Fast Array Multichannel 2D-RLS Based OFDM ...

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Fast Array Multichannel 2D-RLS Based OFDM Channel Estimator Arun Joy · Vijay Kumar Chakka, IEEE Senior member

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Abstract In this paper a Fast Array Multichannel Two-Dimensional Recursive Least Square (FAM 2D-RLS) adaptive filter is proposed for estimating an OFDM channel in frequency domain. This filter makes use of the shift structure of the input data vector. Thus the computational cost of the classical RLS filter which is O(M 2 ) is reduced to O(M ) for each iteration where M is the order of the filter. In order to ensure numerical stability in finite precision, we make use of array based methods for implementing FAM 2D-RLS. The adaptive filters illustrated in the standard literature consist of a weight vector and a scalar desired data. But in our scenario of OFDM channel estimation the weight is a matrix while the desired data is a vector. Hence the algorithm for matrix form of FAM-2D RLS and its steady state equations are derived. Numerical stability, steady state and convergence performance are verified using MATLAB simulations. Keywords Fast Array RLS · Multichannel RLS · 2D-RLS · channel estimation · array RLS

1 Introduction In the present scenario of wireless communication, OFDM is a prominent modulation technique due to its ability to provide high data rate, robustness to Inter Symbol Interference (ISI) and ease of implementation [1]. For coherent detection of data, the channel information is required at the receiver [2]. There is abundant research material available on the topic of OFDM channel estimation [3]. In [4] channel estimation based on time domain channel Arun Joy . Vijay Kumar Chakka Dhirubhai Ambani Institute of ICT Gandhinagar, Gujarat-382007 India E-mail: arun [email protected], vijaykumar [email protected]

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Arun Joy, Vijay Kumar Chakka, IEEE Senior member

statistics have been introduced. It was shown that Minimum Mean Square Error (MMSE) estimator has lesser Symbol Error Rate (SER) compared to Least Square (LS) estimator especially in the case of low Signal to Noise ratio (SNR). In [5], the performance of LS estimation is improved by performing Inverse Discrete Fourier Transform(IDFT) of the Channel Frequency Response (CFR) and then replacing all elements greater than the multipath length by zeros. This is called the significant path capture method. In [6], a robust MMSE channel estimator which is insensitive to the channel statistics have been proposed. In [7], an OFDM channel estimator for time varying channels have been introduced, where the time variation of the channel is captured by a basis expansion model. In [8], a channel estimation technique for the case of insufficient Channel Prefix (CP) is studied. Channel estimation for various OFDM pilot placement is studied in [9]. Preamble based OFDM channel estimation scheme for the use in two-way relay network is discussed in [10]. But all the methods mentioned above assume that the true channel statistics is available with the receiver at all times. But this is impractical, especially in the case of mobile devices where the channel statistics might vary with time. Thus in [11] an adaptive channel estimator for OFDM based on Normalized Least Square (NLMS) and RLS algorithms have been proposed. This filter estimates the channel by making use of the correlation of the channel coefficients across time. An adaptive channel estimation scheme based on Kalman filter is implemented in [12]. In [2] it is shown that channel coefficients are correlated across both time and frequency. So the optimum estimator in the MMSE sense would be the 2D-Wiener filter [13]. A 2D-RLS channel estimator has been implemented in [14]. In order to combat the numerical instability inherent in the classical RLS algorithm [15], an array based 2D-RLS was implemented in [16]. But the RLS algorithms of [14] and [16] have a computational cost of O(M 2 ) in the case of an M order filter. This makes the 2D-RLS filter virtually unusable for channel estimation for OFDM in the frequency domain. In order to reduce this complexity, a 2D-NLMS [17] filter could be implemented. But generally NLMS has a poorer convergence rate compared to the RLS algorithm [15]. So it might not track fast varying channels. The aim of this paper is to design a 2D adaptive filter for OFDM channel estimation which has similar complexity as 2D-NLMS i.e. O(M ), convergence property of the classical 2D-RLS and having numerical stability in finite precision. In [15], it is shown that by making use of the shift structure of the input data vector, the computational cost of classical RLS is reduced while providing the same convergence rate. In this paper we show the similarity between 2D adaptive estimation and multichannel adaptive filter. Thus fast 2D-RLS adaptive filter could be used for OFDM channel estimation. The performance is evaluated using MATLAB simulations. Notations: Bold face capital letters are used for matrices and bold face lower case letters are used to represent vectors. Scalars are represented by normal font lower case letters. Conjugate transpose is represented by (.)H and (.)T is used to represent transpose operation. The expectation operator, trace and Euclidean norm are represented by E[·], tr{·}, k · k respectively.

