Combined Spatial Multiplexing and Diversity ... - IEEE Xplore

Combined Spatial Multiplexing and Diversity Techniques for Coded MC-CDMA Systems With Suboptimal MMSE-Based Receivers Mikko Vehkaperä, Djordje Tujkovic, Zexian Li and Markku Juntti Centre for Wireless Communications (CWC) Tutkijantie 2E, P.O. Box 4500, FIN-90014 University of Oulu, Finland {mikko.vehkapera,djordje.tujkovic,zexian.li,markku.juntti}@ee.oulu.fi

Abstract— We study several transmission techniques utilizing efficient forward error correction coding (FEC), space-time coding (STC), group-wise layered space-time (GLST) architectures, and, due to the inevitable complexity restrictions, suboptimal minimum mean squared error (MMSE) based receiver interfaces in the context of downlink multicarrier code-division multipleaccess (MC-CDMA). Frequency-selective Rayleigh fading channels are assumed and the effect of spatial correlation on the system performance is considered. The results demonstrated that the recently proposed space-time turbo coded modulation (STTuCM) has a significant advantage over the more conventional transmission schemes combining orthogonal space-time block codes (STBCs) and single-antenna optimized turbo coded modulation also when the proposed GLST architectures and suboptimal, lower complexity receivers are utilized.

I. I NTRODUCTION We study a combination of an efficient multi-carrier technique, known as multi-carrier code-division multiple-access (MC-CDMA) [1], and layered space-time (LST) architectures with space-time coding (STC) technique (e.g., [2], [3] and references therein). We aim at exploiting both the diversity and the spatial multiplexing gain of the MIMO channel [4] by utilizing group-wise LST (GLST) architecture similar to the one proposed in [5]. As in [6] the exponential increase in the complexity of the ML decoding is avoided by using sub-optimal minimum mean squared error (MMSE) based receiver interfaces. However, in this paper, we extend our scope of interest further and consider also layered full-rate space-time block codes (STBCs) for two antennas [7], accompanied with union bound optimized turbo trellis coded modulation (TuTCM) [8] or turbo bit-interleaved coded modulation (BICM) in addition to the the recently proposed space-time turbo coded modulation (STTuCM) [3] studied in [6]. The single-antenna coded schemes, TuTCM and BICM are applied for the MIMO transmission through conventional LST techniques and with additional spatial diversity processing by using STBCs. We also study the performance of these techniques in the presence of spatial fading correlation, anticipated to be encountered in the practical multi-antenna systems. This research was supported by the National Technology Agency of Finland, Nokia, the Finnish Defence Forces, Elektrobit and Instrumentointi.

0-7803-8887-9/05/$20.00 (c)2005 IEEE

II. S YSTEM M ODEL Notations used in this paper are as follows. Boldface slanted letters denote matrices and vectors. Matrix I N denotes an identity matrix of size N × N , X = diag X 1 , X 2 , . . . , X N denotes a block diagonal matrix whose block diagonals are given by the square matrices X n , n = 1, 2, . . . , N and tr(A ) calculates the trace of the matrix A . Notation X = vec(X 1 , X 2 , . . . , X M ) ∈ CM J×K represents stacking of matrices X i ∈ CJ×K “on top of each other” to form a matrix X , whereas X = [X 1 X 2 · · · X M ] ∈ CJ×M K represents a matrix X where X i are set “side-by-side’. Throughout the paper we consider a single cell downlink MC-CDMA system with Nc subcarriers and K active users who all have the same spreading factor G. Antenna arrays with N transmit antennas at the base station and M receive antennas at the mobile terminal are assumed with a requirement M ≥ N . Each spatial layer consists of a group of J0 antennas so that N = JJ0 , and one coded frame is defined to consists of P symbols per layer, i.e., P J symbols in total. Let us assume that we are looking at the transmission between the base station and the terminal of the user k, where k = 1, 2 . . . , K. The binary vector b k = vec(b k,1,1 , b k,1,2 , . . . , b k,j,p , . . . , b k,J,P ) ∈ {0, 1}JP κ , where b k,j,p ∈ {0, 1}κ , j = 1, 2, . . . , J, p = 1, 2, . . . , P is the pth input of the frame to an arbitrary single or multi-antenna channel encoder corresponding to the coded symbol vector (note that STBCs treat encoded bits as consecutive blocks) x k,j,p ∈ MJ0 mapped to the jth layer. The alphabet M has cardinality |M| = 2ζ , κ denotes the number of information bits entering the encoder per symbol instant and ν − κ is the corresponding number of output parity bits. The total spectral efficiency of the system, taking into account the user load, is then given by κζK , (1) Reff = JL νG where L is the internal spatial multiplexing order of the encoder. Note that albeit for STTuCM [3] we have L = 2, for the Alamouti’s simple transmit diversity scheme L = 1 since it takes two time units to transmit two symbols containing

