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Unequal Error Correcting Capability Aware Iterative Receiver for (Parallel) Turbo Coded Communications Xiaoming Dai, Zhongshan Zhang, Keping Long, Shaohui Sun, Yingmin Wang
Abstract—Iterative detection and decoding (IDD) technique has received much attention recently due to its capacity approaching performance and manageable complexity for (parallel turbo) coded systems. The existing IDD receivers process indiscriminately the a priori information of the systematic bits and the parity bits generated by the turbo decoder in the front-end detection. In this work, we first show that the parity bits of the parallel turbo codes are inherently less protected than the systematic bits, which leads to sub-optimality for the IDD receivers. Based on this observation, we then propose an unequal-error-correction-capability-aware technique to mitigate the associated performance loss. As an illustrative example, the minimum mean-squared error filtering and soft interference cancellation (MMSE-SIC) aided iterative receiver for parallel turbo coded multiple-input multiple-output (MIMO) systems is employed in this work. Numerical results manifest that the MMSE-SIC aided iterative receiver based on the proposed paradigm achieves noticeable performance gains over the conventional unequal-error-correction-capability-agnostic one even with reduced computational complexity. The extension of the proposed principle to other iterative algorithms for parallel turbo coded data transmission is straightforward. Index Terms—Iterative detection and decoding (IDD), multiple-input multiple-output (MIMO), minimum meansquared error filtering and soft interference cancellation (MMSESIC).
I. Introduction It is well known that iterative detection and decoding (IDD) can achieve near-optimal performance for (parallel turbo) coded1 data transmission over intersymbol interference (ISI) channel [2], multiuser channel [3], and multiple-input multiple-output (MIMO) channel [4]. The key idea underlying the IDD is to utilize the a posteriori information of the transmitted bits (including systematic bits and parity bits) Copyright (c) 2013 IEEE. Personal use of this material is permitted. However, permission to use this material for any other purposes must be obtained from the IEEE by sending a request to
[email protected]. Manuscript received Oct 27, 2013 and accepted Jan. 16, 2014 with no revision. Xiaoming Dai, Shaohui Sun and Yingming Wang are with State Key Laboratory of Wireless Mobile Communications (CATT), Beijing, 100080, P. R. China (email:
[email protected];
[email protected]). Zhongshan Zhang and Keping Long are with Institute of Advanced Network Technologies and New Services (ANTS) and Beijing Engineering and Technology Research Center for Convergence Networks and Ubiquitous Services, University of Science and Technology Beijing (USTB). This work was supported by the National Natural Science Foundation of China under Grants No. 61172050 and Program for New Century Excellent Talents in University (NECT-12-0774). 1 The parallel turbo code is utilized in this work to exemplify the coded transmission since it is widely used in the third generation partnership project (3GPP) standards [1], digital video broadcasting and deep space communications for its good asymptotic as well as practical properties.