Fast Array Multichannel 2D-RLS Based OFDM Channel Estimator

3

2 2D OFDM Channel Estimation Consider the length of the channel to be L. If the channel prefix CP ≥ L − 1, then ISI is eliminated. Also assume that the channel is constant for at least one OFDM symbol duration. Then the OFDM system can be represented by K parallel Gaussian channels, where K is the IFFT size for generating the OFDM symbols. If xn is the K × 1 transmit data vector, then the received data vector after performing CP removal and K-point DFT is, yn = Xn hn + wn

(1)

where Xn = diag{xn } and wn is AWGN noise in the frequency domain. The channel frequency response of dimension K × 1 is defined as, H

hn = [h(n, 0)∗ , h(n, 1)∗ , · · · , h(n, K − 1)∗ ]

(2)

During the initial phase of data transmission(training phase), OFDM symbols xpn called preambles which are known at the receiver is sent. Then the LS estimate of the channel frequency response is, ¯ n = (Xpn )−1 yn h

(3)

Since the channel is assumed to be slowly varying, the OFDM channel estimator can be assumed to work based on the Decision Directed (DD) principle introduced in [18]. The advantage of using DD estimator is that, the detected OFDM symbols can be used as preamble for estimating the channel frequency response. If x ¯n+1 is the LS estimate of the (n + 1)th transmitted data, then the DD-OFDM channel estimator works as follows, ˆ −1 x ¯n+1 = H n yn+1

(4)

ˆ ˆ ˆ −1 where H n = diag{hn }. The vector hn is the estimate of the channel coefficient obtained by performing a 2D filtering on the LS channel estimate. The vector x ¯n+1 is then passed through a decision device in order to detect the transmit data vector. Assuming that there is no error in the detected data, the LS estimate of (n + 1)th channel coefficient vector is obtained as, ¯ n+1 = X−1 yn+1 h n+1

(5)

Let u ¯ n be an N K × 1 vector that contains the LS channel estimate for the past N time samples. It is defined as,  ∗  ¯ (n, 0) · · · h ¯ ∗ (n, K − 1) · · · h ¯ ∗ (n − N + 1, 0) · · · h ¯ ∗ (n − N + 1, K − 1) H u ¯n = h (6) Assuming that hn is the true channel frequency response vector at time n of dimension K × 1, our aim is to design an adaptive filter to solve the following exponentially weighted regularized least square problem [15],[16],

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Arun Joy, Vijay Kumar Chakka, IEEE Senior member

Fig. 1 Adaptive DD channel estimator for OFDM system

" arg min ¯n W

λ

n+1

¯ n ΠW ¯ nH + W

n X

# λ

n−i

ˆ n k2 khn − h

(7)

i=0

where the estimate of the channel at time n is, ˆn = W ¯ nu h ¯n

(8)

¯ n is of dimension K ×N K. The regularization factor is Π = δIN K×N K . and W The method of selecting the regularization parameter δ and the forgetting factor λ is discussed in the later sections. It is observed in (7), that in order to design the adaptive filter, we require the true channel frequency response. The true channel frequency response acts as the reference signal to the adaptive filter. But satisfying this condition is impossible. Hence we need to obtain an estimate of the reference signal vector. During the training period of the adaptive filter the significant path capture ¯ n in order to obtain a noise reduced estimate of hn technique is performed on h ¯ n is performed. [5]. During significant path capture method, K-point DFT of h Then consider the L largest coefficients while discarding the rest. Perform a K-point DFT on the L coefficients to obtain a noise reduced estimate of hn . After the convergence of the adaptive filter, the reference signal vector can be ˆ n−1 . This is possible since the channel is assumed obtained by assuming hn ≈ h to be slowly varying. The block diagram of the adaptive DD-OFDM channel estimator is shown in Fig.1.