First antenna group

zero mean and unit variance, derived from the time domain tapped delay line presentation of the channel via Fourier transform as discussed in [10]. We write also the channel-spreading matrix for user k in terms of the layers j = 1, 2, . . . , J so that C k = [C k,1 C k,2 · · · C k,J ] ∈ CM G×N , where C k,j = [c k,(j−1)J0 +1 c k,(j−1)J0 +2 · · · c k,(j−1)J0 +J0 ] ∈ CM G×J0 . By using similar notation we can represent also the corresponding transmitted signal vector in terms of spatial layers so that x k = vec(x k,1 , x k,2 , . . . , x k,J ) ∈ CN .

xk,p+1,1

b

xk,p-1,J xk,p,J xk,p+1,J

Encoding, modulation, spatial processing

0

0

0

Serial-to-parallel conversion

xk,p-1,1 xk,p,1 xk,p+1,1

xk,p+1,J

0

xk,p,1

xk,p,J

0

Second antenna group

Fig. 1.

Vertical layering for a system with N = 2J0 .

different blocks of κ information bits. For all studied schemes we use a generalized vertical coding technique of [9] that is illustrated in Fig. 1 for a system with N = 2J0 . We assume that the cyclic prefix is longer than the expected channel delay spread and the multipath components are located at the sampling instants of the transmitted signal so that the system has an equivalent frequency domain presentation [10], [11]. Channel coherence time is also assumed be longer than the coded frame length so that the channel is constant over the transmission of one MC-CDMA frame. Thus, without loss of generality, we can concentrate on a single coded MC-CDMA frame. The received signal can be then expressed in terms of symbol intervals as

r p = C p x p + ηp ,

p = 1, 2, . . . , P,

(2)

where, after omitting p for simplicity, transmitted signal vector, received symbol vector and the noise vector are defined as x T1

x TK

x = [x1,1 · · · x1,N · · · xK,1 · · · xK,N ]T ∈ MN K 1 G T r = [r11 · · · r1G · · · rM · · · rM ] ∈ CM G 1 G T η = [η11 · · · η1G · · · ηM · · · ηM ] ∈ CM G ,

respectively. The elements of η are independent and complex Gaussian with equal power real and imaginary parts, i.e., η ∼ CN (0, N0 I M G ) and represent the frequency domain thermal noise at the receiver. The total power per symbol from the base station is held constant regardless of the number of transmit antennas, phase-shift keying is used for modulation and the signal-to-noise ratio (SNR) per receive antenna is defined as γ = N Es /N0 where Es = x∗k,n xk,n ∀xk,n ∈ M. The combined channel-spreading matrix C can be decomposed as C1

