Blind and Semi-Blind Sparse Channel Identification in MIMO OFDM ...

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Blind and Semi-Blind Sparse Channel Identification in MIMO OFDM Systems ∗ Institut

Abdeldjalil Aïssa-El-Bey∗‡ , Dai Kimura† , Hiroyuki Seki† and Tomohiko Taniguchi† Télécom; Télécom Bretagne; UMR CNRS 3192 Lab-STICC, Technopôle Brest Iroise CS 83818, 29238 Brest, France Laboratories Ltd., YRP R&D Center, 5-5, Hikari-no-Oka, Yokosuka-Shi, Kanagawa 239-0847, Japan ‡ Université européenne de Bretagne, France

† Fujitsu

Abstract—In this paper, we are interested in blind and semiblind identification of multiple-input multiple-output (MIMO) channel for orthogonal frequency division multiplexing (OFDM) systems. Using the sparsity property of wireless channel impulse response, we propose an iterative method which minimizes a cost function to result from the combination of blind or semi-blind criterion and the p norm, where this norm is considered as a good sparsity measure. The simulations show that the proposed methods outperform existing techniques in terms of channel estimation error and robustness to channel order overestimation.

∇ d1 (n, k)

d2 (n, k)

OFDM Modulation OFDM Modulation

.. . dM (n, k) t

r1 (n, k)

∇

∇

s2 (n, k)

r2 (n, k)

∇

OFDM Modulation

∇

s1 (n, k)

sM (n, k) t

OFDM Demodulation OFDM Demodulation

y2 (n, k)

.. .

∇ rM (n, k) r

y1 (n, k)

OFDM Demodulation

yM (n, k) r

Fig. 1. MIMO-OFDM system model with Mt transmit and Mr receive antennas

I. I NTRODUCTION Multimedia applications in wireless communication require very high data rate transmissions. To fulfil this demand, several methods have been proposed. Among them, orthogonal frequency division multiplexing (OFDM) and multipleinput multiple-output (MIMO) have emerged as two main techniques in the Long Term Evolution (LTE) and the future fourth-generation (4G) communications [1]. In MIMO-OFDM systems, a plethora of methods have been developed for getting an accurate estimate of the channel. These methods can be classified into three classes, namely, training-based methods, blind methods [2] and semi-blind methods [3]. Training based methods rely on the periodic transmission of known symbols, entailing the reduction of the system bandwidth efficiency. On the other hand, blind methods, do not need any training symbols, at the expense of a high computational complexity and debasement of estimation quality (intrinsic indeterminations of blind channel estimation). Semi-blind methods emerged as new promising techniques which can allow significant reduction in the number of training symbols while keeping a good quality of the channel estimate. Estimation of sparse and long channels (i.e. channels with a small number of nonzero channel coefficients and a large span of delay) is considered in this paper. Such channels are encountered in many communication applications and in particular in the 3GPP norm [4]. We propose exploiting the sparse nature of the channel through the use of the p norm constraint with 0 < p ≤ 1 which is considered as a good sparsity measure [5], [6]. More precisely, We are combining the p norm with subspace blind criteria and semi-blind criteria to improve the channel estimation performance. Throughout the paper, we adopt the following notations:

⊗ Kronecker product, Element-wise division, (·)T Transpose, (·)∗ Complex conjugate, (·)H Complex conjugate transpose, vec(·) a stacking of the columns of the matrix into a vector, span(·) set of all linear combinations of the vector set. II. C HANNEL ESTIMATION TECHNIQUES In this section, we briefly review the blind channel identification method based on subspace criteria [2], [7]. We also provide an expression for the semi-blind channel estimate. A. Blind channel estimation In this section, we overview the blind channel identification method for MIMO OFDM systems presented in [2] Based on the systems illustrated in Fig. 1, we denote the data symbols before an OFDM modulation as d(n, k) dn

= [d1 (n, k), d2 (n, k), . . . , dMt (n, k)] = [d(n, 0), d(n, 1), . . . , d(n, K − 1)]

