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A Comparison of Blind Channel Estimation schemes for OFDM in Fading Channels Chin Keong Ho1 , B. Farhang-Boroujeny1 and Francois Chin2 1 Department of Electrical Engineering, National University of Singapore,Singapore 119260 Tel: (++65)874 6682, Fax: (++65)779 1103, e-mail:{engp9451, elefarhg}@nus.edu.sg 2 Centre for Wireless Communications, 20 Science Park Road, #02-34/37 Tele Tech Park Science Park II, Singapore 117674, e-mail:
[email protected] Abstract— In this paper, we bring together three recently proposed blind detection schemes which are specific to orthogonal frequency division multiplexing (OFDM) systems. Our aim is to study the performance of these schemes in fading channels and thus give a critical comparison of them in the environment where they are most expected to be used. We also propose a modification to one of the methods to improve upon its performance. The proposed new algorithm is found to be outperforming the other two algorithms both with respect to convergence rate and achievable mean square error.
I. Introduction Training symbols are commonly used in wireless communications for channel identification so as to mitigate channel distortion. The use of blind channel equalization schemes appears attractive since it avoids the use of training symbols which consumes valuable channel capacity. Recent works have focused on using second-order moment techniques [1] based on oversampling or spatial diversity which offers faster convergence and lower complexity as compared to techniques exploiting higher order statistics [2]. Orthogonal frequency division multiplexing (OFDM) [3] [4] has generated considerable interest as a highly suitable technique for high-bit-rate transmission. It have been adopted as the European standard for digital audio broadcasting (DAB) [5] and digital video broadcasting for terrestrial systems (DVB-T) [6]. In OFDM systems the problem of channel distortion is taken care of by repeating the last few samples of each OFDM symbol at its beginning, prior to its transmission. This guard interval added is known as the cyclic prefix. The length of the cyclic prefix should be chosen greater than or equal to the duration of the channel impulse response. This allows equalization of the channel distortion in the frequency domain by using a single tap equalizer for each carrier independently. The addition of the cyclic prefix in OFDM symbols creates some redundancy which may be exploited for blind channel identification. This can be carried out without resorting to over-sampling by using the sample-rate correlation information of the received signal at the receiver input. Channel identifications of this nature have only recently been reported [7]-[11]. Our aim in this paper is multifold. We bring together the
three algorithms proposed in [7]-[9] and present a critical comparison of them when applied to identifying a class of fading channels. Comparisons are made in terms of computational complexity and also tracking behaviour. We also observe some of the potential problems of the Cholesky decomposition algorithm of [7] and propose solutions to that. Moreover, the use of similar notations to formulate the three algorithms facilitates further study in this area. In section II, we briefly described the channel model that is used for our study and consider a block modeling of an OFDM system. Using such a model allows us to observe that channel information is isolated in part of the autocorrelation matrix of the received OFDM symbol. Hence, as shown in the section, our problem is reduced to how to extract the channel coefficients appropriately. In section III, we show how the block system model allows us to construct an over-determined system that is suitable for applying subspace methods for channel estimation. In section IV, making explicitly the use of the cyclostationary property of the OFDM symbol, we investigate how blind channel estimation is carried out. The simulations results are shown for time varying Rayleigh channels in section V. For tracking purposes, a forgetting factor is used to update a set of matrix estimations. In our paper, we use the following for the notations: mae trices are written as C, vectors as c, sub-matrices as C, th sub-vectors as e c, the (i, j) element of matrix C as [C]ij . The time index for vectors and matrices, where applicable, is dropped when there is no ambiguity. We use the superscript T , H and ∗ to denote the operation matrix transpose, Hermitian and conjugation, respectively. II. Blind Identification based on Submatrix of Autocorrelation Matrix A. Block System Modeling We define the elementary period of the OFDM system to be T , the number of subcarriers to be N , number of cyclic prefix taps to be D and the total taps of 1 OFDM symbol to be P = N + D. The complex baseband representation [12] of the mobile wireless channel impulse response can be described by c(t, τ ) =
X k
γk (t)δ(τ − τk (t))
(1)
2
where τk (t) and γk (t) are the delay and complex amplitude of the k th path, respectively. We assume an exponential power delay profile with normalized root mean square delay spread τrms /T and normalized maximum spread delay τmax /T . Due to the mobility of the receiver, an maximum Doppler spread of fD will occur in the received signal. Hence, the normalized maximum Doppler spread is fD Ts by assuming that the channel coefficients are time invariant over each OFDM symbol period Ts = P T . Let the discrete channel model of order L(< D) be discretized as cp (n) = c(nTs , pT ) (and similarly for other discrete signal), and represented as a D × 1 vector as follows c = [c0 (n)...cD−1 (n)]T = [c0 (n)...cL (n)0...0]T .