generated by the turbo decoder as the a priori knowledge to aid the front-end detection of transmitted data bits. The additional knowledge allows the detector to obtain a more accurate set of soft outputs, which are then utilized by the decoder as updated a priori information for further decoding, thus completing one whole iterative detection and decoding cycle. The encoder of the parallel concatenated convolutional code (PCCC) [5] is composed of two (or more) constituent systematic recursive convolutional encoders joined by an interleaver. The input information bits feed the first component encoder and, after having been interleaved by the interleaver, enter the second component encoder. The codeword of the parallel concatenated code consists of the information bits followed by the parity check bits of both component encoders. At the decoder, the log-likelihood ratios (LLRs) of the systematic and parity bits are normally obtained in an iterative fashion via the BCJR algorithm based soft-input soft-output (SISO) decoding [5]. The a priori information of the systematic bits provided by the other component decoder is exploited to aid the decoding (of the systematic bits). However, there is no direct a priori information concerning the parity bits generated by the other component decoder (since there is no constraint between the parity bit stream 1 and the parity stream 2 [5]) in the decoding of parity bits. As a result, the parity bits exhibit intrinsically worse error correcting performance than the systematic bits and leads to sub-optimality for IDD receivers, which has not been observed in the existing literature [4]–[8]. Based on this analysis, we introduce a scaling based correcting operation to the IDD receiver to account for the intrinsic unequal-error correction capability between the systematic bit stream and the parity bit stream for parallel turbo coded communications. As an illustrative example, the iterative receiver using linear minimum mean-squared error filtering and soft interference cancellation (MMSE-SIC) [7] for MIMO systems is employed in this work. (For detailed exposition of MMSE-SIC iterative algorithm, the readers are referred to [7] and references therein.) By scaling down the LLR of the parity bit accounting for its statistical characteristics, a more balanced match between the reliability of the systematic bit and the parity bit is achieved. Therefore, a more accurate computation of the soft symbol is obtained and leads to a more precise interference cancellation. More accurate a posteriori information on the transmitted bits (derived from the soft interference cancellation) is then fed into the decoder to produce a more precise extrinsic information (updated a priori LLR information for the ensuing front-end detection). This
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cycle proceeds as the iterative detection and decoding carries on. The performance of the MMSE-SIC iterative receiver is thus greatly enhanced. The associated computational complexity of the proposed scaling operation is shown to be negligible compared with the complexity reduction due to the decrement in the iterative detection-decoding cycle. The rest of this work is structured as follows: Section II introduces the system model and briefly reviews basic principle of the IDD and the conventional MMSE-SIC algorithm. In Section III, we propose an unequal-error-correction-capabilityaware technique and apply it to the MMSE-SIC aided iterative receiver. The performance and complexity of the proposed method is compared with the conventional one in the same Section. Finally, we conclude this work in Section IV.
where LSA (xi,b ) denotes the a priori LLR of the systematic P bit stream and LA1/2 (xi,b ) stand for the a priori LLR of the parity bit stream 1 and 2, respectively. Then LE (xi,b ) is deinterleaved to become the a priori input to the outer softin soft-out decoder which calculates extrinsic (new a priori) LLR (It is shown in [7] that using a posteriori information as feedback performs better using extrinsic information for the SIC-MMSE aided iterative receiver. In this work, the a posteriori information is utilized.) of the transmitted bits. Then the knowledge gleaned from the decoding process is reinterleaved and fed back as updated a priori LLR LA (xi,b ) to the inner detector, thus completing one cycle or “iteration” of the IDD.
II. Conventional MMSE-SIC Iterative Receiver
