coded performance for combined spatial multiplexing ... - IEEE Xplore

CODED PERFORMANCE FOR COMBINED SPATIAL MULTIPLEXING/SPACE-TIME BLOCK CODING WIRELESS SYSTEMS Guillem Femenias and Felip Riera-Palou Mobile Communications Group Dept. of Mathematics and Informatics University of the Balearic Islands Crta. Valldemossa km 7,5, 07122 Palma (Mallorca, Spain) {guillem.femenias,felip.riera}@uib.es

ABSTRACT

streams in parallel through separate sets of antennas. One especially interesting case of combined SDM/STBC system, This paper presents novel detection strategies for hybrid sysusually referred to as double space time transmit diversity tems combining spatial division multiplexing (SDM) and space(DSTTD) [4], results when the number of transmit antennas is time block coding (STBC). Group-based approaches in which set to four and, consequently, there are two Alamouti STBC STBC streams are individually targeted as well as direct apoperating in parallel. This scheme, DSTTD, already forms proaches where the STBC structure is not taken into account part of the IEEE 802.11n standard draft [8]. in the detection process are considered. We introduce two The different detection strategies for these combined systems types of group-based detectors and compare their performance can be classified in two categories: block-based or direct eswith that of previous group-based and direct proposals in both, timation. In the first case the receiver attempts to isolate the coded and uncoded setups, showing the superiority of one of different STBC streams by means of a block equaliser and our proposals (linear). Despite the generality of the proposed then decode each block independently following the usually receivers, the presented simulations results focus on the IEEE simple STBC decoding rule [5, 6]. Alternatively, the second 802.11n wireless local area network (WLAN) as this is one type of receivers directly estimate all the transmitted symbols of the first commercial standards to explicitly support hybrid without explicitly exploiting the group-based space-time enSDM/STBC systems. coding [4, 7, 9, 10]. Direct estimation algorithms outperform the proposed group-based techniques although at the cost of 1. INTRODUCTION a higher computational complexity. Particularly interesting is the detector proposed in [7] where the transmitted symThe capabilities of wireless systems have been greatly imbols are directly estimated by means of a maximum likeliproved by the introduction in the late nineties of spatial diverhood (ML) estimator. This detector serves as a lower bound sity mechanisms at the transmitter side (i.e. multiple transin terms of BER against which to measure the rest of promit antennas). The two most common forms of spatial transposed techniques. The penalty paid is a very high computamit diversity are spatial division multiplexing (SDM) [1] and tional complexity which grows exponentially with the numspace-time block coding (STBC) [2, 3]. While SDM increases ber of transmit antennas. It is shown in [7] that when using a the throughput by sending independent data streams through clever implementation of the ML algorithm (sphere decoding the different transmit antennas, in STBC, the throughput rate [11]), the complexity greatly reduces when operating at high is kept constant or even reduced but the performance (i.e. SNRs, however, this is an unlikely condition given the excelBER) is improved by means of an space-time code which diclent performance this detector obtains at low SNRs (where tates the transmission pattern of a block of symbols over the complexity is still very high). available transmit antennas and several time periods. Among In this paper we propose novel linear block equaliser-based the different space-time block coding techniques, the Alamdetectors that close the performance gap with respect to the outi coding scheme [2] has achieved great popularity due to its ML-based approach while maintaining a low computational simple optimal decoding and the large performance improvecomplexity. The first group of receivers introduced in this pament it offers. In the last few years several authors [4, 5, 6, 7] per combines the previously proposed detectors in [5] with a have addressed the issue of combining the two approaches, noise whitening filter to ensure that the different STBC stream SDM and STBC, with the aim of achieving high transmisestimates are perturbed by uncorrelated noise. This noise sion rates at low BERs. The idea is to transmit various STBC decorrelation optimises the performance of the subsequent STBC decoding step with a modest complexity increase. AdThis work has been supported in part by the MEC and FEDER under project MARIMBA (TEC2005-0997), Govern de les Illes Balears under ditionally, we introduce a new technique, the linear direct project XISPES and grant PCTIB-2005GC1-09, and a Ramon y Cajal felgroup-based detector, which can be seen as an efficient implelowship.

