WLC36-2: Convergence Acceleration of Iterative ... - Semantic Scholar
Convergence Acceleration of Iterative Signal Detection for MIMO System with Belief Propagation † Graduate
Satoshi Gounai †
Tomoaki Ohtsuki††
School of Electrical Engineering, Tokyo University of Science of Information and Computer Science, Keio University † 2641 Yamazaki, Noda, Chiba 278–8510 Japan †† 3-14-1 Hiyoshi, Kohoku, Yokohama, 223-8522 Japan E-mail :†† [email protected]
†† Department
Abstract— In Multiple-Input Multiple-Output (MIMO) wireless systems, the receiver must extract each transmitted signal from received signals. An iterative signal detection with Belief Propagation (BP) is an attractive technique for MIMO systems. This technique can improve the error rate performance with increasing the number of detection and decoding iterations. This number is, however, limited in actual systems because each additional iteration increases the latency, receiver size, and so on. This paper proposes a convergence acceleration technique that can achieve better error rate performance with fewer iterations than the conventional iterative signal detection. Since the LogLikelihood Ratio (LLR) or a probability of one bit propagates to all other bits with BP, improving some LLRs improves overall decoder performance. In our proposal, all the coded bits are divided into groups and only one group is detected in each iterative signal detection whereas in the conventional approach, each iterative signal detection run processes all coded bits, simultaneously. Our proposal increases the frequency of initial LLR update (for BP) by increasing the number of iterative signal detections and decreasing the number of coded bits that the receiver detects in one iterative signal detection. Computer simulations show that our proposal achieves better error rate performance with fewer detection and decoding iterations than the conventional approach.
I. I NTRODUCTION In a Multiple-Input Multiple-Output (MIMO) system, the receiver must detect each transmitted signal from among the signals received. Iterative signal detection with error correction has attracted much attention. In current iterative signal detection schemes [1], [2], conventional signal detection (ex. Maximum Likelihood Detection (MLD), Minimum Mean Square Error (MMSE), and so on) is performed in the first iteration. In the second or later iterative signal detection, the decoder output is processed to realize signal detection. In Iterative Interference Cancellation (IIC), the decoder output is used to make an interference replica. Iterative MLD uses the decoder output as a priori information for the metric calculation. Using a turbo code and a Low-Density Parity Check (LDPC) code together with Belief Propagation (BP) can achieve excellent error rate performance with linear processing time. BP is a decoding algorithm that updates and exchanges the Log-Likelihood Ratio (LLR) of each bit iteratively. BP can approach the performance of Maximum A posterior Probability (MAP) decoding. However, since BP is not exact MAP decoding, decoding is not guaranteed to converge within a fixed number of iterations.
To achieve a reasonable degree of convergence, BP demands quite a few detection and decoding iterations. Increasing the number of iterations yields improved error rate performance but the improvements tend to saturate with iteration number. Unfortunately, in actual systems, the numbers of detections and decoding iterations are restricted to minimize latency, receiver size, and so on. In addition, BP needs several decoding iterations to propagate the LLR. Thus, we need a convergence acceleration technique that can converge the performance or achieve acceptable performance with only a few iterations. Sequential BP can, at the cost of higher decoding latency, converge with a fewer decoding iterations than parallel BP [3], [4]. Since one bit carries all the other bits’ information, LLR propagation is faster with sequential updating than with parallel updating. In particular, sequential BP can achieve better BER performance than the parallel BP with small numbers of decoding iterations. In this paper, we propose a convergence acceleration technique for MIMO systems that use BP. The conventional and proposed approaches perform signal detection and decoding, alternately. These perform decoding on all bits simultaneously. In the proposed, to utilize the BP characteristic of achieving approximate MAP decoding, the proposed technique divides the coded bits into several groups based on transmit time. Proposal performs detection on groups of coded bits, and detects one group only in one iterative signal detection, unlike the conventional approach that detects all coded bits in one iterative signal detection. Since the process of the signal detection is performed independently for time slot, with the same number of detections per coded bit, the proposed technique and the conventional approach yield the same detection complexity, but the former provides a larger number of detection iterations. In addition, when the decoding iterations of the conventional approach is divided by the number of groups in the proposal, with the same number of detections per coded bit, the proposed technique and the conventional approach yield the same numbers of decoding iterations. At the same maximum numbers of detections per coded bit and decoding, the proposed technique offers higher signal detection and LLR update frequencies than the conventional approach. In addition, in BP, the improvement of some LLRs leads to the improvement of the decoder performance. Since the proposed technique updates initial LLRs more frequently than the conventional approach, at the same maximum numbers of detections and decoding iterations, the proposed technique improves the BP performance more
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This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE GLOBECOM 2006 proceedings.
r
~ F
Signal
Fig. 1.
Soft Desicion
xtl ylt rt − H˜
Decoder
Detector
~ x or x
by subtracting x ˜tl from the received signal.
Replica or A priori probability Generator
= H(xt − x ˜tl ) + n,
~c
where H and n represent the channel matrix and the noise vector, respectively. The covariance matrix ∆tl is defined as follows.
System model of receiver
than the conventional approach. Computer simulations show that the proposed technique can achieve better BER with fewer detections per coded bit and fewer decoding iterations than the conventional approach.
