Multiuser Detection and Channel Estimation in MIMO OFDM Systems ...

Multiuser Detection and Channel Estimation in MIMO OFDM Systems via Blind Source Separation Luciano Sarperi
, Asoke K. Nandi, and Xu Zhu Signal Processing and Communications Group, Department of Electrical Engineering and Electronics, The University of Liverpool, Brownlow Hill, Liverpool, L69 3GJ, UK {lsarperi,a.nandi,xuzhu}@liverpool.ac.uk Abstract. This paper proposes a blind multiuser detection and channel estimation method for multiple-input multiple-output (MIMO) orthogonal frequency-division multiplexing (OFDM) systems based on blind source separation (BSS). Using multiple antennas for transmission and reception the multiuser detection problem can be cast as a BSS problem of linear instantaneous mixtures. The proposed method uses BSS in each subcarrier to detect the user signals. The source order and scaling indeterminacies inherent to BSS methods are overcome by exploiting the cross correlation between the subcarriers introduced by convolutional encoding. Additionally, the method provides an estimate of the channels. It is shown that this method provides significant performance enhancement over previous work.

1

Introduction

OFDM systems have recently attracted great interest. Some of the reasons for this are low complexity of implementation, the fact that the orthogonality of the subcarriers is maintained when transmitted through a linear multipath channel and ease of equalisation. OFDM has been adopted in digital audio/video broadcasting standards in Europe and has been proposed for digital cable television systems and wireless networks such as IEEE802.11a [2]. In a multipleinput multiple-output (MIMO) OFDM system, multiple transmit antennas and multiple receive antennas are employed and the same subcarriers are used by all transmitters. The use of MIMO systems is motivated by the significant capacity gains over single-input single-output (SISO) systems [5], [6], [7], [9]. BSS methods allow the recovery of source signals from the observation of the mixtures only, with unknown source signals and mixing transformation. However, some assumptions must be fulfilled by the source signals and the mixing transformation [8]. In a MIMO OFDM system these assumptions are met if the source signals are mutually statistically independent and at least as many receive antennas as transmit antennas are employed.


This work is supported by the Overseas Research Studentship (ORS) Awards Committee UK and the University of Liverpool

C.G. Puntonet and A. Prieto (Eds.): ICA 2004, LNCS 3195, pp. 1189–1196, 2004. c Springer-Verlag Berlin Heidelberg 2004


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Luciano Sarperi, Asoke K. Nandi, and Xu Zhu

In [1] a BSS based OFDM MIMO multiuser detection method was proposed, which employs BSS only in one subcarrier and unmixes the remaining subcarriers using a minimum mean square error (MMSE) approach. The advantage of this BSS-MMSE method is the low computational cost, however, errors tend to propagate from subcarrier to subcarrier since the previous subcarrier is used as a reference to unmix the current subcarrier. Our work is different in that we use BSS in all subcarriers to perform multiuser detection for MIMO OFDM systems. The separation of the multiple users is carried out in the same fashion as the separation of convolved mixtures in the frequency domain, see for example [3]. It is shown that the proposed method can obtain significant performance enhancement compared to the method in [1]. This paper is structured as follows. In section 2 the system model is introduced, section 3 proposes the BSS based multiuser detection method, simulation results are given in section 4 and conclusions are drawn in section 5.

2

System Model

In OFDM systems data are transmitted in blocks using inverse FFT (IFFT) at the transmitter and FFT at the receiver. By adding a guard interval between each block, inter-block interference (IBI) can be eliminated. Usually, during the guard interval a cyclic prefix (CP), consisting of redundant symbols, is transmitted at the beginning of each block [2]. At the transmitter, binary source data du (n) with identical independent distribution (i.i.d.) and unit variance is encoded using a convolutional encoder with real valued impulse response c(n) and length F to obtain the encoded signal for transmit antenna u F −1
su (n) = c(l)du (n − l). (1) l=0

