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Frequency Domain Equalization for Doubly Selective Channels YuQin Chen JongYoung Han Jae Moung Kim river4416@sina.com, fanaticey@hotmail.com, jaekim@inha.ac.kr Graduate school of Information Technology & Telecommunications Inha University Abstract- In this paper, a kind of conventional linear frequency-domain equalizer for doubly selective (time-and frequency-selective) channels is proposed. We use a basis expansion model (BEM) to approximate the doubly selective channels. The time-varying impulse response of rapidly fading mobile communication channels is expanded over a basis of complex exponentials that arise due to Doppler effects encountered with multipath propagation. This model allows us to turn a complicated equalization problem into a simpler one, containing only the BEM coefficients of the doubly selective channels and the frequency domain equalizer. Both Zero-Forcing (ZF) and Minimum Mean Square Error (MMSE) equalization are considered. I. INTRODUCTION Orthogonal frequency division multiplexing (OFDM) is an attractive technique for high-speed data transmission in mobile communications, since it can avoid intersymbol interference (ISI). The Doppler effect caused by time-variation in conjunction with ISI give rise to a so-called doubly selective (frequency- and time-selective) channel. In a doubly selective channel, the channel variation over an OFDM block destroys the orthogonality between the subcarriers resulting into so-called inter-carrier interference (ICI)[1]. In order to mitigate ICI, many techniques, e.g., time-domain windowing combined with iterative MMSE estimation, ICI self-cancellation, have been proposed[2][3]. These approaches require a high computational complexity or at a price of sacrificing the spectral efficiency. Moreover, these works assume perfect knowledge of the TV channel at the receiver, which is rather difficult if not impossible to obtain. In this paper, a kind of frequency domain equalization technique which can compensate for the effects of channel variation in a multipath fading channel is described by approximating that the channel impulse response (CIR) is expanded over a basis of complex exponentials that arise due to Doppler effects encountered with multipath propagation. This allows us to turn a large time-variant (TV) problem into an equivalent small time-invariant (TIV) problem, containing only the BEM coefficients of the doubly selective fading channel[4]. II.FADING CHANNELS: RANDOM MODLES In some communication schemes, unpredictable changes in the medium warrant modeling the TV impulse response (TVIR) h c(t;τ )as a stochastic process in the time variable t. Since the components of the multipath signal arise from a large number of reflections and scattering from rough or granular surfaces, then by virtue of the central limit theorem, the TVIR can be modeled as a complex Gaussian process. A simple way to model the fading channel is wide-sense stationary uncorrelated scattering (WSSUS). It assumes the channel to be wide sense stationary for a fixed lag τ and uncorrelated for different lags. The channel spectral density for a fixed τ is called the scattering function S(ω ;τ );and it provides a single measure of the average power output of the channel as a function of the delayτ and the Doppler frequency ω . The delay-power profile, sometimes also called as multipath intensity profile, which is defined as the integral , ∫ ∞ −∞ p ( τ ) = S( τ , w) dw (1) represents the average received power as function of delay τ ,and the length of its support is called the multipath spread and is a measure of the average extent of the multipath. Another function called Doppler power spectrum ∫ ∞ −∞ S ( w) = S( τ , w) dτ (2) is derived from the scattering function and its extent, the Doppler spread, is a measure of the channel’s time variation[2][5]. s(l) ( ) f tr c ( l) sc(t ) ( f ch c ) ( t; τ) vc(t ) rc (l) f (l) rec c xc (t) nTs Fig. 1. Continuous-time TV communication system. x(n) Consider the fading communication system model of Fig. 1 before the sampler with the input/output (I/O) relationship ∑ ∞ = l= −∞ x c ( t) s( l) hc ( t; t − lTs ) + vc ( t) (3)