(2)
The input digital data stream is divided into blocks of size N . Each block at time n is represented by a N × 1 vector x(n). Let the inverse discrete Fourier Transform (IDFT) of x(n) be e s(n) = Fx(n), where F is the N × N IDFT matrix and FH the DFT matrix. The OFDM modulation consists of an IDFT and the appending of a cyclic prefix. This operation can be repree T FT ]T , where F e is the sented by a P × N matrix K = [F last D rows of F. Hence, the transmitted OFDM symbol is represented by a P × 1 vector: s(n) = Kx(n) = [xN −D (n)...xN −1 (n)xT (n)]T .
(3)
The ISI resulting from the OFDM symbol s(n − 1) will be contained within the next OFDM symbol s(n) (more accurately within the cyclic prefix of the next symbol). With AWGN represented as a noise vector b(n), the received P × 1 vector r(n) consists of the convolution of the channel c with s(n − 1) and s(n): r(n)
= [C1 C0 ]
s(n − 1) s(n)
+ b(n)
(4)
where
C0 =
c0 (n) c1 (n) .. .
0
c0 (n) .. . cP −1 (n) ... 0 cP −1 (n) .. . 0 and C1 = . . .. .. 0 ...
··· 0 .. .. . . .. . 0 c1 (n) c0 (n) ··· .. . .. . ...
c1 (n) .. .
. cP −1 (n) 0
Without losing generality, we consider the case N = 4D e 0 to be the D × D upper left for simplicity. We define C e 1 the D × D upper right submatrix of sub-matrix of C0 , C e i (n) to be D × 1 sub-vectors obtained C1 , and e si (n) and b from the (D × (i − 1) + 1)th to (D × i)th elements of e s(n) and b(n), respectively. Hence, under noiseless conditions,
r(n) can be written as
r(n) = |
e1 C 0 0 0 0
0 e0 C e1 C 0 0
0 0 e0 C e1 C 0 {z C
0 0 0 e C0 e1 C
e0 C e1 C 0 0 e0 C
e s (n − 1) 4 s1 (n) e s2 (n) e e s3 (n) e s4 (n) {z } | s
.(5) }
B. Algorithm From (5), we obtain the received signal autocorrelation matrix Rrr as below: Rrr = E[r(n)r(n)H ] = CRss CH + σb2 I = CCH + σb2 I e 0,0 C 0 0 e 0,0 C = 0 0 0 0 e C0,0 0 e 1,1 C e 0,1 C C e e C 1,0 1,1 e 1,0 C 0 0 0 e 0,1 0 C
0 0 e 0,0 C 0 0
0 e 0,1 C e 1,1 C e 1,0 C 0
0 0 0 e 0,0 C 0
0 0 e C0,1 e 1,1 C e 1,0 C
e 0,0 C 0 0 0 e C0,0
+
0 e 1,0 C 0 + σb2 I (6) e 0,1 C e 1,1 C
e i,j = C e iC e H and σ 2 as the where we define the matrix C j b noise variance. Rss = I is the autocorrelation matrix of s, assuming a white unit variance data input. Hence, alternative to [7] which is based on the use of z-transform, we show that Rrr is of the form
Rrr
× = × e 0,0 C
e 0,0 × C × × + σb2 I. × ×
(7)
We note that in (7), the last D elements of the first column is c0 (n)∗ [c0 (n)...cD−1 (n)]T . Therefore, we may obtain the channel information directly from Rrr with a complex constant ambiguity. This we call the direct based algorithm. A more complex algorithm based on a Cholesky decomposition is introduced in [7] by making use of the submatrix e 0,0 = C e 0C e H . Since C e 0 is an lower triangular matrix, we C 0 e 0,0 to obtain may perform an Cholesky decomposition on C e C0 . However, the Cholesky decomposition may fail at cere 0,0 (denoted as C b 0,0 ) tain iterations since the estimate of C b rr ). We is obtained from an estimate of Rrr (denoted as R thus propose a switch based algorithm which switches to the direct based algorithm when the Cholesky based one fails.