3 Fast Array Multichannel 2D-RLS ¯n The computational cost for estimating a single row of weight matrix W using classical algorithm based 2D-RLS [14],[15] is O((N K)2 ) where N K is the order of the filter, e.g for the case of 256 point OFDM, assuming time correlation for 3 time samples i.e. N = 3, K = 256 the cost is O(7682 ). But by making use of the shift structure of the input vector, the computational cost can be reduced to O(N K). In [19], a fast array based RLS has been used to

Fast Array Multichannel 2D-RLS Based OFDM Channel Estimator

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Fig. 2 2D OFDM Adaptive Channel Estimator as a Multichannel Adaptive Filter

design an adaptive transmultiplexer while in [20], a block based fast RLS has been used in radar imaging application. In this paper we extend the 1D fast multichannel RLS of [15],[19],[20] into 2D and apply it for the channel estimation of OFDM systems. In order to bring in the notion of shift structure, we rewrite u ¯ n as,   ¯ 0)∗ · · · h(n ¯ − N + 1, 0)∗ · · · h(n, ¯ K − 1)∗ · · · h(n ¯ − N + 1, K − 1)∗ H un = h(n, (9) and define,  ∗  ¯ (n, k), h ¯ ∗ (n − 1, k) · · · , h ¯ ∗ (n − N + 1, k) H , k = 0 · · · K − 1 (10) cn,k = h Even though un does not have a shift structure in the real sense it is seen that each subvector cn,k has a shift structure. So we can consider each cn,k as the input vector of K parallel N -tap adaptive filters. This is equivalent to stating that individual subcarrier of the OFDM symbol is provided with an N tap adaptive filter and they work parallely to estimate hn as shown in Fig.2. Then (8) is rewritten as, ˆ n = Wn un h

(11)

¯ n in accordance with where Wn is obtained by rearranging the terms in W the changes that where made in u ¯ n , to obtain un . Hence we have shown that

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Arun Joy, Vijay Kumar Chakka, IEEE Senior member

an adaptive OFDM channel estimation that makes use of the correlation of channel coefficients in frequency and time can be converted to multichannel adaptive filtering problem with time shift structure.

3.1 Derivation of the FAM 2D-RLS The derivation of the FAM 2D-RLS is based on modifying the classical RLS algorithm by making use of the shift structure of the input data. Hence we reproduce the 2D-RLS derived in [14],

gn = λ−1 Pn−1 un γ(n), (N K × 1) 1 , (1 × 1) γ(n) = 1 + λ−1 uH n Pn−1 un gn gnH Pn = Pn−1 − , (N K × N K) γ(n) ˆn , en = hn − h (K × 1) Wn = Wn−1 + en gnH ,

(K × N K)

(12) (13) (14) (15) (16)

where gn is the gain vector , γ(n) is the conversion factor and Pn is the instantaneous covariance matrix. Due to the time shift structure in un we can relate it with un−1 as ,  H ∗ H ∗ H ¯ 0)∗ , cH ¯ ¯ h(n, = n−1,0 , h(n, 1) , cn−1,1 , · · · , h(n, K − 1) , cn−1,K−1  H  ∗ H ¯ − N, 0)∗ , cH , h(n ¯ − N, 1)∗ , · · · , cH ¯ cn,0 , h(n (17) n,1 n,K−1 , h(n − N, K − 1) The gain vector is partitioned as, h iH gn = gn(0)H gn(1)H . . . gn(K−1)H , (N K × 1)

