# An Equalization Technique for Orthogonal Frequency-Division Multiplexing Systems in Time-Variant Multipath Channels

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2 28 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL 47, NO 1, JANUARY 1999 and represent the complex random variable for the th path of the CIR and additive white Gaussian noise (AWGN) at time, respectively A cyclic extension of length, not shown in Fig 1 for reason of simplicity, is used to avoid ISI and to preserve the orthogonality of subchannels It is also assumed that the entire CIR lies inside the guard interval, is the sampling interval The demodulated signal in the frequency domain is obtained by taking the FFT of as denotes the FFT of Also, represents the FFT of a time-variant multipath channel as follows: Here, and represent the multiplicative distortion at the desired subchannel and the ICI, respectively [6] If the channel is assumed to be time-invariant during a block period, in (4) vanishes, implying that there exists no ICI for time-invariant channels In this case, in (3) contains only the multiplicative distortion, which can be easily compensated for by a one-tap frequency-domain equalizer In the general case the multipath channel cannot be regarded as time-invariant during a block period, (3) can be expressed in vector form as, and (3) (4) (5) (6) Here, in (6) is defined as In order to solve for in (5), we need to estimate the channel matrix and calculate its matrix inverse Since can have a large size, it is difficult to process in real time In this paper, it is assumed that the multipath fading channel is slowly time varying (eg, ), so that time variations of the CIR,, for all paths, can be approximated by straight lines with low slopes during a block period Here, the relative Doppler frequency change, which indicates the degree of time variation of the CIR, is defined by the ratio of the block period to the inverse of the Doppler frequency, ie, For the channel with, the assumption that the CIR varies in a linear fashion during a block period no longer holds and gives rise to an error floor When the multipath fading channel is slowly time-varying, the matrix equation in (5) can be greatly simplified Since most energy of the straight line with a low slope is concentrated in the neighborhood of the dc component in the frequency domain, the ICI terms which do not significantly affect in (6) can be ignored, ie, (7) for (8) denotes the number of dominant ICI terms Fig 2 shows the time variation of the CIR in a block and the corresponding magnitude response [absolute value of the Fourier transform of the signal in Fig 2(a)] for three different Doppler frequencies From this figure, one can see that the time variation of the CIR can be modeled as a straight line, and most of the energy is concentrated in the neighborhood of the dc component Substituting the approximation in (8) into (6) can be rewritten as (9), shown at the bottom of the page The matrix in (9) represents a band matrix half of the width of the band is Since the matrix becomes a sparse matrix [7] for, it is not efficient to calculate the matrix inverse of (9) to estimate the transmitted sequence By transforming the matrix of order to a blockdiagonal matrix of order (9)

3 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL 47, NO 1, JANUARY Fig 3 Transformation of matrix H 0 to H (a) input output relationship of the multipath channel is expressed in a vector form as (12) (b) Compensation of both multiplicative distortion and ICI is accomplished by multiplying the inverse of, which is estimated value of, to (12) The resulting signal can be expressed as follows: (13) Fig 2 The characteristics of a CIR in a slowly time-varying environment (a) Time variation of the CIR for different Doppler frequencies within a block period and (b) corresponding magnitude responses for different Doppler frequencies as shown in Fig 3, we obtain (10) is shown in (11) at the bottom of the page Then, the Also, (13) can be rewritten as (14) Finally, the transmitted symbols are estimated by selecting the elements in the middle of The remaining symbols are estimated by taking the first (last) elements of (11)

4 30 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL 47, NO 1, JANUARY 1999 (a) (b) Fig 4 (c) BER comparisons when the conventional proposed equalizers are applied to time-variant channels with three different Doppler frequencies (a) fd = 20 Hz (b) fd = 100 Hz (c) fd = 200 Hz Notice that the size of the matrix inverse is lowered to, implying that the transmitted sequence can be obtained with a moderate amount of computational complexity for a small value of Thus, the linear approximation assumption discussed in this paper has transformed a large-size matrix inverse problem into small-size matrix inverse problems The solution for the transmitted sequence in a multipath fading channel is listed in Table I The required multiplications and additions for are and, respectively In order to construct the matrix equation in (12), it is necessary to estimate the channel matrix, which consists of various transfer functions However, accurate estimation of a transfer function requires complete knowledge of the timevariation of the CIR, for each block, which is not usually available By utilizing the above assumption that the CIR varies in a linear fashion during a block period, the estimation problem of can also be greatly simplified, since, in this situation, the value of the slope of a straight line uniquely determines the ICI The procedure for the estimation of the transfer function is given as follows First, a time-domain pilot signal is inserted at the end of every th symbol [8] The pilot symbol is composed of samples, all with zero values except for a delta function in the middle The first half samples with zero values are put to avoid ISI, while the second half samples are inserted for the CIR estimation As the relative Doppler frequency change increases, the value of needs to decrease, resulting in the expansion of transmitted signal bandwidth For example, for a time-varying channel with a relatively high, the approximate value approaches one Then, by comparing the CIR changes between the received signals corresponding to the th pilot symbol and th pilot symbol for each path, the CIR variation during the block period is estimated using linear interpolation Finally, the components of the channel matrix in the th period are obtained by (15) and denote the CIR s corresponding to the th and th pilot symbols for the th path, respectively

6 32 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL 47, NO 1, JANUARY 1999 [4] J A C Bingham, Multicarrier modulation for data transmission: An idea whose time has come, IEEE Commun Mag, vol 28, no 5, pp 5 14, May 1990 [5] J Ahn and H S Lee, Frequency domain equalization of OFDM signals over frequency nonselective Rayleigh fading channels, Electron Lett, vol 29, no 16, pp , Aug 1993 [6] M Russell and G L Stüber, Interchannel interference analysis of OFDM in a mobile environment, in Proc IEEE VTC 95, Chicago, IL, July 1995, pp [7] G Strang, Linear Algebra and Its Applications, 3rd ed Philadelphia, PA: Saunders, 1988 [8] H K Lau and S W Cheung, A pilot symbol-aided technique used for digital signals in multipath environments, in Proc IEEE ICC 94, New Orleans, LA, May 1994, pp

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