Time-Varying FIR Decision Feedback Equalization of Doubly-Selective Channels

Transcription

1 Time-Varying FIR Decision Feedback Equalization of Doubly-Selective Channels Imad Barhumi, Geert Leus and Marc Moonen K.U.Leuven-ESAT, Kasteelpark Arenberg 10, 3001 Leuven, Belgium Abstract In this paper we propose a minimum mean-square error (MMSE) decision feedback (DF) time-varying (TV) finite impulse response (FIR) equalizer for doubly-selective (time- and frequency-selective) channels. We use the basis expansion model (BEM) to approximate the doubly-selective channel and to design the TV FIR DFE. This allows us to turn a large design problem into an equivalent small design problem, containing only the BEM coefficients of the doubly-selective channel and the BEM coefficients of the TV FIR feedforward and feedback equalization filters. Through computer simulations we show that the performance of the proposed TV FIR DF equalizer approaches the performance of the block DF equalizer, while the equalization as well as the design complexity are much lower. I. INTRODUCTION THE wireless communication industry has experienced rapid growth in recent years, and digital cellular systems are currently designed to provide high data rates at high speeds. High data rates give rise to intersymbol interference (ISI) due to multipath fading. Such an ISI channel is called frequencyselective. On the other hand, due to mobility and/or frequency offset the received signal is subject to frequency shifts (Doppler shifts). The Doppler effect in conjunction with ISI gives rise to a so-called doubly-selective channel (frequency- and timeselective). Equalization is required to compensate for these channel effects. Equalization can be of different types. Well-known types are i) linear equalization (LE), ii) decision feedback equalization (DFE) iii) maximum likelihood sequence estimation (MLSE). LE is used because of its simplicity and cost effectiveness, but may also sacrifice performance. MLSE is extremely complex, and gives optimum or near optimum performance. DFE on the other hand, provides a good compromise between complexity and performance. The basic idea behind DFE is that once information has been detected, the ISI that it induces on future symbols can be subtracted out before detection of subsequent symbols. DFE has been previously proposed in literature for the case of frequency-selective channels, some based on block processing This research work was carried out at the ESAT laboratory of the Katholieke Universiteit Leuven, in the frame of the Belgian State, Prime Minister s Office - Federal Office for Scientific, Technical and Cultural Affairs - Interuniversity Poles of Attraction Programme ( ) - IUAP P5/22 ( Dynamical Systems and Control: Computation, Identification and Modeling ) and P5/11 ( Mobile multimedia communication systems and networks ), the Concerted Research Action GOA-MEFISTO-666 (Mathematical Engineering for Information and Communication Systems Technology) of the Flemish Government, Research Project FWO nr.g ( Design of efficient communication techniques for wireless timedispersive multi-user MIMO systems ) and was partially sponsored by IMEC (Flemish Interuniversity Microelectronics Center). The scientific responsibility is assumed by its authors. Partly supported by the Palestinian European Academic Cooperation in Education (PEACE). Postdoctoral Fellow of the F.W.O. Vlaanderen. [1], [2] and others based on serial processing by means of finite impulse response (FIR) filters [3]. In previous work we proposed time-varying (TV) FIR LEs to compensate for the doubly-selective channel effects [4], [5], [6]. In this paper we extend this idea to design a TV FIR DFE, which shows to give better performance and to be more robust against channel imperfections. The proposed TV FIR DFE consists of a TV