
−2
−3
−4
−5
6
7
8
9
10
Load K (a) 5 iterations
10
10
10
BER
11
12
13
14
15
16
Load K
10
10
10
(b) 10 iterations
0
PIC IPIC FPIC RBF
−1
−2
−3
−4
−5
6
7
8
9
10
11
12
13
14
15
16
Load K (c) 20 iterations
Fig. 5.
BER performance vs number of users at Eb /N0 = 5 dB with N = 8.
supported at only a small number of iterations above 20, providing an extra user at a processing gain of N = 8. It can, however, also be observed that 16 users can most likely not be supported even for a large number of iterations. All of the above results have been for Eb /N0 = 5 dB. The following results consider the convergence behavior for a range of Eb /N0 and a number of iterations. In Figure 7(a), the performance of the PIC as a function of the Eb /N0 with K = 9 users is shown for m = 1 − 5, 7, 10, 20 iterations, respectively. The BER performance improves with the number of iterations up to 7 iterations, where close to single user performance is achieved at Eb /N0 = 5 dB. For an increasing number of iterations, single user performance is achieved for lower Eb /N0 . After 10 iterations, single user performance is achieved at Eb /N0 = 4.5 dB, and after 20 iterations at Eb /N0 = 4.0 dB. This behavior can potentially continue
Submitted to IEEE Trans. Inform. Theory 30 December 2002. Revised 29 May 2003 and 12 September 2003.
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10
10
BER
10
10
10
0
10
−1
10 K=8 K=9 K=10 K=11
−2
10
BER
10
−3
−4
1
10
10
2
4
6
8
10
12
14
16
18
0
−1
−2
−3
K=10 K=11 K=12 K=13 K=14 K=15
−4
20
1
2
4
6
8
Iterations (a) PIC
10
BER
10
10
10
0
10
−1
10
K=11 K=12 K=13 K=14 K=15
−2
10
−3
−4
1
10
10
2
4
6
8
10
14
16
18
20
12
14
16
18
20
18
20
0
−1
K=12 K=13 K=14 K=15 K=16
−2
−3
−4
1
2
4
6
Iterations (c) FPIC
Fig. 6.
12
(b) IPIC
BER
10
10
Iterations
8
10
12
14
16
Iterations (d) RF
BER performance vs number of iterations at Eb /N0 = 5 dB with N = 8.
for an increasing number of iterations down to Eb /N0 =3.0–3.5 dB where the threshold for the waterfall region appears to be. Similar behavior is observed for the IPIC in Figure 7(b) for K = 12 users. Here, single user performance is achieved at Eb /N0 = 6.0 dB after 7 iterations, at Eb /N0 = 5.0 dB after 10 iterations and at Eb /N0 = 4.0 dB after 20 iterations. Again, it is likely that single user performance can be achieved down to Eb /N0 = 3.0 − 3.5 dB as the number of iterations increase. In Figure 7(c) and Figure 7(d), respectively, the SNR convergence behavior for the FPIC and the RF are shown for K = 14 users. As previously noted, similar behavior is observed for 10 and 20 iterations where single user performance is achieved at Eb /N0 = 5.0 dB and Eb /N0 = 4.5 dB, respectively. It is, however, interesting to see
Submitted to IEEE Trans. Inform. Theory 30 December 2002. Revised 29 May 2003 and 12 September 2003.
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that after 7 iterations, the RF is at single user performance at Eb /N0 = 5.5 dB and very close at Eb /N0 = 5.0 dB, where the FPIC is not at single user performance before Eb /N0 = 6.0 dB. The advantage of the RF at a low number of iterations increases as the Eb /N0 increases as can be seen by the steeper BER slop at Eb /N0 = 6.0 dB after 5 iterations.
10
BER
10
10
10
10
10
0
10
−1
10
−2
10
−3
BER
10
m=1 m=2 m=3 m=4 m=5 m=7 m=10 m=20
−4
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−6
2
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3
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4
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m=1 m=2 m=3 m=4 m=5 m=7 m=10 m=20
−5
−6
2
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3
3.5
Eb /N0 (dB) (a) PIC, K = 9
10
BER
10
10
10
10
10
0
10
−1
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−2
10
−3
m=1 m=2 m=3 m=4 m=5 m=7 m=10 m=20
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2
2.5
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0
−1
−2
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m=1 m=2 m=3 m=4 m=5 m=7 m=10 m=20
−5
−6
2
2.5
3
Eb /N0 (dB)
3.5
4
4.5
Eb /N0 (dB)
(c) FPIC, K = 14
Fig. 7.
4.5
(b) IPIC, K = 12
BER
10
4
Eb /N0 (dB)
(d) RF, K = 14
BER performance vs Eb /N0 dB, m =1–5,7,10,20.
V. C ONCLUDING R EMARKS A Kalman filter has been described for use in iterative multiuser decoding. Subject to certain statistical assumptions, it delivers the LMMSE estimate of each user based on the received signal and the entire history of outputs from the single user decoders. Since estimates of the code symbols pertaining to user k is excluded from Submitted to IEEE Trans. Inform. Theory 30 December 2002. Revised 29 May 2003 and 12 September 2003.