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C. Implementation considerations e 0,0 be C b t and C b b corLet the first matrix estimate of C responding to the top right and bottom left submatrix of b rr , respectively. R bt = C b b , which is not true in Because of (7), ideally, C general. To improve the estimation, we may obtain the b 0,0 as the average of C b t and C b b . Since new estimate C H bt = C b due to the fact that R b rr is Hermitian, this is C b b t and C b H (or C b b and equivalent to taking the average of C t H th b Cb ). The (i, j) element of the improved estimate is thus obtained as b 0,0 ] = {[C b t] + [C ij ij
b t ]∗ )}/2, [C ji
∀ i, j = 1, ..., P, i 6= j. (8)
The diagonal element is equivalent to kh(n, o)k2 . Thus it is appropriate to replace the diagonal elements by their respective absolute values: b 0,0 ] = |[C b t ] |2 (= |[C b b ] |2 ) ∀ i = 1, ..., P. [C ii ii ii
III. Blind identification based on Subspace Methods
In [8], a subspace based method is used for channel identification by trying to formulate an over-determined system of the form ¯r(n) = H¯ s(n). We let e ri (n) to be a D × 1 sub-vector obtained from the (D × (i − 1) + 1)th to (D × i)th elements of r(n).We also define the following notations: ¯ r(n) = [e rT2 (n − 1), ..., e rT5 (n − 1), e rT1 (n), ..., e rT5 (n)]T ¯ s(n) = [e sT1 (n − 1), ..., e sT4 (n − 1), e sT1 (n), ..., e sT4 (n)]T e T (n − 1), ..., b e T (n − 1), b e T (n), ..., b e T (n)]T ¯ b(n) = [b 2 5 1 5 Making use of (5), it can be shown that
¯ ¯ r(n) = H¯ s(n) + b(n). (10) C0 0N ×N where H = and C0 is the 0P ×(N −D) C (P +N )×2N N × N bottom right submatrix of C. The (2N + D) × (2N + D) autocorrelation matrix of ¯ r(n) is H
= E[¯ r(n)¯ r(n) ] = HRs¯s¯HH + σb2 I(2N +D) .
gi H = 0
for 0 ≤ i ≤ D − 1.
(11)
We assume that H is full column rank (true if the coefficients of the Fourier transform of c is non-zero), and that
(12)
b r¯r¯ is obtained by estimation. Hence, g bi In practice, the R b obtained from Rr¯r¯ does not satisfy (12) exactly. We solve (12) in the least-squares sense by minimizing the following quadratic form: q(H) =
D−1 X
||giH H||2
i=0
(9)
b 0,0 should be To perform a Cholesky decomposition, C Hermitian positive definite. This is not guaranteed in the estimation process. However, the above two procedures b 0,0 is at least Hermitian and improve the ensures that C success rate of the Cholesky decomposition. b 0,0 is obtained, assuming After the matrix estimate C that each element is independently estimated, the least square estimate of cb(0) is obtained by averaging the main b 0,0 , and b diagonal elements of C c(i), i = 1, ...D − 1, by averaging the ith diagonal elements below the main diagonal.
Rr¯r¯
Rs¯s¯ = I. The first term (and hence the signal subspace1 ) has rank 2N . Thus, the noise subspace of Rr¯r¯ has dimension D and is spanned by D vectors g0 , ..., gD−1 . gi is given by the eigenvectors corresponding to the D minimal eigenvalues (σb2 ) of Rr¯r¯. It is assumed that σb2 is small (compared to signal variance). The signal subspace of Rr¯r¯ is spanned by the columns of H and is orthogonal to the noise subspace. Thus, H is uniquely determined by
=
D−1 X
giH HHH gi .