FIR feedforward filter, a TV FIR feedback filter, and a decision device. We use the basis expansion model (BEM) to model the TV FIR feedforward and feedback filters. The beauty of using the BEM is that it allows us to turn a large design problem into a small design problem, containing only the BEM coefficients of the channel, and the BEM coefficients of the feedforward and feedback TV FIR filters. We also show that this approach extends the results obtained for the case of purely frequency-selective channels. This paper is organized as follows. The system model is described in section II. The block decision feedback equalizer (BDFE) is introduced in section III. The TV FIR DFE equalizer is introduced in section IV. In section V, we show through computer simulations the performance of the proposed equalizer. Finally, our conclusions are drawn in section VI. Notations: We use upper (lower) bold face letters to denote matrices (column vectors). Superscripts, T, and H represent conjugate, transpose, and Hermitian, respectively. We denote the 1- and 2-dimensional Kronecker delta as δ n and δ n,m, respectively. We denote the N N identity matrix as I N and the M N all-zero matrix as 0 M N. Finally, diag{x} denotes the diagonal matrix with x on the diagonal. II. SYSTEM MODEL The system under consideration is depicted in Figure 1. We assume a single-input multiple-output (SIMO) system, where N r receive antennas are used. Focusing on a baseband-equivalent description, when transmitting a symbol sequence s(n) at rate 1/T and sampling each receive antenna at the same rate 1/T, the received sample sequence at the rth receive antenna can be written as: y (r) (n) = (n;ν)s(n ν) η (r) (n) (1) ν= where η (r) (n) is the additive noise at the rth receive antenna, and (n;ν) is the doubly-selective channel (time- and frequencyselective) from the transmitter to the rth receive antenna. Here, we model the doubly-selective channel (n;ν) using the basis expansion model (BEM) [7], [8], [9], [10], which is shown to accurately approximate the well-known Jakes model. In the BEM, the channel is modeled as an FIR filter where the

2 s(n) s(n) h (1) (n; ν) h (Nr) (n; ν) η (1) (n) η (Nr) (n) y (1) (n) y (Nr) (n) η (1) (n) η (Nr) (n) S/P S/P y (Nr) (n) y (1) N 1 y (Nr) N 1 G (1) G (Nr) (a) Block DFE h (1) y (n; ν) (1) (n) g (1) (n; ν) h (Nr) (n; ν) Fig. 1. g (Nr) (n; ν) (b) TV FIR DFE System Model s(n d) s B b(n; ν) δd ŝ ŝ(n d) taps are expressed as a superposition of complex exponential basis functions. Before we proceed to define the BEM for doublyselective channels, we make the following assumptions: A1) The delay-spread is bounded by τ max ; A2) The Doppler-spread is bounded by f max. Under assumptions A1) and A2), it is always possible to model the channel between the transmitter and the rth receive antenna in discrete time as: (n;ν) = L l=0 δ ν l q= q,l ej2πqn/n, (2) where N is the block size over which this channel model holds, and L and Q satisfy the following conditions: C1) LT τ max ; C2) Q/(NT) 2f max, where T is the symbol period. In this expansion model, L represents the delay-spread (expressed in multiples of T, the delay resolution of the model), and Q represents the Doppler-spread (expressed in multiples of 1/(NT), the Doppler resolution of the model). Note that the coefficients q,l remain invariant over a period of length NT, and may change from block to block. Substiute (2) in (1), the received sample at the rth receive antenna at time index n can be written as: y (r) (n) = L l=0 q= e j2πqn/n q,l s(n l) η(r) (n), (3) Suppose we want to transmit a symbol burst s(n), where s(n) 0 for n / {0,1,...,M 1}, M = N L zp and L zp L (zp refers to zero-padding). The input-output relation in (3) can also be written in block form. Defining the M 1 symbol block as s := [s(0),...,s(m 1)] T, the received sample block