where subscript c denotes continuous time, h c(t;T) is the convolution of the spectral pulse f c (tr) (t), the TV impulse response f c (ch) (t;τ ), the receive-filter fc (rec) (t), s(l) is the sequence of input symbols, and vc(t) is the noise process. h c(t;T) can be truncated to an order L, which is common practice in communication applications. On approach in characterizing the variation of the impulse response h(n;l) is to consider it as a stochastic process in the time index n. In communication, tracking the variations of the channel taps is of importance. To this end, linear TV channel is assumed in [6]. III. FADING CHANNELS: BASIS EXPANSION MODELS In this paper, we model the doubly selective channel h(n;v) using a basis expansion model (BEM). In this BEM, the channel is modeled as a TV FIR filter, where each tap is expressed as a superposition of complex exponential basis functions with the frequencies on a DFT grid, as described next. Let us first make the following assumptions: A1) The delay spread is bounded by τ max. A2) The Doppler spread is bounded by fmax. Under assumption A1) and A2), it is possible to accurately model the doubly selective channel h(n;v) for n∈{0,1,..., N −1} as (see Fig.2) h( n; v) = L ∑ v−l Q / 2 j 2πqn / N ∑e hq, l l= 0 q= −Q / 2 δ (4) where L and Q satisfy the following conditions: C1) LT≥τ max; C2) Q/(NT) ≥2f max; h− Q /2,0 h− Q /2,1 h− Q /2,2 Q/2, L j2 Q/2 n/ N h Q /2,0 h Q /2,1 h Q /2,2 hQ/2, L e π h − Fig. 2. BEM channel. e − j2 πQ/2 n/ N where T is the symbol period. In this expansion model, L represents the discrete delay spread (expressed in multiples of T, the delay resolution of the model), and Q/2 represents the discrete Doppler spread (expressed in multiples of 1/(NT), the Doppler resolution of the model). Note that the coefficients h q,l remain invariant for n∈{0,1,..., N −1} and hence are the BEM coefficients of interest[7]. IV. BLOCK DATA MODEL. The discrete-time baseband equivalent model of the OFDM system under consideration is shown in Fig. 3. In the transmitter, the transmitted high-speed data is first converted into parallel data of N subcarriers by the serial-to-parallel converter (S/P). After each symbol is modulated by the corresponding subcarrier, it is sampled and D/A converted. The sampled symbols, implemented by an inverse fast Fourier transform (IFFT), can be expressed as follows: x ∑ − = = 1 N n m 0 X e j 2πnm / N m , 0 ≤ n ≤ N (5) where xn represents the nth sample of the output of the IFFT[6]. Input Bit Stream Output Bit Stream S/P P/S X m ~ IFFT Equalizer Y m Xm FFT yn P/S xn D/A S/P A/D AWGN wn Multipath Channel Fig. 3. A baseband block diagram for an OFDM system. Assuming that the multipath fading channel consists of L discrete paths, the received signal can be written as L−1 ∑ y = h x + ω = h x + h x + ... + h x 1+ ωn n n, l n−l n n,0 n n,1 n−1 n, L−1 n− L+ l= 0 where hn,l and ω n represent the complex random variable for the lth path of the CIR and additive white Gaussian noise (AWGN) at time n, respectively. For reason of simplicity, a cyclic extension of length N G, used to avoid ISI and to preserve the orthogonality of subcarriers, is not shown in Fig.4 and not used in this paper’s simulation. The demodulated signal in the frequency domain is obtained by taking the FFT of y n as m N−1 L−1 ∑∑ k= 0 l= 0 k ( m−k) l − j2 πlk/ N Y = X H e + W L−1 ⎡ 0 − j2 πlm/ N⎤ ∑ He l Xm l= 0 = ⎢ ⎥ ⎣ ⎦ N−1 L−1 ∑∑ + XH e + W k≠ m l= 0 ( m−k) − j2 πlk/ N k l m = α X + β + W , 0≤m≤ N −1 m m m m ( m k) where Wm denotes the FFT of wn . Also, H represents the l − FFT of a time-variant multipath channel hn,l , which is modeled using BEM. It can be represented by 1 H h e N −1 ( m−k) − j2 π n( m−k)/ N l = ∑ n, l N n= 0 N −1 Q /2 ⎡ ∑ ∑ j2 πqn/ N⎤ − j2 πn( m−k)/ N hql , e e n= 0 q=−Q/2 1 = ⎢ ⎥ N ⎣ ⎦ m + (6) (7) (8)
Page 6 and 7: here, αm and β m represents the m

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