(18)

and the covariance martix is partitioned as,  (0,0) (0,K−1) Pn , · · · , Pn   .. .. ..  Pn =  . . .   (0,K−1) (K−1,K−1) Pn , · · · , Pn 

(k)

(l,k)

(19)

where each gn is of dimension N ×1 and Pn is of dimension N ×N . In order to bring in the time shift structure relation of (17) into the RLS algorithm, we will modify (12) as,

Fast Array Multichannel 2D-RLS Based OFDM Channel Estimator



(0)

gn 0 (1) gn 0 .. .

7



            = γ −1 (n)         0   (K−1)    gn 0   (0,K−1)     (0,0)  ¯ 0) Pn−1 0N ×1 Pn−1 h(n, 0N ×1 · · ·    cn−1,0  01×N 0 01×N 0       .. .. .. .. λ−1     . . . .       (K−1,K−1) ¯  P(K−1,0) 0  h(n, K − 1) P 0 N ×1 N ×1 n−1 n−1 ··· c n−1,K−1 01×N 0 01×N 0

(20)

Similarly we can write, 

0



 g(0)   n−1   0     (1)   g n−1   γ −1 (n − 1)   0 =   ..     .    0  (K−1) gn        ¯ 0) 0 01×N 0 01×N h(n, · · · (0,0) (0,K−1)    cn−1,0  0N ×1 Pn−2 0N ×1 Pn−2      . . .. −1  . .. .. .. λ    .       ¯    0 01×N 0 01×N h(n, K − 1)  · · · (0,K−1) (K−1,K−1) cn−1,K−1 0N ×1 Pn−2 0N ×1 Pn−2 substracting (21) from (20) and utilizing (17), we obtain, 

(0)

gn 0 (1) gn 0 .. .





0



   g(0)     n−1     0         (1)      g n−1  − γ −1 (n − 1)   γ −1 (n)     0      ..        0  .  (K−1)     gn   0  (K−1) 0 gn

(21)

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Arun Joy, Vijay Kumar Chakka, IEEE Senior member

 

(0,0)

∆Pn−1 .. .

··· .. .

(0,K−1)

∆Pn−1 .. .



¯ 0) h(n, cn−1,0 .. .



          (0,K−1) (K−1,K−1)  h(n, ¯ K − 1)  ∆Pn−1 · · · ∆Pn−1 cn−1,K−1   ¯ h(n, 0)  cn−1,0      .. = λ−1 ∆Pn−1   .   ¯ K − 1)   h(n, cn−1,K−1  = λ−1  

(22)

where ∆Pn−1 is of dimension K(N + 1) × K(N + 1). Hence we have derived an update equation for the gain vector that depends only on the difference between the correlation matrix at time n and n − 1, defined as ∆Pn−1 . Now the update equation of conversion factor in terms of ∆Pn−1 is obtained by manipulating (13) as,   H γ(n)−1 − γ(n − 1)−1 = λ−1 uH n Pn−1 un − un−1 Pn−2 un−1

(23)

The RHS of (23) can be expressed as, 

¯ 0) h(n, cn−1,0 .. .



      ¯ λ ∆Pn−1   n−1,0 , · · · , h(n, K − 1)   ¯ K − 1)   h(n, cn−1,K−1 (24) The regularization factor Π should be selected in a special way so that an efficient calculation of ∆Pn−1 is possible. This is because the initial value of ∆Pn−1 is defined as, −1

 ¯ 0)∗ , cH h(n,

∗

  ∆P−1

(0,0)

P−1 0N ×1 01×N 0 .. .

