Spatial Mapping for MIMO systems - CiteSeerX

Spatial Mapping for MIMO systems Frederik Simoens, Henk Wymeersch and Marc Moeneclaey DIGCOM research group, TELIN department Ghent University, Sint-Pietersnieuwstraat 41, 9000 GENT, BELGIUM E-mail: {fsimoens,hwymeers,mm}@telin.ugent.be Abstract — This contribution proposes a new spatial mapping technique for bit-interleaved coded modulation (BICM) over multiple-input multipleoutput (MIMO) channels. It is an extension of conventional BICM whereby the serial-to-parallel converter is replaced with a more general mapping strategy. Genie performance analysis of spatial mapping schemes results in very simple design criteria for the mapping function. Using an iterative detector, a significant performance gain is observed compared to a conventional BICM scheme, at no significant increase in computational complexity. The predicted performance gains are verified through computer simulations.

I. Introduction It is well known that by transmitting independent information streams in parallel over different antennas, data rates over MIMO channels can be increased [1]. However, this so-called spatial multiplexing often results in a loss in spatial diversity. Bit-interleaved coded modulation schemes (BICM) for MIMO systems, on the other hand, have recently been addressed as an effective means to achieve high data rates while maintaining high diversity [2]. BICM schemes were originally designed for SISO channels to combat the effect of fading [3]. A typical system consist of a (convolutional) encoder, a bitinterleaver and a symbol mapper. This idea was later extended to BICM with iterative decoding (BICM-ID), whereby the decoder and the demapper exchange information in order to improve the performance [4]. It was soon recognized that the choice of mapping (i.e., how a group of bits is mapped onto a complex constellation symbol) is a crucial design parameter [5, 6]. BICM-ID can be used on a MIMO channel in a straightforward way [7]: the convolutional encoder, bitinterleaver and symbol mapper are now followed by a serial-to-parallel (S/P) converter, which passes the coded symbols to the different antennas in a roundrobin fashion. This form of vertical encoding has the potential of achieving full diversity as each information bit can be spread across all the transmit antennas. Since maximum likelihood decoding is prohibitively complex, near-optimal iterative detection strategies, that are similar to the SISO algorithms from [4], have been devised for this MIMO configuration [7]. This contribution deals with spatial mapping strategies for BICM-ID over MIMO channels. Optimized per

antenna mapping schemes have been presented for conventional configurations with a S/P converter in [8, 9]. In the present paper, we propose a new scheme replacing the conventional S/P converter with a more general bit-to-symbol-vector mapper. We show how good spatial mappings may be obtained, based on the analysis of a genie receiver [8]. Through computer simulation we demonstrate that spatial mapping is able to achieve a significant performance gain as compared to conventional S/P schemes. This paper is organized as follows: in section II the system model is explained, while section III deals with finding a criterion for optimized mappings. In section IV we derive optimized mappings for BPSK transmission. The obtained mapping is compared to S/P conversion through computer simulations. Apart from carrying out an EXIT chart analysis, we compare bit- and frame-error rate performance. We end with conclusions in section V. II. System Model A general BICM transmission scheme for a MIMO system with NT transmit and NR receive antennas is illustrated in Fig. 1. We assume signaling using an M -point constellation Ω (e.g., M-PSK or M-QAM). A burst of L information bits is convolutionally encoded and interleaved, resulting in a sequence of N coded bits, (c0 , . . . , cN −1 ). This sequence is broken down in blocks . of mNT coded bits1 , with m = log2 M ; the k-th block is denoted ck = (ck [0], ck [1], . . . , ck [mNT − 1]). Next, ck (k = 0, . . . , K − 1, with K = N/ (mNT )), is mapped to a NT × 1 vector ak having elements in Ω using a mapping function M:   a0,k   .. ak =  (1)  = M (ck ) . . aNT −1,k

In general, each ai,k (i = 0, . . . , NT − 1) is a function of all mNT coded bits within ck . It is important to note that the S/P conversion applied in conventional spatial multiplexing corresponds to a particular mapping function: i.e., one whereby ai,k depends only on the m coded bits (ck [im], . . . , ck [(i + 1)m − 1]). The space-time matrix A = [a0 , . . . , aK−1 ] is transmitted over the channel H. The channel is assumed to be a quasi-static block fading channel. The channel coefficients have independent real and imaginary 1 we

assume N to be a multiple of mNT

SPATIAL PE→D (ck [l]) b

c ENCODER

MODULATION GROUPING

Π

EQUALIZER

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a M ˆ b iter 0

iter > 0

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Π PD→E (ck [l])

x

Figure 1: BICM transmitter and receiver part, each having a zero-mean Gaussian distribution with variance 1/2. The received NR × K space-time matrix Y is given by