∆tl cov{xt − x ˜tl } t t t − |˜ xtl−1 |2 , 1, Sl+1 − |˜ xtl+1 |2 , .., SN = diag{S1t − |˜xt1 |2 , ..., Sl−1
t
−1 T wlt = el HH H∆tl HH + σ 2 I
Fig. 1 shows the receiver of a MIMO system with iterative signal detection. In this paper, we evaluate the error rate performance of iterative MMSE Soft Interference Cancellation (SIC) [5] and iterative MLD as the iterative signal detection schemes. In the first signal detection, since priori information is not available, the conventional signal detection (MLD or MMSE) is performed using a soft decision decoder. Note that in this paper, we assume that signal detection and the derivation of the initial LLR are performed simultaneously. When decoding yields a valid codeword, both iterative signal detection and iterative decoding stop, and the valid codeword is output. On the other hand, if no valid codeword is obtained up to the maximum number of decoding iterations, iterative signal detection is then performed. In iterative MMSE-SIC, a posterior probability LLR is used to make a soft replica. Interference cancellation is performed by subtracting the soft replica from the received signal to suppress the interference. The resulting signal is then input to the MMSE module. The detected signal is decoded by the soft decision decoder again. On the other hand, in iterative MLD, a posterior probability LLR is used as priori information. For iterative MLD, we use this prior information in the metric calculation. In this paper, we use the same soft decision decoders for both iterative MLD and MMSE-SIC. A. MMSE-SIC In MMSE-SIC, soft replica x ˜tl of signal xtl transmitted from the lth antenna at time t is derived from a posterior probability LLR c˜tl,m as follows. x ˜tl = E[xtl ] x ˆP (xtl = x ˆ) = x ˆ∈Ω
x ˆ
(3)
(4)
where el is the vector whose lth element is 1 and all other elements are 0. σ 2 is the noise variance. ylt , the estimated T t signal of xtl , is detected as ylt = wlt ylt . The initial LLR F˜l,m t is derived from yl . B. Iterative MLD With iterative MLD, we derive a priori probability, the occurrence probability of the transmitted vector x ¯, from a posterior probability LLR c˜tl,m , as follows. x ¯t =
Nt log 2 |Ω|
[1 + exp(−{¯ x}l,m · c˜tl,m )]−1 ,
(5)
l=1 m=1
where {¯ x}l,m , which corresponds to the mth bit of the lth antenna’s symbol in vector x ¯, is either 1 or −1. The initial LLR of the mth bit of the symbol transmitted from the lth antenna is derived as follows. t r −H¯ x0l,m 0,t · x ¯ max exp 2 l,m σ ¯0 x¯0l,m ∈X l,m t − c˜tl,m , (6) F˜l,m = ln t r −H¯ x1l,m 1,t · x ¯ max exp l,m σ2 ¯1 x ¯1l,m ∈X l,m
¯ 0 and X ¯ 1 are the sets of transmitted vectors where X l,m l,m wherein the mth bit of the symbol transmitted from the lth antenna is 0 and 1, respectively. x ¯0l,m and x ¯1l,m are the transmitted vectors wherein the mth bit of the symbol transmitted from the lth antenna is 0 and 1, respectively. x ¯0,t ¯1,t l,m and x l,m 0 1 are the priori probabilities that x ¯l,m and x ¯l,m occur at time t, respectively. C. Update of Initial LLR
x ˆ∈Ω
=
− |˜ xtNt |2 }
Slt is the signal energy of the symbol that is nearest to x ˜tl . tT t MMSE weight wl for yl is obtained as follows.
II. S YSTEM M ODEL
log2 |Ω|
(2)
[1 + exp(−{ˆ x}m · c˜tl,m )]−1 ,
(1)
m=1
where m is the bit index in the symbol. Ω shows the constellation. |Ω| shows the size of the constellation. The interference replica vector x ˜tl for detecting xtl is defined t t t t as x ˜l [˜ ˜l−1 , 0, x x1 , ..., x ˜l+1 , ..., x ˜tNt ]T where T shows transposition. Interference cancellation to detect xtl is performed
The a posterior probability of the nth bit cd,b n in the dth iterative signal detection and the bth decoding iteration is d d,b d shown as cd,b n = Fn +Ln where Fn is the initial LLR for the soft decision decoder; Ld,b is the extrinsic information derived n by the decoder. Note that in this section, for convenience, we omit the time index t, the antenna index l, and the symbol t index m of both c˜tl,m and F˜l,m , and cn and Fn correspond to t t ˜ c˜l,m and Fl,m , respectively.
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Fig. 2.
Iterations for All the Bits
Bmax/4 times Decoding
Iterations for All the Bits
Bmax/4 times Decoding
Iterations for All the Bits
Bmax/4 times Decoding
Iterations for All the Bits
Bmax/4 times Decoding
Iterations for All the Bits
Detect Group 4
for All the Bits
Detect Group 3
Iterations
Detect Group 4
Detect Group 1
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Detect Group 3
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for All the Bits
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Iterations
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...
...
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All the Bits
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Decoding
Detect Group 3
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Iterations for
for All the Bits
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Iterations for
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Bmax times Decoding Iterations for All the bits
Detect Group 1
Bmax/4 times Decoding
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for All the Bits
Decoding
Detect Group 4
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Bmax/2 times
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Decoding
Detect Group 2
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Bmax times Decoding Iterations for All the bits
Detect Group 1
Detect Group 4 Detect Group 4 Detect Group 4
Detect Group 3 Detect Group 3 Detect Group 3
Detect Group 2 Detect Group 2 Detect Group 2
.... .... ....
G=4
Detect Group 1
G=2
Detect Group 1
G=1 (Conv.)
Detect Group 1
Detection and Decoding time
...