, cH n−1,K−1



 ···



(0,K−1)

P−1 01×N

0N ×1 0

 

      .. .. =  . .    (K−1,K−1)   P(K−1,0) 0 P−1 0N ×1  N ×1 −1 ··· 01×N 0 01×N 0       0 01×N 0 01×N ··· (0,0) (0,K−1)   0N ×1 P−2 0N ×1 P−2     . . . .. .. .. −         0 01×N 0 01×N (0,K−1) · · · (K−1,K−1) 0N ×1 P−2 0N ×1 P−2

(25)

Fast Array Multichannel 2D-RLS Based OFDM Channel Estimator

9

The inverse of (l, k)th block of Π is assigned as the value of the (l, k)th block of P−1 . It is selected as, i−1 h  1 (l,k) P−1 = Π(l,k) = · diag λ2 , λ3 , . . . , λN +1 δ

(26)

and, i−1 h  1 (l,k) = · diag λ3 , λ4 , . . . , λN +2 , P−2 = λ Π(l,k) δ where δ is the regularization parameter and is selected as [21],[22]; δ = σu2 (1 − λ) The variance of the input signal to the adaptive filter (8) is forgetting factor. Hence each (l, k)th block of ∆p−1 is,

(27)

(28) σu2

and λ is the

 λ2 (29) diag 1, 0, · · · , 0, −λN δ Thus we observe that each N × N block of P−1 is reduced to a rank-2 matrix. (l,k) It is proved in [15] that the rank-2 property of P−1 is invariant over time (this is the case of single channel fast array RLS ). Hence we claim that this property holds for our scenario of FAM 2D-RLS. Factorizing ∆P−1 as, (l,k)

∆p−1 =

¯ −1 S−1 L ¯H ∆P−1 = λL −1 ¯ −1 of dimension K(N + 1) × 2 is defined as, where L   ¯ (0) L −1  ¯ (1)   L−1   ¯ L−1 =  ..     . (K−1) ¯ L−1

(30)

(31)

and  1 0 0 0   p  . .  = ηλ  .. ..    0 0  0 λN/2 

¯ (k) L −1

(32)

is of dimension (N +1)×2. The signature matrix is defined as S−1 = diag{1, −1}. Due to the rank preserving property of ∆Pn−1 the signature matrix remains a constant across time. Then at time n we can write, ¯ n−1 SL ¯H ∆Pn−1 = L n−1

(33)

Substituting (33) in (23),(24) and writing them in the array form we obtain the pre-array and post-array [15]. Defining J = diag{1, S}, the objective is to find

10

Arun Joy, Vijay Kumar Chakka, IEEE Senior member

a J-unitary matrix Θn that annihilates the last two elements in the first row of the pre-array(LHS of 34). This could be done by making use of Householer or givens rotations [23]. Then gn is propagated iteratively as shown (34),[15].



γ −1/2 (n − 1)

    0    (0) −1/2 (n − 1)    gn−1 γ    ..    .    0  (K−1) −1/2 gn−1 γ (n − 1) 

   ∗ H ¯ 0)∗ , cH ¯ ¯ h(n, n−1,0 . . . h(n, K − 1) cn−1,K−1 Ln−1     (0)   ¯ L  n−1  Θn  ¯ (1)    Ln−1     ..       .  (K−1) ¯ L n−1

γ −1/2 (n)

01×2



       (0) −1/2   (0) gn γ (n)   ¯ L   n−1  ..  ¯ (1)   =   .  √   L n−1       λ .. 0       (K−1) −1/2   .   gn  γ (n) ¯ (K−1) L n−1 0

(34)

The weight vector Wn is updated as, h iH h i Wn = Wn−1 + (hn − Wn un ) gn γ −1/2 (n) γ −1/2 (n)−1

(35)

where gn γ −1/2 (n) and γ −1/2 (i)−1 are read from the post-array in (34). 4 Steady State Analysis of FAM 2D-RLS Let the optimal weight matrix in the MMSE sense be Wo . The aim of this section is to analyze the proximity of the weight matrix obtained by FAM 2D-RLS in steady state to that of Wo . The analysis is based on the energy conservation method described in [15]. But unlike in [15], where the weight is a vector and the desired data is scalar, our scenario consist of a weight matrix and the desired data is a vector. Since gn = Pn un we can rewrite (16) as, H Wn = Wn−1 + en uH n Pn