We summarize the derivation of the MMSE-SIC algorithm [10] in the following four paragraphs. 1) Computation of Soft Symbols: Since the bits resulting from the channel code are independent among spatial streams i and bits b for BICM systems, the estimated soft symbols sˆi (i = 1, · · · , NT ) are computed according to [3] X sˆi = E[si ] = Prob[si = s]s, (3)
Consider a bit-interleaved coded modulation (BICM) MIMO multiplexing system with NT transmit and NR receive antennas [1], [9]. (NR = NT is assumed in this work for analytical simplicity.) The turbo encoder contains of two identical recursive systematic binary convolutional encoders (henceforth denoted as component encoders) [5]. The turbo encoder receives information bits b = [b1 , b2 , · · · , bK ]T and outputs the systematic bit stream bS = [bS1 , bS2 , · · · , bSK ]T , parity bit stream 1 bP1 = [b1P1 , b2P1 , · · · , bPK1 ]T and parity P2 T P2 P2 ] of the component , · · · , b2K , bk+2 bit stream 2 bP2 = [bk+1 encoder 1 and 2, respectively. (For ease of exposition, we do not consider the tail bits due to trellis termination. Higher code rates are obtained by puncturing the coded bits.) The resulting coded bit stream c = [bS , bP1 , bP2 ] is passed through a bitwise interleaver Π. The interleaved bits cΠ = x = [x1 , x2 , · · · , x3K ]T is mapped (using Gray mapping) to a sequence of transmit vectors s = [s1 , · · · , sNT ]T ∈ XNT , where X refers the set of complex-valued quadrature amplitude modulation (QAM) constellation points with the size |X| of Q [9]. The standard complex baseband model between the transmitted and received signals is expressed as y = Hs + n
(1)
where y ∈ CNR ×1 denotes the received vector, H ∈ CNR ×NT represents the channel matrix, s ∈ CNT ×1 is the transmitted vector, and n ∈ CNR ×1 is a vector of independent zero-mean complex Gaussian noise entries with variance σ2 . Slightly abusing common terminology, we denote the entries of x as xi,b (i = 1, · · · , NT , b = 1, · · · , Mc ), where the indices i and b stands for the b-th bit in the label of the constellation point corresponding to the i-th entry of s and Mc is number of bits per constellation symbol. A. Basic Principle of the IDD The front-end detector of the IDD receiver takes channel observations y and a priori LLR LA (xi,b ) LA (xi,b ) (= P [LSA (xi,b ), LA1/2 (xi,b )]) feedback from the channel decoder and computes the extrinsic LLR LE (xi,b ) as [3], [4] LE (xi,b ) = ln
Prob[xi,b = 1|y] − LA (xi,b ), Prob[xi,b = 0|y]
(2)
B. MMSE-SIC Algorithm
s∈X
Q Mc
where Prob[si = s] = b=1 Prob[xi,b = x] denotes the a priori probability of the symbol s ∈ X with xi,b = [s]b referring to the b-th bit associated with the symbol s. The probability of the transmitted bit xi,b is calculated by [3] !! 1 1 1+(2x − 1) tanh LA (xi,b ) . (4) Prob[xi,b = x] = 2 2 The a priori LLR LA (xi,b ) is set to zero in the first iteration, i.e., LA (xi,b ) = 0, ∀i, b. The error between the transmitted symbol si and the estimated soft symbol sˆi is defined as ei = si − sˆi . The reliability of each soft symbol sˆi is determined by its variance Ei = Var[si ] = E[|ei |2 ]. (5) 2) Soft Instantaneous Interference Cancellation: The MMSE-SIC front-end detector perceives each stream separately and cancels the interference in y induced by all other streams j , i as follows [3] X yˆ i = y − h j sˆ j = hi si + n˜ i (6) j, j,i P where h j stands for the j-th column of H and n˜ i = j, j,i h j e j + n denotes the residual interference plus noise. 3) MMSE Filtering: The distribution of the residual interference-plus-noise at the output of a linear detector is well approximated by a Gaussian distribution [3]. Based on (6), the NT MMSE filter vectors minimizing the mean-squared error (MSE) between the vector yˆ i and the transmitted symbol on the i-th stream are computed as [12] ˜ i HH + σ2 INT )−1 = hiH A−1 , ˜ iH = hiH (HΛ w
(7)
˜ i is a n real-valued NT × NT diagonal matrix having where Λ ˜ i,i = Ei , j,i . The instantaneous MMSE filter forms entries Λ 1, j=i a maximum-ratio combining with the corresponding column
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vector of the channel matrix. The output of the soft instantaneous MMSE filter of i-th data layer is given by (8)