Example of the proposed iterative signal detection technique with G = 1, 2, and 4
III. P ROPOSED I TERATIVE S IGNAL D ETECTION T ECHNIQUE The iterative signal detections performs signal detection and decoding, alternately and perform decoding on all bits simultaneously. In the conventional approach also detects all the coded bits, simultaneously. To utilize the MAP decoding characteristic of BP, however, our proposal divides all the coded bits into several groups based on the transmitted time, and only one group is detected in each signal detection, unlike the conventional approach. Since BP achieves only approximate MAP decoding, improving one LLR improves the overall decoding performance. In addition, in iterative signal detection, improving the decoding performance improves overall system performance. In the proposed technique, the number of coded bits detected in each iteration is reduced while the frequencies of signal detection and initial LLR updating are increased. In the proposed technique, we divide all symbols transmitted, M , into G groups based on the transmitted time. G is the number of groups and g ∈ {0, ..., G − 1} is the target group for signal detection and detect signals group by group. Dmax and Bmax are the maximum numbers of detections per coded bit and iterative decoding. Bperiod = Bmax /G is the number of decoding iterations performed between group based signal detections. Our proposal performs Bperiod decoding iterations and group based signal detection, alternately. Note that the proposed technique with G = 1 corresponds to the conventional iterative signal detection method. In the first signal detection, all coded bits are detected simultaneously in the conventional manner. BP is then initiated using initial LLR Fn1 . Irrespective of the number of groups G, the proposed technique is the same as the conventional approach until the number of decoding iterations b reaches Bperiod . At this point, the proposed iterative technique starts the second iterative signal detection. 1. Initialize: We set the initial detection number, d, to 1, the initial decoding iteration number, b, to 1, and the target group in signal detection, g, to 0 2. For t = 0, ..., M/Nt , we detect yt from rt . 3. Derive the initial LLR Fn1 for n = 0, ..., N 4. Start iterative decoding. 5. Proposed iterative detection technique, 5-a. When b is smaller than Bperiod , we iterate the decoding process. 5-b. When b reaches Bperiod , we make a soft replica,
x ˜t or calculate the a priori information for symbols, x ¯t included in the gth group x ˜t (t ∈ g) using a posterior probability LLR. In addition, when g is 0, we increment d. 5-c. Detect coded bits included in the gth group. 5-d. The initial LLRs of the bits included in the gth group are updated. 5-e. When g is smaller than G − 1, we increment g. When g = G − 1, we set g to 0 5-f. When d ≤ Dmax , we reset b to 0 and return to 5-a. 5-g. When d > Dmax , we do not perform any additional signal detection. We iterate the decoding process until b reaches Bmax . 5-h. When a valid codeword is obtained, we regard that valid codeword as the estimated codeword. If d and b reach Dmax and Bmax , respectively, we regard decoding as having failed. Note that with error detection (ex. the LDPC parity check matrix, CRC, and so on), when the estimated codeword is regarded as the valid codeword, we halt detection and decoding irrespective of b and d. In the proposed technique, when the number of decoding iterations, b, reaches Bperiod , we perform signal detection, and signal detection is performed on M /G symbols. In the proposed technique, we increment the number of detections, d, when all groups (or all coded bits) have been detected once. Note that the proposed technique performs group based signal detection when the number of decoding iterations, b, reaches Bperiod . Irrespective of G, there are an even number of decoding iterations if the number of detections per coded bit is even. Fig. 2 shows the block diagram of the proposed iterative signal detection technique with G = 1, 2, and 4. Note that, in this paper, when d is greater than Dmax , we halt iterative signal detection at step 5-h. This is done to ensure that the conventional and proposed iterative signal detection techniques have the same maximum numbers of detection and decoding iterations. IV. S IMULATION R ESULTS
A. LDPC Codes Figs. 3–11 show the BER performances of the conventional and proposed iterative signal detection techniques. All simulations assume that channel estimation is perfect and that the channel gain varies independently symbol by symbol. In Figs. 3 – 10, QPSK modulation is used. In Fig. 11, the modulation is
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16QAM. “Conv.” shows the BER of the conventional iterative signal detection technique. Figs. 3 and 4 show the BER performances of the conventional and proposed iterative signal detection techniques with the irregular LDPC code and the regular LDPC code for 4 × 4 MIMO systems with MMSESIC, respectively. Figs. 5 and 6 show, respectively, the average number of detections per coded bit and decoding iterations of the conventional and proposed iterative signal detection techniques with the irregular LDPC code for 4 × 4 MIMO systems with MMSE-SIC. In Figs. 3–6, the code length and the code rate are 8000 and 0.5, respectively, and the maximum number of detections per coded bit, Dmax , and decoding iterations, Bmax , are 5 and 20, respectively. For the proposed technique, G is set to 1, 2, 4 and 10. Note that G = 1 is equivalent to the conventional iterative signal detection. We also show the BER performances of the conventional iterative signal detection technique with Dmax values of 10 and 100. Figs. 3 and 4 show that for the same Dmax value, the proposed technique offers better BER performance than the conventional approach. Since the proposed technique updates the initial LLRs more frequently than the conventional approach, the proposed technique offers better decoding performance than the conventional approach. The BER performance of the proposed technique is improved with increasing the number of