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the observation vector input to the filter for user k, the resulting filter is called an extrinsic Kalman filter. This filter was shown to perform better than other suggested structures in all cases investigated. At a processing gain of N = 8, an extra user can be supported for a large number of iterations, as compared to the LMMSE filtered parallel interference cancellation. This is interesting as such a gain is expected to increase for larger processing gains. For a small number of iterations, the new filter also provides better BER performance. This advantage becomes significant for increasing Eb /N0 as the slope of the BER curves for a low number of iterations is steeper than for the other linear estimators. This is expected since it has previously been observed that the estimates provided by the single user decoders are loosely correlated for the first few iterations, thus the output of each iteration can potentially provide additional information, which otherwise would have been ignored. The principles described here are also applicable to other system components and technologies such as spacetime coding, equalization, coded modulation, and joint channel estimation/decoding. Another interesting avenue of further investigation would be the development of other, simpler structures which approximate the extrinsic Kalman filter, yet retain the benefit due to the use of all, or part of the iteration history. R EFERENCES [1] S. Verd´u, “Minimum probability of error for asynchronous Gaussian multiple–access channels,” IEEE Trans. Inform. Theory, vol. 32, no. 1, pp. 85–96, Jan. 1986. [2] ——, “Computational complexity of optimum multiuser detection,” Algorithmica, pp. 303–312, May 1989. [3] T. M. Cover and J. A. Thomas, Elements of Information Theory.
New York: John Wiley, 1991.
[4] T. R. Giallorenzi and S. G. Wilson, “Multiuser ML sequence estimator for convolutionally coded asynchronous DS-CDMA systems,” IEEE Trans. Commun., vol. 44, no. 8, pp. 997–1008, Aug. 1996. [5] J. Hagenauer, “Forward error correcting for CDMA systems,” in IEEE Int. Symp. Spread Spectrum Techn. App., Mainz, Germany, Sept. 1996, pp. 566–569. [6] P. D. Alexander, A. J. Grant, and M. C. Reed, “Iterative detection on code-division multiple-access with error control coding,” European Trans. Telecommun., vol. 9, no. 5, pp. 419–426, Sept.-Oct. 1998. [7] X. Wang and V. Poor, “Iterative (turbo) soft interference cancellation and decoding for coded CDMA,” IEEE Trans. Commun., vol. 47, no. 7, pp. 1046–1061, July 1999. [8] H. El-Gamal and E. Geraniotis, “Iterative multiuser detection for coded CDMA signals in AWGN and fading channels,” IEEE J. Selected Areas Commun., vol. 18, no. 1, pp. 30–41, Jan. 2000. [9] J. Boutros and G. Caire, “Iterative multiuser joint decoding: Unified framework and asymptotic analysis,” IEEE Trans. Inform. Theory, pp. 1772–1793, July 2002. [10] S. Marinkovic, B. S. Vucetic, and J. Evans, “Improved iterative parallel interference cancellation for coded CDMA systems,” in IEEE Int. Symp. Inform. Theory, Washington D.C., USA, June 2001, p. 34. [11] S. Marinkovic, B. S. Vucetic, and A. Ushirokawa, “Space-time iterative and multistage receiver structures for CDMA mobile communication systems,” IEEE J. Selected Areas Commun., vol. 19, pp. 1594–1604, Aug. 2001. [12] D. Divsalar, M. K. Simon, and D. Raphaeli, “Improved parallel interference cancellation for CDMA,” IEEE Trans. Commun., vol. 46, no. 2, pp. 258–268, Feb. 1998. [13] L. K. Rasmussen, P. D. Alexander, and T. J. Lim, “A linear model for CDMA signals received with multiple antennas over multipath fading channels,” Chapt. 2 in: CDMA Techniques for 3rd Generation Mobile Systems, edited by F. Swarts, P. van Rooyen, I. Oppermann and M. L¨otter, Kluwer Academic Publisher, Sept. 1998. [14] M. C. Reed, C. B. Schlegel, P. D. Alexander, and J. Asenstorfer, “Iterative multiuser detection for CDMA with FEC: Near-single-user performance,” IEEE Trans. Commun., vol. 46, pp. 1693–1699, Dec. 1998.
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[15] M. Moher, “An iterative multiuser decoder for near-capacity communications,” IEEE Trans. Commun., vol. 46, no. 7, pp. 870–880, July 1998. [16] P. D. Alexander, M. C. Reed, J. Asenstorfer, and C. B. Schlegel, “Iterative multiuser interference reduction: Turbo CDMA,” IEEE Trans. Commun., vol. 47, pp. 1008–1014, July 1999. [17] L. Brunel and J. Boutros, “Code division multiple access based on independent codes and turbo decoding,” Annales des Telecommunications, vol. 54, no. 7-8, pp. 401–410, Jul.-Aug. 1999. [18] L. R. Bahl, J. Cocke, F. Jelinek, and J. Raviv, “Optimal decoding of linear codes for minimizing symbol error rate,” IEEE Trans. Inform. Theory, vol. 20, pp. 284–287, Mar. 1974. [19] M. L. Honig, G. Woodward, and P. D. Alexander, “Adaptive multiuser parallel-decision-feedback with iterative decoding,” in IEEE Int. Symp. Inform. Theory, Sorrento, Italy, June 2000, p. 335. [20] G. Woodward and M. L. Honig, “Performance of adaptive iterative multiuser parallel decision feedbackwith different code rates,” in IEEE Int. Conf. Commun., Helsinki, Finland, June 2001, pp. 852–856. [21] N. Ibrahim and G. K. Kaleh, “Iterative decoding and soft interference cancellation for the Gaussian multiple access channel,” in Int. Symp. on Signals, Systems, and Electronics, Pisa, Italy, 1998, pp. 156–161. [22] M. Kobayashi, J. Boutros, and G. Caire, “Successive interference cancellation with SISO decoding and EM channel estimation,” IEEE J. Selected Areas Commun., vol. 19, pp. 1450–1460, Aug. 2001. [23] Z. Shi and C. Schlegel, “Joint iterative decoding of serially concatenated error control coded CDMA,” IEEE J. Selected Areas Commun., vol. 19, pp. 1646–1653, Aug. 2001. [24] S. Haykin, Adaptive Filter Theory.
