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here, αm and β m represents the multiplicative distortion at the desired subcarrier and the ICI, respectively. If the channel keeps invariant during a block period, β m will be zero, which means no ICI. In the general case where the multipath cannot be regarded as time-invariant during a block period, (7) can be expressed in vector form as: Y = H X + W (9) where Y=[Y 0,…,Y N-1] T , X=[X 0,…,X N-1] T , W=[W 0,…,W N-1] T , and ⎡a0,0 a0,1La0, N −1 ⎤ ⎢ ⎥ a1,0 a1,1 a1, N −1 H ⎢ L = ⎥ ⎢M O M ⎥ ⎢ ⎥ ⎢a a a ⎥ ⎣ N−1,0 N−1,1 N−1, N−1⎦ here, a m,k in (10) is defined as ( m−k) − j2 π k( L−1)/ N + HL1 e ,0 ≤( m, k) ≤ N −1. * * * * = + (14) ⎤ ⎦ (10) ( m−k) ( m−k) − j2 π k/ N amk , = H0 + H1 e + K (11) − In this paper, we assume the multipath fading channel is slowly-varying, so most energy concentrate in the dominant adjacent subcarriers, the ICI terms which do not significantly affect Y m in (10) can be ignored. And then by transforming the matrix H of order N× N to a block-diagonal matrix H * of order × (N-q)(q+1) (N-q)(q+1). We obtain: where, ⎡Aq/2 0 ⎤ ⎢ ⎥ A * q /2+ 1 H = ⎢ ⎥ ⎢ O ⎥ ⎢ ⎥ ⎢⎣0AN−− 1 q/2⎥⎦ ⎛an−q/2, n−q/2 L an−q/2, n L 0 ⎞ ⎜ ⎟ ⎜an− q/2+ 1, n−q/2 ⎟ ⎜ M M ⎟ ⎜ ⎟ ⎜ann , −q/2 O ⎟ An = ⎜ ⎟ ⎜ O an+ 1, n+ q/ 2 ⎟ ⎜ M M ⎟ ⎜ ⎟ ⎜ an+ q/2− 1, n+ q/2⎟ ⎜ ⎟ 0 an+ q/2, n a ⎟ ⎝ L L n+ q/2, n+ q/2 ⎠ (12) (13) We can locate some of the unused subcarriers at the first q/2 and the last q/2 IFFT grids to get the estimation of all symbols. Then, the input-output relationship of the multipath channel is expressed in a vector form as: where, Y H X W * T X ⎡xq/2 xq/2+ 1 L xN−1 −q/2 = ⎣ xn = ⎡ ⎣Xn−q/2 Xn− q/2+ 1 L X ⎤ n+ q/2⎦ * T Y ⎡yq/2 yq/2+ 1 L y ⎤ N−1 −q/2 = ⎣ ⎦ yn = ⎡ ⎣Xn−q/2 Xn− q/2+ 1 L X ⎤ n+ q/2⎦ * T W ⎡wq/2 wq/2+ 1 L w ⎤ N−1 −q/2 = ⎣ ⎦ wn = ⎡ ⎣Wn−q/2 Wn− q/2+ 1 L W ⎤ n+ q/2⎦ T T T (15) V. FREQUENCY DOMAIN EQUALIZER A. Zero-Forcing Equalizer Zero-Forcing equalization compensates both multiplicative distortion and ICI by multiplying the inverse of , which is the * estimated value of H , to (14). The resulting signal can be expressed as: * 1 * − % % (16) X = H Y where X % is the estimated signal, and H% * −1 −1 ⎡ A% 0 0 ⎤ ⎢ ⎥ −1 ⎢ A% 1 ⎥ = ⎢ O ⎥ ⎢ ⎥ −1 ⎢0 A% ⎣ ⎥ N−− 1 q⎦ Also, (16) can be rewritten as 1 X% − = A% y , q/2≤ n≤ N −1 −q/ 2. (18) n n n Finally, the transmitted symbols are estimated by selecting the elements in the middle of X . n % B. MMSE Equalizer H % * (17) Using the fact that the ICI power is localized to the dominant adjacent subcarriers, only a few neighborhood subcarriers can be used for equalization without much performance penalty. We find the solution for each desired subcarrier individually[4]. The problem of MMSE is to find the equalizer coefficient vector g m=[g m,0,…,g m,q-1] which minimizes the mean-squared error 2 ˆ { m − m } E X X where ˆ T X m = gmy and , y m y ⎡Y , Y ⎤ where Hm=A m. The MMSE solution is m ( m− q/2) L m is then ( m+ q/2) = ⎣ ⎦ ym = Hmx+ w (19) m g R R − = (20) 1 m X , mym ym assuming H is known, x is a zero-mean i.i.d. random vector with variance 2 σ x , and w is an AWGN vector with variance { } 2 H H X mym m m 2 σ w ,then R = E X y = σ xh (21) m where hm is the q/2 th column of the matrix H m, i.e., Also, we can get hm = ⎡ ⎣am− q/2, q/2 , L , a ⎤ (22) m+ q/2, q/2⎦ H 2 H 2 { } σ σ R = E y y = H H + wIq (23) ymm m x m m After inserting (21) and (23) into (20), the q-tap equalizer vector g m becomes T