groups. As the number of groups increases, the initial LLR is updated more frequently. In addition, at a low number of decoding iterations, even if the soft replica is wrong, the effect of error propagation is likely to be small, since the LLR of each bit is likely to be small. Fig. 3 shows that the BER performances of the proposed technique with G = 4 and 10 match that of the conventional approach with Dmax = 10. However, the conventional approach with Dmax = 10 needs 5 more detections per coded bit and 100 more decoding iterations than the proposed technique with G = 4 and 10. In addition, the conventional approach with Dmax = 100 achieves better BER performance than the proposed technique with G = 4 and 10. However, the conventional approach with Dmax = 100 uses 95 more detections per coded bit and 1900 more decoding iterations than the proposed technique. Fig. 3 shows that the BER performance is improved with increasing the number of detection and decoding iterations. Figs. 5 and 6 confirm that the proposed technique needs fewer (average) detections per coded bit and decoding iterations than the conventional approach. Since LDPC codes can realize error detection, the average numbers of detections per coded bit and decoding iterations fall as decoder performance is improved. Since the proposed technique can achieve better BER performance than the conventional approach even though it uses fewer (average) detection and decoding iterations with G = 4 and 10, the proposed technique converges faster than the conventional approach. B. Turbo Codes Figs. 7 shows the BER performance of the conventional and proposed techniques in 4×4 MIMO systems with QPSK modulation, MMSE-SIC, and the turbo code. Figs. 8 and 9 show the average numbers of detections per coded bit and decoding iterations of the conventional and proposed techniques in 4×4 MIMO systems with QPSK modulation, MMSE-SIC, and the
turbo code. Note that, in Fig. 9, one run through the component decoder is counted as 0.5 decoding iterations. In Figs. 7– 9, “with CRC” shows the performance of the turbo code concatenated with CRC for the error detection [6]. “w/o CRC” shows the performance of the straight turbo code. The turbo encoder is constructed by the parallel concatenation of two (31, 27)8 RSC encoders. Subscript 8 represents octal notation. We use CCITT CRC-16. The code rate of the turbo encoder is 0.5, and the code length of the turbo code is 8000; that of the CRC concatenated turbo code is 8032. The maximum number of detections, Dmax , and that of decoding iterations, Bmax , are 5 and 10, respectively. The number of groups G is 1, 2, and 5. G = 1 corresponds to the conventional approach. From Fig. 7, the proposed technique achieves better BER performance with the turbo code than the conventional approach. Since the turbo code undergoes BP decoding, the update of some initial LLRs improves overall decoder performance. In addition, with the same number of groups, “w/o CRC” achieves better BER performance than “with CRC”. Since the concatenation of CRC lowers the code rate, “with CRC” yields smaller SNR than “w/o CRC” at the same Eb /N0 . “with CRC” is weak for channel error. However, Fig. 7 shows that the performance improvement offered by the proposed technique exceeds the performance degradation created by the concatenation of CRC. In addition, Figs. 8 and 9 show that since CRC yields error detection, the concatenation of CRC to the turbo code reduces the average number of detections per coded bit and the decoding iterations. In particular, in the high Eb /N0 region, the average numbers of detections per coded bit and decoding iterations are smaller than one half of the maximum numbers of detections per coded bit and decoding iterations. With the turbo code, increasing the number of groups raises the BER performance of the proposed technique while lowering the average number of iterations. Fig. 7 confirms that the proposed technique is also effective with the turbo code. From Figs. 7–9, we can also see that using error detection makes the proposed technique even more effective with the turbo code. While we use CRC for error detection in this paper, other error detection schemes can be applied. C. Extension of Proposed Signal Detection Technique for MLD and 16QAM Fig. 10 shows the BER performance of the irregular LDPC code for a 4 × 4 MIMO system with QPSK modulation and iterative MLD. The maximum numbers of detections per coded bit and decoding iterations are 5 and 20, respectively. In the proposed technique, the numbers of groups G are 1, 2, 4, and 10. In Fig. 10, as is the case with MMSE-SIC, increasing the number of groups improves the BER performance. Fig. 10 shows that the proposed technique is effective for iterative MLD. Fig. 11 shows the BER performance of the irregular LDPC code for 4 × 4 MIMO systems with 16QAM modulation and MMSE-SIC. In Fig. 11 “Information” shows the BER performance of the mapping scheme in which information bits are mapped to MSB [7]. “Random” shows the performance of the mapping scheme in which information bits and parity bits are mapped randomly. The maximum number of detections per coded bit and decoding iterations are 5 and 20, respec-
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This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE GLOBECOM 2006 proceedings.
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tively. From Fig. 11, irrespective of the mapping scheme, the proposed technique is effective for 16QAM. V. C ONCLUSION In this paper, we proposed a convergence acceleration technique for MIMO systems with BP. To utilize the characteristic of BP, the proposed iterative signal detection technique divides the transmitted signals into several groups. The proposed technique detects coded bits group by group and increases the frequency of initial LLR updating compared to the conventional approach. Computer simulations showed that when the proposed technique and the conventional approach have equal maximum number of detections and decoding iterations, the former can achieve better BER performance than the latter with fewer detection and decoding iterations.