Prentice Hall, 2002.
[25] T. J. Lim and Y. Ma, “The Kalman filter as the optimal linear minimum mean-squared error multiuser CDMA detector,” IEEE Trans. Inform. Theory, vol. 46, pp. 2561–2566, Nov. 2000. [26] D. E. Catlin, Estimation, Control, and the Discrete Kalman Filter.
Springer Verlag, 1989.
Submitted to IEEE Trans. Inform. Theory 30 December 2002. Revised 29 May 2003 and 12 September 2003.
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An Extrinsic Kalman Filter for Iterative Multiuser Decoding Lars K. Rasmussen, Alex J. Grant, and Paul D. Alexander
Abstract One powerful approach for multiuser decoding is to iterate between a linear multiuser filter (which ignores coding constraints) and individual decoders (which ignore multiple-access interference). Subject to clearly formulated statistical assumptions and the history of input signals provided by the outer decoders over all previous iterations, an extrinsic Kalman filter is suggested. This approach is motivated by the recent observation that decoder outputs are loosely correlated during initial iterations. Numerical results show that iterative decoding using this filter provides better performance in terms of the supportable load and convergence speed as compared to previously suggested linear-filter-based iterative decoders. Index Terms Multiple-access, multiuser detection, iterative decoding, recursive filters
I. I NTRODUCTION A multiple-access channel is a communications channel in which several independent users transmit information to a common receiver. In certain cases it is not possible, or undesirable to provide orthogonality between the users. This causes multiple-access interference (MAI), which limits the performance of any receiver which does not take it into account. The optimal (error probability minimizing) multiple-access receiver must make joint decisions, rather than treating the multiple-access interference as uncontrollable noise. The multiple-user detection problem is difficult. In general, optimal detection [1] is NP-complete [2]. To make things worse, information theory tells us that multiple-user coding strategies should in fact be used [3, Ch. 14]. Once again, complexity is the limiting factor. Brute force joint decoding of a random K user code is exponentially complex in both the codeword length and the number of users. Optimal decoding for convolutionally encoded users This work was supported in part by the Swedish Research Council for Engineering Sciences under grants no. 217-1997-538, 621-2001-2976, 621-2002-4533 and the Australian Government under ARC Grant DP0344856. L. K. Rasmussen has dual appointments with the Institute for Telecommunications Research, University of South Australia, Mawson Lakes, SA 5095, Australia (email: [email protected], FAX:+61 8 8302 3873) and with the Department of Computer Engineering, Chalmers University of Technology, SE-412 96 G¨oteborg, Sweden (email: [email protected], FAX: +46 31 772 3663). A. J. Grant and P. D. Alexander are with the Institute for Telecommunications Research, University of South Australia, Mawson Lakes, SA 5095, Australia (email: [email protected], [email protected], FAX:+61 8 8302 3873).
Submitted to IEEE Trans. Inform. Theory 30 December 2002. Revised 29 May 2003 and 12 September 2003.
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was considered in [4], resulting in a trellis with complexity growing exponentially with both the number of users and the constraint length of the code. One approach to the problem of coding for the multiple-access channel is the use of single-user codes together with sub-optimal, reduced complexity joint decoding methods which attempt to approximate the effect of optimal joint decoding. Of particular interest is iterative multiuser decoding. The basic principle behind iterative decoding is to decode independently with respect to the various constraints imposed on the received signal, rather than considering them jointly. The overall constraint is accommodated by iteratively passing extrinsic information between the individual decoders. For multiuser decoding, there are constraints due to the multiple-access channel and due to the individual users’ encoders. Iterative multiuser decoding therefore separates multiuser channel decoding and error control code decoding. Multiuser channel decoding refers to an estimator that performs inference on the users’ signals without regard to the time domain structure imposed by forward error correction coding. The term “multiuser detection” is avoided in this context, since that term is more properly associated with hypothesis testing, in which one of a number of decisions must be made. The information usually exchanged is a posteriori probability (APP) on bits or symbols, determined by separate APP decoders. Computationally efficient algorithms exist for APP decoding of good codes with suitable trellis structure, but the complexity of the multiuser channel APP decoder remains exponential in the number of users. From a practical point of view, decoders, or estimators with linear (or at worst polynomial) growth in complexity are preferred. A variety of reduced-complexity iterative decoders exist in the literature, mostly based on different types of multiuser channel decoding. Of particular interest is the use of linear filters and cancellation strategies. The linear soft parallel interference cancellation (PIC) strategy suggested in [5] was further developed in [6]. Cancellation followed by instantaneous linear minimum mean squared error (LMMSE) filtering was proposed in [7]. This structure is identical to the feed-forward feed-back LMMSE filter in [8]. An attempt to create a unifying framework for describing these approaches has been presented in [9]. An interesting improvement to the PIC is given in [10, 11] where partial cancellation as proposed in [12] for uncoded transmission is introduced. The main result of this paper, is an extrinsic Kalman filter with potentially superior characteristics. This recursive filter yields estimates based on the received signal and all the successive outputs provided by the error control decoders over all previous iterations. The approach is motivated by the recent observation that these estimates are loosely correlated during initial iterations [10, 11]. Notations: x ∈ S n shall be a column n-vector with elements xi = (x)i , chosen from the set S, commonly the reals R. The space of probability n-vectors (length n non-negative vectors that sum to 1) is denoted Pn . Similarly, X ∈ S m×n is a m by n matrix with elements xij ∈ S. The superscript
t
denotes the transpose operator. For
random vectors x and y, E [x] is the expectation, var x = E [xt x] the variance, and cov x = hx, xi = E [xxt ] the covariance, respectively. Likewise cov(x, y) = hx, yi = E [xyt ].
