Improved LAS detector in the Iterative Large MIMO Systems with LDPC Codes Chen Zheng, Yabo Li, Jie Zhong, Minjian Zhao, Xiaomu Zhao Department of Information Science and Electronic Engineering Zhejiang University, Hangzhou, Zhejiang, P. R. China, 310027 Email:
[email protected]
Abstract—In this paper, low complexity iterative detection and decoding algorithms applied in large MIMO systems are considered. We propose an iterative scheme consists of likelihood ascent search (LAS) detection and low density parity check (LDPC) decoder in the uplink of large MIMO systems. During the iterative process, the check candidate set of LAS detector is decided by the feedback extrinsic information of the LDPC decoder. The feedback extrinsic information can be enlarged when the output codeword satisfies the parity-check equations, which can further improve the performance. Through simulation results we can see the proposed scheme achieves approximately 0.6dB gain compared with direct iterative scheme. When compared with MMSE-LDPC iterative scheme, the BER performance is only 0.2dB worse with the complexity much lower. Index Terms—Joint iterative detection-decoding algorithm, LAS detection, large MIMO systems, LDPC code.
I. I NTRODUCTION Large MIMO system has drawn more and more attention nowadays due to its high capacity and spectral efficiency [1]. In the uplink of large MIMO systems, tens to hundreds of antennas are employed at the receiver, thus low complexity detection algorithms are required. The optimal maximum likelihood(ML) detector is infeasible for large MIMO systems since its complexity grows exponentially with the number of transmitted antennas and the modulation order. Therefore, research on the low complexity detection algorithm has been studied extensively recently [2]. LAS detector, due to its low complexity and excellent BER performance, is proposed in [3] for large MIMO systems. The average per-bit complexity of LAS detector is O(Nt Nr ), with Nt and Nr denote the number of transmit and receive antennas. With large number of antennas, the BER performance of LAS detector under fading channel is almost the same as the performance under single in single out (SISO) AWGN channel [4]. Although a good performance is obtained through the detection algorithm, channel coding technique is usually applied in MIMO systems [5]. By iterating with the detector, a great coding gain can be obtained, which makes the performance gap to the capacity much smaller. LDPC codes are widely used in these systems due to their near shannon limit performance and low decoding complexity [6]. In this paper, we propose an iterative detection and decoding algorithm with LAS detector and LDPC decoder. The LAS detector in the first iteration is initialized by MF detector with
low complexity. In the following iterations, the initial vector is obtained by the hard decisions of LDPC decoder and the check candidate set is chosen by the feedback extrinsic information of the LDPC decoder. Further, the feedback extrinsic information can be modified by checking the parity-check equations of LDPC codes. When the output codeword satisfies the paritycheck equations, the decoding results are rather reliable, and we could enlarge the corresponding extrinsic information. By doing this, the performance of the iterative system outperforms the direct iterative system by about 0.6dB at BER equals 10−4 . When compared with MMSE-LDPC iterative system, the performance is only 0.2dB worse at BER equals 10−4 with the complexity much lower. The paper is organized as follows. Section II briefly introduces the system model. Section III describes the conventional algorithms of the LAS detector. In section IV, iterative detection and decoding algorithm with LAS detector and LDPC decoder is presented. Numerical results and complexity analysis are presented and compared in section V and VI. Finally, conclusions are drawn in Section VII. II. S YSTEM M ODEL In this section, we present the system model of the proposed iterative detection and decoding algorithm in the uplink of large MIMO system. The block diagram of the system is shown in figure 1. Here we consider a multiuser MIMO system with Nt users of one transmit antenna each and Nr receive antennas, Nt ≤ Nr . At the transmitter of each user, k bits are encoded to n bits by a (n, k) LDPC code. For the jth user, j = 1, 2, · · · , Nt , the codeword is denoted as cj = [cj1 , cj2 , · · · , cjn ],
(1)
where cji ∈ {0, 1} for i = 1, 2, · · · , n. The modulator maps the encoded bits to constellations before transmission. Without loss of generality, assume BPSK modulation is applied. The transmitted symbol vector is given by bj = [bj1 , bj2 , · · · , bjn ],
(2)
where bji ∈ {+1, −1} for i = 1, 2, · · · , n. After modulation, the symbols are transmitted through the wireless channel to arrive at each of the Nr receive antennas.
978-1-4799-0308-5/13/$31.00 © 2013 IEEE
Fig. 1.
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Block diagram of an iterative system with LAS detector and LDPC decoder.