CK

C = [c 1,1 · · · c 1,N · · · c K,1 · · · c K,N ] ∈ CM G×N K c k,n = vec(c 1,n,k , c 2,n,k , . . . , c M,n,k ) ∈ CM G c(p+1) 1 c(p+2) 2 c(p+G) G T c m,n,k = [Hm,n sk Hm,n sk · · · Hm,n sk ] ∈ CG , T G where s k = [s1k s2k · · · sG k ] ∈ S is the signature sequence of H user k, s k s k = 1, S denotes the chip alphabet and c(p + g) ≡ (p−1)G+g (mod Nc ) is the frequency order of the subcarrier. c(p+g) g For simplicity of notation we also denote Hm,n Hm,n . g The channel coefficients Hm,n are complex Gaussian with

III. L INEAR R ECEIVERS FOR C ODED MIMO MC-CDMA A. Symbol-Level Joint SF-MMSE Detector Let us now consider the system model of (2) and omit the explicit presentation of the symbol interval index p. As shown in [6], [12], the matrix filter W = [W 1 W 2 · · · W K ] ∈ CM G×N K that can simultaneously estimate the transmissions from all antennas and for all users k = 1, 2, . . . , K is given by the well-known Wiener solution [13] −1 W = C R xx C H + R ηη C R xx , (3) when the receiver has perfect CSI and the noise is uncorrelated with the transmitted signals and fading processes. Because the space-frequency-MMSE (SF-MMSE) detector has no prior knowledge of the channel code structure, we assume R xx = Es I N K . Furthermore, the thermal noise between the receive antennas and subcarriers may be considered to be uncorrelated and, thus, R ηη = N0 I M G . When the Alamouti’s simple transmit diversity scheme [7] is combined with spatial multiplexing, the symbols within one layer in x p are transmitted twice from two transmit antennas within each of the antenna groups j = 1, 2, . . . , J. Thus, we have to rewrite our system model slightly to use the above presented front-end for space-time block coded systems. Let us now assume for simplicity N = 4, that is, we have spatially multiplexed two Alamouti’s STBCs (which code symbols in time) and that the channel is constant during the consecutive OFDM symbols. We can write the equivalent system model for the system utilizing layered STBCs as C1 C2 · · · CK η p1 r r p = p∗1 = x + (4) K p 2 ··· C r p2 η ∗p2 C1 C C eq

where the equivalent combined channel-spreading matrices are

C k = c k,1 c k,2 c k,3 c k,4 ∈ CM G×4 (5)

∗ M G×4 ∗ ∗ ∗ k = c −c c −c ∈C C . (6) k,2

k,1

k,4

k,3

We can then derive a symbol-level detector for the MC-CDMA system utilizing layered STBCs and the solution for the linear case (i.e., no soft feedback from decoder) is found to be very similar to (3) except that we have to replace C by C eq . The iterative receivers discussed in [6], [12] could be similarly transformed for the MC-CDMA system utilizing STBCs and spatial layering. However, as seen from the simulation results, the STTuCM gives a superior performance over all STBC schemes in the considered linear cases, and, thus, the iterative receiver for the layered STBCs is not studied in this paper.

B. MAP Decoding and Gaussian Approximation of the SFMMSE Output Using Mahalanobis Distance We discuss now the iterative decoding of the underlying transmission schemes when a posteriori probability modules [14] are used. In [6] we used a simplified Euclidean distance when the transition probabilities were calculated. In this paper we use Mahalanobis-distance instead to take into account the correlation between the spatial symbols within a layer after the MMSE filtering. We assume for simplicity of notation R xx = I N and write the equivalent system model, as seen by the decoder, after the SF-MMSE filter as

xˆ k = W Hk r = Ξk x k + ϕk ∈ CN ,

(7)

where xˆ k is an MMSE estimate of x k . The residual multipleaccess interference (MAI)-plus-noise term ϕk ∈ CN is assumed to be a Gaussian random variable with PDF CN (0, R ϕϕ ). Extending the work of [15] for ST-coded MCCDMA system, the “bias” of estimate xˆ k can be written as   A1 Ωk,1   H .. Ξk = E{ˆ x k x Hk } =  (8)  = W k C k, . A2

