IEEE TRANSACTIONS ON COMMUNICATIONS, VOL 47, NO 1, JANUARY 1999 27 An Equalization Technique for Orthogonal Frequency-Division Multiplexing Systems in Time-Variant Multipath Channels Won Gi Jeon, Student Member, IEEE, Kyung Hi Chang, Senior Member, IEEE, and Yong Soo Cho, Member, IEEE Abstract A loss of subchannel orthogonality due to timevariant multipath channels in orthogonal frequency-division multiplexing (OFDM) systems leads to interchannel interference (ICI) which increases the error floor in proportion to the Doppler frequency In this paper, a simple frequency-domain equalization technique which can compensate for the effect of ICI in a multipath fading channel is proposed In this technique, the equalization of the received OFDM signal is achieved by using the assumption that the channel impulse response (CIR) varies in a linear fashion during a block period and by compensating for the ICI terms that significantly affect the bit-error rate (BER) performance Index Terms Equalization, interchannel interference, orthogonal frequency-division multiplexing, time-variant multipath channels I INTRODUCTION ORTHOGONAL frequency-division multiplexing (OFDM) is generally known as an effective technique for high bit rate applications such as digital audio broadcasting (DAB), digital video broadcasting (DVB), and digital high-definition television (HDTV) broadcasting, since it can prevent intersymbol interference (ISI) by inserting a guard interval and can mitigate frequency selectivity of a multipath channel using a simple one-tap equalizer [1] [4] In an OFDM system, although the degree of channel variation over the sampling period becomes smaller as data rates increase, the time variation of a fading channel over an OFDM block period causes a loss of subchannel orthogonality, resulting in an error floor that increases with the Doppler frequency The performance degradation due to the interchannel interference (ICI) becomes significant as the carrier frequency, block size, and vehicle velocity increase The time-domain compensation technique, which can reduce the fading distortion in a flat (not frequencyselective) Rayleigh fading channel by correcting gain and phase distortions of the received time-domain signal using a pilot symbol, is proposed in [1] In [5], the frequencydomain equalization technique is proposed to compensate for the fading distortion with less noise enhancement in a flat Rayleigh fading channel This latter approach is based on the Paper approved by S B Gelfand, the Editor for Transmission Systems of the IEEE Communications Society Manuscript received April 30, 1997; revised February 6,1 998 This paper was presented in part at the International Conference on Acoustics, Speech, and Signal Processing, ICASSP 97, Munich, Germany, April 21 24, 1997 W G Jeon and Y S Cho are with the Department of Electronic Engineering, Chung-Ang University, Seoul 156-756, Korea (e-mail: yscho@dsplabeecauackr) K H Chang is with the Electronics and Telecommunications Research Institute (ETRI), Taejon 305-350, Korea Publisher Item Identifier: S 0090-6778(99)00799-0 Fig 1 A baseband block diagram for an OFDM system assumption that the pattern of ICI, corresponding to the fading distortion in the time domain is invariant for all subchannels due to the frequency nonselectivity assumption, which is not the case in a multipath fading channel In this paper, a new frequency-domain equalizer which can compensate for the effects of channel variation in a multipath fading channel is described by assuming that the channel impulse response (CIR) varies in a linear fashion during a block period The ICI terms significantly affecting the loss of subchannel orthogonality are then compensated for by a frequency-domain equalizer with a minimum computational complexity II AN EQUALIZATION TECHNIQUE FOR TIME-VARIANT MULTIPATH CHANNELS The discrete-time baseband equivalent model of the OFDM system under consideration is shown in Fig 1 In an OFDM system, several input bits are first encoded into one symbol, and then symbols are transferred by the serial-toparallel converter (S/P) to the OFDM modulator After each symbol is modulated by the corresponding subcarrier, it is sampled and D/A converted Samples of an OFDM signal, implemented by an inverse fast Fourier transform (IFFT), can be expressed as follows: represents the th sample of the output of the IFFT Assuming that the multipath fading channel consists of discrete paths, the received signal can be written as (1) (2) 0090 6778/99$1000 1999 IEEE
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)
IEEE TRANSACTIONS ON COMMUNICATIONS, VOL 47, NO 1, JANUARY 1999 29 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)