Submitted to IEEE Trans. Inform. Theory 30 December 2002. Revised 29 May 2003 and 12 September 2003.
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II. S YSTEM M ODEL AND I TERATIVE D ECODER Consider the K-user linear multiple-access system of Figure 1. Let user k = 1, 2, . . . , K encode their binary information sequence bk [l] using a rate R code C, to produce the coded binary sequence dk [i]. Transmission occurs st1
s1
b1
- Encoder C
d1
- π1
.. .
bK
Fig. 1.
- Encoder C
u1
-
.. .
dK
- πK uK -
Mapper .. .
Mapper
? - × J J .. JJ ^
? - × - y1
x1
sK
.
+
6
? xK n
- ×
•
stK
.. .
? - × - yK
The transmission system model.
in blocks of L code bits per user, corresponding to KLR information bits. Each user independently permutes their encoded sequence with an interleaver πk . The sequence output from the interleaver of user k is uk [i], where i = 1, 2, ..., L is the symbol time index. The interleaved code bits uk [i] are then memorylessly mapped onto a binary phase-shift keyed (BPSK) constellation, B = {−1, 1}, giving sequences of modulated code symbols xk [i]. Extension to higher order signalling constellations is straightforward, BPSK is chosen only for simplicity. At symbol time i, each user transmits sk [i]xk [i], the multiplication of xk [i] with the real N -chip spreading N
sequence1 , sk [i] ∈ {−1, 1} . Spreading sequences with period much longer than the data symbol duration are modelled by letting each element of sk [i] be i.i.d. over users and time. For conceptual ease only, users are symbol synchronized, transmit over an additive white Gaussian noise (AWGN) channel, and are received at the same power level. These assumptions however are not required for the following development. Given symbol-synchronicity, the chip-match filtered received vector r[i] ∈ RN at symbol time i = 1, 2, . . . , L is r[i] = S[i]x[i] + n[i],
(1)
where S[i] = (s1 [i], s2 [i], . . . , sK [i]), is a N × K matrix with the spreading sequence for user k as column k. The vector x[i] ∈ B K has elements xk [i] and the vector n[i] ∈ RN is a sampled i.i.d. Gaussian noise process, with cov n[i] = σ 2 I. It will not be required to identify specific symbol intervals, so symbol indices will be omitted. For later use, t
define the following notation, xk¯ = (x1 , x2 , ...xk−1 , xk+1 , ..., xK ) to indicate deletion of user k from x. Likewise, Ik¯ denotes a K − 1 × K matrix formed by deleting row k from the K × K identity matrix. 1 Although
this model is cast in terms of direct-sequence code-division multiple-access, the sequences sk [i] may be used to represent any
type of complex vector modulation, thus admitting a general linear multiple-access system model [13].
Submitted to IEEE Trans. Inform. Theory 30 December 2002. Revised 29 May 2003 and 12 September 2003.
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Application of the “turbo-principle” to the coded multiple-access system just described, where for each user, the error control code is considered as one constraint and the multiuser channel (1) as the other constraint, results in the canonical iterative multiuser of Fig. 2 [14–17]. This is a user-by-user structure where each user has a separate decoder branch. The outputs following each full iteration are however shared among all the decoder branches. r
?
(n)
p1
-
(n−1) q1
-
Multiuser APP
π1−1
-
.. .
-
APP C
.. .
- q(n) 1
π1
.. .
(n)
Fig. 2.
pK
-
(n−1) qK
-
−1 πK
-
-
APP C
- q(n) K
πK
Canonical iterative multiuser decoder.
At each iteration, the multiuser APP decoder takes as input the received signal r and the set of extrinsic (n−1)
from user k = 1, 2, . . . , K calculated in the previous iteration. The extrinsic probability
probabilities qk distribution
(n−1) qk [i]
∈ P|B| is on the transmitted symbols xk [i] ∈ B of user k. With a small abuse of the (n)
(n)
vector subscripting notation, (qk [i])xk is the element of the vector qk [i] corresponding to user k transmitting the symbol xk ∈ B at time i (technically these subscripts should be integers corresponding to an index mapping between B and 0, 1). (n)
The multiuser APP calculates the updated extrinsic probability vector pk [i] for user k and after appropriate (n)
de-interleaving, the extrinsics pk
are used as priors for independent APP decoding of the code C by each user.
