Turbo Performance of Low-Complexity CDMA

Turbo Performance of Low-Complexity CDMA Iterative Multiuser Detection Christian Schlegel

Zhenning Shi Department of Electrical Engineering University of Utah, Salt Lake City, UT

Department of Electrical Engineering University of Alberta, Edmonton, CANADA

Abstract: Low complexity multistage filters are introduced for iterative reception of error control coded CDMA signals. It is shown that a first-order stationary filter approaches the performance of the complex per-user MMSE filter with a small number of stages. It is observed that iterative decoding has two basic phases, each having distinctive fundamental limiting factors. The first phase is limited by the level of multiple-access interference, while the other is limited by the FEC code performance. The well-known turbo cliff accompanies successful decoding of the first phase.

1.

Introduction

Code-Division Multiple-Access (CDMA) is a method where a number of users access the wireless medium concurrently with waveforms that overlap in both time and frequency. Conventional correlation reception of CDMA signals is limited by the multipleaccess interference, and joint detection methods have been studied recently to improve the spectral efficiency. Due to the prohibitive computation power required by he optimal detector [12], linear filters have been studied as lower-complexity alternatives [6], [14]. It is shown in [3], [9] that such multiuser detectors can achieve a significant portion of the channel capacity in unsaturated scenarios, but their effectiveness diminishes rapidly as the system load increases. Nonlinear receiver structures [4], [7], [10], [13], which follow the turbo decoding principle [1], perform significantly better with the iterative information processing between an interference resolution device and individual error control decoders. Among all these interference resolution devices, the per-user MMSE filter [13] delivers the best joint linear system performance, at the cost of consuming much more computation power than the simple interference canceller. In this paper we investigate low-complexity alternatives, in particular the first-order stationary method [15], for the per-user MMSE filter in the range of loads of interest [8], [10]. We use a variance transfer (VT) analysis [4], [10] to study the dynamics of such multistage receivers over iterations, and show that they can achieve the performance of the MMSE de-

tector with a few number of stages. We show that the iterative decoding processing for such a system has two basic phases. The performance of the first phase is limited by the multiple-access interference, while the second phase is bounded by the channel noise level and the capability of the FEC code to overcome it. The turbo cliff phenomenon, which typically accompanies the iterative decoding, is observed when the first decoding phase succeeds.

2.

System Model

CDMA deals with a linear channel where the signals of different users superpose and cause mutual interference due to a lack of orthogonality between their spreading sequences. The system model considered is that of an asynchronous CDMA with K transmitters which generate independent binary information bits uk ∈ {0, 1}, k = 1, . . . , K. These K streams of bits are FEC coded, and then spread with user specific sequences. Assuming equal powers, the signal from the k-th spreader is xk (t) =

L−1 X

dk [j]sk,j (t − jT − τk ),

(1)

j=0

In (1), L is the number of data symbols per frame, τk (< T ) is the time delay of user k, sk,j (t), supported on the interval [0, T ], is the spreading sequence waveform for user k during symbol time j: sk,j (t) =

N −1 X

sk,j [l]g(t − lTc ),

(2)

l=0

where N is the spreading gain, Tc is the chip interval, , √1N } is the l-th spreading chip for user sk,j [l] ∈ { √−1 N k during symbol j, and g(t) is the normalized chip waveform. We consider an AWGN channel and the received signal is y(t) =

K X

xk (t) + z(t),

(3)

k=1

where z(t) is zero mean white Gaussian noise with variance N20 . The signal after the chip matched filtering is r = Sd + z, (4)

Soft-Decision Decoder

r

CDMA

FEC Soft-Decoder Π−1 1

APP

Π1

Π−1 2

APP

Π2

Π−1 K

APP

ΠK

where Pk is the transmission power of user k, Kk,j = Stk,j Dk Sk,j + σ 2 I, Dk is a diagonal matrix of the residual power of the interfering users, and Sk,j = [s1,1 , · · · , sk−1,j , sk+1,j , · · · , sK,L ] are spreading sequences of interfering users. The MMSE filter in (6) is too complex for implementation due to the inverse of Kk,j , which has to be calculated for every user in each iteration, and, in the case of random CDMA, also for every symbol time. One way to reduce the complexity incurred in (6) is to use computationally efficient iterative methods [2], [5], [15]. Given a linear algebraic equation x = M−1 b,

tanh()

the first-order stationary iteration ³ ´ x(n+1) = x(n) − τ Mx(n) − b

tanh() tanh()

Figure 1: Block diagram of the joint iterative decoder. where z is a length-(L+1)N vector of noise samples of variance N20 , S is a matrix size of (L+1)N×LK with jK+k-th column sk,j = [0jN + τk , sk,j [0], · · · , sk,j [N− Tc

1], 0(L−j)N − τk ]T , 0l is a length-l all-zero row vecTc tor, and d = [d1 [1], · · · , dK [1], d1 [2], · · · , dK [L]] is a length-LK vector of encoded symbols.