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rier are schematically shown in Fig. 1. Since perfect synchronisation is assumed, no intercarrier interference occurs and to all effects we can focus on an arbitrary subcarrier. Following the notation introduced in [5, 7], we define hn (j) with n = 1, . . . , Ns as the NR × 2 channel matrix for the jth (0 ≤ j ≤ Nc − 1) subcarrier associated with the nth data stream whose entries are independent zero-mean complex Gaussian variables with unit variance. Furthermore, we assume that the channel remains static over the two-symbol block. For each data stream, we define a signal transmission matrix constructed according to the Alamouti encoding rule defined as   s2n−1 (j) s∗2n (j) (1) S n (j) = ∗ s2n (j) −s2n−1 (j)

Fig. 1. Combined SDM/STBC (uncoded) with either Linear Group Based or Direct Estimation receivers. mentation of the direct estimation detector achieving the same BER performance at a fraction of its computational complexity. The performance of the existing and proposed schemes is considered when error correction coding is included in the system. Example results are shown in the context of the new wireless local area networks (WLANs) standard IEEE 802.11n which is one of the first commercial systems to contemplate hybrid SDM/STBC schemes. We conclude this introduction with a notational remark: vector and matrices are denoted by lower- and upper-case bold letters, respectively. Lower-case non-bold letters serve to denote scalar values. Superindices T , ∗ and H will be used to denote transpose, conjugate and conjugate transpose, respectively, and IK is used to denote the K-dimensional identity matrix. Finally, we define the Alamouti transform of a K × 2 matrix x = [x1 x2 ] with x1 , x2 being K × 1 vectors as A (x)  [−x∗2 x∗1 ]. 2. SYSTEM MODEL We focus on a multicarrier system using Nc orthogonal subcarriers equipped with NT and NR (≥ NT ) transmit and receive (uncorrelated) antennas, respectively. The information to be transmitted is split into Ns independent data streams which are individually encoded according to the Alamouti scheme [2] implying that Ns = NT /2. For clarity of exposition, and similarly to [4, 5, 7], we restrict our attention to the case NT = NR = 4, Ns = 2 (e.g. DSTTD setup) although the results shown here can be extended to configurations with a higher number of antennas/data streams as long as it holds Ns = NT /2. At this stage, and for the sake of clarity to present the different detection techniques, we restrict our attention to an uncoded system, leaving the coding effects to be considered later in the paper (see Section 6). The transmission and reception schemes for the jth subcar-

where s2n−1 (j) and s2n (j) denote symbols of the nth data stream from consecutive OFDM symbols that will be transmitted on the same subcarrier and which are drawn from an M -ary modulation alphabet. For completeness, we also deT fine at this point sn (j) = [s2n−1 (j) s2n (j)] the vector of symbols to be transmitted for the nth data stream on the jth subcarrier during a space-time block. Given the independence among subcarriers, and to simplify notation, the subcarrier index j will be dropped from subsequent equations although it should be borne in mind that each expression applies to every subcarrier. It is now possible to write the reception equation as: Ns  hi S i + υ (2) y = HS + υ = i=1

T  and υ is where H = [h1 · · · hNs ], S = S T1 · · · S TNs an NR × 2 noise vector whose entries are zero-mean complex Gaussian variables with variance συ2 . 3. EXISTING DETECTION SCHEMES This section reviews various detection approaches both, groupbased and direct estimation-based which have been previously proposed. It should be noted that the different linear techniques presented here could incorporate some form of interference cancellation (IC, see for example [12]) which, for the sake of brevity, is not considered in this paper, however, its inclusion in the reviewed and proposed algorithms is straightforward. 3.1. Group-based schemes As seen on the top receiver on the right hand side of Fig. 1, group-based reception takes place in two steps: first, the block equaliser (linear group receiver) takes care of separating the different streams and then Alamouti decoding is applied on each stream independently. In the particular case of linear detectors, this operation can be formally expressed as: y ˜n = W n y.

