Performance and Complexity Tradeoffs of Space-Time Modulation and Coding Schemes

Performance and Complexity Tradeoffs of Space-Time Modulation and Coding Schemes The MIT Faculty has made this article openly available. Please share how this access benefits you. Your story matters. Citation As Published Publisher Chang, N.B., A.R. Margetts, and A.L. McKellips. Performance and complexity tradeoffs of space-time modulation and coding schemes. Signals, Systems and Computers, 2009 Conference Record of the Forty-Third Asilomar Conference on. 2009. 1446-1450. Copyright 2009 IEEE http://dx.doi.org/10.1109/acssc.2009.5469836 Institute of Electrical and Electronics Engineers Version Final published version Accessed Sat Dec 29 00:05:54 EST 2018 Citable Link Terms of Use Detailed Terms http://hdl.handle.net/1721.1/58942 Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use.

Performance and Complexity Tradeoffs of Space-Time Modulation and Coding Schemes Nicholas B. Chang, Adam R. Margetts, Andrew L. McKellips Abstract Wireless communication using multiple-input multiple-output (MIMO) systems improves throughput and enhances reliability for a given total transmit power. Achieving a higher data rate in MIMO systems requires utilizing an effective space-time coding and modulation scheme. The appropriate algorithm to use for a system will depend on parameters such as the number of transmit/receive antennas, target spectral efficiency, complexity limitations, channel environment, and other factors. In this paper, we examine the performance of various two-transmit and four-transmit space-time coding schemes under different channel types and target data rates. We compare the performance of state of the art space-time coding schemes including direct non-binary LDPC GF(q) modulation, bit interleaved coded modulation using iterative detection, and space-time trellis coded modulation. We obtain a tradeoff between performance and complexity of these various schemes. I. INTRODUCTION Wireless communication using multiple-input multipleoutput (MIMO) schemes dramatically increases system capacity and reliability when information symbols are appropriately coded and modulated across transmit antennas [1, p. 332]. The main goal of this paper is to examine both the performance and decoding complexity tradeoffs of various spacetime codes for 2x2 and 4x4 MIMO system configurations under various target spectral efficiencies. We primarily consider three groups of space-time schemes, described in Section III. Each scheme assumes no channel feedback to the transmitter, i.e. an uninformed transmitter. The main contributions of this work are as follows: We propose an efficient LDPC GF (q) code which has low encoding complexity and is later shown to have very strong performance. Using outage probability as our baseline metric, we compare the performance of various space-time coding schemes for different target spectral efficiencies, antenna configurations, and channel models. We show empirically for various space-time coding schemes that the excess SNR (db), defined for a fixed * Please direct all correspondence to Nicholas B. Chang, MIT Lincoln Laboratory, 244 Wood St., Lexington, MA 02420, e-mail: nchang@ll.mit.edu, phone 781.981.0606, fax 781.981.3905. The authors are with the Advanced Sensor Techniques Group, MIT Lincoln Laboratory, Lexington, MA. This work was sponsored by the Department of Defense under Air Force contract FA8721-05-C-0002. Opinions, interpretations, conclusions, and recommendations are those of the authors and are not necessarily endorsed by the United States Government. spectral efficiency as the additional SNR needed for a given code to achieve the same frame error rate (FER) performance as the best possible code, decays linearly as the decoding complexity increases exponentially. This important relationship demonstrates the amount of extra computation required to decrease power consumption. For a fixed target spectral efficiency, we examine how the complexity per successfully decoded bit (goodbit) decreases with SNR for various space-time coding schemes. This analysis simultaneously measures the performance and complexity for different codes as a function of SNR. II. SYSTEM MODEL For narrowband MIMO systems, the coupling between transmit and receive antennas can be modeled as: ρ z n = H n x n + w n, (1) n T where n =1, 2,...,n S (number of space-time symbols), H