−1 2 H ⎛ H σ ⎞ w m = m ⎜ m m + 2 q⎟ σ x g h H H I ⎝ ⎠ where I q is the q-by-q identity matrix. And moreover, we have N 1 2 σ x ( 1 m m 0 − ∑ = MMSE = −gh m) (24) (25) By reducing the number of equalizer’s taps, the calculation complexity is reduced largely. VI. SIMULATION RESULT In our simulations, we investigate the performance of both the ZF equalizer and MMSE equalizer. A three-path BEM channel was considered. Other parameters are listed below: Tab. 1. Simulation parameter. Doppler spread fmax 100 Hz Delay spread τmax 65.1 µ s Block size N 768 Symbol/sample period T 21.7 µ s Discrete Doppler spread Q/2= ⎡⎢ fmax NT ⎤⎥ 2 Discrete delay spread L= max /T τ ⎡⎢ ⎤⎥ 3 Note that the maximum Doppler spread of 100Hz corresponds to a vehicle speed of 60 km/h and a carrier frequency of 1.35GHz. BER BER 1.00E+00 1.00E-01 1.00E-02 1.00E-03 1.00E-04 1.00E-05 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 1.00E+00 1.00E-01 1.00E-02 1.00E-03 1.00E-04 1.00E-05 Ebn0(dB) ICI+no equalization ZF(tap=3) ZF(tap=5) no ICI+ZF equalization Fig.4. BER performance of the Zero-Forcing equalizer. 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Ebn0(dB) ICI+no equalization MMSE(tap=3) MMSE(tap=5) no ICI+MMSE equalization Fig. 5 BER performance of the MMSE equalizer. Fig.4 and Fig.5 illustrate the BER performance of ZF and MMSE equalization respectively. The number of subcarrier is 1024, and 3-tap, 5-tap ZF and MMSE equalizers are considered. The figures show that the 5-tap equalizer has better performance than 3-tap one. So we can change the number of equalizer’s tap to get the tradeoff between the complexity and performance. VII. CONCLUSION In this paper, we have utilized the basis expansion model (BEM) to get a good comprise between complexity and performance in conventional frequency domain equalizer. Both ZF and MMSE equalization are considered. This research was supported by University IT Research Center Project (INHA UWB-ITRC), Korea REFERENCE [1] S. Kim, and G. J. Pottie, “Robust OFDM in Fast Fading Channels,” Global Telecommunications Conference, vol. 2, pp. 1074-1078, Dec. 2003. [2] P. Schniter and S. D’silva, “Low-Complexity Detection of OFDM in Doubly-Dispersive channels,” in Proc. Asilomar Conf. on Signals, Systems, and Computers, (Pacific Grove, CA), Nov. 2002. [3] J. Armstrong, “Analysis of New and Existing Methods of Reducing Intercarrier Interference Due to Carrier Frequency Offset in OFDM,” IEEE Trans. Commun., vol. 47, pp. 365-369, Mar.1999. [4] G. B. Giannakis and C. Tepedelenlioglu, “Basis Expansion Models and Diversity Techniques for Blind Identification and Equalization of Time-Varying Channels,” Proc. IEEE, vol. 86, no. 10, pp. 1969-1986, Oct.1998. [5] M. C. Jeruchim, P. Balaban, and K. S. Shanmugan, “Simulation of Communication Systems Modeling, Methodology, and Techniques, (Second Edition)” KLUWER ACADEMIC/PLENUM PUBLISHERS, second edition, pp.550-554, 2000. [6] W. G. Jeon, K.H. Chang, and Y. S. Cho, “An Equalization Technique for Orthogonal Frequency-Division Multiplexing Systems in Time-Variant Multipath Channels,” IEEE Trans. Commun., vol. 47, no. 1, pp. 27-32, Jan. 1999. [7] I. Barhumi, G. Leus and M. Moonen, “Time-Varying FIR Equalization for Doubly Selective Channels,” IEEE Trans. Commun., vol. 4, no. 1, pp.202-214, Jan. 2005.
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