(1) (2)

where dj (n, k) is a data symbol loaded on the k th subcarrier in the nth OFDM symbol to be transmitted from the j th antenna. By collecting J consecutive OFDM symbols from Mt transmit antennas, the data symbol vector d(n) is constructed by d(n) = [dn , dn−1 , . . . , dn−J+1 ]

978-1-61284-233-2/11/$26.00 ©2011 IEEE

T


When we define the matrices F (k), F , and F associated with the IFFT as, respectively 1 j2πk/K 1, e , . . . , ej2πk(K−1)/K F (k) √ K F F (K − 1)T , . . . , F (0)T , T F (K − 1)T , . . . , F (K − Cp )T F

s(n)

= [s1 (n, k), . . . , sMt (n, k)] = s(n, K − 1), . . . , s(n, 0), s(n, K − 1), . . . , s(n, K − Cp ) =

T

[sn , sn−1 , . . . , sn−J+1 ]

we obtain the relationship given as s(n) = F d(n) we express the received signal at Mr receive antennas as r(n, k) rn

= =

[r1 (n, k), . . . , rMr (n, k)] [r(n, Q − 1), . . . , r(n, 0)]

where Q = K + Cp . By collecting J consecutively received OFDM symbols, the received signal vector r(n) is given as T (3) r(n) = rn , rn−1 , . . . , rn−J+1 [1 : (Q − L) Mr ] We assume that the discrete channels between Mt transmit antennas and Mr receive antennas are modeled as an Mr ×Mt finite impulse response filter with L as the upper bound on the orders of these channels. ⎤ ⎡ h11 (l) . . . h1Mt (l) ⎥ ⎢ .. .. H(l) = ⎣ ⎦ . . hMr 1 (l) . . . hMr Mt (l) T Let H = H(0)T , . . . , H(L)T and h = vec (H). The received signal vector r(n) in (3) can be written in a matrix form as r(n)

= I JQ (H) s(n) + w(n) =

ΠH n Ξ=0 In practice, Rrr is estimated over Nb blocks by

I J ⊗ F ⊗ I Mt

where Cp is the length of the cyclic prefix. Denote the timedomain signal vector s(n) to be transmitted after the OFDM modulation as s(n, k) sn

2 are all equal to the noise variance σw . Denote by Πn = πJKMt +1 , . . . , π(JQ−L)Mr the vectors spanning the noise subspace, where πk is k th eigenvector of the matrix Rrr . Since span(Ξ) is orthogonal to span(Π), we have the following orthogonal relationship [8]:

I JQ (H) F d(n) + w(n) Ξ d(n) + w(n)

where I JQ (H) is the (J Q − L) Mr × J Q Mt block Toeplitz matrix channel. The covariance matrix of the received signal can be expressed as: 2 I(JQ−L)Mr Rrr = E r(n)r(n)H = Ξ ΞH + σw Based on the eigenvalues of the covariance matrix, one can perform the decomposition into signal and noise subspaces. The signal subspace is spanned by eigenvectors corresponding to the largest J K Mt eigenvalues, whereas the noise subspace is spanned by the remaining eigenvalues which

N b −1 rr = 1 r(n)r(n)H R Nb n=0

by minimizing We can obtain the channel matrix estimate H 2 H C (H) = Π n Ξ 2

k with dimension Partitioning the eigenvector estimate π (JQ − L)Mr into JQ − L equal segments as given in T (k)T (k)T (k)T JQ−L 2 k = v 1 v ...v π k as described in constructing the (L + 1)Mr × JQ matrix V [2] and defining the matrix Ψ as (JQ−L)Mr

Ψ

H k IJ ⊗ F ∗ F T V V k

k=JKMt +1

Mt H we can write a cost function i=1 hi Ψhi equivalent to C (H). By imposing the constraints such as hi 2 = 1 for of the 1 ≤ i ≤ Mt to avoid trivial solutions, the estimate H channel coefficient matrix H is obtained by M t