3.

Low-Complexity Filters

Figure 1 shows an iterative joint detector, consisting of a soft-output CDMA channel “decoder” and a bank of a posteriori probability (APP) error control decoders. Each individual decoder processes the received signal and generates soft-values of the transˆ k [j] of mitted symbols dk [j]. Once soft estimates d these symbols have been generated via the tanh() functions, a cancelled received signal can be calculated as ˆ k,j + z, rk,j = Sd − Sd (5) ˆ k,j = [d ˆ 1 [1], · · · , d ˆ k−1 [j], 0, d ˆ k+1 [j], · · · , d ˆ k [K]]. where d Hence the error control codes assume a two-fold role in the reception of CDMA signals. On one hand, they are supposed to provide good estimates of the ˆ k [j] in (5). On the other hand, FEC coded symbols d codes should be able to overcome the additive noise efficiently and identify the transmitted message with low error probabilities. These two requirements have an inherently contradicting nature, and lead to two distinctive decoding phases in the iterative detection, which will be discussed in Section 4. The per-user MMSE approach proposed in [13] minimizes the mean-square error of the output via the filter wk,j

=

1+

Pk t Pk sk,j K−1 k,j sk,j

(7)

K−1 k,j sk,j ,

(6)

(8)

converges to the correct solution with an exponentially vanishing error as long as M is invertible and the spectral radius of matrix (I − τ M) is smaller than unity. The explicit equation of x(n+1) , using as initial vector x(0) = b, can be calculated as à ! n−1 X n l (n+1) (I − τ M) + (I − τ M) τ b x = l=0

=

Φn (M)b.

(9)

For large systems, K, N → ∞, the MMSE filter generates Gaussian-like outputs. Therefore, the normalization factor Pk /(1 + Pk stk,j K−1 k,j sk,j ) in (6) can be dropped, and the filter of interest reduces to wk,j = K−1 k,j sk,j .

(10)

We now apply the stationary inversion method (8) to K−1 k,j , which leads to a multistage filter implementation (n+1) wk,j = Φn (Kk,j )sk,j . (11) As N, K → ∞ and K N is fixed, it can be shown, using the arguments in [11], that the random signal power and noise variance of outputs of the first-order filter in (11) converge in probability to SP =

¡

¢2 P stk Φn (K)sk →

µZ

¶2

∞

φn (λ)fλ (λ)dλ 0

NP + IP = stk Φn (K)KΦn (K)sk Z ∞ P → φn (λ)2 κ(λ)fλ (λ)dλ,

(12)

0

where φn (λ), κ(λ) and λ are eigenvalues of Φn (K), K and Sk Sk t respectively. A similar approach was used to analyze the SNR performance of multistage filters for the non-iterative LMMSE receiver [15]. For the iterative receiver, however, the so-called effective interference [11] keeps

2.5 per userMMSE

K/N = 2 n=4

normalized output variance σ2k

2

n=2 n=1 interference cancellation

1.5

Es /N0 = 0 dB

1

n=10

0.5 n=5 n=2 n=1

0

0

0.2

Es /N0 = 20 dB

0.4 0.6 residual bit variance σd2

0.8

1

Figure 2: Asymptotic VT for multistage approximaEs tions to the MMSE cancellation detectors at N = 0 Es K 0dB and N0 = 20dB for a double-loaded ( N = 2) CDMA.

of the iterative decoding. The first phase is limited by the multiple-access interference embedded in the CDMA signals, which is represented by the narrow channel between the VT curves of FEC code and the first-order multistage filter. Weak FEC codes usually possess a ”shallow” VT slope that allows the decoding go through the first phase even for heavily loaded CDMA systems. The second phase occurs near the point σd2 = 0, where the performance is mainly limited by the channel noise level. Strong FEC codes are typically optimized to provide good performance in this range. Figure 4 illustrates this two-phase phenomenon from the BER perspective. At low SNRs (≤ 4.5 dB), the BER is limited by the large amount of mutual interference in the system Eb (K = 45, N = 15). The turbo cliff at N ≈ 4.6 0 dB indicates that the first decoding phase succeeds. After that, the decoding enters the second phase, and the BER is bounded by that of the individual FEC codes, in this case, the rate- 13 convolutional code. 2 σIC