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R EFERENCES [1] B. Steingrimsson, Z. Q. Luo, and K. M. Wong, “Soft quasi-maximumlikelihood detection for multiple-antenna wireless channels,” IEEE Trans. Signal Proc., vol. 51, no. 11, pp. 2710–2719, Nov. 2003. [2] B. Lu, G. Yue and X. Wang, “Performance analysis and design optimization of LDPC-coded MIMO OFDM system,” IEEE Trans. Signal Proc., vol. 52, pp. 348–361, Feb. 2004. [3] H. Kfir and I. Kanter, “Parallel versus sequential updating for belief propagation decoding,” Physica A, vol. 330, pp. 259-270, 2003. [4] J. Xhang and M. Fossorier, “Shuffled Belief Propagation Decoding,” IEEE Trans. Commun., Feb. 2005. [5] X. Wang and H. V. Poor, “Iterative (turbo) soft interference cancellation and decoding for coded CDMA,” IEEE Trans. Commun., vol. 47, pp. 1046–1061, July 1999. [6] A. Shibutani, H. Suda, and F. Adachi, “Complexity reduction of turbo decoding,” IEEE VTC 1999-Fall vol. 3, pp. 1570 – 1574, Sept. 1999. [7] Y. Li and W. E. Ryan, “Bit-reliability mapping in LDPC-coded modulation systems,” IEEE Trans. Commun., Lett., vol. 9, Jan. 2005.
1-4244-0357-X/06/$20.00 ©2006 IEEE
This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE GLOBECOM 2006 proceedings.
Satoshi Gounai †
Tomoaki Ohtsuki††
School of Electrical Engineering, Tokyo University of Science of Information and Computer Science, Keio University † 2641 Yamazaki, Noda, Chiba 278–8510 Japan †† 3-14-1 Hiyoshi, Kohoku, Yokohama, 223-8522 Japan E-mail :†† [email protected]
†† Department
Abstract— In Multiple-Input Multiple-Output (MIMO) wireless systems, the receiver must extract each transmitted signal from received signals. An iterative signal detection with Belief Propagation (BP) is an attractive technique for MIMO systems. This technique can improve the error rate performance with increasing the number of detection and decoding iterations. This number is, however, limited in actual systems because each additional iteration increases the latency, receiver size, and so on. This paper proposes a convergence acceleration technique that can achieve better error rate performance with fewer iterations than the conventional iterative signal detection. Since the LogLikelihood Ratio (LLR) or a probability of one bit propagates to all other bits with BP, improving some LLRs improves overall decoder performance. In our proposal, all the coded bits are divided into groups and only one group is detected in each iterative signal detection whereas in the conventional approach, each iterative signal detection run processes all coded bits, simultaneously. Our proposal increases the frequency of initial LLR update (for BP) by increasing the number of iterative signal detections and decreasing the number of coded bits that the receiver detects in one iterative signal detection. Computer simulations show that our proposal achieves better error rate performance with fewer detection and decoding iterations than the conventional approach.
I. I NTRODUCTION In a Multiple-Input Multiple-Output (MIMO) system, the receiver must detect each transmitted signal from among the signals received. Iterative signal detection with error correction has attracted much attention. In current iterative signal detection schemes [1], [2], conventional signal detection (ex. Maximum Likelihood Detection (MLD), Minimum Mean Square Error (MMSE), and so on) is performed in the first iteration. In the second or later iterative signal detection, the decoder output is processed to realize signal detection. In Iterative Interference Cancellation (IIC), the decoder output is used to make an interference replica. Iterative MLD uses the decoder output as a priori information for the metric calculation. Using a turbo code and a Low-Density Parity Check (LDPC) code together with Belief Propagation (BP) can achieve excellent error rate performance with linear processing time. BP is a decoding algorithm that updates and exchanges the Log-Likelihood Ratio (LLR) of each bit iteratively. BP can approach the performance of Maximum A posterior Probability (MAP) decoding. However, since BP is not exact MAP decoding, decoding is not guaranteed to converge within a fixed number of iterations.
To achieve a reasonable degree of convergence, BP demands quite a few detection and decoding iterations. Increasing the number of iterations yields improved error rate performance but the improvements tend to saturate with iteration number. Unfortunately, in actual systems, the numbers of detections and decoding iterations are restricted to minimize latency, receiver size, and so on. In addition, BP needs several decoding iterations to propagate the LLR. Thus, we need a convergence acceleration technique that can converge the performance or achieve acceptable performance with only a few iterations. Sequential BP can, at the cost of higher decoding latency, converge with a fewer decoding iterations than parallel BP [3], [4]. Since one bit carries all the other bits’ information, LLR propagation is faster with sequential updating than with parallel updating. In particular, sequential BP can achieve better BER performance than the parallel BP with small numbers of decoding iterations. In this paper, we propose a convergence acceleration technique for MIMO systems that use BP. The conventional and proposed approaches perform signal detection and decoding, alternately. These perform decoding on all bits simultaneously. In the proposed, to utilize the BP characteristic of achieving approximate MAP decoding, the proposed technique divides the coded bits into several groups based on transmit time. Proposal performs detection on groups of coded bits, and detects one group only in one iterative signal detection, unlike the conventional approach that detects all coded bits in one iterative signal detection. Since the process of the signal detection is performed independently for time slot, with the same number of detections per coded bit, the proposed technique and the conventional approach yield the same detection complexity, but the former provides a larger number of detection iterations. In addition, when the decoding iterations of the conventional approach is divided by the number of groups in the proposal, with the same number of detections per coded bit, the proposed technique and the conventional approach yield the same numbers of decoding iterations. At the same maximum numbers of detections per coded bit and decoding, the proposed technique offers higher signal detection and LLR update frequencies than the conventional approach. In addition, in BP, the improvement of some LLRs leads to the improvement of the decoder performance. Since the proposed technique updates initial LLRs more frequently than the conventional approach, at the same maximum numbers of detections and decoding iterations, the proposed technique improves the BP performance more
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r
~ F
Signal
Fig. 1.