n is the n R n T (number of receive by number of transmit antenna) channel matrix containing the complex attenuation between each transmit and receive antenna over the nth spacetime symbol, x n is an n T 1 space-time symbol with unit energy elements, ρ n is the average total transmit energy for the nth space-time symbol, z n is the n R 1 complex receivearray output, and w n is an n R 1 vector containing temporally and spatially white zero-mean complex Gaussian noise, i.e., E[w n w n] = I, where I is the identity matrix, E[ ] is the expectation operator and denotes the conjugate transpose. The term ρ can be interpreted as the SNR per receive antenna for each space-time symbol. We study performance over a block fading channel model, in which the channel varies (fades) periodically within a codeword. Under this model, the channel changes randomly between fades, and the length of each fade (block length) is assumed to be constant. The block length is a parameter that can be tuned to match different environments. Note that block lengths of 1 and n S correspond to the fast fading and flat fading (or quasi-static) channel models, respectively. In this paper, elements of the channel attenuation matrix for each fading block are modeled as i.i.d. complex circularly symmetric Gaussian random variables with unit variance. Additionally, channel realizations are independent between different codewords (frames). The channel is known only by the receiver. The FER is the expected rate at which codewords are incorrectly decoded. In this case, outage probability [1, 978-1-4244-5827-1/09/$26.00 2009 IEEE 1446 Asilomar 2009

p. 366] serves as the lower bound on FER performance for infinitely long, isotropically transmitted codewords. III. SYSTEM DESIGN In this section, we provide an overview of the coding and modulation schemes which are compared in this study. A. Bit Interleaved-Coded Modulation with Iterative Detection Bit-interleaved coded modulation with iterative detection (BICM-ID) schemes utilize strong binary error-correcting codes, including low-density parity-check (LDPC) and turbo codes, for MIMO Rayleigh fast-fading channels [2], [3], [4] and for quasi-static channels [5]. Encoded bits are interleaved and mapped to space-time symbols. Due to the binary code, BICM-ID schemes must iteratively pass information between the demodulator and decoder to effectively separate the bits transmitted over multiple antennas [6] on the same space-time symbol. In this paper, for best performance we consider fullcomplexity demapping schemes. Lower complexity list-sphere decoding techniques trade performance with complexity and are discussed in [6]. B. Direct GF(q) LDPC Modulation The second set of space-time candidate codes utilizes LDPC codes over binary field extensions (higher-order fields) [7], [8], which have been shown to outperform their binary counterparts [9] in designing a space-time coding scheme. In [7], a direct space-time GF(256) LDPC modulation scheme outperformed binary LDPC-encoded bit-interleaved codedmodulation iterative detection (BICM-ID) schemes over a 2 2 fast-fading MIMO channel and in [5], results for quasi-static channels were provided for single receive antenna systems. For direct LDPC GF(q), each LDPC code symbol over GF(q), q =2 m, is directly mapped to L space-time symbols, where L is a positive integer. The binary representation of each GF(2 m ) code symbol is partitioned into L n T sets of m L n T -tuples such that each set maps to a single-antenna constellation point using a Gray-mapping scheme. The mapped constellation points are then transmitted simultaneously from the n T transmit antennas over L space-time symbols. For example, one could map symbols from a GF(256) LDPC code to a four-transmit antenna system employing a QPSK constellation at each transmitter. The receiver generates a set of likelihoods for each spacetime symbol and passes this information to the GF(q) LDPC code decoder. Bayesian belief networks [10] and graphical decoding techniques can be used to formulate decoders for LDPC codes and turbo codes. Some simplifications are possible, but in general the GF(q) decoding complexity per iteration grows as O(n S q log q). The reader is referred to [11] for details regarding GF(q) LDPC decoding and reducedcomplexity implementations, and to [7], [8], [12] for more details on the direct LDPC space-time modulation and coding. While LDPC