4 3.5

(1): VT for rate−1/3 CC (v=2) (2): VT for interference canceller

σd2 ,

3

(4): VT for 1st−order filter (n = 2)

2

(4) K = 45, N=15 L = 5000 u

N

+1

chosen to guarantee the convergence of the iterations. Figure 3 shows the VT characteristics of the iterative reception of signals for a CDMA system K = 45, N = 15. It is observed that the first-order filter significantly outperforms the interference canceller, and behaves almost as well as the per-user MMSE filter. Simulation points sit right on the VT curves of the FEC code and first-order second-stage filter for a frame length Lu = 5000, which shows the accuracy of the VT analysis of iterative decoding. Figure 3 also shows the two-phase characteristics

(5)

º·

1.5

¹¸ interference limited

1 º· 0.5 ¹¸ noise limited 0 0

Two Phases of Iterative Decoding

N0 +

(3)

2.5

k

We use a numerical example to illustrate the decoding process of an iterative CDMA receiver. The FEC code used for error correction is a rate- 13 fourstate convolutional code G(D) = [1 + D2 , 1 + D + D2 , 1 + D + D2 ]t . A two-stage first-order stationary method is used to suppress the interference over ³√ 2 ´2 is iterations. The parameter τ = K−1

(2)

(5): VT for per−user MMSE filter

parameterized by the number of filter stages. One or at most two stages gain the majority of the performance advantage that the MMSE filter has to offer over simple cancellation.

4.

(1)

(3): simulation points

changing as the error control decoders produce more reliable results over iterations. Variance transfer analysis of the first-order stationary method is conducted to show the dynamics of its output variance over the input symbol variance. Figure 2 shows the asymp2 totic normalized (SP=1) interference variance σIC (= 1 ) as a function of the residual symbol variance SNR∗

0.2

0.4

0.6

0.8

1

σd2

Figure 3: Variance transfer curves for simulating an iterative receiver using a two-stage first-order stationary method.

5.

Performance of Iterative CDMA Receivers with Coding

Figure 5 shows the capacities that are achieved by a strong SCCC code and a weak parity check code, both of rate- 31 , at their system convergence thresholds that lead to low error rates, and for the multistage interference cancellation filter at n = 1, 3. It is observed that the iterative reception using linear interference resolution methods can achieve a considerable portion of the channel capacity in a wide range of SNRs (≤ 8 dB). The low-complexity multistage filter can approach the complex per-user MMSE per-

10 bits/dimension

0

10 BER

−1

10

MultiStage Filter

1 iteration

n=2

5 iterations

K=45 N=15

10 iterations

rate 1/3 CC (v=2)

optimal processing −2

10

(8)

AWGN capacity Single User 15 iterations

−3

(5)

1 (2)

20 iterations

10

(6)

(7)

(4)

(3)

(1)

30 iterations (1) MF interference cancellation (2) MMSE cancellation (3) Single-stage iterative filter (4) Three-stage iterative filter

−4

10

2.5

3

3.5

4

4.5

5 5.5 E /N [dB] b

0.1 -2

0

Figure 4: BER Performance of an iterative receiver using a two-stage first-order stationary method. formance with a few number of stages. The iterative decoding process for a CDMA system shows a two-phase phenomenon. The first phase is limited by the interference, while the second one is limited by the channel noise. Error control codes that are appropriate for CDMA scenarios should be able to efficiently overcome both the interference and the additive noise. We have demonstrated that simple FEC codes can overcome a large amount of multiaccess interference with per-user MMSE receivers or its low-complexity multistage approximations. Therefore, high spectral efficiency can be achieved in realistic CDMA networks with manageable computation power.

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12 14 16 Eb /N0 [dB]

Figure 5: Achievable capacities of iterative receivers for strong (dash curves) and weak FEC codes (solid curves), for MF, MMSE, and multistage filters. [7] M. Moher, “An Iterative Multiuser Decoder for [8] [9] [10] [11]

[12]

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