The encoded signal is then transmitted in blocks su (i) = [su (iN ), su (iN + 1), · · · , su (iN + N − 1)]T where i is the block index and the block length N corresponds to the number of subcarriers. Using a CP of sufficient length, the frequency-selective channel is transformed into a flat-fading channel for each subcarrier [2]. Considering Nt transmit antennas and Nr receive antennas (see Fig. 1) we can build blocks of received signals per subcarrier k for k = 0, 1, · · · , (N − 1): r(k) = [r1 (iN + k), r2 (iN + k), · · · , rNr (iN + k)]T where rv (·) is the signal from receive antenna v. The total received signal per subcarrier k becomes now r(k) = H(k)s(k) + n(k) with

(2)



 H11 (k) · · · H1Nt (k)  H21 (k) · · · H2Nt (k)    H(k) =   .. .. ..   . . . HNr 1 (k) · · · HNr Nt (k)

(3)

Multiuser Detection and Channel Estimation in MIMO OFDM Systems

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H11 H12 S1

R1

H21

S2

H22

R2

Fig. 1. MIMO OFDM system with Nt = Nr = 2.

where Hvu (k) is the k-th DFT coefficient of the channel between transmit antenna u and receive antenna v. The signal transmitted per subcarrier k is s(k) = [s1 (iN + k), s2 (iN + k), · · · , sNt (iN + k)]T where su (·) is the signal transmitted by transmit antenna u and n(k) = [n1 (iN + k), n2 (iN + k), · · · , nNr (iN + k)]T is the additive white gaussian noise (AWGN) vector with zero mean and variance N0 . In [1] it was recognised that (2) corresponds to a BSS problem of linear instantaneous mixtures with the complex valued mixing matrix M(k) = H(k), sources s(k) and mixtures r(k).

3

BSS Based Multiuser Detection

In the proposed BSS-only multiuser detection and channel estimation method BSS is used in each subcarrier to obtain an estimate of the sources ˆ s(k) and the ˆ mixing matrix M(k) from the received mixtures r(k). The order and scaling indeterminacies inherent in BSS methods necessitate a post-BSS processing to order the estimated sources and mixing matrices in the same way for all subcarriers k = 0, 1, · · · , (N − 1). The ordering and scaling of the estimated sources and mixing matrices rely on the correlation introduced by the convolutional encoder with impulse response c(n) in (1). As presented in [1], the cross correlation between the transmitted signals from transmit antenna p in one subcarrier and transmit antenna q in a neighbouring subcarrier is  F −2  
c(p)c(p + 1) when p = q ∗ (4) γpq = E[sp (iN + k)sq (iN + k ± 1) ] = p=0   0 when p
= q where E[·] is the expectation with respect to i. The following method provides reordering and scaling of the estimated sources ˆ s(k) = [ˆ sA (iN +k)ˆ sB (iN +k)]T and ˆ mixing matrix M(k) for an Nt = Nr = 2 MIMO system, which can be extended to more transmit and receive antennas. First, the sources in subcarrier k = 0 are separated. Next, to reorder and scale the remaining subcarriers k = 1, 2, · · · , (N − 1) the following cross correlations

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Luciano Sarperi, Asoke K. Nandi, and Xu Zhu

ρ1 = E[ˆ sA (iN + k)ˆ s1 (iN + k − 1)∗ ] sB (iN + k)ˆ s2 (iN + k − 1)∗ ] ρ2 = E[ˆ

(5) (6)

ρ3 = E[ˆ sA (iN + k)ˆ s2 (iN + k − 1)∗ ] sB (iN + k)ˆ s1 (iN + k − 1)∗ ] ρ4 = E[ˆ

(7) (8)

of the estimated sources are obtained (see Fig. 2), where E[·] is the expectation with respect to i. Then, the following metrics based on the difference between the cross correlations of the estimated sources and the true cross correlations in (4) are obtained δ1 = (|ρ1 | − γ11 )2 + (|ρ2 | − γ22 )2

(9)

δ2 = (|ρ4 | − γ11 ) + (|ρ3 | − γ22 )

(10)