GF(q) codes are defined by sparse parity check matrices, their corresponding generator matrices are generally not sparse and encoding thus requires a full matrix multiplication. Structured codes have been studied to decrease encoding complexity for binary LDPC codes [13], but their extension to higher-order fields and performance in MIMO systems have not been well-studied. To reduce the encoding complexity, we form a staircase parity-check matrix via concatenation of two matrices, [W V ], where W is a sparse n k by k matrix while V is an n k by n k matrix described in Figure 1. Note that Fig. 1. Right portion of LDPC GF(q) staircase parity check matrices. V contains two nonzero diagonals, whose coefficients {α j } and {β j } are nonzero GF (q) elements chosen randomly. If a codeword has the form [s 1,s 2,,s k,p 1,,p n k ], where {s j } are the systematic information symbols and {p j } are the parity symbols, then multiplying the parity check matrix by the codeword symbols results in the following equations for the parity symbols: p 1 =. k j=1 W 1,j s j α 1 p i = β i 1 α i p i 1 +. k j=1 p n k = β n k 1 α n k p n k 1 + W i,j s j α i k j=1 W n k,j s j α n k where W l,j is the element in the lth row and jth column of W. Thus, encoding can be implemented by an accumulator which consists of two steps described in Figure 2: (1) compute a vector dot product between information symbols and a normalized row of the sparse matrix W and (2) pass the multiplication output to an accumulator to compute the parity symbols. We use the term staircase LDPC code to describe a code whose parity check matrices take the form given in Figure 1. Other LDPC codes with low encoding complexity have also been recently proposed [12]. 1447

Fig. 2. Accumulator implementation of encoder for staircase LDPC codes. Note that encoding complexity scales linearly with codeword length. Fig. 3. Throughput vs. SNR per receive antenna at FER of 10% for various 2 2 schemes, each using 6000 space-time symbols, in a flat fading channel. Theoretical bounds are given by the solid curve and calculated from p out in (2). C. Space-Time Trellis Coded (STTC) Modulation The third group of candidate codes are space-time trellis codes (STTC) [14], [15] which sacrifice some performance for lower decoding complexity. STTC simultaneously offer coding gain with spectral efficiency and full diversity over fading channels. Powerful space-time trellis codes for QPSK modulated schemes were derived in [16], [15] for different number of states. The reader is referred to [14], [16], [15] for more detailed information on STTC schemes. IV. RESULTS We first present the performance results obtained for the various signaling schemes described, and then use these results to study the performance-complexity tradeoffs. A. Performance The following results demonstrate the performance of the aforementioned space-time schemes for a wide range of antenna configurations, signaling constellations, and SNR levels. The channel matrix was perfectly known to the receiver; in practice, channel estimation errors will impact performance. Fig. 4. Throughput vs. SNR per receive antenna at FER of 10% for various 2 2 schemes, each using 6000 space-time symbols, in a block fading channel with the block length fixed at 5 spacetime symbols. Theoretical bounds are given by the solid curve and calculated from p out in (2). For the direct modulation schemes using GF(q) LDPC codes, decoding is halted if a maximum of 100 iterations is reached, if the parity-check equations are satisfied after an iteration, or if the detected codeword remains unchanged for 10 consecutive iterations. The BICM-ID scheme is allowed to use up to 10 iterations for passing information between the demodulator and decoder. To compare the performances of the proposed space-time coding schemes to theoretical limits, we first define the average total transmit energy over all space-time symbols as ρ. Then the outage probability as a function of ρ is given by [1, p. 367]: [ ns ( p out (ρ) =P log 2 det I + ρ ) ] H k H k <R, (2) n T k=1 where R is the spectral efficiency (b/s/hz) specified by the system parameters, and P [ ] is the probability measure. Equation (2) was evaluated using Monte Carlo methods. The performances of the direct LDPC GF(q) schemes, a BICM-ID scheme [6], and STTC schemes [15] are compared by measuring the 10% outage throughput vs. SNR in Fig. 3 1448