Ωk,J

where C k = [c k,1 c k,2 · · · c k,N ] ∈ CM G×N , the block diagonals of Ξk ∈ CN ×N are Ωk,j ∈ CJ0 ×J0 and “triangles” A1 , A2 contain the rest of the elements of Ξk . The elements contained by A1 and A2 will be now neglected since the ST decoder is able to process spatially only those J0 symbols that belong to the layer to be decoded. Thus, the equivalent channel and residual MAI-plus-noise covariance matrices for the decoder at considered symbol interval p are Ωk = diag (Ωk,1 , Ωk,2 , . . . , Ωk,J ) R ϕϕ = diag R 1ϕϕ , R 2ϕϕ , . . . , R Jϕϕ , respectively, where Ωk,j are Hermitian matrices. The covariance matrix for the residual interference for layer j is also Hermitian and given by

R jϕϕ

=

E{ˆ x k,j xˆ Hk,j }

−

Ωk,j ΩH k,j

= Ωk,j −

Ωk,j ΩH k,j .

(9)

Assuming the decoder uses the a posteriori probability modules to perform the iterative decoding, the conditional log x k,j,p ] log p[b k,j,p |ˆ x k,j,p ] for each likelihoods L[b k,j,p |ˆ p = 1, 2, . . . , P are calculated as [14]

x k,j,p ] L[b k,j,p |ˆ = log :b p ()=b p

exp Ap−1 [ξ S ()] + Γp [ˆ x k,j,p , b k,j,p ()] +Bp [ξ E ()] ,

where defines an edge of a trellis section, ξ S () the starting state of the edge, ξ E () the ending state of the edge and b p () the input bits evoking the trellis transition ξ S () → ξ E (). The state probabilities Ap and Bp are calculated recursively as shown in [14]. Using the Mahalanobis distance, the trellis transition likelihood in Γp , invoked by b k,j () creating the

symbol output x k,j ∈ MJ0 , is for the user k and layer j = 1, 2, . . . , J of the form x k,j | b k,j ()] log p [ˆ

x k,j − Ωk,j x k,j )H [R jϕϕ ]−1 (ˆ x k,j − Ωk,j x k,j ) = (ˆ in the case of coded modulation. The above method can be applied also for the soft detection in the case of BICM. C. Chip-Level MMSE Based Receiver One problem with the SF-MMSE detector is that the knowledge of the other users’ signature sequences must be available at the receiver. If, however, the spatial filtering and chip combining are separated, only the knowledge of the spreading sequence of the desired user is required. In addition the inversion of a M G×M G Hermitian matrix can be reduced to an inversion of a M × M Hermitian matrix. To begin, let us again omit the explicit presentation of index p and rewrite the system model of (2) as

r g = H g z g + ηg ,

g = 1, 2, . . . , G,

(10)

where the received, channel and noise vectors are r g = g T ] , H g = [H g1 H g2 · · · H gJ ] ∈ CM ×N and [r1g r2g · · · rM η g ∼ CN (0, N0 I M ), respectively. The channel matrix for g ] ∈ CM ×J0 layer j = 1, 2, . . . , J is given by H gj = [Hm,i and the transmitted symbols are considered to be chips z g = g T ] , in which zng = k sgk xk,n , n = 1, 2, . . . , N . [z1g z2g · · · zN The equation (10) can be further decomposed as co-antenna interference

multiple-access interference

desired g g g g g g g r = H j zk,j + H j zk,j + H j zk ,j +η g , j =j

k =k

j

(11) and the spatial matrix filter that we use to eliminate the second RHS term in (11) is found by using again the Wiener solution. Thus, for the gth subcarrier of the pth symbol the spatial filtering is performed by using the matrix −1 g