Soft Desicion
xtl ylt rt − H˜
Decoder
Detector
~ x or x
by subtracting x ˜tl from the received signal.
Replica or A priori probability Generator
= H(xt − x ˜tl ) + n,
~c
where H and n represent the channel matrix and the noise vector, respectively. The covariance matrix ∆tl is defined as follows.
System model of receiver
than the conventional approach. Computer simulations show that the proposed technique can achieve better BER with fewer detections per coded bit and fewer decoding iterations than the conventional approach.
∆tl cov{xt − x ˜tl } t t t − |˜ xtl−1 |2 , 1, Sl+1 − |˜ xtl+1 |2 , .., SN = diag{S1t − |˜xt1 |2 , ..., Sl−1
t
−1 T wlt = el HH H∆tl HH + σ 2 I
Fig. 1 shows the receiver of a MIMO system with iterative signal detection. In this paper, we evaluate the error rate performance of iterative MMSE Soft Interference Cancellation (SIC) [5] and iterative MLD as the iterative signal detection schemes. In the first signal detection, since priori information is not available, the conventional signal detection (MLD or MMSE) is performed using a soft decision decoder. Note that in this paper, we assume that signal detection and the derivation of the initial LLR are performed simultaneously. When decoding yields a valid codeword, both iterative signal detection and iterative decoding stop, and the valid codeword is output. On the other hand, if no valid codeword is obtained up to the maximum number of decoding iterations, iterative signal detection is then performed. In iterative MMSE-SIC, a posterior probability LLR is used to make a soft replica. Interference cancellation is performed by subtracting the soft replica from the received signal to suppress the interference. The resulting signal is then input to the MMSE module. The detected signal is decoded by the soft decision decoder again. On the other hand, in iterative MLD, a posterior probability LLR is used as priori information. For iterative MLD, we use this prior information in the metric calculation. In this paper, we use the same soft decision decoders for both iterative MLD and MMSE-SIC. A. MMSE-SIC In MMSE-SIC, soft replica x ˜tl of signal xtl transmitted from the lth antenna at time t is derived from a posterior probability LLR c˜tl,m as follows. x ˜tl = E[xtl ] x ˆP (xtl = x ˆ) = x ˆ∈Ω
x ˆ
(3)
(4)
where el is the vector whose lth element is 1 and all other elements are 0. σ 2 is the noise variance. ylt , the estimated T t signal of xtl , is detected as ylt = wlt ylt . The initial LLR F˜l,m t is derived from yl . B. Iterative MLD With iterative MLD, we derive a priori probability, the occurrence probability of the transmitted vector x ¯, from a posterior probability LLR c˜tl,m , as follows. x ¯t =
Nt log 2 |Ω|
[1 + exp(−{¯ x}l,m · c˜tl,m )]−1 ,
(5)
l=1 m=1
where {¯ x}l,m , which corresponds to the mth bit of the lth antenna’s symbol in vector x ¯, is either 1 or −1. The initial LLR of the mth bit of the symbol transmitted from the lth antenna is derived as follows. t r −H¯ x0l,m 0,t · x ¯ max exp 2 l,m σ ¯0 x¯0l,m ∈X l,m t − c˜tl,m , (6) F˜l,m = ln t r −H¯ x1l,m 1,t · x ¯ max exp l,m σ2 ¯1 x ¯1l,m ∈X l,m
¯ 0 and X ¯ 1 are the sets of transmitted vectors where X l,m l,m wherein the mth bit of the symbol transmitted from the lth antenna is 0 and 1, respectively. x ¯0l,m and x ¯1l,m are the transmitted vectors wherein the mth bit of the symbol transmitted from the lth antenna is 0 and 1, respectively. x ¯0,t ¯1,t l,m and x l,m 0 1 are the priori probabilities that x ¯l,m and x ¯l,m occur at time t, respectively. C. Update of Initial LLR
x ˆ∈Ω
=
− |˜ xtNt |2 }
Slt is the signal energy of the symbol that is nearest to x ˜tl . tT t MMSE weight wl for yl is obtained as follows.
II. S YSTEM M ODEL
log2 |Ω|
(2)
[1 + exp(−{ˆ x}m · c˜tl,m )]−1 ,
(1)
m=1
where m is the bit index in the symbol. Ω shows the constellation. |Ω| shows the size of the constellation. The interference replica vector x ˜tl for detecting xtl is defined t t t t as x ˜l [˜ ˜l−1 , 0, x x1 , ..., x ˜l+1 , ..., x ˜tNt ]T where T shows transposition. Interference cancellation to detect xtl is performed
The a posterior probability of the nth bit cd,b n in the dth iterative signal detection and the bth decoding iteration is d d,b d shown as cd,b n = Fn +Ln where Fn is the initial LLR for the soft decision decoder; Ld,b is the extrinsic information derived n by the decoder. Note that in this section, for convenience, we omit the time index t, the antenna index l, and the symbol t index m of both c˜tl,m and F˜l,m , and cn and Fn correspond to t t ˜ c˜l,m and Fl,m , respectively.
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Fig. 2.
Iterations for All the Bits
Bmax/4 times Decoding
Iterations for All the Bits
Bmax/4 times Decoding
Iterations for All the Bits
Bmax/4 times Decoding
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Bmax/4 times Decoding
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Detect Group 4
for All the Bits
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Detect Group 4
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Detect Group 3
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for All the Bits
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...
...