˜ iH hi . where µ˜ i = w 4) Extrinsic LLR Computation: The MMSE-SIC iterative receiver approximates the a posteriori LLR by assuming that NT single-input single-output systems in (8) are statistically independent and that the weighted residual interference plus (1) ˜ iH n˜ i is Gaussian distributed [7]. Let X(0) noise term w i,b and Xi,b stand for the sets of candidate symbol vectors corresponding to xi,b = 0 and xi,b = 1, respectively. The a posterior LLRs of the transmitted bits are then computed as |˜zi − µ˜ i s|2 X Mc (2xi,k − 1) + LD (xi,b ) ≈ min L (x ) A i,k − 2 k=1 2 v˜ i s∈X(0) i,b |˜zi − µ˜ i s|2 X Mc (2xi,k − 1) LA (xi,k ) + min 2 (1) k=1 2 v˜ i s∈Xi,b (9) where v˜ 2i denotes the variance of the weighted residual interference plus noise, i.e., v˜ 2i = Var[˜zi ] = P H 2 ˜ ˜ iH w . h h + σ I w i NT j, j,i j j Finally, the transmitted data bits’s extrinsic LLR LE (xi,b ) is computed as LE (xi,b ) = LD (xi,b ) − LA (xi,b ). The MMSE-SIC front-end detection performs a combination of suppressions and cancellations. The amount of suppression realized by the detector is determined by the quality of the canceled soft symbols. The accuracy of the soft symbols hinges solely on the reliability of the a priori LLRs [12] P (LA (xi,b ) = [LSA (xi,b ), LA1/2 (xi,b )]), which ultimately dictates the performance of the MMSE-SIC iterative detector. III. Proposed Method A. A Closer Observation at MMSE-SIC The implicit premise of the (3) and (4) is that the a priori P LLRs LA (xi,b ) (= [LSA (xi,b ), LA1/2 (xi,b )]) of all transmitted bits, i.e., systematic and parity bits, are equally reliable in the computation of the soft symbols sˆi . The parity bit stream 1 and parity bit stream 2 are not related to each other (due to interleaving) as that of the systematic bit stream to the parity bit streams. Therefore, the parity stream 1/2 can not exploit the a priori information provided by component decoder 2/1 (due to the mutual constraint) in the iterative decoding process as the systematic bit stream [8]. The LLRs of parity bits are, thus, inherently less reliable than those of the systematic bits. As a result, the soft symbols sˆi calculated based on (3) are not precise as originally expected, which in turn degrades the subsequent parallel interference cancellation performance [cf. (6)]. The less-accurate extrinsic LLR output (a priori input to the decoder) calculated based on (9) leads to suboptimal a posterior information of the transmitted bits from the decoder. This vicious cycle continues as the iterative detection and decoding carries on, which has not been observed before [4], [6], [7]. To give a preliminary explanation of this observation, we investigated the bit error rate (BER) performance of the
−2
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b , R =1/3 c
bP1, Rc=1/3 −3
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˜ iH n˜ i , ˜ iH yˆ i = µ˜ i si + w z˜i = w
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b , Rc=2/3
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Eb /Nparity 0 (dB) bits and systematic Fig. 1: BER comparisons of the bits in the AWGN channel for different code rates.
systematic bit stream and the parity bit streams during the turbo decoding process for different code rates in the additive white Gaussian noise (AWGN) channel.2 Binary information data bits of length 1024 are turbo encoded by a mother code of rate Rc = 1/3 (with generator polynomials [13, 15]8) and then punctured to code rate Rc = 2/3, 3/4, 5/6 as specified in the 3GPP standard [1]. The improved Max-Log-MAP decoding algorithm [14] with six iterations and scaling factor of 0.75 was applied in the SISO channel decoders. Eb /N0 denotes the ratio of energy-per-bit to the noise power spectral density. Fig. 1 illustrates that the parity bits exhibit worse BER performance than the systematic bits for code rate-1/3, which is consistent with the analysis of Section I. It is also shown in Fig. 1 that the parity bit stream 2 performs slightly worse than the parity bit stream 1. This is because the extrinsic information used to decode the parity stream 2 (provided by component decoder 1) executes half-iteration less than that of the parity stream 1. This discrepancy between the reliability of the parity bit stream 1 and 2 can not be remedied by unequal decoding iterations for component decoder 1 and component decoder 2 (i.e., one more decoding iteration for component decoder 1) due to the inherent unequally reliable a priori input to the respective component decoders. For high code rates, e.g., Rc = 3/4, the BER performance gap between the systematic bits and the parity bits is even more pronounced, about 0.2 dB (versus 0.12 dB for low code rates) at BER of 10−2 due to puncturing as shown in Fig. 1. B. Unequal-Error-Correction-Capability-Aware Method Based on the analysis of the unequal-error correction capability of the systematic bits and parity bits in Section I and III-A, we introduce a scaling technique to modify the LLR of 2 Note, the bit errors of the systematic bit stream and the parity bit streams are based on the post-rate-matching results in contrast with the conventional pre-rate-matching one.