W g = H g R zz (H g )H + N0 I M H ∈ CM ×N , (12) where R zz is the covariance matrix of vector z g . We assumed at the receive in the simulations that the signature sequences are random and the coded symbols independent between all transmit antennas, which leads to R zz = Es · (K/G) · I N after some derivations. Although this is not true for the WalshHadamard sequences the transmitter used, the results showed that the chip-level receiver obtains performance close to the symbol-level receiver also with this simplifying assumption. Estimates of the transmitted chips corresponding to the symbol x at subcarrier g can be calculated after spatial filtering as zˆg = (W g )H r g ∈ CN and the symbol estimates for user k and antenna n = 1, 2, . . . , N are achieved via simple g g ˆn . As opposed to the SF-MMSE combining x ˆk,n = g sk z receiver presented in previous section, only the knowledge of the noise power and the signature sequence of the desired user were required. With additional computational cost, the instantaneous MAI-plus-noise power at the output of the spatial filter and chip combiner can be calculated in a similar manner as discussed in Section III-B.

IV. S IMULATION R ESULTS In this section, the numerical results illustrating the performance of the discussed transmission schemes are provided. A downlink single-cell MC-CDMA system with Nc = 64 subcarriers is assumed and the physical layer parameters are adopted from the HIPERLAN/2 standard [16]. One coded frame consists of P = 4Nc modulated symbols per antenna and BICM, TuTCM or STTuCM for J0 = 2 [3] is used as a FEC method. Puncturing and modulation are selected so that Reff = N · (K/G) bps/Hz is achieved after spatial processing as summarized in Table I for M = N = 4. The spreading codes are real-valued Walsh-Hadamard and the system is fully loaded, i.e., G = K in all cases. The power delay profile conforms the ETSI BRAN channel model A and the delay spread of the channel is shorter than the cyclic prefix. For spatial correlation we use the “Kronecker”-model [17] with two scenarios: uncorrelated and correlated for which the correlation matrices are given in Table II. The FER performance of an MC-CDMA system with uncorrelated antenna setup of 2×2 and single- or multi-antenna FEC is presented in Fig. 2. In this scenario the transmission schemes utilizing spatial diversity do not require layering, whereas the BICM with QPSK has two vertically coded layers. In both, uncorrelated and correlated (not shown here) cases, the performance of the BICM is degraded when compared to the other techniques. It is also clear from the figures that whereas both STTuCM and STBCs provide spatial diversity of order 4, for the layered BICM spatial diversity is not directly available. This is because the SF-MMSE interface is used before the decoder and the transmission method does not provide inherent transmit diversity. It should be noted, however, that coding over the antennas provides some level diversity in decoding. Within the techniques that utilize space-frequency processing, STTuCM provides the best performance with a 1 dB difference to the optimized TuTCM with STBCs. It should be also noted that due to the orthogonality constraint on the space-frequency transmit signal matrix, the maximal mutual information achievable with Alamouti’s diversity scheme is equal to the full MIMO channel capacity only at the special case of M = 1 [18]. For STTuCM we have looser spatial constraints and we suspect that this results in higher maximal mutual information in the 2 × 2 case, shown also by the better performance of the simulation results. We also notice that the performance in the multiuser case with G = K = 8 is degraded only about 0.5 dB compared to the single-user case and both the symbol-level and chip-level MMSE detectors provide the same performance in the considered scenario. Fig. 3 demonstrates the FER performance of an MC-CDMA system with N = M = 4 where both spatially uncorrelated and correlated cases are shown. Since N > J0 , spatial layering is used for all transmission techniques and the antennas are divided into two groups according to Fig. 1 where J0 = 2. The previously noted difference between STTuCM and the best Alamouti coded scheme remains. This can be explained by the fact that after the proposed SF-MMSE filtering the

