A SUBSPACE BASED BLIND AND SEMI-BLIND CHANNEL IDENTIFICATION METHOD FOR OFDM SYSTEMS Bertrand Muquet1 , Marc de Courville1 , Pierre Duhamel2 and V´eronique Buzenac1 1 Centre
de Recherche Motorola Paris, Espace Technologique Saint-Aubin 91193 Gif-sur-Yvette France - tel: +33 (0)1 69 35 25 18 nationale Sup´erieure des T´el´ecommunications, 46 rue Barrault 75013 Paris France - tel: +33 (0)1 45 81 73 65
2 Ecole ´
ABSTRACT A new subspace method performing the blind and semi-blind identification of the transmission channel suited to multicarrier systems with cyclic prefix (OFDM) is proposed in this paper. This technique has the advantage to preserve the classical OFDM emitter structure based on a cyclic prefix insertion. Therefore it applies to all existing standardized multicarrier systems (DAB, ADSL, etc.) and does not prevent the use of the classical very simple equalization scheme. Moreover the method detailed provides an unbiased channel estimation and is robust to channel order overdetermination provided that the channel frequency response has no zeroes located on a subcarrier. 1. INTRODUCTION Though unnoticed for some time, there has recently been a large increase in interest towards Orthogonal Frequency Division Multiplexing (OFDM) modulation schemes [1] for digital broadcasting (Terrestrial Digital Audio, Video Broadcasting: DAB, DVB) but also for high speed modems over twisted pairs (Digital Subscriber Line: xDSL) and newly for mobile wireless broadband systems (ETSI BRAN, IEEE802.11a and MMAC). In parallel, numerous identification and equalization methods operating in a blind context (i.e. without requiring training symbols) have been proposed in the literature (cf. [2] and the references therein). In this paper, a new subspace [3] algorithm able to perform the identification of the transmission channel for OFDM systems in a blind and semi-blind context is detailed. Under the assumption of a noiseless received signal, this method, as any other subspace method, is theoretically able to provide in a finite number of iteration a perfect channel estimation. In the noisy case, the estimator is granted to be unbiased. Moreover, the method is naturally robust to channel order overestimation and is able to identify any channel whose frequency response has no zeroes located on a subcarrier. Basically subspace methods need to consider a number of received signal observations larger than the number of emitted symbols. Solving the resulting overdetermined linear system provides the channel coefficients. Producing more outputs than the corresponding number of transmitted symbols can be achieved using various mechanisms. One way is to introduce a time or spatial diversity at the receiver [3] but this approach prevents the identification of a particular class of channels [4]. Another method consists in artificially introducing overdetermination at the emitter, for example by inserting zeroes between blocks of symbols to
transmit (Trailing Zeroes: TZ precoder) [5] or by duplicating them [6]. However these modifications are not compatible with most of existing multicarrier systems. Here we propose to exploit the inherently induced cyclostationarity in OFDM transmitters by the cyclic prefix insertion in order to obtain an overdetermined system without modifying the emitter structure (as opposed to the TZ precoder). This has the advantage of preserving the small arithmetical complexity OFDM equalization scheme and grants that the method can be applied to all existing multicarrier systems. By construction, the cyclic prefix has a length smaller than the block size (corresponds typically to 25% of overhead) and allows the use of a fractional oversampling rate as opposed to what is proposed in [6] which in comparison decreases the overall receiver/emitter complexity. 2. NOTATIONS AND DESCRIPTION OF THE OFDM SYSTEM