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0.035
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Fig. 2: The histograms of a priori information LA (xi,b ) of the turbo decoding in the AWGN channel for different code rates.
the parity bits of conventional iterative receiver given by LAPk (xi,b ) = αk · LAPk (xi,b )
(10)
where αk denote the scaling factor (αk < 1) for parity stream k. We will elaborate the determination of αk later in Section III-C. The primary target of (10) is to balance the reliability between the systematic bit stream and the parity bit stream with regard to their LLR magnitude. After scaling, we obtain more accurate soft symbols [cf. (4)] and as a direct consequence this leads to more precise interference cancellation [cf. (6)]. More accurate extrinsic LLRs of the transmitted bits (updated a priori input to the decoder) are then produced by the front-detector [cf. (9)] and result in more accurate extrinsic information on the transmitted bits generated by the channel decoder. C. Determination of αk It is apparent that the optimal scaling factor is determined by the reliability gap between the systematic bits and the parity bits of the specific code utilized. The reliability gap between the systematic bits and the parity bits is mainly determined by the puncturing pattern, generator polynomials, data block length, and interleaver. On close examination of (3) and (4), we observe that the optimized αk is also related to the distribution P of LSA (xi,b ) and LA1/2 (xi,b ) in the iterative decoding process, not just their last iteration (hard-decision) BER performance. We first examine the empirical distributions of the a priori LLR LA (xi,b ) in the standard turbo decoding process and in the MMSE-SIC iterative detection process. We consider a turbo coded NT = NR = 4 MIMO orthogonal frequency division multiplexing (OFDM) system with 2048-point fast Fourier transform (FFT) and 15 KHz subcarrier spacing. (The simulation parameters conforms to the 3GPP standard [1], [15].) The modulation scheme was quadrature phase shift keying (QPSK). The simulations were carried out based on the
extended Pedestrian A model (EPA) model and the maximum Doppler frequency is set to fD = 5 Hz corresponding to the speed of approximately 2.7 km/h at the carrier frequency of 2.0 GHz. Fig. 2 depicts the histograms of of the a priori LLR LA (xi,b ) P (= [LSA (xi,b ), LA1/2 (xi,b )]) in the turbo decoding process with 6 inner decoding iterations in the AWGN channel. (The left halves parts of the sub-figures in Fig. 3 are omitted due to their symmetrical nature to save space for the legends.) Fig. 2 P illustrates that the expected value of the LA1/2 (xi,b ) are higher than that of the LSA (xi,b ) for code rate-1/3. It is shown in Fig. 2 that component decoder 1 generates a slightly higher output of LAP1 (xi,b ) on average than LAP2 (xi,b ) of component decoder 2. This is attributed to a half more iteration decoding for the a priori input of component decoder 1. For high code rates, the right part of Fig. 2 shows that the P expected value of LA1/2 (xi,b ) is lower than that of the LSA (xi,b ) due to puncturing. The expected value of the LAP2 (xi,b ) of the high code rate-5/6 is also slightly lower than that of the LAP1 (xi,b ) for the same reason as that of the low code rate. Similar phenomena as those of the AWGN channels are obtained (not shown due to space constraints) for MMSE-SIC iterative receiver with Out-it = 2 and In-it = 6 for a 4-by4 MIMO multiplexing systems, where Out-it stands for the number of MMSE-SIC outer detection and In-it refers to the number of the inner iterations of the turbo decoder. With these observations in mind, we now investigate the parametrization of αk on the BER performance of the MMSESIC aided