These decoders are standard implementations of the forward-backward algorithm [18] on the trellis associated with (n)
C. These decoders produce (after interleaving) the extrinsics qk , which serve as priors for the subsequent iteration. (n)
Calculation of pk [i] requires summation over |B|K−1 terms, which limits the practical application of this receiver. Because of this prohibitive complexity, many lower-complexity alternatives have been proposed while retaining the same basic architecture. Of particular interest are structures that replace the multiuser APP block (n)
with a bank of linear filters as shown in Fig. 3. In this structure, there is a bank of filters Λk , one for each user. The coefficients of these filters may be re-computed every iteration. For the first iteration, n = 1, the input (1)
to each filter Λk
is just the received signal r. For subsequent iterations n = 2, 3, ..., the input to the filter
for user k is the received signal r and a set of signal estimates for all the other users from previous iterations, (m)
{ˆ xk′
: k ′ 6= k, m ∈ M}, where M ⊆ {1, 2, . . . , n − 1} is a set defining the memory order of the iteration.
Submitted to IEEE Trans. Inform. Theory 30 December 2002. Revised 29 May 2003 and 12 September 2003.
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(m)
{ˆ xk
: k 6= 1, m ∈ M}
-
(n)
(m)
{ˆ xk
(n)
- T
.. .
r
Fig. 3.
x ˜1
(n) Λ1
p1
.. . (n)
-
.. .
- π1
APP C
.. .
- T
q1
- U
.. .
(n)
x ˜K
(n) ΛK
(n)
- π1−1
- xˆ(n) 1
.. . (n)
pK
−1 - πK
-
- πK
APP C
qK
- U
- xˆ(n) K
: k 6= K, m ∈ M}
Iterative multiuser decoder with linear multiuser estimation.
Typically in the literature, M = {n − 1}, although recently M = {n − 1, n − 2} has been considered [10, 11]. (m)
These input signals can in some sense be thought of as extrinsic information since the estimates x ˆk
for user k,
have been excluded. (n)
The output of the filter Λk
(n)
is an updated sequence of estimates x ˜k
of the corresponding code symbol for user
k. These estimates are then mapped from the signal space onto the probability vector space using a symbol-wise (n)
mapping T : R 7→ P|B| . The resulting sequence of probability vectors pk
are used as priors for individual APP (n)
decoding of the code C. These decoders can output either posterior or extrinsic probabilities qk . The sequence of (n)
probability vectors qk
is in turn mapped back onto the signal space by a symbol-wise function U : P|B| 7→ R. (n)
(n)
Typically, T calculates the vectors pk assuming that x ˜k is Gaussian distributed with known mean and variance, (n) ˜(n) (n) x ˜k ∼ N µ ˜k , ζk . Likewise, a common choice for U is the conditional mean. Design of improved maps T
and U is an interesting topic, but is outside the scope of this paper, where the focus is on the exploitation of all
the successive outputs from the decoders, rather than just the most recent. (n)
The use of posterior and extrinsic probabilities qk as feedback information has been investigated in the literature, e.g. [9, 11]. When sub-optimal multiuser detection strategies are used in place of multiuser APP detection, it is however still an open problem whether posterior or extrinsic probabilities should be used. For the numerical examples presented here, posterior probabilities have been observed to provide better performance in conjunction with linear filters. The conclusive solution to this problem is however, outside the scope of this paper. III. E XTRINSIC K ALMAN F ILTER F OR M ULTIUSER E STIMATION In the literature, work on linear filters for iterative decoding has focussed on LMMSE filtering [7, 8, 19, 20], linear interference cancellation [5, 6, 9, 21–23], and linear weighted cancellation [10, 11]. Furthermore, these filters have been designed based on the received signal r and the most current code symbol estimates of the interfering users (n)
(1)
(2)
(n)
ˆ k¯ . After n iterations, there are however a sequence of such estimates available, namely {ˆ ˆ k¯ , ..., x ˆ k¯ } x xk¯ , x together with r. It has been observed that the estimates are not strongly correlated during the initial iterations [10,
Submitted to IEEE Trans. Inform. Theory 30 December 2002. Revised 29 May 2003 and 12 September 2003.
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11], so in principle, using all the previous estimates in the filter design could improve performance. The main result (n)
of this paper is the derivation of an extrinsic Kalman filter Λk
for use in the iterative decoder of Fig. 3 that
outputs the LMMSE estimate for xk , given r and the output of all the APP decoders for every previous iteration. We call this filter an extrinsic Kalman filter, since it does not use previous estimates of xk in its update procedure. (n)
Consider the following recursively defined vector of observations as input to the filter Λk , r n=1 (n) (n−1) (2) ck = ck n = 2, 3, . . . (n−1) x ˆ k¯ D ED E−1 (n) (n) (n) (n) Direct application of the LMMSE design criterion results in Λk = xk , ck ck , ck . It is clear however (n)
that Λk
grows in dimension with n, which is impractical. Application of the principles of Kalman filtering [24,
25], or recursive Bayesian LMMSE estimation [26], solves this dimensionality problem by giving a recursive form (n)
for Λk , subject to the following assumptions. The received signal is r = Sx + n, according to (1) where n is Gaussian with cov n = σ 2 I, and σ 2 and
A1
S are known. (n)
ˆ k¯ The interleaved code symbol estimates of the interfering users x
A2
coming out of the single user APP
decoders can be written (n)
x ˆk (n)
where it is assumed that vˆk
(n)
= xk + vˆk
(3)
is uncorrelated with x and also uncorrelated over time and iterations, but
not over users at a given iteration,
(n)
The matrix Qk
A3
(n)
Let ck
(n)
defined as Qk
E D (n) =0 x, vˆk E 0 n 6= m D (n) (m) = vˆk , vˆj q n=m kj D E (n) (n) ˆ k¯ , v ˆ k¯ = v with elements determined by (5), is known.