H 2, . . . , h M = arg min 1, h = h (4) h Ψhi H t

hi 2 =1

i

i=1

B. Least-square channel estimation We assume that the channel estimation at the receiver side is conducted over Np OFDM symbols, each OFDM symbol containing Kp pilot samples (i.e pilot subcarriers). Let define the matrix Y (n) and D(n) as ⎡ ⎤ ⎤ ⎡ y(n, 0) d(n, 0) ⎢ ⎥ ⎥ ⎢ .. .. Y (n) = ⎣ D(n) = ⎣ ⎦, ⎦ . . y(n, Kp − 1)

d(n, Kp − 1)

where y(n, k) is define in the same manner that d(n, k) by collecting the pilot symbol after OFDM demodulation, and let define Φ = diag e−j2πk/Kp k=0,...,Kp −1 . Then, received signals can be expressed in a simple form: Y (n) =

L

+ W (n) Φl D(n)H(l)

l=0

where the matrix H(l) contains channel impulse response = H(l)T . For purposes of channel coefficients, such as H(l) estimation, it is convenient to express the received signal as: + W (n) Y (n) = Δ(n)H


T = T T , . . . , H(L) where H and Δ(n) = H(0) L D(n), ΦD(n), . . . , Φ D(n) . By collecting the Np pilot OFDM symbols, we can write + W, Y = ΔH where

⎡

⎢ ⎢ Y =⎢ ⎣

Y (0) Y (1) .. .

⎤

⎡

⎥ ⎥ ⎥, ⎦

⎢ ⎢ Δ=⎢ ⎣

⎤

Δ(0) Δ(1) .. .

⎥ ⎥ ⎥ ⎦

Δ(Np − 1) Y (Np − 1) = vec H and y = vec (Y ). The least-square estimate Let h minimizes the following criterion: 2 2 (5) min y − IMr ⊗ Δh = min y − Δh h

h

2

2

= (IM ⊗ Δ) E and E is the permutation matrix where Δ r that transforms h into h. C. Semi-blind channel estimation To solve the intrinsic indetermination of blind channel estimation techniques and improve their performance, [3] to combine linearly the training sequence criterion (5) with the blind criterion (4), thus leading to the following cost function: 2 (6) C (h) = y − Δh + α hH Ψh 2

= IM ⊗ Ψ and α is a regularizing constant. The where Ψ r semi-blind estimator as given in [3] by: −1 = Δ + αΨ H y HΔ (7) Δ h It was shown in [3] that the choice of the regularizing constant has a great impact on the channel estimation error. As the weight for a weighted least-square (WLS) minimization problem can usually be determined according to the variance of the individual estimation error [9]. III. S PARSE C HANNEL ESTIMATION TECHNIQUES In this section, we introduce a sparse channel estimation techniques based on the previous blind and semi-blind subspace algorithm. A. Blind sparse channel estimation In order to exploit a priori information on the sparsity of the channel impulse response [10], [11], we introduce an additional cost function based on the Generalized Gaussian Distribution (GGD) model of channel h. The GGD model, can be mathematically represented under the assumption that all the component of h are i.i.d, as follows : ⎞−Mr Mt (L+1) ⎛ hpp p ⎠ ⎝ (8) exp − p g(h) = β 2β Γ 1 p

where β > 0 is a scale parameter, 0 < p ≤ 1 and ∞ Γ(z) = 0 tz−1 e−t dt, z > 0, is the Gamma function. This

prior distribution gives more weight to values that are close to zero, thus encouraging the model to set many variables to (or close to) zero. This makes it ideal for learning sparse representations. The combination of equation (4) and the log of equation (8) leads to the following objective function : h + λ hp J (h) = hH Ψ p

(9)

where λ is a weighting parameter which controls the tradeoff between approximation error and sparsity. The first term is the subspace criterion and the second term is the penalty term, which minimizes the p norm of the channel impulse response h. Therefor, the desired solution of h is determined by minimizing the cost function J (h) under the unit norm constraint h2 = 1 : ! = arg min hH Ψ h + λ hp h p h2 =1