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for All the Bits
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Bmax times Decoding Iterations for All the bits
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Decoding
Detect Group 4
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Bmax times Decoding Iterations for All the bits
Detect Group 1
Detect Group 4 Detect Group 4 Detect Group 4
Detect Group 3 Detect Group 3 Detect Group 3
Detect Group 2 Detect Group 2 Detect Group 2
.... .... ....
G=4
Detect Group 1
G=2
Detect Group 1
G=1 (Conv.)
Detect Group 1
Detection and Decoding time
...
Example of the proposed iterative signal detection technique with G = 1, 2, and 4
III. P ROPOSED I TERATIVE S IGNAL D ETECTION T ECHNIQUE The iterative signal detections performs signal detection and decoding, alternately and perform decoding on all bits simultaneously. In the conventional approach also detects all the coded bits, simultaneously. To utilize the MAP decoding characteristic of BP, however, our proposal divides all the coded bits into several groups based on the transmitted time, and only one group is detected in each signal detection, unlike the conventional approach. Since BP achieves only approximate MAP decoding, improving one LLR improves the overall decoding performance. In addition, in iterative signal detection, improving the decoding performance improves overall system performance. In the proposed technique, the number of coded bits detected in each iteration is reduced while the frequencies of signal detection and initial LLR updating are increased. In the proposed technique, we divide all symbols transmitted, M , into G groups based on the transmitted time. G is the number of groups and g ∈ {0, ..., G − 1} is the target group for signal detection and detect signals group by group. Dmax and Bmax are the maximum numbers of detections per coded bit and iterative decoding. Bperiod = Bmax /G is the number of decoding iterations performed between group based signal detections. Our proposal performs Bperiod decoding iterations and group based signal detection, alternately. Note that the proposed technique with G = 1 corresponds to the conventional iterative signal detection method. In the first signal detection, all coded bits are detected simultaneously in the conventional manner. BP is then initiated using initial LLR Fn1 . Irrespective of the number of groups G, the proposed technique is the same as the conventional approach until the number of decoding iterations b reaches Bperiod . At this point, the proposed iterative technique starts the second iterative signal detection. 1. Initialize: We set the initial detection number, d, to 1, the initial decoding iteration number, b, to 1, and the target group in signal detection, g, to 0 2. For t = 0, ..., M/Nt , we detect yt from rt . 3. Derive the initial LLR Fn1 for n = 0, ..., N 4. Start iterative decoding. 5. Proposed iterative detection technique, 5-a. When b is smaller than Bperiod , we iterate the decoding process. 5-b. When b reaches Bperiod , we make a soft replica,
x ˜t or calculate the a priori information for symbols, x ¯t included in the gth group x ˜t (t ∈ g) using a posterior probability LLR. In addition, when g is 0, we increment d. 5-c. Detect coded bits included in the gth group. 5-d. The initial LLRs of the bits included in the gth group are updated. 5-e. When g is smaller than G − 1, we increment g. When g = G − 1, we set g to 0 5-f. When d ≤ Dmax , we reset b to 0 and return to 5-a. 5-g. When d > Dmax , we do not perform any additional signal detection. We iterate the decoding process until b reaches Bmax . 5-h. When a valid codeword is obtained, we regard that valid codeword as the estimated codeword. If d and b reach Dmax and Bmax , respectively, we regard decoding as having failed. Note that with error detection (ex. the LDPC parity check matrix, CRC, and so on), when the estimated codeword is regarded as the valid codeword, we halt detection and decoding irrespective of b and d. In the proposed technique, when the number of decoding iterations, b, reaches Bperiod , we perform signal detection, and signal detection is performed on M /G symbols. In the proposed technique, we increment the number of detections, d, when all groups (or all coded bits) have been detected once. Note that the proposed technique performs group based signal detection when the number of decoding iterations, b, reaches Bperiod . Irrespective of G, there are an even number of decoding iterations if the number of detections per coded bit is even. Fig. 2 shows the block diagram of the proposed iterative signal detection technique with G = 1, 2, and 4. Note that, in this paper, when d is greater than Dmax , we halt iterative signal detection at step 5-h. This is done to ensure that the conventional and proposed iterative signal detection techniques have the same maximum numbers of detection and decoding iterations. IV. S IMULATION R ESULTS
A. LDPC Codes Figs. 3–11 show the BER performances of the conventional and proposed iterative signal detection techniques. All simulations assume that channel estimation is perfect and that the channel gain varies independently symbol by symbol. In Figs. 3 – 10, QPSK modulation is used. In Fig. 11, the modulation is
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16QAM. “Conv.” shows the BER of the conventional iterative signal detection technique. Figs. 3 and 4 show the BER performances of the conventional and proposed iterative signal detection techniques with the irregular LDPC code and the regular LDPC code for 4 × 4 MIMO systems with MMSESIC, respectively. Figs. 5 and 6 show, respectively, the average number of detections per coded bit and decoding iterations of the conventional and proposed iterative signal detection techniques with the irregular LDPC code for 4 × 4 MIMO systems with MMSE-SIC. In Figs. 3–6, the code length and the code rate are 8000 and 0.5, respectively, and the maximum number of detections per coded bit, Dmax , and decoding iterations, Bmax , are 5 and 20, respectively. For the proposed technique, G is set to 1, 2, 4 and 10. Note that G = 1 is equivalent to the conventional iterative signal detection. We also show the BER performances of the conventional iterative signal detection technique with Dmax values of 10 and 100. Figs. 3 and 4 show that for the same Dmax value, the proposed technique offers better BER performance than the conventional approach. Since the proposed technique updates the initial LLRs more frequently than the conventional approach, the proposed technique offers better