(36) o

subtracting both sides of the above equation by W , fn = W f n−1 − en uH PH W n n

(37)

multiplying both sides by un from the right, 2 eaps = eapr n n − en kun kPH n

(38)

Fast Array Multichannel 2D-RLS Based OFDM Channel Estimator

11

where eaps is the a posteriori error vector , eapr is the a priori error vector n n 2 H and kun kPH = uH P u is the weighted norm of un . Substituting for en in n n n n (37) by making use of (38), H H H H f n + eapr un Pn = W f n−1 + eaps un Pn W n n kun k2PH kun k2PH n

(39)

n

−1 Taking Frobenius norm weighted by (PH on both sides of (39), n)

2

H H u P

f n n

Wn + eapr

n 2

kun kPH n

−1 F,(PH n )

2

H H u P

f n n = Wn−1 + eaps

n 2

kun kPH n

(40)

−1 F,(PH n )

where kAk2F,K = tr{AKAH }

(41)

Expanding (40) by making use of (41), and cancelling out cross product terms by substituting eaps = Wn un and eapr = Wn−1 un we obtain, n n f n k2 H −1 + kW F,(Pn )

2 apr 2 keaps n k f n−1 k2 H −1 + ken k = k W F,(Pn ) kun k2PH kun k2PH n

(42)

n

This equation is the energy conservation relation for the case of an RLS filter having a weight matrix. In order to obtain the variance relation [15] we have to find the average of the covariance matrix PH n . This is defined in [15] as, −1 E[PH n ] = (1 − λ)Ru = P

(43)

f n k2 H −1 ] = E[kW f n−1 k2 H −1 ] E[kW F,(Pn ) F,(Pn )

(44)

Since at steady state,

(42) can be reduced to, " # " # 2 2 keaps keapr n k n k E =E kun k2PH kun k2PH n

n→∞

(45)

n

substituting for eaps from (38) into (45) and expanding we can rewrite (45) n as, n o " # 2 2 4 keapr − 2Re eapr,H en kun k2PH apr 2 n k + ken k kun kPH n k ke n n n =E E kun k2PH kun k2PH 

n

n

(46) cancelling equal terms in (46), we obtain the variance relation [15] as, E[kun k2PH ken k2 ] = 2Re{E[eapr,H en ]} n n

(47)

12

Arun Joy, Vijay Kumar Chakka, IEEE Senior member

Table 1 Computational Complexity per Iteration in terms of complex multiplication and complex addition for estimating a single row of weight matrix. Algorithm

×

+

FAM 2D-RLS 2D-RLS 2D-NLMS

6N K + 10 (N K)2 + 5N K + 2 3N K + 2

10N K + 16 (N K)2 + 3N K 3N K

Now assume a linear regressor model, hn = Wo un + vn

(48)

where vn is zero mean i.i.d random vector with covariance matrix Rv = σv2 IK×K . Also assume that vi is independent of all uj for all i, j and the initial weight matrix W−1 is independent of all {hn , vn , un }. The above assumptions lead to the following relation [15], en = eapr n + vn

(49)

Making use of (49), we can rewrite (47) as, 2 apr 2 RLS tr{Rv }E[kun k2PH ] + E[kun k2PH keapr n k ] = 2E[ken k ] = 2ξ n n

(50)

where ξ RLS is the Excess Mean Square Error (EMSE). Making use of (43) and the separtion property i.e assuming that at steady state, kun k2PH is indepenn dent of eapr [15], an expression for EMSE is obtained from (50) as, n ξ RLS =

tr{Rv }(1 − λ)N K 2 − (1 − λ)N K

(51)

when λ ≈ 1 (which is normally the case), tr{Rv }(1 − λ)N K 2 Hence the Mean Square Error (MSE) at steady state is, ξ RLS =

M SE = ξ RLS + tr{Rv }

(52)

(53)