TABLE I S IMULATED SCHEMES FOR N = M = 4, Reff = 4K/G FEC and diversity scheme

κ

ν

ζ

J

L

BICM BICM + STBC TuTCM + STBC STTuCM

1 1 2 2

2 2 3 4

2 4 3 4

4 2 2 2

1 1 1 2

TABLE II T RANSMITTER AND RECEIVER CORRELATION MATRICES Correlation matrices for transmit and receice antennas   1 0.7544 0.4109 0.23.5 1 0.7544 0.4109  0.7544 R TX =  0.4109 0.7544 1 0.7544 0.23.5 0.4109 0.7544 1   1 −0.3043 0.2203 −0.1812 −0.3043 1 −0.3043 0.2203   R RX =  0.2203 −0.3043 1 −0.3043 −0.1812 0.2203 −0.3043 1

system reduces essentially to a 2 × 2 system with additional co-antenna interference but increased diversity due to coding over independent antennas. With spatial correlation however, the TuTCM with STBCs offers nearly the same performance as STTuCM. In addition to the linear cases, examples of the effect of soft feedback [6], [12], is presented are Fig. 3. The normalized throughput with the assumption of an ideal automatic repeat request (ARQ) for a single-user MIMO MCCDMA system is presented in Fig. 4. Correlated antennas with N = M = 4 is assumed and the 1 % outage capacity for the channel is also plotted in the figure. We neglect the overhead introduced by the ARQ and assume that all flawed packets are re-transmitted until successfully received. Because adaptive coding and/or modulation was not utilized in the system, the hard limit for the throughput is 4 bps/Hz which is depicted by the solid flat line. This limit could be increased by using higher order modulation schemes although with nonadaptive transmission degradation in FER performance would be unavoidable. It is evident that the more robust schemes utilizing STCs offer a superior performance compared to the pure spatial multiplexing with BICM. This is due to the fact that the most severe correlation is between the adjacent antennas, i.e., within the layer, and there the code structure can be used to separate the transmissions. The best scheme with spatially layered STTuCM and two receiver iterations is roughly within 2.5 dB from the outage capacity in the correlated scenario. When uncorrelated antennas are assumed (figure not shown here) this gap diminishes to about 2 dB and the difference between the single-antenna techniques with and without STBCs was noted to be relatively minor. V. C ONCLUSIONS Downlink MIMO MC-CDMA system utilizing STC and GLST architectures was considered in this paper. The results indicated that the error rate performance of the pure spatial multiplexing is severely degraded in both single and multiuser scenarios when antenna correlation is present whereas the techniques utilizing spatial diversity processing offered robust

10

6

0

STBC

FER

STTuCM 10

10

normalized throughput [bps/Hz]

5

−1

STTuCM (QPSK) STTuCM (QPSK), SF−MMSE, G = K = 8 STTuCM (QPSK), Chip−MMSE, G = K = 8 BICM (QPSK) BICM (16−QAM) + STBC TuTCM (8−PSK) + STBC

−2

2

3

4

5

6

7

8

FER

10

3

STTuCM STBC 2

1% Outage capacity STTuCM (QPSK), LMMSE BICM (QPSK), LMMSE BICM (16−QAM) + STBC, LMMSE TTCM (8−PSK) + STBC, LMMSE STTuCM (QPSK), 2 iterations

0

4

5

6

7

8

9

10

11

SNR [dB]

Fig. 2. Performance of a MIMO MC-CDMA system with single- or multiantenna channel coding and linear receivers. Uncorrelated antennas with N = M = 2.

10

2.5dB

1

SNR [dB]

10

4

Fig. 4. Normalized throughput of a single-user MIMO MC-CDMA system with single- or multi-antenna channel coding and ideal ARQ. Correlated antennas with N = M = 4.

0

No corr.

−1

High corr.

STTuCM, linear, NC STTuCM, 2 iter., NC STTuCM, linear, HC STTuCM, 2 iter., HC TTCM + STBC, NC TTCM + STBC, HC

−2

2

3

2 iter.

4

2 iter.

5

6

7

8

9

SNR [dB]

Fig. 3. Performance of a fully loaded (G = K = 8) MIMO MC-CDMA system with single- or multi-antenna channel coding and linear and iterative receivers. Uncorrelated and correlated antennas with N = M = 4.

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