2

2

sA (·)] = assuming that the encoded signals su (·) are real valued and var[ˆ var[ˆ sB (·)] = var[ˆ s1 (·)] = var[ˆ s2 (·)] = var[su (·)], where var[·] is the variance. The estimated sources sˆA (iN + k) and sˆB (iN + k) are now reordered and scaled as follows to obtain the same order and scaling as in the previous subcarrier (k − 1): 

|ρ1 |  s ˆ (iN + k) A  ρ 1  when δ1 < δ2  |ρ2 |    sˆB (iN + k) ρ2 (11) ˆ s(k) =

 |ρ4 |   sˆB (iN + k) ρ  4  when δ1 ≥ δ2   sˆ (iN + k) |ρ3 | A ρ3 ˆ The estimate of the channel H(k) can be obtained by reordering and scaling the ˆ estimated mixing matrix M(k) as follows  ρ1 ρ2 when δ1 < δ2  M(k) diag( |ρ1 | , |ρ2 | ) ˆ H(k) = (12)  M(k) J diag( |ρρ44 | , |ρρ33 | ) when δ1 ≥ δ2 where diag(d1 , d2 ) is a diagonal matrix with diagonal elements d1 , d2 and J is the 2x2 exchange matrix with ones on the antidiagonal and zeros elsewhere.

4

Simulations

Simulations were carried out for a Nt = Nr = 2 MIMO system. Both our BSS-only method and the BSS-MMSE method in [1] employed the JADE BSS algorithm in [4]. BPSK source data du was used. The encoded and modulated signal was transmitted through channels with the following impulse responses: h11 (n) = h12 (n) = h21 (n) = h22 (n) =

[ 1.00; 0.30; −0.10 ] [ 1.00; −0.60; 0.08 ] [ 1.00; 0.00; −0.25 ] [ 1.00; 0.30; −0.28 ]

(13)

Multiuser Detection and Channel Estimation in MIMO OFDM Systems

R1

Subcarrier k=0 k=1 OFDM RECEIVER

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s1(iN) BSS(k=0) s2(iN)

k=N−1

POST−BSS BSS(k=1)

s1(iN+1)

PROCESSING(k=1) s2(iN+1)

k=0 k=1 R2

OFDM RECEIVER

POST−BSS BSS(k=N−1)

s1(iN+N−1)

PROCESSING(k=N−1) s2(iN+N−1)

k=N−1

Fig. 2. BSS-only OFDM receiver for Nt = Nr = 2.

The impulse response of the convolutional encoder was c(n) = [1; 0.5; −0.25] and the number of subcarriers was N = 20 as in [1]. AWG noise was added at the receiver to obtain the desired SNR level. Both methods used the noisy received signal r(k) only. For perfect separation of the sources in the noiseless case the global matrix ˆ −1 H(k) is of the form diag(α1 , α2 ), where αp is a scalar and assuming G = H(k) that the sources were recovered in the original order. The quality of the source separation was measured using the mean square error (MSE) between a scaled version of G and the identity matrix I.  g11 g12  g11 −I (14) Q = gg11 21 g22 g22 g22

Using (14) the MSE was obtained by averaging 100 runs:  2 2



2 |qpq | MSE = 10 log10 E

(15)

p=1 q=1

Simulations were carried out for Ns = 1000, 5000 and 10000 symbols per subcarrier and at SNR levels of 20 dB and 10 dB. Figs. 3 and 4 were obtained with Ns = 1000 and SNR = 20 dB and 10 dB respectively. They show that both methods obtain better performance at higher SNR levels, but while the MSE of the BSS-MMSE method increases progressively for higher subcarrier numbers k, the MSE of the BSS-only method remains at a low level for all subcarriers. This is because the BSS-MMSE method uses the sources separated in the previous subcarrier as a reference to unmix the current subcarrier while in the BSS-only method the quality of the source separation does not depend on the previous subcarrier. The performance in the first subcarrier k = 0 is similar for both methods, as both use BSS in the first subcarrier. For Ns = 5000 in Fig. 5 the performance of both methods is better than with Ns = 1000 in Fig. 3 and Fig. 4. A further increase to Ns = 10000 in Fig. 6 does for both methods not lead to a significant improvement.