and Fig. 4. The theoretical bounds, given by the solid curves, indicate the minimum ρ such that p out (ρ) is 10% for a given spectral efficiency. In order to achieve various spectral efficiencies, the following parameters are varied: coding rate, galois field size, and signal constellation. All schemes are allowed to use 6000 space-time symbols per frame. Thus, a scheme achieving throughput of 2/3 b/s/hz will encode 4000 information bits in a single frame. For all of the various target spectral efficiencies, the LDPC GF(256) code performs very well, coming within 0.5 1 db of the theoretical bound. In addition, by simply changing the constellation size from BPSK up to 16QAM, strong performance can be achieved by the direct LDPC modulation. This strong performance over the different spectral efficiencies is appealing for designing rate-adaptive systems. The staircase LDPC codes described in Section III-B, which are denoted as Direct Stair LDPC in the figures, have very similar performance to their more random GF(256) counterparts for the block fading model but suffer performance degradation for the flat fading channel in Fig. 3. The performance of BICM-ID is 1 3 db worse than direct LDPC modulation for various spectral efficiencies under both channel models, which can be attributed to the iterative receiver processing described in [6]. The 64-state QPSK-modulated STTC scheme [16], [15] indicated in the figure is the best space-time trellis code at the given rate known to the authors. Deriving strong STTC codes for an arbitrary spectral efficiency requires an exhaustive search or rate-adaptive puncturing methods. Its performance significantly trails direct LDPC modulation and BICM-ID. The performance is also compared for a 4 4 system in Fig. 5. Again, the direct-modulated LDPC GF(256) scheme achieves performance close to the achievable-rate bounds. The plot also includes a 64-state punctured STTC whose throughput matches that of the direct LDPC and BICM schemes, as well as a strong convolutional code with a constraint length of 13. B. Performance Versus Complexity Tradeoff The various space-time coding and modulation schemes described in the previous section achieve a wide range of performance levels but require varying degrees of computational complexity which leads to a performance-complexity tradeoff. In this section, complexity is measured by the total number of multiplications, additions, and lookup table operations. Performance is measured by the excess SNR, which is defined for a fixed spectral efficiency and target FER as the additional SNR needed by a code to match the performance of the best possible code. Fig. 6 depicts the excess SNR with respect to the achievable rate and the corresponding number of operations per information bit for some 4x4 schemes under a flat fading channel. As the excess SNR at target FER 10% approaches 0, the complexity increases exponentially. For example, to achieve this target FER one can either use a 64-State STTC or a Fig. 5. Throughput vs. SNR per receive antenna at FER of 10% for various 4 4 schemes, each using QPSK constellation for 240 spacetime symbols, under a flat fading channel. The theoretical bound, given by the solid curve, is calculated from p out in (2). Fig. 6. Plot of excess SNR for 2 2 and 4 4 systems at 10% outage vs. operations per information bit for space-time trellis coded modulation, binary LDPC BICM, and direct GF(256) LDPC modulation. GF(256) LDPC code which requires approximately 100 more operations per information bit but saves 2 db SNR. The LDPC decoding complexity, however, is an SNRdependent random variable because the number of decoding iterations decreases as the SNR increases. Additionally, FER also decreases as SNR increases. Fig. 6 thus does not compare the complexity and performance of codes at the same SNR level. We thus consider the following metric for each scheme: β f (ρ) γ f (ρ) = β f (ρ) [1 p f (ρ)]r, (3) where β f (ρ) is the average number of operations per frame at SNR ρ, γ f (ρ) is the goodput, e.g. average number of successfully decoded information bits per frame, p f (ρ) is the frame error rate, and r is the number of information bits per frame. The metric (3) indicates the complexity required 1449