5 Results The computational complexity of various 2D adaptive filters is shown in Table.1. The Fast array 2D-RLS algorithm has a computational complexity comparable to that of 2D-NLMS i.e. O(N K). For MATLAB simulations, a SISO 256 point OFDM system is considered. The modulation technique is QPSK. A five path Rayleigh fading channel with tap delay line is considered and the CP length is taken as four so as to eliminate ISI. In Fig.3, the theoretical and

Fast Array Multichannel 2D-RLS Based OFDM Channel Estimator

13

−24 Theory Simulation MSE (dB)

−26

−28

−30

−32 0.98

0.982

0.984

0.986

0.988 0.99 Forgetting factor

0.992

0.994

0.996

Fig. 3 Theoretical and simulated MSE for FAM 2D-RLS

20 2D−NLMS FAM 2D−RLS 2D−RLS

10 0

MSE(dB)

−10 −20 −30 −40 −50 −60

0

10

20

30

40

50 Iterations

60

70

80

90

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Fig. 4 Learning curve for FAM 2D-RLS, 2D-NLMS & 2D-RLS

20

10 2D−RLS FAM 2D−RLS 2D−NLMS

MSE(dB)

0

−10

−20

−30

−40

−50

0

10

20

30

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50 Iterations

60

70

80

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Fig. 5 Numerical stability analysis of FAM 2D-RLS, 2D-NLMS & 2D-RLS for 16 bit quantized data

14

Arun Joy, Vijay Kumar Chakka, IEEE Senior member

simulated MSE is plotted for various values of forgetting factor assuming σu2 = 1. The optimal weight matrix Wo is a K × N K complex Gaussian matrix with zero mean and unit variance. In Fig.3, it is seen that λ = 0.995 gives the best MSE and hence is chosen as the value of forgetting factor for our simulations. The regularization parameter i.e. δ is obtained by substituting λ = 0.995 and σv2 = 0.001 in (28). It is shown in Fig.4 that FAM 2D-RLS and 2D-RLS converges in about 8 iterations while it takes about 80 iterations for the 2D-NLMS to converge. In Fig.5 the numerical stability of FAM 2D-RLS is compared with 2D-RLS [14] and 2D-NLMS [17] by quantizing the input data vector and filter coefficients to 16 bits.It is observed that FAM 2D-RLS and NLMS converges using quantized data, while 2D-RLS does not converge. The better numerical stability of FAM 2D-RLS compared to 2D-RLS is because the former is implemeted using the array method [15] while the latter is implemented using the classical algorithm of RLS [15]. 6 Conclusion In this paper we proposed an adaptive OFDM channel estimation in the frequency domain that makes use of Fast array multichannel 2D-RLS technique. It was shown that the computational cost of the filter is comparable to that of 2D-NLMS while providing the same convergence property as 2D-RLS algorithm. Hence this low complexity filter could be used to track fast varying OFDM channels. Also it was shown that this filter is numerically stable compared to 2D-RLS filter. Thus we conclude that this channel estimation technique for OFDM improves upon the algorithms proposed in [14],[16],[17]. References 1. L.J. Cimini,Jr.,“Analysis and simulation of a digital mobile channel using orthogonal frequency division multiplexing”,IEEE trans. Commun., vol.COM-33, pp.665-675, July 1985. 2. A.Goldsmith,“Wireless Communication”, Cambridge,2002. 3. M.K.Ozdemir and H.Arslan,“Channel estimation for wireless OFDM systems”,IEEE commun. survey and tutorials, vol.9, pp.18-48, Jul 2007. 4. J.-J. Van de Beek, O.Edfors, M.Sandell, S.K.Wilson and P.O.B¨ orjesson,“On channel estimation in OFDM systems”,in Proc. 45th IEEE Vehicular technology Conf., Chicago, IL, pp.915-819, July 1995. 5. H.Minn and V.K.Bhargava,“An investigation into time-domain approah for OFDM channel estimation”, in Proc. IEEE trans. on broadcasting, vol.68, pp.240-248, Dec.2000. 6. Y.(G.)Li,L.J.Cimini,N.R.Sollenberger,“Robust channel estimation for OFDM systems with rapid dispersive fading channels”, in IEEE trans.Commun.,vol.46, no.7, pp.902915, July 1998. 7. Z.Tang, R.C.Cannizzaro,G.Leus,P.Banelli,“Pilot-Assisted Time-Varying Channel Estimation for OFDM Systems ”, in IEEE trans. signal proc., vol.55, pp. 2226-2238, April 2007. 8. J.I.Montojo,L.B.Milstein,“Channel estimation for non-ideal OFDM systems”, in IEEE trans. commun., vol.58, pp. 146-156, Jan 2010. 9. S.Coleri, M.Ergen, A.puri, A.Bahai,“Channel estimation techniques based on pilot arrangement in OFDM systems”, in IEEE trans. broadcasting,vol.48, pp. 223-229, Nov.2002.