denoting transposiIn the following, let T be the operator tion, conjugation and H T . As illustrated in figure 2, a multicarrier system first modulates the size N input digital vector S k using an orthogonal matrix F (which is classically the Inverse FNH of the Discrete Fourier Transform: DFT matrix FN ) and then a cyclic prefix (a.k.a. as Guard Interval: GI) of length D is inserted between each time domain block vector s k generated producing the vector sgi n whose components are finally sent sequentially through the channel. The total number of time domain samples to be transmitted per block is thus P N D. The effects of the channel are modeled by a linear filtering c and the addition of a white noise bn of variance σb 2 . In the following, the channel order L and the number of carriers N are assumed to verify L D and 2D N. Note that any OFDM systems is designed to verify these conditions. For the sake of simplicity, only the case N 4D (which is the classical value used in DAB systems) is treated here but the results presented in this paper can be extended straight to any value of N verifying the constraint N 2D (even if N D is not an integer). Be c : c0 channel im cN 1 c0 cL 0 0 the F c pulse response and C 0 N 0 C N 1 cN 1 its frequency domain coefficients. Define G FgiT F T T as the P N matrix corresponding to the combined modulation and the Guard Interval insertion where the D N matrix F gi stands for the last D rows of F. Finally, denoting by H0 and H the P P Toeplitz filtering 1 T matrices with first column c0 cP 1 , first row c0 0 0
T for H0 and first column 0 0 , first row 0 cP 1 H1 , the expression of the block received signal is [7]:
rgi n
H0 GS n H0 sgi n
H1 GS n 1 b n H1 sgi n 1 b n
c1
Rss H σb 2 I " 2N # D $ . In what follows, the autocorrelation matrix Rss is assumed to be full rank (H1) and the matrix is assumed to be full column rank (H2). Under H1,H2, the left part of Rr¯ r¯ has rank 8D. Therefore its noise subspace has dimension D and is spanned by a set of D vec tors denoted by G0 GD 1 . A particular set of these vectors can be found by noticing that a basis for the noise subspace is provided by the eigenvectors associated to the D minimal eigenvalues σb 2 of the complete autocorrelation matrix Rr¯ r¯ . Moreover the signal subspace of the left part of Rr¯ r¯ is spanned by the columns of and is orthogonal to the noise subspace. Hence it can be shown under assumption H2 that is uniquely determined by the D linear following systems :
for
(1)
3. A SUBSPACE METHOD FOR OFDM SYSTEMS In general, subspace methods rely on a formulation of the input-output relation of the form: r n Hs n where H is a vertical rectangular and full-column rank matrix. However the inputoutput relationship (1) of a classical OFDM system has not exactly the desired structure due to the presence of InterBloc Interference (IBI). One solution to avoid this problem is to use a particular precoder in order to remove the IBI term [5]. Another solution is to modify the modulator in order to introduce a controlled ISI which can easily be removed by block manipulations [6]. We propose here a subspace method that can be directly applied to classical OFDM systems. This method is detailed below. Since N 4D, the channel input and output signal vectors can be split into five blocks of equal length D:rgi n rT1 n rT2 n T gi T T T T T T T T ands r3 n r4 n r5 n n s0 n s1 n s2 n s3 n s4 n and since the block s0 n corresponds to the cyclic prefix adjunc tion, the following relation holds : s0 n s4 n . Intuitively, it is this portion of redundancy introduced at the input of the channel that is used for obtaining an overdetermined system. Denoting by C0 and C1 the D D H0 upper left and H1 upper right submatrices respectively, since L D and N 4D, the block received signal can now be expressed as
C1 0 0 0 0
gi
r
n
0 C0 C1 0 0
0 0 C0 C1 0
0 0 0 C0 C1
C0 C1 0 0 C0
s4 n 1 s1 n s2 n s3 n s4 n
bn
T r5 n T s4 n T b5 n n verifies: r¯ n
rT2 n 1 sT1 n 1 bT2 n 1
T
T 1 rT1 n r5 n T T T 1 s1 n s n 4 T T 1 bT1 n b5 n
i D 1
(2)
In practice Rr¯ r¯ is estimated by an averaging over time of the form 1 ¯ n r¯ n H which is equivalent in a noiseless conRˆ r¯ r¯ 1 K ∑K n% 0 r text to compute the following summation :
T
Rˆ r¯ r¯
1 K
K 1