(4) (5)
be according to (2). Then the system has trivial state evolution x(n) = x and observations Sx + n n=1 (n) (n) (n) y = Hk x + z = (n−1) I¯ x + I¯ v n≥2 k kˆ (n)
The usual Kalman equations, under A1-A3, yield the LMMSE estimate x ˜k
(n)
of xk given ck
as the k-th element
of (n)
˜ (n) = x ˜ (n−1) + Mk x
(n)
˜ (n−1) y(n) − Hk x
Submitted to IEEE Trans. Inform. Theory 30 December 2002. Revised 29 May 2003 and 12 September 2003.
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where (n) Mk (n)
Wk
(n)
Rk (0)
=
(n−1) Wk
(n) Hk
t
(n) (n−1) Hk Wk
(n) Hk
t
+
(n) Rk
−1
(n−1) (n) (n) Wk = I − Mk Hk σ 2 I n = 1, = cov z(n) = Q(n) n > 1. k
˜ (0) = 0. with initial conditions Wk = I and x
Figure 4 shows how the filter exploits feedback to avoid growing dimensionality. Observe that the filter for each user is required to estimate the code symbols for the interfering users based on the user-specific input signals to the user-specific filter. These estimates are for internal filtering use only and are not available in the external iterative structure.
(n = 1)
r ˆ (n−1) x ¯ k
H H•- + • − 6 (n > 1) •H
(n)
Mk
- x˜(n) k
de - + - mux 6
˜n x
? z
(n)
Hk
Fig. 4.
−1
˜ n−1 x
(n)
(n−1)
ˆ k¯ The extrinsic Kalman filter Λk . For n = 1 the input signal is r while for n ≥ 2 the input signal is x
.
Assumptions A1-A3 yield a tractable filter design problem and are the key to obtaining the convenient iterative structure. In practice, these assumptions hold only approximately, and some performance loss can be expected due to this inaccuracy in modelling. The filter is referred to as the extrinsic Kalman filter since the estimate for user k from the error control decoders is excluded by choice from the observation vector input to the filter. This is not a requirement of the Kalman filter derivation, rather it is neccessary for good performance of the iterative decoder. Similar deletion of user k has been widely used in the litertature with cancellation/filtering based receivers. It is envisaged that many other simpler filter structures exploiting all or part of the iteration history of the decoder could be investigated. The Kalman filter just described provides however a useful benchmark, just as the post-cancellation LMMSE based iterative decoder provides a reference for first-order iterative decoders. Submitted to IEEE Trans. Inform. Theory 30 December 2002. Revised 29 May 2003 and 12 September 2003.
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IV. N UMERICAL R ESULTS In this section numerical results are used to illustrate the performance benefits obtained by the proposed filter. For the purposes of simulation, each user applies the maximum free distance four state convolutional code (with generators (5, 7)) mapped onto BPSK. Each user therefore transmits one bit per two channel uses. Binary spreading sequences with length N = 8 were generated i.i.d. at each symbol for each user. Transmission is symbol(n)
synchronous and all users are received at the same power level. The matrix Qk
is determined based on decision-
directed estimation in the simulations. The performance of the new recursive filter (RF) is compared to the parallel interference canceller (PIC) [6], the improved parallel interference canceller (IPIC) [10, 11], and the LMMSE filtered parallel interference canceller (FPIC) [7, 8]. In Figure 5(a), the bit error rate (BER) performances for the four cases are shown as functions of the load in terms of the number of active users. Here, the number of iterations is limited to five and the performance is captured for Eb /N0 = 5 dB. Observe that the PIC can support up to 8 users and still provide close to single user performance. The IPIC can support 9–10 users, while the FPIC and the RF can support 12–13 users. As predicted, the RF provides better performance than other linear estimators, although it is only marginally better than the FPIC after five iterations. In Figure 5(b), the BER performances are shown after ten iterations as functions of the load. The PIC can now support 9 users, the IPIC 12–13 users, while the FPIC and the RF both can support up to 14 users. After ten iterations, the advantage of the RF over the FPIC is still marginal, although becoming more significant. The BER performances after twenty iterations are shown in Figure 5(c) as functions of the load. The PIC can still only support 9 users, however, for more iterations, 10 users may be supported. The IPIC can comfortably support 13 users, while the FPIC has not improved with iterations and thus can still support 14 users. In contrast, the RF has improved with iterations and can now support 14–15 users with close to single user performance. In order to investigate the convergence behavior for each system, the BER performance is now given as a function of the number of iterations at Eb /N0 = 5 dB for a load close to the maximum for the estimator in question. In Figure 6(a), the BER performance for the PIC as a function of the number of iterations is shown. As was the case in the previous figures, the PIC can support 8 users after 5 iterations, 9 users after 7–8 iterations and possibly 10 users for a large number of iterations. However, 11 users cannot be supported for any number of iterations. In Figure 6(b), the corresponding plot for the IPIC is shown. Again, the supported loads observed in previous figures are confirmed. The IPIC can support 9–10 users after 5 iterations, 11 users after 7 iterations, 12 users after 9 iterations and 13 users after 11 iterations. It is also possible that 14 users can be supported for a large number of iterations, however, 15 users cannot be supported even for a large number of iterations. The convergence behavior for the FPIC is shown in Figure 6(c). Here, it is observed that 11 users can be supported after only 4 iterations, 12 users after 5 iterations, 13 users after 6 iterations and 14 users after 9 iterations. It is, however, not likely that the FPIC can support 15 users even for a large number of iterations. Finally, the convergence behavior for the RF is demonstrated in Figure 6(d). As for the FPIC, 12 users can be supported after 5 iterations, 13 users after 6 iterations and 14 users after 9 iterations. Note that 15 users can be Submitted to IEEE Trans. Inform. Theory 30 December 2002. Revised 29 May 2003 and 12 September 2003.