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(3)

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where W n denotes the equalising matrix for the nth data stream and y ˜n is an 4 × 2 matrix with the resulting nth stream data samples to be supplied to the Alamouti decoder. The channel matrix H can be rewritten as   A B H= (4) C D where A, B, C and D are 2 × 2 matrices. The ZF group detectors for each of the two groups can easily be shown to be  −1  −1   B A −D−1 −C −1 ZF W ZF = = , W . −1 −1 −1 −1 1 2 B −D A −C The ZF group-equalising matrices have rank 2 and, therefore, the resulting equalised samples, y˜1 and y˜2 , have only 2 × 2 and independent samples each. The rank deficiency of W ZF 1 allows a more compact expression for the equalised W ZF 2 samples as [5]    −1  y ˆ1 B −D−1 = W ZF y = y (5) y ˆ2 A−1 −C −1 ˆ2 are used to denote the independent samples where yˆ1 and y ˜2 , respectively. in y˜1 and y It is well-known that ZF equalisation often leads to excessive noise enhancement and MMSE is usually the preferred criterion to design filters leading to lower BERs. The group-based MMSE equaliser can be shown to be [5, 7]  −1 H W MMSE = hn hH + συ2 INR . n n HH

(6)

3.2. Direct detection schemes In direct detection schemes [4, 7, 9, 10], shown at the bottom on the right hand side of Fig. 1 the receiver directly estimates the transmitted symbols without explicitily exploiting the structure of the Alamouti encoding. It is easy to check that the reception equation (2) can be rewritten as       y1 H υ1 = s + (7) HA y ∗2 υ ∗2 where y 1 , y 2 and υ 1 , υ 2 are used to denote each of the indi

T vidual columns of y and υ, respectively, s = sT1 . . . sTNs and H A = [A (h1 ) . . . A (hNs )]. The expression in (7) can be written in more compact form as ¯ +υ ¯ = Hs ¯ y

(8)

¯ = [H H A ]T and υ ¯ = [y 1 y ∗2 ]T , H ¯ = [υ 1 υ ∗2 ]T . with y The reception equation in (8) is now in the standard form of MIMO systems and therefore, any of the standard MIMO detection techniques can be applied (e.g. ZF, MMSE, OC, IC-based, ML). Particularly important is the optimal receiver proposed in [7] based on ML detection as this has to be taken as the lower bound in terms of BER and the upper bound in terms of complexity for the rest of receivers. Two points are

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worth noting with respect to (8): 1) proposed direct receivers ([4, 7, 9, 10]) are computed to recover the transmitted symbols and do not take into account the Alamouti structure and ¯ has dimensions 8 × 4 and therefore, 2) the system matrix, H, the matrix (pseudo)inverses and products usually required to compute the reception filters are substantially more complex than for group-based equalisers. 4. IMPROVED GROUP-BASED SCHEMES In the next subsections we introduce two improvements that enhance the performance of group-based detectors closing the gap with respect to direct estimation while keeping the computational complexity low. 4.1. Whitening group-based detectors The linear group-based detectors introduced in [5] do not take into account that the noise colouring introduced by the group equaliser causes the subsequent Alamouti decoding to loose its optimality. A simple improvement of these detectors is accomplished by complementing the equaliser with a noisewhitening filter. In the case of ZF, it is easy to see from (5) that  H the filtered noise has covariance Ω = W ZF W ZF συ2 . Since any covariance matrix is hermitian and positive definite, it is possible to obtain its eigen-decomposition Ω = V DV H