At the receiver, the Nr × n received signal vector is denoted by Y = HS + N, (3) where H ∈ CNr ×Nt is the channel fading matrix, whose entry hkj is assumed to be complex Gaussian random variable with T zero mean and unit variance. S = bT1 , bT2 , · · · bTNt is the T T T Nt × n transmitted signal. N = n1 , n2 , · · · nTNt is the Nr × n noise matrix whose entry is a complex Gaussian noise with zero mean and variance σn2 per real component. The iterative joint detection and decoding algorithm is based on the exchange of soft information between the detector and the decoder. For the matrix Y received at the receiver, every time the ith column of it, denoted as yi , i = 1, 2, · · · , n, is sent to the LAS detector. The Nr × 1 signal vector is detected by LAS detection. Nt × 1 hard decisions are obtained after LAS detection and the corresponding soft output values are calculated. After all the n columns of matrix Y are detected, the n output Nt ×1 soft value vectors are combined to one Nt × n matrix. Then every row of the matrix is sent to one LDPC decoder to do decoding. After the Nt LDPC decoders finish decoding, the Nt × n extrinsic information or hard decision matrix is fed back to the LAS detector to continue iteration. III. C ONVENTIONAL LAS A LGORITHM The conventional LAS algotithm starts from an initial vector which can be the output vector of any known detector, such as matched filter (MF), zero forcing (ZF) and minimum mean square error (MMSE) detector. It then searches through a sequence of solution vectors to refine the solution with monotonic likelihood ascent. The principle is to choose the vector b with larger ML cost function which is given by ∗ Λ(b) = bT HH y + bT HH y − bT HH Hb, (4) where y is the Nr × 1 soft input vector of the LAS detector. Assume the maximum iteration number is T . At the tth iteration, t = 1, 2, · · · , T , the LAS update algorithm is shown as follows. Given the check candidate set L(t) ⊆ {1, 2, · · · , Nt } , ∀n ≥ 0 and an initial bit vector b (0) ∈
Nt
{−1, +1}
, the jth bit of b(t) is updated by +1, if j ∈ L (t) , bj (t) = −1 and Λ (b (t + 1)) > Λ (b (t)) , −1, if j ∈ L (t) , bj (t) = +1 bj (t + 1) = and Λ (b (t + 1)) > Λ (b (t)) , bj (t) , otherwise.
(5) The LAS algorithm checks the bits in the candidate set L(t) and updates b(t) according to (5). After all the bits in L(t) are check, one iteration finishes. The iteration terminates when no bit is flipped for one entire iteration or the maximum iteration number is reached. After LAS detection, soft values for all the bits are generated based on the output of the detector. Assume the output vector from the LAS detector is b = (b1 , b2 , · · · , bNt ). Define vectors bj+ and bj− to be b vector with its jth entry forced to +1 and −1,respectively. Then the soft value of the jth bit can be calculated as
bllr,j
= log
e
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2 −ky−Hbj− k /2σn
j+ 2 2 e(−ky−Hb k /2σn )
y − Hbj− 2 − y − Hbj+ 2 = . 2σn2
(6) (7)
These soft outputs are then sent to the LDPC decoder. IV. I TERATIVE D ETECTION AND D ECODING A LGORITHM A. Iterative LAS Detector and LDPC Decoder In the iterative system, the received signal Y is an Nr × n matrix with n denoting the length of LDPC code. The signal is sent to LAS detector to start the iteration between the detector and decoder. In the first iteration, the LAS detector is the same as the conventional LAS detector. Every time one column of Y is sent to the LAS detector and Nt × 1 soft values are obtained. After all the n columns are detected, the Nt × n soft value matrix is obtained, and every row of it is sent to one LDPC decoder. For the direct iterative algorithm, after LDPC decoding, Nt × n hard decision matrix is fed back to the LAS detector,.
They are updated according to (5) and the following iterations are the same as the first. For the soft iterative algorithm, both hard decision matrix D(t) and extrinsic information matrix E(t) of the decoder are fed back in the tth (t = 1, 2, · · · , T ) iteration. The LAS algorithm is as follows. First, denote the ith column of the matrix E(t) as ei (t) = [e1i (t), e2i (t), · · · , eNt i (t)], i = 1, 2, · · · , n. Choose a threshold M to decide the check candidate set, which is composed of bits whose absolute value of the extrinsic information is lower than M . The check candidate set of the ith column of E(t) is given by Li (t) = {j| |eji (t)| < M, j = 1, 2, · · · , n} .