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Load K (c) 20 iterations
Fig. 5.
BER performance vs number of users at Eb /N0 = 5 dB with N = 8.
supported at only a small number of iterations above 20, providing an extra user at a processing gain of N = 8. It can, however, also be observed that 16 users can most likely not be supported even for a large number of iterations. All of the above results have been for Eb /N0 = 5 dB. The following results consider the convergence behavior for a range of Eb /N0 and a number of iterations. In Figure 7(a), the performance of the PIC as a function of the Eb /N0 with K = 9 users is shown for m = 1 − 5, 7, 10, 20 iterations, respectively. The BER performance improves with the number of iterations up to 7 iterations, where close to single user performance is achieved at Eb /N0 = 5 dB. For an increasing number of iterations, single user performance is achieved for lower Eb /N0 . After 10 iterations, single user performance is achieved at Eb /N0 = 4.5 dB, and after 20 iterations at Eb /N0 = 4.0 dB. This behavior can potentially continue
Submitted to IEEE Trans. Inform. Theory 30 December 2002. Revised 29 May 2003 and 12 September 2003.
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Iterations (c) FPIC
Fig. 6.
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(b) IPIC
BER
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8
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Iterations (d) RF
BER performance vs number of iterations at Eb /N0 = 5 dB with N = 8.
for an increasing number of iterations down to Eb /N0 =3.0–3.5 dB where the threshold for the waterfall region appears to be. Similar behavior is observed for the IPIC in Figure 7(b) for K = 12 users. Here, single user performance is achieved at Eb /N0 = 6.0 dB after 7 iterations, at Eb /N0 = 5.0 dB after 10 iterations and at Eb /N0 = 4.0 dB after 20 iterations. Again, it is likely that single user performance can be achieved down to Eb /N0 = 3.0 − 3.5 dB as the number of iterations increase. In Figure 7(c) and Figure 7(d), respectively, the SNR convergence behavior for the FPIC and the RF are shown for K = 14 users. As previously noted, similar behavior is observed for 10 and 20 iterations where single user performance is achieved at Eb /N0 = 5.0 dB and Eb /N0 = 4.5 dB, respectively. It is, however, interesting to see
Submitted to IEEE Trans. Inform. Theory 30 December 2002. Revised 29 May 2003 and 12 September 2003.
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that after 7 iterations, the RF is at single user performance at Eb /N0 = 5.5 dB and very close at Eb /N0 = 5.0 dB, where the FPIC is not at single user performance before Eb /N0 = 6.0 dB. The advantage of the RF at a low number of iterations increases as the Eb /N0 increases as can be seen by the steeper BER slop at Eb /N0 = 6.0 dB after 5 iterations.
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Fig. 7.
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(b) IPIC, K = 12
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(d) RF, K = 14
BER performance vs Eb /N0 dB, m =1–5,7,10,20.
V. C ONCLUDING R EMARKS A Kalman filter has been described for use in iterative multiuser decoding. Subject to certain statistical assumptions, it delivers the LMMSE estimate of each user based on the received signal and the entire history of outputs from the single user decoders. Since estimates of the code symbols pertaining to user k is excluded from Submitted to IEEE Trans. Inform. Theory 30 December 2002. Revised 29 May 2003 and 12 September 2003.
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the observation vector input to the filter for user k, the resulting filter is called an extrinsic Kalman filter. This filter was shown to perform better than other suggested structures in all cases investigated. At a processing gain of N = 8, an extra user can be supported for a large number of iterations, as compared to the LMMSE filtered parallel interference cancellation. This is interesting as such a gain is expected to increase for larger processing gains. For a small number of iterations, the new filter also provides better BER performance. This advantage becomes significant for increasing Eb /N0 as the slope of the BER curves for a low number of iterations is steeper than for the other linear estimators. This is expected since it has previously been observed that the estimates provided by the single user decoders are loosely correlated for the first few iterations, thus the output of each iteration can potentially provide additional information, which otherwise would have been ignored. The principles described here are also applicable to other system components and technologies such as spacetime coding, equalization, coded modulation, and joint channel estimation/decoding. Another interesting avenue of further investigation would be the development of other, simpler structures which approximate the extrinsic Kalman filter, yet retain the benefit due to the use of all, or part of the iteration history. R EFERENCES [1] S. Verd´u, “Minimum probability of error for asynchronous Gaussian multiple–access channels,” IEEE Trans. Inform. Theory, vol. 32, no. 1, pp. 85–96, Jan. 1986. [2] ——, “Computational complexity of optimum multiuser detection,” Algorithmica, pp. 303–312, May 1989. [3] T. M. Cover and J. A. Thomas, Elements of Information Theory.