PE→D (ck [l]) [11]. Based on these APPs, final decisions on the coded bits are made. III. Selection of Spatial Mapping

Y = HA + N

(2)

where N is a NR × K matrix of independent complex AWGN random variables with variance σ 2 . The receiver, also depicted in Fig. 1, is based on [7]. It iterates between MAP decoding and equalization. The MAP decoder operates according to the BCJR algorithm [10]. We refer to [7] for a detailed description of this detector. The key property of the detector is that it exchanges so-called extrinsic information between the decoder and the equalizer. The extrinsic information computed by the equalizer (resp. decoder) will be used as a priori information by the decoder (resp. equalizer). We will focus on the l-th bit (for l = 0, . . . , mNT − 1) of ck . The extrinsic information from the decoder to the equalizer regarding this bit will be denoted by PD→E (ck [l]). Similarly, the extrinsic information from the equalizer to the decoder will be denoted by PE→D (ck [l]): this information about ck [l] is computed from the received signal and the information (supplied by the decoder) regarding the other bits contained in ck , namely PD→E (ck [l0 ]) , l0 6= l. More specifically, PE→D (ck [l] = b) = p (yk |ck [l] = b ) X Y =C p (yk |ak ) PD→E (ck [l0 ]) ak ∈χlb

(3)

In this section, we investigate the effect of the mapping function M on the pre-decoding bit error rate (BER) of the genie method [8], and derive a criterion for selecting good mapping functions. The post-decoding BER of the transmission scheme is very hard to evaluate. However, since the reliability of the decoder outputs (PD→E (ck [l])) is directly related to its inputs (PE→D (ck [l])), it makes sense to optimize the reliability of the decoder inputs. More specifically, we determine a lower bound on the pre-decoding BER of the bit ck [l], using a genie argument. This means that we consider the bits contained in ck to be statistically independent, and compute the pre-decoding error probability related to the bit ck [l], under the assumption that all other bits from ck are known. This pre-decoding error probability is a measure of the reliability of the information PE→D (ck [l]) that the equalizer supplies to the decoder. At high SNR, the equalizer receives very reliable information from the decoder, in which case the actual pre-decoding BER is close to the lower bound computed below. The genie argument is as follows: assume we want to transmit a particular ˜k at time k. This vector is mapped to a bit-vector c space-vector ak = M (˜ ck ) and sent over the channel. At the receiver, we observe yk = Hak + nk . Using the genie argument, the information about ck [l], computed by the equalizer, is given by

l0 6=l

χlb

where C is a normalizing constant and denotes the set of symbol-vectors in ΩNT whose l-th bit under inverse mapping M−1 equals b ∈ {0, 1}. The detection starts with uniform information for the equalizer, i.e., PD→E (ck [l]) = 0.5. The equalizer computes PE→D (ck [l]) based solely on the received signal Y. Based on PE→D (ck [l]), the MAP decoder updates PD→E (ck [l]), which will in turn lead to more reliable PE→D (ck [l]) etc. After convergence, the a posteriori probabilities (APP) of ck [l] are simply given by (up to an irrelevant multiplicative constant) PD→E (ck [l]) ×

PE→D (ck [l] = α) = C exp −

°2 1 ° ° ° l − Ha (α) ° (4) °y k k 2σ 2

where C is a normalizing constant and alk (α) denotes the symbol-vector corresponding to the actual transmitted bits ck , except for the l-th bit, which is set to α ∈ {0, 1}. Hence, alk (α) = (˜ ck [0], . . . , c˜k [l − 1], α, c˜k [l + 1], . . . , c˜k [mNT − 1]). It is readily seen that the pre-decoding BER2 using a genie 2 The pre-decoding BER (5) is defined as the probability that PE→D (ck [l] = c˜k [l]) ≤ PE→D (ck [l] = 1 − c˜k [l]), with c˜k [l] ∈ {0, 1}.

bits

argument is given by

Pe (ck [l])

s

= Q

° ¡ ¢°  °H al (1) − al (0) °2 k k  (5) 4σ 2

√ R +∞ ¡ ¢ where Q (x) = 1/ 2π x exp −t2 /2 dt. Upper bounding and averaging over the channel matrix H yields, after some calculations,

EH [Pe (ck [l])] ≤

Ã

1+

° l ° !−NR °a (1) − al (0)°2 k

k

8σ 2

.