(8)
When the absolute value is larger than M , it means this bit is very reliable and doesn’t need to be updated in the LAS algorithm. Then, bits in the candidate set are checked and updated. It keeps updating the bits in each step based on the given rule (5) until the bits remain constant for some periods or the maximum iteration number is reached. Hard decisions of the extrinsic information are obtained. At last, soft values of the hard decisions are calculated. For the bits in the candidate set, their soft values are calculated by (6), while for the other bits, their soft values are made directly equal to the extrinsic information, which are considered rather reliable. The output soft values are calculated as ( bllr,ji (t + 1) =
2
j+ kyi −Hbj− i (t+1)k −kyi −Hbi (t+1)k 2 2σn
2
, j ∈ Li (t) ,
eji (t) , j ∈ / Li (t) ,
(9)
After all the n columns finish LAS detection, the resulting Nt × n soft values are sent to the LDPC decoder to continue the iteration. B. Improvement of the Decoding Feedback In our proposed iterative algorithm, the check candidate set is decided by the value of LDPC extrinsic information, which represents the reliability of the hard decision. Since LDPC is a linear block code, the bits of a codeword c must satisfy a set of parity-check equations specified by the rows of parity-check matrix. We can see, when the output of the LDPC decoder satisfies all the parity-check equations, the reliability of the output codeword is rather high and the hard decisions can be treated as the right codeword. From the above analysis, we can check the hard decisions of LDPC decoder by the parity-check matrix. If they satisfy all the parity-check equations, multiply the corresponding extrinsic information by a reliable factor γ(γ > 1) to increase the reliability, else, the extrinsic information is directly fed back to the LAS detector. By doing this, when some of the Nt codewords are successfully decoded, all their feedback extrinsic information is very large. The corresponding bits in the initial vector of LAS detector are supposed to be right, which is benefit for the detection of other bits. The size of the check candidate set is also reduced, which can decrease the complexity of updating the vector.
V. N UMERICAL R ESULTS In this section, we evaluate the performance of the proposed soft iterative algorithm with the feedback improvement through extensive simulations. Performances of the proposed algorithm with different iteration numbers are compared. Also, the proposed algorithm is compared with direct iterative algorithm, soft iterative algorithm without the feedback improvement and MMSE-LDPC iterative algorithm. The bit error rate (BER) and frame error rate (FER) of these systems are shown to evaluate the performance. In the simulated systems, the parameters are chosen as follows. (1024, 512) binary LDPC code is applied for the LDPC encoder and decoder. For the multiuser-MIMO system, the number of users is assumed to be the same as the number of receive antennas, that is Nr = Nt . The channel is fast fading channel with the channel matrix changes every time slot. At the receiver, for the LAS detector, the maximum iteration number is 5, while for the LDPC decoder, the maximum iteration number is 20. The threshold M to choose the check candidate set in the iterations is chosen as 20, and the enlarge coefficient γ is 100. Fig.2 and Fig.4 compare the performances of the proposed algorithm with different iteration numbers when the numbers of antennas are 64 and 128, respectively. While Fig.3 and Fig.5 compare the performances of the four different systems described above with 3 iterations when the numbers of antennas are 64 and 128, respectively. In Fig.2, for Nr = Nt = 64, we can see for the proposed algorithm, the BER performance of two-iteration scheme outperforms one-iteration scheme by about 1dB, while three-iteration scheme further outperforms two-iteration scheme by about 0.3dB at BER equals 10−4 . The FER performance is similar as BER performance with the corresponding performance gaps 0.8dB and 0.25dB at FER equals 10−2 . The increment of the performance decreases as the iteration number increases. That is because as the iteration number increases, the correlation between the output extrinsic information of the LAS detector and the output extrinsic information of the LDPC decoder increases, and for every iteration less soft information can be provided for each other. In Fig.3, BER and FER curves of the proposed algorithms are shown. We represent the direct iterative algorithm, soft iterative algorithm without feedback improvement and soft iterative algorithm with feedback improvement by LAS-D, LAS-S and LAS-I, respectively. The MMSE-LDPC iterative algorithm is represented by MMSE. At BER equals 10−4 , the LAS-S has a performance improvement of 0.3dB compared with LAS-D and the LAS-I outperforms LAS-S by 0.35dB. When compared with MMSE algorithm, LAS-I is worse by approximately 0.2dB, while its complexity is only O (Nr Nt ) per symbol compared with O Nr Nt 2 of MMSE. The FER performances are of the same relationships for the four algorithms. In Fig.4, for Nr = Nt = 128, the performance is similar as Fig.2. The BER performance of two-iteration scheme out-
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Fig. 5. BER and FER performance of the iterative system with Nt = Nr = 128.