at the rth receive antenna y (r) := [y (r) (0),...,y (r) (N 1)] T, can be written as: y (r) = H (r) T zp s η (r), (4) where η (r) is similarly defined as y (r), T zp := [I M,0 M Lzp ] T, and H (r) is an N N lower triangular matrix. Using (2), H (r) can be written as: H (r) = L l=0 q= q,l D qz l (5) where D q := diag{[1,,e j2πq(n 1)/N ] T }, and Z l is an N N lower triangular Toeplitz matrix with first column [0 1 l,1,0 1 (N l 1) ] T. Substituting (5) in (4), the N 1 received sample block at the rth receive antenna can be written as: y (r) = L l=0 q= q,l D qz l T zp s η (r). (6) Stacking the N r received sample blocks: y := [y (1)T,...,y (Nr)T ] T, we obtain y = HT zp s η, where η is similarly defined as y and H := [H (1)T,...,H (Nr)T ] T. III. BLOCK DECISION-FEEDBACK EQUALIZATION In this section we review conventional BDFEs. We focus on the minimum mean-square error (MMSE) BDFE. The BDFE consists of a feedforward equalizer represented by the M N matrix G (r) on the rth receive antenna, the decision making device, and the feedback equalizer represented by the M M matrix B, as shown in Figure 1(a). Hence, a soft estimate of s is computed as: N r s = G (r) y (r) Bŝ, ŝ =Q( s), (7) where Q( ) is the quantizer used by the decision device. The MMSE BDFE is designed to minimize the error at the decision device. Define the error vector e = s s. Assuming past decisions are correct (ŝ = s) we get N r ( e = G (r) H (r) T zp s η (r)) (B I M )s. (8) Defining G := [G (1),...,G (Nr) ], (8) can be written as: e = GHT zp s Gη (B I M )s. (9) Similar to [1], we obtain the MMSE feedforward and the feedback block equalizers under the constraint that (B I M ) is an upper triangular matrix as: G = (B I M) (HT zp) H R 1 η 1 (HT zp) R 1 (HTzp) H R 1 B =L H I M, (10) where L is a lower triangular matrix, obtained by Cholesky factorization of ( ) (HT zp ) H R 1 η (HT zp ) R 1 s = LDL H. The above approach is rather complex in particular for large N. In the next section, we will therefore develop a DFE equalizer, s η,

3 with TV FIR feedforward and TV FIR feedback filters. Doing this we lower the design and the equalization complexity at the cost of a slightly lower performance. IV. TIME-VARYING FIR DECISION-FEEDBACK EQUALIZATION In this section, we will apply a TV FIR feedforward filter (n;ν) on the rth receive antenna and a TV FIR feedback filter b(n;ν), as depicted in Figure 1(b). Hence, a soft estimate of s(n d) is computed as XN r s(n d) = X ν= (n; ν)y (r) (n ν) X ν= b(n; ν)ŝ(n ν), ŝ(n d) =Q( s(n d)), (11) where d is the synchronization delay. Since the doubly-selective channel (n;ν) was modeled by the BEM, it is also convenient to design the TV FIR feedforward and feedback filters, (n;ν) and b(n; ν), using the BEM. This will allow us to turn a large design problem into an equivalent small design problem, containing only the BEM coefficients of the doubly-selective channel and the BEM coefficients of the TV FIR feedforward and feedback filters. Using the BEM, we design the TV FIR feedforward filter (n;ν) to have 1 taps, where the time-variation of each tap is modeled by Q 1 complex exponential basis functions. Similarly we design the TV FIR feedback filter b(n;ν) to have taps, where the time-variation of each tap is captured by Q 1 complex exponential basis functions. Hence, we can write the TV FIR feedforward and feedback filters as: (n;ν) = b(n;ν) = l =0 1 δ ν l δ ν l q = Q /2 q,l e j2πq n/n, b q,l ej2πq n/n, (12a) (12b) where is chosen to be larger than the synchronization delay; i.e. d 1. since we can only feedback the past decisions and not the future ones. Instead of continuing to work on the sample level, it is easier to switch to the block level at this point. On the block level, (11) corresponds to estimating s as in (7) but with G (r) and B constrained to L G (r) =R zp l =0 q = L 1 B =R zp q,l D