(9)

where V is a matrix having as columns the eigenvectors of Ω and D is a diagonal matrix containing the eigenvalues of Ω at its main diagonal, which can be shown to be positive and real. We can now define the zero-forcing noise whitening equaliser as −1  ˜ ZF = V D 1/2 W W ZF . (10) The same procedure can be applied to the MMSE equalising matrices in (6) by using the corresponding covariance matrix for each group  H W MMSE συ2 Ωn = W MMSE n n

n = 1 · · · Ns , (11)

to obtain separated STBC streams affected by decorrelated noise samples. We note that the SVD computation of an N dimensional square matrix has complexity O(N 2 ), which can be deemed negligible in comparison with the cubic complexity required to compute the pseudo-inverse typically needed in most detection techniques. Therefore, the whitening step does not significantly increase the complexity of existing groupbased detectors. 4.2. Linear direct group-based detectors We introduce a new class of linear receivers which we term direct linear group-based detectors and have the objective of achieving the performance of linear direct detectors while reducing their computational complexity. The idea is to exploit

2008 International ITG Workshop on Smart Antennas (WSA 2008)

the Alamouti structure to simplify the equaliser computation which in turns implies performing the detection group-bygroup. Starting from (8), the direct-estimation MMSE detector can be written as −1  ¯ + συ2 I ¯ H. ¯ HH (12) W= H H Now, by exploiting the Alamouti encoding structure the computation of (12) can be greatly simplified. To this end we define the 2 × 2 matrices H

H αij = hH i hj + A (hi ) A (hj ) = αji .

(13)

Assuming NT = NR = 4, Ns = 2, it is straightforward to show that     W 11 W 12 X 11 X 12 ¯H W= (14) = H W 21 W 22 X 21 X 22 where   −1  −1 2 2 X 12 = αH 12 − α22 + 2συ I α12 α11 + 2συ I    −1  −1 X 21 = α12 − α11 + 2συ2 I αH α22 + 2συ2 I 12  X 11 = −X 12 α22 + 2συ2 I α−1 12  −1  X 22 = −X 21 α11 + 2συ2 I αH . 12 The key point to notice in (14) is that thanks to the Alamouti encoding pattern, it holds that W12 = A (W11 ), and W22 = A (W 21 ) allowing the estimated symbols to be computed as 

  T sˆ1 W T11 y 1 + A (W 11 ) y ∗2 ¯= (15) = Wy sˆ2 W T21 y 1 + A (W 21 )T y ∗2 where sˆ1 and sˆ2 are the estimates for the symbols in the two streams. The detection procedure given by (15) can be put in the form of group-based detection as described in Section 3.1 by defining the group equaliser as Wn = [Wn1 A(Wn1 )]. Now the symbol estimates for the nth group can be computed as,   y1 . (16) sˆn = Wn ∗ y2 The attractive feature of the linear-direct group-based detector is that for the computation of W only four different trivial1 inverses of 2 × 2 matrices are required rather than products and inverses involving 4 × 4 and 8 × 4 matrices as required in the direct solution of (12). 5. CODED SYSTEM ARCHITECTURE Next sections present simulation results of the various detection techniques for uncoded (see Fig. 1) and coded setups. 1 Defining