∑snsnH
H
n% 0
Hence in the absence of noise, the orthogonality relations (2) still hold as long as the estimated modulated source autocorrelation matrix is full rank. In that case a perfect identification in a finite number of samples is granted. In the noisy case, the orthogonality is not preserved and therefore the linear system (2) has to be solved in the mean square sense which is equivalent to minimizing the following quadratic form
An overdetermined system can be obtained by expressing the the size 2N D 1 output vector r¯ n as a function of the size 2N 1 input and noise vectors s¯ n and b¯ n defined by : r¯ n s¯ n b¯ n
Gi H 0 for 0
q c :
D 1
∑
i% 0 &
Gi H
2 &
It is worth noticing that this quadratic form can be expressed (in a similar way as it is done in [3]) as a function of the channel vector c instead of the matrix : q c cH Qc where Q is the q D D matrix. Based on this observation, c can thus be obtained (up to a scalar coefficient) as the eigenvector associated to the minimum eigenvalue of Q. 4. SEMI-BLIND IMPLEMENTATION
is the
An inherent problem to blind methods is their rather slow converging rate [2]. This drawback often prevents their use in a practical context where methods based on training sequence are preferred. However it is possible to merge the advantages of both approaches operating in a semi-blind context [8]. The idea is to refine the pilot based estimation along the frame blindly which allows a better tracking of the channel variations. The proposed identification method can easily be used in a semi-blind context in order to exploit the pilot symbols that are usually specified in existing (and future) standard for facilitating the synchronization and channel identification. The corresponding algorithm is detailed below:
We have proved that is full column rank if and only if the chan nel frequency domain coefficients C i 0 i N 1 are all non zeros (i.e. if there is no huge fading located on a subcarrier). The autocorrelation matrix of r¯ n is Rr¯ r¯ : E r¯ n r¯ n H !
1. derive an “pilot-based” estimation of the channel coefficients: cˆ ref from the OFDM reference symbols
Indeed r¯ 2N D 2N following matrix :
C0 C1 0 0 0 0 0 0 0
0 C0 C1 0 0 0 0 0 0
0 0 C0 C1 0 0 0 0 0
C1 0 0 C0 C1 0 0 0 0
s¯ n 0 0 0 0 0 C0 C1 0 0
b¯ n where 0 0 0 0 0 0 C0 C1 0
0 0 0 0 0 0 0 C0 C1
0 0 0 0 C0 C1 0 0 C0
2. generate a “pilot-based” estimation of the structured matrix : ref from the channel estimation obtained and then
build an estimation of the matrix Rr¯ r¯ : '
ref
Rss
ref H
3. refine iteratively using a forgetting factor λ the autocorrelation matrix estimation each time a new OFDM symbol is received : "
N# Rˆ rr
1$
" N$
: λRˆ rr
1 λ rgi N 1 rgi N 1
H
"
0$ with Rˆ rr : Rˆ ref r¯ r¯
4. perform the subspace algorithm on the semi-blind estima" N$ tion of Rr¯ r¯ : Rˆ rr . It is important to notice that the noise term appearing in the real autocorrelation matrix Rr¯ r¯ ( Rss H σb 2 I is not taken into account in step 2 since it does not change the signal and noise subspaces of Rr¯ r¯ . Thus our semi-blind algorithm does not require an estimation of the noise variance. As highlighted next section, this procedure practically enhances the channel estimation convergence rate.
Semi blind MSE for SNR =10 dB
0
10
Blind estimation Semi−blind estimation
−1
10
Channel MSE
Rˆ ref r¯ r¯
6. CONCLUSIONS In this paper, a new subspace channel identification method suited to OFDM systems using a cyclic prefix has been presented. The advantage of the proposed approach is to keep the classical OFDM emitters structure which enables its use on existing systems and to preserve a very simple equalization scheme. Finally, note that it is possible to improve the subspace method proposed in this paper by increasing its channel variation tracking capabilities. Actually the channel can be reestimated more often: each time a block of D samples is received (at a rate of N D i.e without waiting for the reception of the complete next OFDM symbol). The cost for this improvement is an increase in the size of the matrix .