Fast Array Multichannel 2D-RLS Based OFDM Channel Estimator

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10. B.Jiang, H.Wang, X.Gao, S.Jin, K.Wong,“ Preamble-Based Channel Estimation for Amplify-and-Forward OFDM Relay Networks”,in proc. of GLOBECOM 2009,pp. 1-5, Mar.2010. 11. D.Schafhuber, G.Matz and F. Hlawatsch,“Adaptive wiener filters for time varying channel estimation in wireless OFDM systems”,in Proc. IEEE Int. Conf. Acoust., Speech, Signal Process., vol. 4, pp.688691, Apr.2003. 12. T.Natori, N.Tanabe, H.Matsue, T.Furukawa,“Practical OFDM channel estimation method with Kalman filter theory ”,in proc. IEEE DSP/SPE 2011,pp.18-23, Mar.2011. 13. P. Hoeher, S. Kaiser, and P. Robertson, “Two-dimensional pilot-symbol aided channel estimation by Wiener filtering”, in Proc. 1997 IEEE Int.Conf. Acoustics, Speech, and Signal Processing, Munich, Germany, pp. 18451848, Apr.1997. 14. X.Hou, S.Li, C.Yin and G.Yue,“Two-dimensional recursive least square adaptive channel estimation for OFDM systems”, in Int. Conf. on Wireless Commun,Networking and Mobile Computing, pp.232-236, 2005. 15. A.H.Sayed, Fundamentals of Adaptive Filtering, New York:Wiley, 2003. 16. A.Soni, T.Sharma and V.Chakka,“Inverse QR 2D-RLS Adaptive Channel Estimation for OFDM Systems”, IEICE Transactions, pp.2822-2825, 2010. 17. X.Hou, S.Li, D.Liu, C.Yin and G.Yue,“On two-dimensional adaptive channel estimation in OFDM systems”, in proc.IEEE Vehicular tech. conf, Los Angeles, Vol.1, pp.498-502, Sept. 2004. 18. V.Mignone and A.Morello,“CD3-OFDM:A novel demodulation scheme for fixed and mobile receivers”,IEEE trans.commun., vol.44, pp.1144-1151, Sept 1996. 19. D.C.Chang and H.C.Chiu,“A stabilized multichannel fast RLS algorithm for adaptive transmultiplexer receivers”,Circuits Syst. Signal Proc., vol.28,pp.845-867, Sept. 2009. 20. R.West, T.moon and J.Gunther,“A novel block fast array RLS algorithm applied to linear flight strip-map SAR imaging”,in proc. ASILOMAR 2010, pp.979-983, Nov. 2010. 21. G.V.Moustakides,“ Study of the transient phase of the forgetting factor RLS,”, in IEEE trans. Signal Procees., vol. 45, pp. 2468-2476, 1997. 22. S. Haykin,“ Adaptive Filter theory”, New York:Wiley, 2002. 23. D.S.Watkins,“Fundamentals of Matrix Computations”,New York:Wiley, 2002.