(6)

It is obvious that the pair of vectors alk (1) and alk (0) ° °2 corresponding to smallest values of °alk (1) − alk (0)° will dominate the total error probability at high SNR. Therefore, according to this criterion, the optimal mapˆ should maximize ping (which we will denote by M) the minimal squared Euclidean distance d2 . We drop the time index k, since we only consider time-invariant mapping schemes. This yields © ª ˆ = arg max d2 M

(7)

° °2 d2 = min °al (1) − al (0)°

(8)

M

where

a,l

and minimization is performed over all a ∈ ΩNT and l ∈ {0, , . . . , mNT − 1}. Note that for a conventional BICM scheme with S/Pconversion, maximization in (7) is performed over those mappings whereby ai , the i-th element in a, depends only on the m coded bits (c[im], . . . , c[(i + 1)m − 1]) from c. Restricting the mapping function in a S/P BICM scheme results in an optimization per antenna, rather than over all antennas. The criterion for a S/P BICM scheme becomes ° °2 d2 = min °al (1) − al (0)° . (9) a,l

In [8], optimal per antenna mapping schemes were proposed to minimize the pre-decoding error probability. IV. An example: BPSK mapping

In the previous section, we showed that for conventional S/P BICM configurations the mapping criterion is symbol-based. Hence, it is irrelevant to consider different mappings within a BPSK signaling set, since the conventional S/P BPSK-system yields d2 = 4, irrespective of the mapping and the number of transmit antennas. In the new spatial mapping scheme, however, different mappings result in different performances, even for a BPSK constellation. Additionally, when using a

000 001 010 011 100 101 110 111

Conventional S/P £ ¤T −1 −1 −1 £ ¤T −1 −1 +1 £ ¤T −1 +1 −1 £ ¤T −1 +1 +1 £ ¤T +1 −1 −1 £ ¤T +1 −1 +1 £ ¤T +1 +1 −1 £ ¤T +1 +1 +1

New £ −1 £ +1 £ +1 £ −1 £ −1 £ +1 £ +1 £ −1

Spatial Map ¤T −1 −1 ¤T +1 +1 ¤T −1 +1 ¤T +1 −1 ¤T +1 +1 ¤T −1 −1 ¤T +1 −1 ¤T −1 +1

Table 1: An optimal spatial mapping for BPSK signalling with 3 transmit antennas BPSK-signaling set, optimization is fairly straightforward, since the number of possible symbol-vectors a is limited. Restricting ourselves to ’good’ mappings (i.e., mappings where different bit-patterns are mapped to different symbol-vectors), it can easily be seen that d2 ≥ 4. Moreover, the conventional S/P BPSK-system yields d2 = 4, irrespective of the mapping and the number of transmit antennas. Hence, S/P conversion is the worst possible ’good’ mapping. As each point a in the space N {−1, 1} T has exactly NT neighbors at squared distance 4 (NT − 1) and exactly one neighbor at squared distance 4NT , the squared distance d2 will always be less than ½ 4 (NT − 1) NT > 1 d2opt = 4NT NT = 1. This means that all mappings will have a squared distance d2 such that 4 ≤ d2 ≤ d2opt . We will now show how optimized mappings with d2 = d2opt can be constructed. Note that these mappings are not unique. A potential performance gain of 10 log ((NT − 1)) dB at high SNR can be expected, as compared to S/P conversion. Optimized mappings for BPSK A sequence of NT bits c = (c0 , . . . , cNT −1 ) is mapped onto a vector a of NT BPSK symbols. To lighten the notation we will write both a and c as row-vectors. The optimized mapping will be denoted MNT with a = MNT (c). We further denote the Hamming weight of c by wH (c). • We first consider NT even. We propose the following ’good’ mapping ½ 2c − 1 wH (c) = E a= (10) 1 − 2c wH (c) = O where ’E’ stands for even and ’O’ for odd. This scheme uniquely defines MNT . It is readily seen that d2 = 4 (NT − 1) = d2opt .

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BI Spat. Map. 1 it. BI Spat. Map. 15 it. BI Spat. Map. genie BI S/P 1 it. BI S/P 15 it. BI S/P genie

BI Spat. Map. 1 it. BI Spat. Map. 15 it. BI Spat. Map. genie BI S/P 1 it. BI S/P 15 it. BI S/P genie

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¡ ¢ Figure 2: FER and BER performance of the S/P and spatial mapping scheme as a function of E b /N0 = 1/ 2Rσ 2 with R denoting the code rate. Both genie performance and the iterative detector (at the first and the 15-th iteration) are illustrated

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Figure 3: EXIT charts at SNR -1 dB (left) and -6 dB (right)