(13)
i=0
Moreover, we can represent q(H) as a function of c explicitly such that q(c) = c
H
D−1 X i=0
g ¯i g ¯i
!
c
(14)
where g ¯i is a modified form of gi that can be obtained after simplifications; see [8]. The vector c is thus obtained as P the eigenvector corresponding to the smallest eigenvalue D−1 of i=0 g ¯i g ¯i . IV. Blind Identification based on Input Cyclostationarity
We define the time-varying correlation of sp (n) as cs (p; τ ) = E[sp (n0 )s∗p+τ (n0 )] = cs (nP + p; τ )
(15)
where the last equality is added as a result of its cyclostationarity property (that is, its autocorrelation is periodic with period P). The Fourier series expansion of cs (p; τ ) with respect to p as shown in [9] is known as the cyclic correlation: Cs (k; τ )
=
P −1 X
cs (p; τ ) exp(−j2πkp/P )/P
p=0
= σs2 N/P {δ(τ )δ(k) + [δ(τ − N ) δ(τ + N ) exp(−j2πkN/P )]f (k)} (16) where f (k) = exp(−jπk(L−1)/P ) sin(πkL/P )/ sin(πk/P ). The z transform of Cs (k; τ ) with respect to τ is known as the cyclic spectrum Ss (k; z). For rp (n), its cyclic correlation Cr (k; τ ) and cyclic spectrum Sr (k; z) are defined 1 [13] provides more discussion on the topic of noise and signal subspace
4
similarly. Cr (k; τ ) is found to be :
=
l,q=−∞ ∞ X
c(l)e−j2πkl/P
∞ X
l=−∞
q=−∞
0 X
N X
=
c(−l)ej2πkl/P
l=−D
Cs (k; q)c∗ (τ + l − q) Cs (k; q)c∗ (τ − l − q)
q=−N
= c(−τ )ej2πkτ /P ⊗ Cs (k; τ ) ⊗ c∗ (τ ) ↔ Sr (k; z) = H(ej2πk/P z −1 )H ∗ (z ∗ )Ss (k; z). (17) where ⊗ and ↔ denote the operations convolution and ztransform, respectively. Thus, for any two appropriate cycles k1 and k2 such that f (k1 ), f (k2 ) 6= 0, and k1 6= k2 , we have the following relationship: Sr (k1 ; z)Ss (k2 ; z)H(exp(j2πk2 /P )z −1 ) = Sr (k2 ; z)Ss (k1 ; z)H(exp(j2πk1 /P )z −1 ).
(18)
The inverse z-transform of the left hand side of the equation is Cr (k1 ; τ ) ⊗ Cs (k2 ; τ ) ⊗ c(−τ )ej2πk2 τ /P , which in matrix notation, can three be expressed as the multiplication of Cr (k1 ; −N − L) 0 .. .. . . matrices: Cr (k1 ; N + L) . . . Cr (k1 ; −N − L) × .. .. . . 0 C (k ; N + L) r 1 Cs (k2 ; −N ) 0 e−j2πk2 L/P cL (n) .. .. .. . . . . −j2πk l/P 2 . . C (k ; −N ) e cp (n) Cs (k2 ; N ) . s 2 .. . . .. .. . c (n) 0 0 Cs (k2 ; N ) We denote the first matrix as Trk1 , the second as Tsk2 , and the third as Dk2 h, where h = [cL (n)...c0 (n)]T and Dk = diag (exp(−j2πkL/P ), ..., 1). Using these notations, (18) can be expressed as Th = 0,
T = [Trk2 Tsk1 Dk1 − Trk1 Tsk2 Dk2 ].