q Z l, (13) Q /2 b q,l D q Z l T zp, (14) where R zp := [0 M d I M 0 M (Lzp d)]. It is clear that this requires 0 < d L zp. In practice, we will take L zp = max{l,d}. Hence we can write N r s =R zp r N r R zp R zp q = l =0 r q = l =0 Q /2 q,l D q Z l 1 L q= l=0 q,l D q Z l η (r) q,l D qz l T zp s b q,l D q Z l T zpŝ. (15) Defining p = q q and k = l l, and using the property Z l D q = e j2πql /N D q Z l, (15) can be written as: where s =R zp f p,k := (QQ )/2 LL p= (QQ )/2 k=0 N r R zp R zp N r q = l =0 Q /2 q = l =0 1 We can further rewrite (16) as f p,k D p Z k T zp s q,l D q Z l η (r) b q,l D q Z l T zpŝ, (16) e j2π(p q )l /N q,l p q,k l. (17) N r s = (f T I M )Ãs (T I M ) Cη (r) ( b T I M )Ãŝ = (f T I M )Ãs (gt I M )(I Nr C)η ( b T I M )Ãŝ, (18) where we have f := [f,0,...,f,l,..., f Q /2,L]T, := [,0,...,g(r),L,...,,L ] T, g := [g (1)T,...,g (Nr)T ] T, and the augmented vector b := [0 (QQ Q )/2 (L 1) 0 1 b T Q /2 0 u 1,..., 0 1 b T Q /2 0 u 1 0 (QQ Q )/2 (L 1)] T, where u = L 1 and b q := [b q,,,b q, 1] T, i.e. we construct b in a way to match f in dimension. Defining the matrices A and C as D Q /2Z 0 D Q /2Z 0.. A := D Z L. D Q /2Z L, C := D Z. D Q /2Z, the matrices à and C in (18) are à := (I (QQ 1)(L 1) R zp )AT zp and C := (I (Q 1)( 1) R zp )C, respectively. Note that the term in f p,k corresponding to the rth receive antenna is related to a 2-dimensional convolution of the BEM coefficients of the doubly-selective channel for the rth receive antenna and the BEM coefficients of the TV FIR feedforward filter for the

4 rth receive antenna. This allows us to derive a linear relationship between f and g. We first define the ( 1) ( L 1) Toeplitz matrix T l,l 1( q,l ) := q,0... q,l q,0... q,l. We then define H q (r) := Ω q T l, 1( q,l ), where Ω q := diag{[1, e j2πq/n,...,e j2πql /N ] T }, and introduce the (Q 1)( 1) (Q Q 1)(L 1) block Toeplitz matrix T q,q 1(H q (r) ) := H (r)... H (r) H (r)... H (r). Introducing the definitions H (r) := T q,q 1(H (r) q ) and H := [H (1)T,...,H (Nr)T ] T, we can derive from (17) that f T = g T H. (19) Assuming past decisions are correct, the error at the decision device e = s s can be written as: e =(g T H I M )Ãs (gt I M )(I Nr C)η (( b T I M )Ã I M)s. (20) Now we can write the mean-square error MSE = E{e H e} as: MSE = tr{(g T H I M )ÃR sãh (H H g I M )} tr{(g T I M )(I Nr C)R η (I Nr C H )(g I M )} tr{( b T I M )ÃR sãh ( b I M )} 2R{tr{(g T H I M )ÃR sãh ( b I M )}} 2R{tr{(g T H I M )ÃR s}} 2R{tr{( b T I M )ÃR s}} tr{r s }. (21) Introducing the properties tr{(x T I M )X} = x T red(x), tr{(x T I M )X(x I M )} = x T red(x)x, where red( ) splits the matrix up into M M submatrices and replaces each submatrix by its trace, then (21) can be rewritten as: ( ) MSE = g T HRÃH H R C g ( b T e T d )RÃ( b e d ) 2R{g T HRÃ b } 2R{g T HrÃ}, (22) where RÃ := red(ãr sãh ), R C := red((i Nr C)R η (I Nr C H )) and rã := red(ãr s). Note also that we have used the fact that tr{r s } = e T d red(ãr sã)e d, where e d is a (Q Q 1)(L 1) 1 unit vector with the 1 in position d(qq 1) (Q Q )/2 1. By solving MSE/ g = 0, we then obtain: g T = ( b T e T d )RÃH H ( HRÃH H R C) 1, (23) where we used the fact that r H = Ã et d R Ã, substituting (23) in (22) we get: where MSE = ( b T e T d )R ( b e d ), (24) ( 1 R = RÃ RÃH H HRÃH H R C) HRÃ ( ) 1 = R 1 Ã HH R 1 C H. (25) The last line in (25) is obtained by applying the matrix inversion Lemma. By defining the selection matrix P as: I Q (e T /2 I L P = ) 0 0 0k v k v 0 0 I Q /21 (e T I ) (26) where k = Q 1, v = (Q Q Q )(L 1)/2 and e is an L 1 unit vector with the 1 in position ( 1), we can write (24) as: MSE = [b T Q /2,,bT 1, 1, b T 0,,b T Q /2 }{{} ] R b, (27) b T where R = P R P H. The formula in (27) is a quadratic form that is minimized by choosing the TV FIR feedback filter coefficients as: b = R 1 e x e T xr 1 e (28) x where