Direct minimization is computationally intensive and may be even intractable when the channel impulse responses are long and the number of channels is large. Therefore, we propose to use a gradient technique to solve this minimization problem efficiently. Unfortunately, the criterion (9) will be non-convex if 0 ≤ p < 1. Thus, the gradient convergence will be ensure only for p = 1. The gradient solution is compute iteratively by : k − μ∇J (hk ) , k+1 = h (10) h where μ is a small positive step size and ∇ is a gradient operator. The gradient of J (h) is given by : ∂J (h) + λ sign (h) , = 2 Ψh ∂h The unit norm constraint is to ensure that the iterative algorithm do not converge to a trivial solution with all zero elements. Therefore, the update equation is given by : k + λ sign h k − μ 2 Ψ k h h k+1 = . (11) h k + λ sign h k h hk − μ 2 Ψ ∇J (h) =

2

1) Optimal step size: In order to avoid divergence, a conservatively small μ is usually used, which inevitably sacrifices the convergence speed of the recursive algorithm. In this section, we will derive an optimal step size for the gradient optimization method and hence propose a variable step size recursive algorithm. To find an optimal step size μ for each iteration we propose to use a line search method. More precisely, we choose a line search, in which μ is chosen to minimize J ! (12) μopt = arg min J h − μ ∇J (h) . μ

The criterion in the (k + 1)th iteration is written as follow : J (hk+1 ) = hH k+1 Ψ hk+1 + λ hk+1 1

(13)

by replacing hk+1 by (10) we rewrite equation (13) as : (hk − μ∇J (hk )) J (hk+1 ) = (hk − μ∇J (hk )) Ψ +λ hk − μ∇J (hk )1 H


− λ hH ) − 2hH Ψ ∇J (hk ) , G(μ) = μ(2∇J (hk )H Ψ k k

where hk = sign (hk − μ∇J (hk ))). Therefore, the optimal step size in each iteration is obtained in the form : μk = μk−1 − G(μk−1 )

μk−1 − μk−2 , G(μk−1 ) − G(μk−2 )

where we use an approximate Newton approach for solving equation (12). 2) Reweighted 1 criterion: Motivated by reweighted in compressive sampling [12], we propose a heuristic approach to reinforce the zero attractor based on reweighted 1 penalty term. The new proposed cost function is given by: h+λ J (h) = hH Ψ

Mr Mt (L+1)

log (|h(i)| + ε)

(14)

i=1

The log-sum penalty has been introduced as it behaves more similarly to the 0 norm the 1 penalty cost function [12]. Thus, the updating equation (11) becomes: k | + ε1 k + λ sign h k − μ 2 Ψ k |h h h k+1 = . h k | + ε1 k + λ sign h k |h h hk − μ 2 Ψ 2

A. Blind channel estimation In blind channel estimation context, the QPSK modulation is used and a channel modeled by a 33-tap MIMO FIR filter is assumed, in which each tap corresponds to a 2 × 2 random matrix whose elements are independent identically distributed (i.i.d.) with Bernoulli-Gaussian distribution [5] : 1 exp −h2i /2σi2 f (hi ) = pi δ(hi ) + (1 − pi ) " 2 2πσi generated by the MATLAB function SPRANDN. We used the parameters pi = 0.1 and σi = 1. In figure 2, the Blind MIMO: Mt=2, Mr=2, N=2000, Sparse channel pi=0.1, L=100 0 Blind subspace Blind subspace l1 Blind subspace reweighted l1

−10

Normalized Mean Square Error (dB)

we take a derivative of J (hk+1 ) with respect to μ, ∂J (hk+1 ) = G(μ) such as : ∂μ

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B. Semi-blind sparse channel estimation Fig. 2.