(19)
By using (19) and choosing appropriate k1 , k2 , we can obtain the channel coefficients h by finding the null space of T. However, the estimation will result in some error of the matrix T. The optimum estimate for h (or c) in the least-squares sense is obtained as b opt h
0.95
c(l)c∗ (τ + l − q)Cs (k; q)e−j2πkl/P
= arg min kThk2 = arg min hH TH Th
(20)
b opt is obtained as the eigenvector which corresponds Thus, h to the minimum eigenvalue of the matrix TH T. Tski can be calculated a priori from (16). The estibr (ki ; τ ), for i = 1, 2, is required to form the matrix mate C
Probablity of success
∞ X X
Cr (k; τ ) =
1
0.9
0.85
0.8
λ= 0.99 λ= 0.97 λ= 0.94
0.75
0.7
50
100
150
200
250
300
350
400
450
500
iterations
Fig. 1. Probability of successful Cholesky decomposition (K=1000)
Trki . This can be obtained from the autocorrelation matrix Rr0 r0 (n) where r0 (n) = [r(n − 1)T r(n)T r(n + 1)T ]T . Since [Rr0 r0 (n)]P +p,P +p+τ = cr (p; τ ) for p = 0, ..., P − 1, and τ = −P, ...P , Cr (ki ; τ ) can be estimated according to the definition of cyclic correlation. Also, we only consider the case when one pair of cycles k1 and k2 is used and we set k1 = −k2 as in [9]. This, based on simulations, gives the best performance. V. Simulation For our simulations, the common system parameters are set as N = 32, D = 8, L = 7, τrms = 0.5, τmax = 8 and SNR=20dB. We assume that the channel order is matched. The 2-norm of the channel vector and its estimates are normalized to be 1. The performance index at the nth iteration is the mean square error (MSE) of the channel estimate defined as PK−1 c0,k (n)...ˆ cL,k (n)]k2 /(KL), where k=0 k[c0 (n)...cL (n)] − [ˆ K is the total number of simulations performed. In practical implementation, the autocorrelation matrix Rrr is estimated by time averaging over M OFDM symbols b rr (n) = PM −1 r(n)r(n)H /M. as follows: R n=0 The above algorithm incurs processing delay and is suitable for a channel that is at least quasi-static over M OFDM symbols. If the channel is time varying, we may use a forgetting factor to track the channel as follows: b rr (n) = λR b rr (n − 1) + (1 − λ)r(n)r(n)H . R
(21)
This form of updating the estimation will be used for our simulations whenever it is applicable. We initialize b rr (0) = I. R As mentioned earlier, the Cholesky based method may fail. Fig. 1 shows the probability (P), obtained from simulations, that the Cholesky decomposition is successfully performed at each iteration. P is small initially when only a few symbols are used for estimation. We also observed that P becomes smaller for decreasing λ, since the memory of the estimation is 1/(1 − λ) and is an indication of the range of OFDM symbols used for averaging.
5
−1
10
MSE
Direct based Switch based Cholesky based
−2
10
50
100
150
200
250 300 iterations
350
400
450
500
Fig. 2. Based on Autocorrelation of Received Symbol with λ = 0.97
Cyclostationarity based with λo = 0.995 Subspace based with λ = 0.98 o Switch based with λ = 0.97 o
−1
MSE
10
a eigenvector corresponding to the smallest eigenvalue of a D × D matrix. In addition, the subspace based algorithm also needs to extract D minimal eigenvalues from a (P + N ) × (P + N ) matrix Rr¯r¯. In general, the complexity of each process is dominated by the size of the matrix. Hence, the subspace based algorithm is the most complex, followed by the Cholesky based and the cyclostationarity based algorithm. In terms of convergence, from Fig. 3 we see that the subspace and switch based algorithms have the same rate of convergence; however, the former one requires more than 2N iterations to obtain a full column rank estimation matrix before the MSE starts to decrease. The cyclostationarity based algorithm appears to have a much slower convergence. VI. Conclusion In this paper, we introduced a switch based algorithm with some modifications to [7]. The various algorithms were presented and their tracking ability were investigated. The simulations results, complexity and convergence behaviour were briefly discussed. We found that the switch based scheme performs the best in time variant Rayleigh fading channels while keeping up a moderate complexity. References
−2
10
0
50
100
150
200 iterations
250
300
350
400
Fig. 3. Comparison of MSE at SNR=20dB for various schemes
Fig. 2 shows the MSE of various algorithms introduced in section II for λ = 0.97. The MSE for the Cholesky based algorithm is obtained by performing ensemble average over MSE only when Cholesky decomposition is successful. Hence, due to practical implementations, we need to use the switch based algorithm for comparison purposes. In our simulations, the best MSE performance for the switch based algorithm is given when λ = 0.97. Fig. 3 compares the three algorithms with their respective optimum forgetting factor λo . For the cyclostationarity based scheme the best performance is given for k1 = 2 = −k2 . It is seen that the switch based algorithm performs better than the other schemes when a simple approach of using a forgetting factor is used for tracking a channel. Next, we consider briefly how complex each algorithm is. Under the most computationally demanding situation, the switch based algorithm requires a Cholesky decomposition of a D × D matrix to be performed for every iteration, since the direct based algorithm requires negligible complexity. Both the other two algorithms require extracting
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