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Luciano Sarperi, Asoke K. Nandi, and Xu Zhu MSE vs k, Ns = 1000, SNR = 20 dB 10 BSS−only BSS−MMSE 5

0

MSE(dB)

−5

−10

−15

−20

−25

−30

−35

0

2

4

6

8

10 subcarrier k

12

14

16

18

20

Fig. 3. MSE (dB) vs subcarrier k at SNR = 20 dB for Ns = 1000 symbols per subcarrier.

MSE vs k, Ns = 1000, SNR = 10 dB 10 BSS−only BSS−MMSE 5

0

MSE(dB)

−5

−10

−15

−20

−25

−30

−35

0

2

4

6

8

10 subcarrier k

12

14

16

18

20

Fig. 4. MSE (dB) vs subcarrier k at SNR = 10 dB for Ns = 1000 symbols per subcarrier.

Multiuser Detection and Channel Estimation in MIMO OFDM Systems

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MSE vs k, Ns = 5000, SNR = 10 and 20 dB 10 BSS−only BSS−MMSE 5

0

MSE(dB)

−5

−10 10dB −15

−20

−25 20dB −30

−35

0

2

4

6

8

10 subcarrier k

12

14

16

18

20

Fig. 5. MSE (dB) vs subcarrier k at SNR = 10 and 20 dB for Ns = 5000 symbols per subcarrier. MSE vs k, Ns = 10000, SNR = 10 and 20 dB 10 BSS−only BSS−MMSE 5

0

MSE(dB)

−5

−10 10dB −15

−20

−25 20dB −30

−35

0

2

4

6

8

10 subcarrier k

12

14

16

18

20

Fig. 6. MSE (dB) vs subcarrier k at SNR = 10 and 20 dB for Ns = 10000 symbols per subcarrier.

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Luciano Sarperi, Asoke K. Nandi, and Xu Zhu

The simulations show that the performance improvement of the BSS-only method is significant, however, the computational requirements are higher. Depending on the source signals and the channels, the choice of a particular BSS method may play an important role for the successful separation in both approaches.

5

Conclusions

The BSS-only multiuser detection and channel estimation method proposed in this paper has been found to clearly outperform the BSS-MMSE method in [1]. The BSS-only method obtains a nearly constant performance for all subcarriers while the performance of the BSS-MMSE method degrades progressively for increasing subcarrier numbers. Future work will focus on reducing the complexity of our BSS based receiver.

References 1. D. Iglesia, A. Dapena, C. J. Escudero, “Multiuser Detection in MIMO OFDM Systems Using Blind Source Separation”, in Proc. Sixth Baiona Workshop on Signal Processing in Communications (WSPC03), Baiona, Spain, 2003, pp. 41–46. 2. Z .Wang, G. B. Giannakis, “Wireless Multicarrier Communications”, IEEE Signal Processing Magazine, Vol. 17, No. 3, May 2000, pp. 29–48. 3. P. Smaragdis, “Blind Separation of Convolved Mixtures in the Frequency Domain”, Neurocomputing, Vol. 22, No. 1–3, November 1998, pp. 21–34. 4. J. F. Cardoso, A. Souloumiac, “Blind Beamforming for Non-Gaussian Signals”, IEE Proceedings F, Vol. 140, No. 6, December 1993, pp. 362–370. 5. H. Bolcskei, D. Gesbert, A. J. Paulraj, “On the Capacity of OFDM-Based MultiAntenna Systems”, in Proc. IEEE Int. Conf. Acoust., Speech, Sig. (ICASSP), Istanbul, Turkey, June 2000, pp. 2569-2572. 6. Z. Liu, G. B. Giannakis, “Space-Time Block Coded Multiple Access Through Frequency-Selective Fading Channels”, IEEE Transactions on Communications, Vol. 49, No. 6, June 2001, pp. 1033-1044. 7. B. Lu, G. Yue, X. Wang, “Performance Analysis and Design Optimization of LDPC-Coded MIMO OFDM Systems”, IEEE Transactions on Signal Processing, Vol. 52, No. 2, February 2004, pp. 348–361. 8. A. K. Nandi (ed.), “Blind Estimation using Higher-Order Statistics”, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1999. 9. S. Verdu, “Multiuser Detection”, Cambridge Press, Cambridge, U.K., 1998.