In the same way that presented in section II-C, we define a 1 -sparse semi-blind criterion by combining the linearly the training sequence criterion (5) with the 1 -sparse blind criterion (9) (for p = 1), thus leading to the following cost function: 2 + λ h1 (15) Cs (h) = y − Δh + α hH Ψh 2

The 1 -sparse semi-blind estimator can be found iteratively by applying gradient optimization on the criterion (15). Therefore, the update equation is given by : k −μ Δ k + λ sign h k − y + α Ψ k+1 = h k h h H Δ h Also, by combining the linearly the training sequence criterion (5) with the reweighted 1 -sparse blind criterion (14), we obtain a new update equation such as: k − μ Δ k k+1 = h k − y + α Ψ h h H Δ h k | + ε1 k |h +λ sign h IV. S IMULATIONS We consider a MIMO-OFDM system with Mt = 2 transmit and Mr = 2 receive antennas. The number of subcarriers is set to K = 512 and the length of cyclic prefix is CP = 36.

Blind MIMO OFDM Channel Estimation Performance

normalized mean-square error is plotted versus the SNR for the proposed 1 -sparse and reweighted 1 -sparse algorithms and subspace algorithm [2] for N = 2000 OFDM symbols. It is clearly shown that our algorithms perform better in terms of the normalized mean-square error especially for moderate and high SNR. Also, we have observed that the proposed algorithms are robust against the channel order overestimtion errors contrary to subspace algorithm. Unfortunately, we have omited to add the simulation results due to lack of space. B. Semi-blind channel estimation In semi-blind channel estimation context, the QPSK modulation and 3GPP ETU channel model are used [4]. The symbol pilot mapping is used as described for LTE norm in [13] with pilot frequency rate 16 and pilot time rate 27 . In figure 3, the NMSE is plotted versus the SNR for the proposed semiblind 1 -sparse and reweighted 1 -sparse algorithms and semiblind subspace algorithm for N = 2000 OFDM symbols. It is clearly shown that our algorithms outperform the subspace algorithm in term of NMSE. Figures 4 and 5 compare the performance in term of NMSE of the proposed semi-blind 1 sparse and reweighted 1 -sparse algorithms with semi-blind subspace algorithm. The NMSE is plotted versus the pilot time rate (symbol pilot density) for N = 14 OFDM symbols and SNR=10dB and SNR=30dB respectively. Note that, N = 14


Semi−Blind MIMO: Mt=2, Mr=2, 3GPP ETU channel, LTE pilot symbol, N=2000

Semi−Blind MIMO: Mt=2, Mr=2, 3GPP ETU channel, N=14, SNR=30dB −34

Least Square Solution Semi−blind subspace Semi−blind subspace l1 Semi−blind subspace reweighted l1

−30

Semi−blind subspace Semi−blind subspace l1 Semi−blind subspace reweighted l1

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Fig. 3.

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Semi-Blind MIMO OFDM Channel Estimation Performance

Fig. 5. Semi-Blind MIMO OFDM Channel Estimation Performance: NMSE versus Pilot Time Rate for N = 14 and SNR = 30dB

Semi−Blind MIMO: Mt=2, Mr=2, 3GPP ETU channel, N=14, SNR=10dB −14

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overestimation errors in blind channel estimation context and reduce pilot rate while keeping channel estimation quality in semi-blind channel estimation context.

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Semi−blind subspace Semi−blind subspace l1 Semi−blind subspace reweighted l1


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Fig. 4. Semi-Blind MIMO OFDM Channel Estimation Performance: NMSE versus Pilot Time Rate for N = 14 and SNR = 10dB

corresponds to 1 packet size (1ms) in LTE downlink system. We also observe in this figures that the proposed algorithms outperform the semi-blind subspace algorithm in term of NMSE. In other words, we can reduce the pilot rate while keeping channel estimation quality by using the proposed sparse algorithms. V. C ONCLUSION This paper deals with a simple and efficient algorithms for blind and semi-blind channel estimation in MIMO OFDM systems. In the proposed methods, we use a channel sparsity measure together with the blind and semi-blind criteria to improve the estimation quality and to take into account the sparsity of the channel. A gradient type technique with optimized step size has been considered for the optimization of the proposed cost functions. Besides its improved performance, the new sparse methods are robust against channel order

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