performs one-iteration scheme by about 0.9dB, while threeiteration scheme further outperforms two-iteration scheme by about 0.3dB at BER equals 10−4 . The corresponding FER performance gaps are 0.85dB and 0.3dB at FER equals 10−2 . In Fig.5, the performances of the four systems are similar as Fig.3. The BER performance of LAS-I is 0.45dB better than LAS-S and 0.65dB better than LAS-D at BER equals 10−4 . Compared with MMSE, the BER performance of LAS-I is only 0.25dB worse at BER equals 10−4 .
soft output generation. Here for simplicity we only analyze the case for Nr = Nt = N . TABLE I
T HE NUMBER OF FLOPS FOR SOME OPERATIONS Operation Real multiplication Complex multiplication Real addition Complex addition
Number of flops 1 3 1 1
VI. C OMPLEXITY A NALYSIS In this section we will analyze the complexity of the proposed LAS-LDPC and MMSE-LDPC algorithms according to Table I [8]. For the algorithms in Section V, the same LDPC code is applied and the complexity is the same for the decoding process. Thus, only the complexity of the detector is compared and analyzed which comprises two steps: 1) detection and 2)
A. Proposed LAS-LDPC Detector For the LAS detector, the check candidate set L(t) consists all the Nt bits in the first iteration. For the direct iterative algorithm, L(t) remains unchanged in the following iterations, while for the soft iterative algorithm with and without feedback improvements, L(t) can be decreased by (8). Here we calculate
the flops of LAS detection with full L(t) for all the iterations which is the upper bound of the real situations. The flops of LAS detector in one detector and decoder iteration is 23N 3 + 23N 2 + 4N . B. MMSE-LDPC Detector For the MMSE detector, in the first iteration, only one matrix inversion is needed, while in the following iterations, one matrix inversion is necessary for every symbol detected. Since the matrix inversion complexity is 16N 3 −8N 2 +4N [9] [10], the flops for the first detection and the following iterations to detect Nt × 1 symbols are calculated by 22N 3 − 5N 2 + 12N and 19N 4 + 11N 2 + 15N , respectively. The complexity of both the proposed LAS detector and MMSE detector are summarized in Table II in terms of flops. Here the iteration number of detection and decoding is 3. When the number of antennas N is chosen as 64, the number of flops required is 18, 371, 328 for LAS detection and 643, 373, 696 for MMSE detection. The complexity of LAS is only 2.86% compared with MMSE. When N equals 128, the number of flops becomes 145, 835, 520 for LAS and 10, 246, 968, 576 for MMSE. The complexity of LAS becomes only 1.42% compared with MMSE. Thus, by using the proposed detector, the computational complexity of large MIMO detection is strongly reduced. TABLE II
T HE NUMBER OF FLOPS FOR MIMO DETECTION Type LAS detector MMSE detector
Number of flops 69N 3 + 69N 2 + 12N 38N 4 + 22N 3 + 17N 2 + 42N
VII. C ONCLUSION In this paper, we propose an iterative detection and decoding algorithm based on the LAS detector and LDPC decoder. The check candidate set of the LAS detector is chosen according to the feedback extrinsic information of the LDPC decoder, which is enlarged by checking the parity relationships of the output codeword. The proposed algorithm can achieve high performance in large MIMO systems with low complexity. Simulation results verify that the proposed algorithm outperforms the direct iterative algorithm by about 0.6dB. When compared with MMSE-LDPC iterative system, the performance gap is only about 0.2dB while the complexity is much lower. R EFERENCES [1] F. Rusek, D. Persson, B. K. Lau, E. G. Larsson, T. L. Marzetta, O. Edfors, and F. Tufvesson, “Scaling up MIMO: Opportunities and challenges with very large arrays,” IEEE Signal Processing Magazine, vol. 30, no. 1, pp. 40-60, 2013. [2] A. Chockalingam “Low-complexity algorithms for large-MIMO detection,” 2010 4th International Symposium on Communications, Control and Signal Processing (ISCCSP), pp. 1-6, 2010. [3] K. Vishnu Vardhan, S. Mohammed, A. Chockalingam, and B. Sundar Rajan “A low-complexity detector for large MIMO systems and multicarrier CDMA systems,” IEEE Journal on Selected Areas in Communications, vol. 26, no. 3, pp. 473-485, 2008.
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