decoding performance than the conventional approach. The BER performance of the proposed technique is improved with increasing the number of groups. As the number of groups increases, the initial LLR is updated more frequently. In addition, at a low number of decoding iterations, even if the soft replica is wrong, the effect of error propagation is likely to be small, since the LLR of each bit is likely to be small. Fig. 3 shows that the BER performances of the proposed technique with G = 4 and 10 match that of the conventional approach with Dmax = 10. However, the conventional approach with Dmax = 10 needs 5 more detections per coded bit and 100 more decoding iterations than the proposed technique with G = 4 and 10. In addition, the conventional approach with Dmax = 100 achieves better BER performance than the proposed technique with G = 4 and 10. However, the conventional approach with Dmax = 100 uses 95 more detections per coded bit and 1900 more decoding iterations than the proposed technique. Fig. 3 shows that the BER performance is improved with increasing the number of detection and decoding iterations. Figs. 5 and 6 confirm that the proposed technique needs fewer (average) detections per coded bit and decoding iterations than the conventional approach. Since LDPC codes can realize error detection, the average numbers of detections per coded bit and decoding iterations fall as decoder performance is improved. Since the proposed technique can achieve better BER performance than the conventional approach even though it uses fewer (average) detection and decoding iterations with G = 4 and 10, the proposed technique converges faster than the conventional approach. B. Turbo Codes Figs. 7 shows the BER performance of the conventional and proposed techniques in 4×4 MIMO systems with QPSK modulation, MMSE-SIC, and the turbo code. Figs. 8 and 9 show the average numbers of detections per coded bit and decoding iterations of the conventional and proposed techniques in 4×4 MIMO systems with QPSK modulation, MMSE-SIC, and the
turbo code. Note that, in Fig. 9, one run through the component decoder is counted as 0.5 decoding iterations. In Figs. 7– 9, “with CRC” shows the performance of the turbo code concatenated with CRC for the error detection [6]. “w/o CRC” shows the performance of the straight turbo code. The turbo encoder is constructed by the parallel concatenation of two (31, 27)8 RSC encoders. Subscript 8 represents octal notation. We use CCITT CRC-16. The code rate of the turbo encoder is 0.5, and the code length of the turbo code is 8000; that of the CRC concatenated turbo code is 8032. The maximum number of detections, Dmax , and that of decoding iterations, Bmax , are 5 and 10, respectively. The number of groups G is 1, 2, and 5. G = 1 corresponds to the conventional approach. From Fig. 7, the proposed technique achieves better BER performance with the turbo code than the conventional approach. Since the turbo code undergoes BP decoding, the update of some initial LLRs improves overall decoder performance. In addition, with the same number of groups, “w/o CRC” achieves better BER performance than “with CRC”. Since the concatenation of CRC lowers the code rate, “with CRC” yields smaller SNR than “w/o CRC” at the same Eb /N0 . “with CRC” is weak for channel error. However, Fig. 7 shows that the performance improvement offered by the proposed technique exceeds the performance degradation created by the concatenation of CRC. In addition, Figs. 8 and 9 show that since CRC yields error detection, the concatenation of CRC to the turbo code reduces the average number of detections per coded bit and the decoding iterations. In particular, in the high Eb /N0 region, the average numbers of detections per coded bit and decoding iterations are smaller than one half of the maximum numbers of detections per coded bit and decoding iterations. With the turbo code, increasing the number of groups raises the BER performance of the proposed technique while lowering the average number of iterations. Fig. 7 confirms that the proposed technique is also effective with the turbo code. From Figs. 7–9, we can also see that using error detection makes the proposed technique even more effective with the turbo code. While we use CRC for error detection in this paper, other error detection schemes can be applied. C. Extension of Proposed Signal Detection Technique for MLD and 16QAM Fig. 10 shows the BER performance of the irregular LDPC code for a 4 × 4 MIMO system with QPSK modulation and iterative MLD. The maximum numbers of detections per coded bit and decoding iterations are 5 and 20, respectively. In the proposed technique, the numbers of groups G are 1, 2, 4, and 10. In Fig. 10, as is the case with MMSE-SIC, increasing the number of groups improves the BER performance. Fig. 10 shows that the proposed technique is effective for iterative MLD. Fig. 11 shows the BER performance of the irregular LDPC code for 4 × 4 MIMO systems with 16QAM modulation and MMSE-SIC. In Fig. 11 “Information” shows the BER performance of the mapping scheme in which information bits are mapped to MSB [7]. “Random” shows the performance of the mapping scheme in which information bits and parity bits are mapped randomly. The maximum number of detections per coded bit and decoding iterations are 5 and 20, respec-
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This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE GLOBECOM 2006 proceedings.
10
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tively. From Fig. 11, irrespective of the mapping scheme, the proposed technique is effective for 16QAM. V. C ONCLUSION In this paper, we proposed a convergence acceleration technique for MIMO systems with BP. To utilize the characteristic of BP, the proposed iterative signal detection technique divides the transmitted signals into several groups. The proposed technique detects coded bits group by group and increases the frequency of initial LLR updating compared to the conventional approach. Computer simulations showed that when the proposed technique and the conventional approach have equal maximum number of detections and decoding iterations, the former can achieve better BER performance than the latter with fewer detection and decoding iterations.
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Fig. 11. BER versus Eb /N0 dB of 4 × 4 MIMO system with 16QAM and irregular LDPC codes using MMSE-SIC: Bmax = 20, and Dmax = 5
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This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE GLOBECOM 2006 proceedings.