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
IEEE TRANSACTIONS ON COMMUNICATIONS, VOL 47, NO 1, JANUARY 1999 31 TABLE I FREQUENCY-DOMAIN EQUALIZATION (q =2) FOR AN OFDM SYSTEM IN A MULTIPATH FADING CHANNEL the multipath channel, to be used in the conventional equalizer, is estimated for every block by taking Fourier transform of the CIR, estimated with a pilot symbol As is the case with the conventional frequency-domain equalizer for a multipath fading channel, the estimate of the CIR is updated for every block, assuming that the CIR does not vary during a block period [3] From Fig 4(a), one can see that there is no big advantage when using the proposed equalizer for Hz However, as the Doppler frequency increases, the gap between the conventional equalizer and the proposed equalizer becomes larger The error floor occurs with the conventional equalizer for the higher due to the time-invariant assumption of the CIR during a block period, as the proposed equalizer successfully compensates for the multipath fading distortion Notice that the proposed equalizers, even with a small matrix size (one or three), are effective in compensating for the ICI and the multiplicative distortion at the desired subchannel Also note that, although the proposed equalizer with matrix size one corresponds to the conventional one-tap equalizer, the proposed equalizer still performs better than the conventional one due to the different approaches for the CIR estimation The CIR estimation for the conventional equalizer is made at the end (or beginning) of an OFDM block, while the CIR estimation for the proposed equalizer is made by averaging the consecutive CIR estimates obtained by the conventional approach, implying that the actual CIR estimation is made in the middle of an OFDM block [see (15)] III COMPUTER SIMULATION To demonstrate the effectiveness of the proposed approach for time-variant multipath channels, the following simulations were performed A two-path fading channel with a multipath spread of 2 s was considered for a mobile radio channel A bandwidth of 500 khz with a carrier frequency at 1 GHz was assumed for transmission The total bandwidth was divided into 64 subbands, hence the size of FFT became 64 An OFDM block was composed of 68 samples, one for a cyclic prefix and three for pilot signals The modulation scheme used for this simulation was 16-QAM Thus, the data rate for this OFDM system was about 188 Mb/s (64 subcarriers 4 bits per symbol/136 s) The Doppler frequency was taken into account up to 200 Hz, resulting in equal to 0272% for 20 Hz, 136% for 100 Hz, and 272% for 200 Hz Perfect carrier and symbol synchronizations were assumed Fig 4 shows the BER comparison when a conventional equalizer, denoted by C, and the proposed equalizers, denoted by P, with different matrix sizes were used to reduce the performance degradation due to the frequency selectivity and time-variation of the multipath fading channel In the conventional equalizer, the frequency selectivity of the multipath channel is compensated for by using simple one-tap frequency-domain equalizers The channel transfer function of IV CONCLUSION The conventional frequency-domain equalizer with one tap in an OFDM system compensates for the frequency-selectivity of a multipath fading channel, assuming that the channel is stationary over the period of an FFT block In this paper, a new frequency-domain equalization technique to reduce the time-variation effect of a multipath fading channel is described by assuming that the CIR varies in linear fashion during a block period It is shown through simulation that the loss of orthogonality caused by the time-variation of a multipath fading channel can be compensated effectively by the proposed equalizer if the relative Doppler frequency change is in the range of It is also shown that improved BER results can be obtained with the proposed approach by compensating for the multiplicative distortion and for the ICI from only a few neighboring subchannels REFERENCES [1] L J Cimini Jr, Analysis and simulation of a digital mobile channel using orthogonal frequency-division multiplexing, IEEE Trans Commun, vol 33, pp 665 675, July 1985 [2] M Alard and R Lassalle, Principles of modulation and channel coding for digital broadcasting for mobile receivers, EBU Tech Rev, no 224, pp 3 25, Aug 1987 [3] H Sari, G Karam, and I Jeanclaude, Transmission techniques for digital terrestrial TV broadcasting, IEEE Commun Mag, vol 33, pp 100 109, Feb 1995
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 146 147, 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 820 824 [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 1126 1130