Fig. 7. Number of operations per goodbit for a 2x2 system in a block fading environment. per successfully decoded information bit (goodbit), assuming packets are retransmitted until successful. Fig. 7 depicts the performance of three schemes with respect to (3). The STTC has fixed complexity per decoded frame; however, at low SNR (i.e., high FER), multiple retransmissions lead to high complexity per good bit. The LDPC GF(256) and BICM-ID decoding complexities decrease rapidly as the SNR increases since fewer decoding iterations are required for successful decoding. At high SNR, the complexity requirements of all three schemes lie within the same order of magnitude. Previous works have studied the complexity versus performance tradeoff for specific coding schemes. For example, [17] studies the relationship between complexity and excess SNR for binary LDPC codes. Our results, by contrast, compare the complexity between different schemes. [4] J. Zheng and B. D. Rao, LDPC-coded MIMO systems with unknown block fading channels: Soft MIMO detector design, channel estimation, and code optimization, IEEE Trans. on Signal Processing, vol. 54, pp. 1504 1518, Apr. 2006. [5] G. Li, I. J. Fair, and W. A. Krzymień, Low-density parity-check codes for space-time wireless transmission, IEEE Trans. on Wireless Communications, vol. 5, pp. 312 322, Feb. 2006. [6] B. M. Hochwald and S. ten Brink, Achieving near-capacity on a multiple-antenna channel, IEEE Trans. on Communications, vol. 51, pp. 389 399, Mar. 2003. [7] R. Peng and R.-R. Chen, Design of nonbinary LDPC codes over GF(q) for multiple-antenna transmission, Proc. IEEE Military Communication Conf., pp. 1 7, Oct. 2006. [8] A. R. Margetts, K. W. Forsyth, and D. W. Bliss, Direct space-time GF(q) LDPC modulation, Proc. Asilomar Conf. on Signals, Systems and Computers, pp. 1264 1268, Oct. 2006. [9] M. C. Davey and D. J. MacKay, Low-density parity check codes over GF(q), IEEE Communications Letters, vol. 2, pp. 165 167, Jun. 1998. [10] R. J. McEliece, D. J. C. MacKay, and J.-F. Cheng, Turbo decoding as an instance of Pearl s belief propagation algorithm, IEEE Journal on Selected Areas in Communications, vol. 16, pp. 140 152, Feb. 1998. [11] H. Song and J. Cruz, Reduced-complexity decoding of Q-ary LDPC codes for magnetic recording, IEEE Trans. on Magnetics, vol. 39, pp. 1081 1087, Mar. 2003. [12] R. Peng and R.-R. Chen, Application of nonbinary LDPC cycle codes to MIMO channels, IEEE Trans. on Wireless Communications, vol. 7, pp. 2020 2026, Jun. 2008. [13] T. Richardson and R. Urbanke, Efficient encoding of low-density paritycheck codes, IEEE Trans. on Information Theory, vol. 47, pp. 638 656, Feb. 2001. [14] A. V.Tarokh, N. Seshadri, Space-time codes for high data rate wireless communication: Performance criterion and code construction, IEEE Transactions on Information Theory, vol. 44, pp. 744 765, Mar 1998. [15] Z. Chen, B. S. Vucetic, J. Yuan, and K. L. Lo, Space-time trellis codes for 4-PSK with three and four transmit antennas in quasi-static flat fading channels, IEEE Communications Letters, vol. 6, pp. 67 69, Feb. 2002. [16] Z. Chen, J. Yuan, and B. S. Vucetic, Improved space-time trellis coded modulation scheme on slow Rayleigh fading channels, Electronics Letters, vol. 37, pp. 440 441, Mar. 2001. [17] A. Khandekar, Graph-based codes and iterative decoding, PhD Thesis, California Insitute of Technology, 2002. Pasadena, CA. V. CONCLUSION This work studied the performance and complexity tradeoff of various space-time coding schemes at different spectral efficiencies and channel models. Results were obtained for schemes including direct GF(q) LDPC space-time coding, the bit interleaved coded modulation (BICM), and space-time trellis coded modulation. It was shown empirically that for various space-time codes, the excess SNR decreases linearly as the decoding complexity increases exponentially. Additionally, the complexity per goodbit was measured for various codes. This performance measure resulted a comparison in the effective complexity for different space-time coding schemes at the same SNR levels. REFERENCES [1] D. Tse and P. Viswanath, Fundamentals of Wireless Communication. New York: Cambridge University Press, 2005. [2] S. ten Brink, G. Kramer, and A. Ashikhmin, Design of low-density parity-check codes for modulation and detection, IEEE Trans. on Communications, vol. 52, pp. 670 678, Apr. 2004. [3] F. Guo and L. Hanzo, Low complexity non-binary LDPC and modulation schemes communicating over MIMO channels, Proc. IEEE Vehicular Technology Conference, vol. 2, pp. 1294 1298, Sep. 2004. 1450