a,b,c,d as scalars, it holds that   −1  1 d a b = c d −c ad − bc

−b a



In the coded configuration, the system has been configured to operate in accordance with the specifications of the IEEE 802.11n current draft [8]. The operating bandwidth is 20 MHz utilized by means of Nc = 64 subcarriers of which 52 are used to transmit data whereas the rest are reserved for pilot tones and guard intervals. Figure 2 shows the complete transmit and receive structure for the IEEE 802.11n model whose operation is now briefly outlined: as shown in this figure, the information bits are processed with an Rc -rate compatible punctured convolutional encoder (RCPPC) to generate the coded sequence which is subsequently split (block S/P) into Ns parallel data streams. Each stream is then interleaved (block Π in Fig. 2) and mapped to symbols (Gray mapping) resulting in vectors sn with 1 ≤ n ≤ Ns . We note that each of these vectors should be long enough so as to contain enough symbols to conform an even number of OFDM symbols in order to make the subsequent Alamouti encoding possible (on a per carrier basis). The STBC encoded symbols on each data stream are then supplied to different OFDM modulators which take care of modulating the subcarriers (typically implemented via IFFT), serialize the resulting transformed symbols and add a sufficiently long cyclic prefix (CP) to avoid any interblock interference at the receiver due to the channel dispersion. Each OFDM modulator has at its disposal two antennas in order to enable transmission according to the the Alamouti encoding rule. Reception starts with the OFDM demodulation process which includes the discarding of the CP and the recovery of the frequency domain symbols via FFT. The resulting samples are then supplied to the SDM/STBC detector (either group-based ˆn can or direct estimation). The estimated symbol streams s then be either quantised and demapped (demapper) to obtain the detected bits or, alternatively, soft information in the form of the likelihood ratio (LLR) for each bit making up the different streams can be computed. The detected bits or LLRs are finally deinterleaved and supplied to a hard- or soft-decision Viterbi decoder, respectively, to yield the estimated information bits. 6. NUMERICAL RESULTS This section presents simulation results of the different detectors for both, uncoded (Fig. 1) and coded (Fig. 2) setups. For both cases we consider a system with NT = NR = 4 transmitting Ns = 2 independent streams (DSTTD setup) with symbols modulated using BPSK. Simulations have been performed using channel model E-NLOS as defined in [13]. For comparison purposes, we have also considered the performance of single stream 2 × 2-STBC and 2 × 4-STBC systems, both configurations having half the transmission rate of the DSTTD setup. Figure 3 summarises the BER results of the different algorithms and configurations in the case of the uncoded scenario. The first noticeable fact, albeit expected, is that the 2 × 4STBC constitutes the BER lower bound for all DSTTD configurations. Remarkably, the ML-based detection in DSTTD as proposed in [7] practically attains this lower bound, im-

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Fig. 2. IEEE 802.11n transmitter and receiver employing combined SDM/STBC.

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as specified in [8, pp. 80]. Figure 4 presents the results for different detection schemes in the coded setup when using hard information is the Viterbi decoding. It can be seen how the BER performance among the different algorithms shrink and remarkably, the group-based direct MMSE detector is less than 1 dB away from the optimum performance provided by the ML detector. It is also interesting to see that the whitened ZF group-detector outperforms the STBC(2,2) in the coded scenario indicating that error detection contributes to extract additional diversity inherent in a 4 × 4 setup. Finally, Fig. 5 show results when using soft information as input to the Viterbi decoder. A gain of 2-3 dB can be observed with respect to the hard-decision system. Performance differences among the detection algorithms reduce even more in the case of soft decisions, and in fact, it can be observed that the group-based direct MMSE detector virtually attains the same BER than the more complex ML detector. 7. CONCLUSIONS

plying that when using this detector, there is no performance penalty in transmitting at a double rate. The MMSE and ZF group-based detectors proposed in [5] are seen to offer a poor performance in comparison with the direct-estimation MMSE receiver. The noise whitening step we have introduced in Section 4.1 significantly improves the performance of these detectors (Group WZF, Group WMMSE) but they are still far from the more computationally complex direct-estimation algorithms. It is interesting to notice that the the noise-whitened group-based ZF equaliser has exactly the same performance as the 2×2-STBC system implying that the introduced decorrelation procedure effectively transforms the DSTTD system into two parallel non-interfering 2×2-STBC systems. Finally, we show the results for the direct-estimation MMSE detector and our proposal group-based direct solution of (14). Obviously, the two solutions attain exactly the same BER although our proposal has lower computational complexity. For the IEEE 802.11n coded scenario, a 1/2-coding rate convolutional encoder with generator polynomial [133 171] has been employed. This configuration of coding, jointly with the selection of BPSK modulation and number of transmitted streams, corresponds to the modulation coding set 8 (MCS-8)