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• Now take NT > 1 and odd. Suppose we have ˜= constructed MNT −1 according to (10). Define c (c0 , . . . , cNT −2 ) so that c = (˜ c, cNT −1 ).  (MNT −1 (˜ c) , −1) cNT −1 = 0, wH (˜ c) = E    (MNT −1 (˜ c) , +1) cNT −1 = 0, wH (˜ c) = O a= (−MNT −1 (˜ c) , +1) cNT −1 = 1, wH (˜ c) = E    (−MNT −1 (˜ c) , −1) cNT −1 = 1, wH (˜ c) = O (11) Again, taking into account the properties of MNT −1 , it can be easily verified that d2 = d2opt and the mapping is ’good’ (note that this condition is not fulfilled in (10) for odd NT ). Performance results As an illustration, we consider a MIMO-system with NT = 3 transmit antennas and NR = 5 receive antennas, along with a rate 1/3 recursive systematic convolutional encoder and BPSK signaling (interleaver size = 900 bits). An optimal mapping is easily found by applying (11) with NT = 3. The result is shown in Table 1. This mapping results in d2 = d2opt = 8. Note a difference in d2 of a factor 2 between the S/P conversion and the optimal mapping. We therefore expect a gain of at least 3 dB in pre-decoding genie performance. Fig. 2 shows plots of the post-decoding frame error rate (FER) and bit error rate (BER) of the genie-method and the iterative detector for S/P and Spatial Mapping. We observe a gain of almost 4 dB in post-decoding genie performance when using optimized spatial mapping instead of the conventional S/P conversion. At the first iteration of the iterative detector (when the equalizer has no a priori information on the coded bits), the conventional S/P configuration outperforms the new mapping scheme. From the figures it is clear that S/P conversion benefits only to a very limited extent from iterating between equalization and decoding. The new mapping, on the other hand, results in a significant performance gain (nearly 3 dB in FER-performance, 1 dB in BERperformance) as compared to S/P conversion after a number of iterations between equalization and decoding. Finally, we have carried out an EXIT chart analysis [12]. In Fig. 3 we show the obtained results. We first notice that the curve for the S/P conversion is nearly flat: the equalizer does not benefit from the extrinsic information computed by the decoder. On the other hand, the optimal spatial mapping results in an EXIT curve that is nearly a straight line and has a significantly higher slope than the curve for S/P conversion. Also, observe that the leftmost (resp. rightmost) point of the EXIT curve for spatial mapping lies beneath (resp. above) the leftmost (resp. rightmost) point of the EXIT curve for S/P conversion. It is interesting to note that conventional S/P mapping scheme optimizes the performance w.r.t. the leftmost point (corresponding to input mutual information 0 bit), while in our ge-

nie analysis, we mainly try to optimize w.r.t. the rightmost point (corresponding to input mutual information 1 bit). We see that at early iterations, the S/P conversion should outperform our spatial mapping scheme (as indeed it does). At higher iterations, spatial mapping is able to benefit from the information computed by the decoder. We have also included the information trajectory in Fig. 3. At low SNR, the spatial mapping scheme gets stuck at a fixed point corresponding to low mutual information. This is also observed in the BER(FER)analysis, since the new mapping scheme does not work well at low SNR. At high SNR, spatial mapping is able to achieve a mutual information above the one of S/P conversion. Consequently, the BER(FER)-performance plots show that the new scheme outperforms the S/P conversion scheme.

V

Conclusions and remarks

A new spatial mapping technique has been proposed for BICM over MIMO channels. It is a simple extension of conventional BICM whereby the serial-to-parallel converter is replaced with a more general mapping. Genie performance analysis of spatial mapping schemes results in very simple design criteria for the mapping function. Using an iterative detector, a significant performance gain (almost 3 dB in FER-performance and 1dB in BER-performance for BPSK signaling with 3 transmit antennas) was observed, compared to a conventional BICM scheme. The proposed technique comes at hardly any computational cost at the receiver as compared to S/P conversion. However, finding optimal mapping functions for higher-order constellations is far from trivial and remains a topic for further research. We finally note that the proposed spatial mapping scheme can be interpreted in a number of ways: it can be seen as a multi-dimensional mapping technique for BICM. On the other hand, it can also be construed as a kind of precoding: groups of mNT bits are transformed with a rate 1 precoder and subsequently presented to a conventional mapper and S/P convertor. These interpretations might aid further analysis. Acknowledgments This work has been supported by the Interuniversity Attraction Poles Program P5/11- Belgian Science Policy. The first author also acknowledges the support from the Institute for the Promotion of Innovation through Science and Technology in Flanders (IWTVlaanderen). References [1] D. Gesbert, M. Shafi, D. Shiu, P.J. Smith and A. Naguib, ”From theory to practice: an overview of MIMO space-time coded wireless systems,” IEEE journal on Selected Areas in Comm., 21(3):281–302, April 2003.

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