iterative receiver. (For clarity, we only provided results for α1 = α2 = α to save space. Based on extensive numerical results not shown due to space constraints, only a marginal performance gain is obtained for an optimized setup of 0 < α1 < α2 < 1.) Fig. 3 illustrates that the proposed unequal-error-correctioncapability-aware detector with α of 0.3 achieves the best performance for code rate-1/3. The slightly underestimated α = 0.2 (with respect to the empirically determined optimal α = 0.3) performs marginally better than the slightly overestimated α = 0.4. We explain this phenomena as follows. Although the slightly underestimated α = 0.2 introduces more P attenuation to LA1/2 (xi,b ) than that of the slightly overestimated α = 0.4, the performance gain due to the unequal-errorcorrection-capability-aware technique outweighs the negative influence caused by slightly excessive (0.2 versus 0.4) attenP uation to LA1/2 (xi,b ). For code rate-5/6, a greater performance gain of approximately 0.45 dB at block error rate (BLER) = 2 · 10−2 (versus 0.24 dB for code rate-1/3 at the same BLER) is achieved by the proposed method with α of 0.5 as shown in Fig. 3. This is attributed to the larger reliability gap between the systematic bits and the parity bits of the high code rate (due to puncturing) as illustrated in Fig. 1. As a result, the proposed method achieves greater performance gains over the conventional one for the high rate code which is more plagued by the unequalerror-correction-capability sub-optimality issue. It also shown in Fig. 3 that the optimized α1/2 = 0.5 for high code rate-5/6 is higher than the α1/2 = 0.3 of the low code rate1/3. This is explained as follows. The standard BCJR based
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Rc = 1/3 −1
BLER
10
Out−it = 1, Conventional Out−it = 1, Proposed Out−it = 2, Conventional Out−it = 2, Proposed Out−it = 3, Conventional Out−it = 3, Proposed Out−it = 4, Conventional Out−it = 4, Proposed
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Fig. 3: BLER comparisons of different αs for MMSE-SIC iterative receiver with code rate 1/3 and 5/6. The number of outer iteration Out-it is 2 and the number of inner iteration In-it is 6.
D. BLER Comparisons with Conventional Method To give a more comprehensive picture of the proposed method, we plot the BLER results through iterations of the proposed method with optimized αs for different code rates along with those of the conventional one in Fig. 4. Fig. 4a shows that the proposed MMSE-SIC iterative receiver with Out-it of 2 achieves a performance gain of approximately 0.32 dB at BLER = 1 · 10−3 over the conventional one (i.e., α = 1). The gap between the conventional MMSESIC iterative receiver with Out-it = 4 and the proposed one with Out-it = 2 is only less than 0.01 dB at BLER of 1 · 10−1. The proposed MMSE-SIC iterative receiver with Out-it of 3 even outperforms the conventional one with Out-it = 4 by
4
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(a)
Rc = 5/6
−1
BLER
turbo decoder generates disproportionate higher LLR estimates of the parity bits (with respect to their BER performance) compared with that of the systematic bits for low code rate-1/3 P as shown in Fig. 2. Whereas the expected value of LA1/2 (xi,b ) of S the high code rate is less than that of the LA (xi,b ) (as illustrated in Fig. 2) which better reflects its inferior error correcting capability. As a result, a smaller α is required to penalize the overly optimistic estimates for the low code rate than for the high code rate. Remark 1: Since the performance of the MMSE-SIC aided iterative receiver varies for different channel models, the optimized scaling factor α also depend on the channel models. For simplicity, we may choose α of 0.3 for low rates (e.g., Rc