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We have introduced two new families of detectors for the DSTTD setup, namely, the whitening group-based detectors and the linear direct group-based detectors. While the first family improves the performance of previous group detectors their performance still lags behind that of the more complex direct detectors. Our second proposal decomposes the linear direct solution into (smaller dimension) group solutions by exploiting the Alamouti encoding structure, reducing in this way the overall computational complexity while attaining the same BER performance. It has been shown how in the context of a coded system like IEEE 802.11n, performance differences among the detection algorithms are greatly reduced and in particular, when employing soft decoding, group-based direct linear detectors perform at the same level as ML-based direct detection. 8. REFERENCES [1] G. Foschini, “Layered space-time architecture for wireless communication in a fading environment when using multielement antennas,” Bell Labs Technical Journal, vol. 1, no. 2, pp. 41–59, 1996.

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[7] Y. Wu and C. Tellambura, “Low-complexity optimal detection for hybrid space-time block coding and spatial multiplexing,” in IEEE Vehicular Technology Conference Fall, Montreal (Canada), Sept. 2006, pp. 1–4.

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[8] S. A. Mujtaba, “TGn sync proposal technical specification. doc.: Ieee 802.11-04/0889r7,” Draft proposal, July 2005.

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[9] M. V. Do, W. H. Chin, and Y. Wu, “Performance study of a hybrid space time block coded system,” in IEEE Int. Symp. on Wireless Pervarsive Computing, Phuket, Thailand, Jan. 2006, pp. 1–4.

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[10] J. Wu, Y. R. Zheng, A. Gumaste, and C. Xiao, “Error performance of double space time transmit diversity system,” IEEE Trans. on Wireless Communications, vol. 6, pp. 3191–3196, 2007.

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[11] U. Fincke and M. Pohst, “Improved methods for calculating vectors of short length in a lattice, including a complexity analysis,” Math. Comput., vol. 44, pp. 463–471, 1985. [12] W. Shin, H. Lee, C. Seo, and J. Kang, “Iterative biased mmse detections for dsttd-ofdm systems over time-varying multipath channels,” in Proc. IEEE Vehic. Tech. Conf. Spring, Dublin (Ireland), April 2007, pp. 1976–1980.

Fig. 4. Coded system (1/2-RCPPC) with hard decisions. 0

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[13] V. Erceg, “Indoor MIMO WLAN Channel Models. doc.: IEEE 802.11-03/871r0,” Draft proposal, November 2003.

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Fig. 5. Coded system (1/2-RCPPC) with soft decisions. [2] A. Alamouti, “A simple transmit diversity technique for wireless communications,” IEEE J. Selec. Areas on Comm., vol. 16, pp. 1451–1458, 1998. [3] V. Tarokh, H. Jafarkhani, and A. Calderbank, “Space-time block codes from orthogonal designs,” IEEE Transactions on Information Theory, vol. 45, pp. 1456–1467, July 1999. [4] E. N. Onggosanusi, A. G. Dabak, and T. M. Schmidl, “High rate space-time block coded scheme: Performance and improvement in correlated fading channels,” in Proc. IEEE Wireless Communications and Networking Conference, Orlando (USA), March 2002, pp. 194–199. [5] L. Zhao and V. Dubey, “Detection schemes for space-time block code and spatial multiplexing combined systems,” IEEE Communications Lett., vol. 9, pp. 49–51, 2005. [6] W. F. Jr., F. Cavalcanti, and R. Lopes, “Hybrid transceiver schemes for spatial multiplexing and diversity in MIMO systems,” Journal of Communication and Information Systems, vol. 20, pp. 141–154, 2005.

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