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[4] T. R. Giallorenzi and S. G. Wilson, “Multiuser ML sequence estimator for convolutionally coded asynchronous DS-CDMA systems,” IEEE Trans. Commun., vol. 44, no. 8, pp. 997–1008, Aug. 1996. [5] J. Hagenauer, “Forward error correcting for CDMA systems,” in IEEE Int. Symp. Spread Spectrum Techn. App., Mainz, Germany, Sept. 1996, pp. 566–569. [6] P. D. Alexander, A. J. Grant, and M. C. Reed, “Iterative detection on code-division multiple-access with error control coding,” European Trans. Telecommun., vol. 9, no. 5, pp. 419–426, Sept.-Oct. 1998. [7] X. Wang and V. Poor, “Iterative (turbo) soft interference cancellation and decoding for coded CDMA,” IEEE Trans. Commun., vol. 47, no. 7, pp. 1046–1061, July 1999. [8] H. El-Gamal and E. Geraniotis, “Iterative multiuser detection for coded CDMA signals in AWGN and fading channels,” IEEE J. Selected Areas Commun., vol. 18, no. 1, pp. 30–41, Jan. 2000. [9] J. Boutros and G. Caire, “Iterative multiuser joint decoding: Unified framework and asymptotic analysis,” IEEE Trans. Inform. Theory, pp. 1772–1793, July 2002. [10] S. Marinkovic, B. S. Vucetic, and J. Evans, “Improved iterative parallel interference cancellation for coded CDMA systems,” in IEEE Int. Symp. Inform. Theory, Washington D.C., USA, June 2001, p. 34. [11] S. Marinkovic, B. S. Vucetic, and A. Ushirokawa, “Space-time iterative and multistage receiver structures for CDMA mobile communication systems,” IEEE J. Selected Areas Commun., vol. 19, pp. 1594–1604, Aug. 2001. [12] D. Divsalar, M. K. Simon, and D. Raphaeli, “Improved parallel interference cancellation for CDMA,” IEEE Trans. Commun., vol. 46, no. 2, pp. 258–268, Feb. 1998. [13] L. K. Rasmussen, P. D. Alexander, and T. J. Lim, “A linear model for CDMA signals received with multiple antennas over multipath fading channels,” Chapt. 2 in: CDMA Techniques for 3rd Generation Mobile Systems, edited by F. Swarts, P. van Rooyen, I. Oppermann and M. L¨otter, Kluwer Academic Publisher, Sept. 1998. [14] M. C. Reed, C. B. Schlegel, P. D. Alexander, and J. Asenstorfer, “Iterative multiuser detection for CDMA with FEC: Near-single-user performance,” IEEE Trans. Commun., vol. 46, pp. 1693–1699, Dec. 1998.
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[15] M. Moher, “An iterative multiuser decoder for near-capacity communications,” IEEE Trans. Commun., vol. 46, no. 7, pp. 870–880, July 1998. [16] P. D. Alexander, M. C. Reed, J. Asenstorfer, and C. B. Schlegel, “Iterative multiuser interference reduction: Turbo CDMA,” IEEE Trans. Commun., vol. 47, pp. 1008–1014, July 1999. [17] L. Brunel and J. Boutros, “Code division multiple access based on independent codes and turbo decoding,” Annales des Telecommunications, vol. 54, no. 7-8, pp. 401–410, Jul.-Aug. 1999. [18] L. R. Bahl, J. Cocke, F. Jelinek, and J. Raviv, “Optimal decoding of linear codes for minimizing symbol error rate,” IEEE Trans. Inform. Theory, vol. 20, pp. 284–287, Mar. 1974. [19] M. L. Honig, G. Woodward, and P. D. Alexander, “Adaptive multiuser parallel-decision-feedback with iterative decoding,” in IEEE Int. Symp. Inform. Theory, Sorrento, Italy, June 2000, p. 335. [20] G. Woodward and M. L. Honig, “Performance of adaptive iterative multiuser parallel decision feedbackwith different code rates,” in IEEE Int. Conf. Commun., Helsinki, Finland, June 2001, pp. 852–856. [21] N. Ibrahim and G. K. Kaleh, “Iterative decoding and soft interference cancellation for the Gaussian multiple access channel,” in Int. Symp. on Signals, Systems, and Electronics, Pisa, Italy, 1998, pp. 156–161. [22] M. Kobayashi, J. Boutros, and G. Caire, “Successive interference cancellation with SISO decoding and EM channel estimation,” IEEE J. Selected Areas Commun., vol. 19, pp. 1450–1460, Aug. 2001. [23] Z. Shi and C. Schlegel, “Joint iterative decoding of serially concatenated error control coded CDMA,” IEEE J. Selected Areas Commun., vol. 19, pp. 1646–1653, Aug. 2001. [24] S. Haykin, Adaptive Filter Theory.
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