−2
5. SIMULATIONS This section presents some simulations illustrating the performance of the new proposed identification method. As a reference, a comparison with the subspace algorithm based on a trailing zeroes precoder described in [5] is provided. All results are obtained running Monte Carlo simulations based on 100 trials for a N 32 carrier OFDM system with a cyclic prefix of D N 4 8 samples. The symbols to be modulated by the IDFT belong to a QPSK constellation with average energy σ2S 1 and are independent and identically distributed. Figure 3 shows the frequency impulse response and the zeroes position of the channel to be estimated. Its time domain response is given by c 0 66 0 46 0 28 0 22 0 12 and its length is assumed to be D (i.e. the channel order is overestimated). The performance of the new blind algorithm ( ) ) compared to the trailing zeroes (TZ) subspace method ( * ) proposed in [5] are displayed figure 4. Two subplots are provided: the first one shows the Mean Square Error (MSE) of the channel estimation versus the number of OFDM symbols received (and used for the autocorrelation matrix estimation) and the second one plots the channel estimation MSE variations as a function of the SNR for a fixed number of iterations (the estimation of the autocorrelation matrix being based on 300 OFDM symbols). This figure underlines the robustness of the two methods towards channel order overestimation (8 instead of 5). The TZ-based subspace method abviously performs slightly better than our OFDM-based method. However this result is not surprising since a smaller matrix is processed by the TZ-based method for the SVD decomposition. For this reason full rank is reached by the TZ algorithm atfter a smaller number of iterations leading to a more accurate channel estimation. However our method can be applied on existing OFDM systems and does not rely on particular precoders. Figure 1 illustrates the improvement brought by working in a semi-blind context: the initialization period of 2N OFDM symbols required to obtain a full column rank estimation matrix is avoided since our semi-blind matrix estimation is immediately full rank.
10
−3
10
0
50
100
150 Iterations
200
250
300
Figure 1: Semi-blind improvement 7. REFERENCES [1] J.A.C. Bingham. Multicarrier Modulation for Data Transmission: An Idea Whose Time Has Come. IEEE Communications Magazine, May 1990. [2] H. Liu, G. Xu, L. Tong, and T. Kailath. Recent developments in blind channel equalization : From cyclostationnarity to subspaces. Signal Processing, 50(1-2):83–99, April 1996. [3] E. Moulines, O. AitAmrane, and Y. Grenier. The Generalized MultiDelay Filter: Structure and Convergence Analysis. IEEE Trans. on Acoustics, Speech and Signal Processing, 43(1):14–28, January 1995. [4] Zhi Ding. Characteristics of Band-Limited Channels Unidentifiable From Second-Order Cyclostationary Statistics. IEEE Signal Processing Letters, 3(5):150–152, May 1996. [5] A. Scaglione, G.B. Giannakis, and S. Barbarossa. Self-recovering multirate equalizers using redundant filterbank precoders. In IEEE International Conference on Acoustics, Speech, and Signal Processing, volume 6, pages 3501–3504, Seattle, USA, April 1998. [6] M. Tsatsanis and G.B. Giannakis. Transmitter Induced Cyclostationarity for Blind Channel Equalization. IEEE Trans. on Signal Processing, 45(7):1785–1794, July 1997. [7] B. Muquet and M. de Courville. Blind and semi-blind channel identification methods using second order statistics for OFDM systems. In IEEE International Conference on Acoustics, Speech, and Signal Processing, volume 5, pages 2745–2748, Phoenix, USA, March 1999. [8] E. de Carvalho and D. Slock. Cramer-Rao bounds for semi-blind, blind and training sequence based channel estimation. In Proceedings of the IEEE Workshop on Signal Processing Advances for Wireless Communications, Paris, France, April 1997.
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20 log |H(f)/Hmax| (dB)
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Figure 3: Channel frequency response and zeroes location
MSE for SNR =20 dB
0
MSE for300 iterations
0
10
10
Cyclic Prefix Precoder TZ Precoder
Cyclic Prefix Precoder TZ Precoder −1
10
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50
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150
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250 Iterations
300
350
400
450
500
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−5
0
Figure 4: Performance of the OFDM subspace algorithm
5 SNR (dB)
10
15
20
25
30