e x is a Q 1 unit vector, with the 1 in the position (Q /2 1). By substituting (28) in (23) we can solve for the TV FIR feedforward equalizer coefficients. In comparing the two approaches, BDFE and TV FIR DFE, we consider two types of complexity: design complexity and implementation complexity. The design complexity is the complexity associated with computing the equalizer (feedforward and feedback) coefficieints. To design the BDFE, we need a matrix inversion of size M M to compute the feedforward matrix G, and a Cholesky factorization of an M M matrix to compute the feedback matrix B. Both require about O(M 3 ) flops. On the other hand, to design the TV FIR DFE equalizer, we need to compute the inverse of a K K matrix to obtain the feedforward filter coefficients, where K = (Q Q 1)(L 1), and we need to compute the inverse of a (Q 1) 1 (Q 1) 1 matrix to obtain the feedback filter coefficients. So, provided that K and (Q 1) 1 are less than M, the design complexity of the TV FIR DFE is less than the design complexity of the BDFE. In practice and also in our simulations, the TV FIR DFE parameters Q,, Q and always satisfy K < M and (Q 1) 1 M. The implementation complexity is the complexity associated with estimating a block of M samples. For the BDFE, to estimate a block of M samples we require M N multiply-add (MA) operations per receive antenna coming from the feedforward part, and (M 1)M/2 MA operations coming from the feedback part. To estimate the same block using the TV FIR DFE we require M( 1) MA operations per receive antenna coming from the feedforward part and M MA operations coming from

5 10 0 BDFE TV FIR DFE TV FIR LE 10 0 BDFE TV FIR DFE TV FIR LE BER BER SNR (db) Fig. 2. BER vs. SNR for Nr = 1 the feedback part. So, provided that is less than the block size N, and is less than M/2, the implementation complexity of the TV FIR DFE is lower than the implementation complexity of the BDFE. In practice and also in our simulations, the TV FIR DFE parameters Q,, Q and always satisfy N and M/2. V. SIMULATIONS In the simulations, we consider a SISO as well as SIMO (N r = 2) system. The channel is assumed to be doubly-selective, simulated according to the BEM with the following parameters: transmitted block size N = 800; symbol period T = 25µsec; maximum delay spread τ max = 75µsec maximum Doppler spread f max = 100Hz; number of basis functions Q = 2 f max NT = 4; channel order L = τ max /T = 3. N r Q Q TABLE I FILTER COEFFICIENTS FOR DIFFERENT RECEIVE ANTENNAS The proposed number of equalizer parameters for different receive antennas is listed in table I. In all simulations, QPSK signaling is used, the delay d = LL 2 1, and = d1. The performance is measured in terms of the BER vs. SNR. In each plot, we draw comparisons between the proposed TV FIR DFE, the TV FIR LE (no feedback case), and the BDFE. As shown in Figure 2 (SISO N r = 1), the SNR loss for the TV FIR DFE compared to BDFE is less than 1dB at BER = On the other hand, the proposed TV FIR DFE outperforms the TV FIR LE, and the latter suffers from an early error floor. Similarly, in Figure 3 (SIMO N r = 2), we see that the performance of the proposed TV FIR DFE almost coincides with that of the BDFE, whilst it significantly outperforms the TV FIR LE equalizer SNR (db) Fig. 3. BER vs. SNR for Nr = 2 VI. CONCLUSION In this paper, we have proposed a TV FIR DFE to compensate for doubly selective channel distortion. The proposed equalizer provides a good compromise between complexity and performance compared to BDFE and the TV FIR LE. We show that by properly choosing the feedforward and feedback filter coefficients, we can reach the performance of the BDFE. We also show that the TV FIR DFE outperforms the TV FIR LE. REFERENCES [1] A. Stamoulis, G. B. Giannakis and A. Scaglione, Block FIR Decision- Feedbak Equalizers for Filterbank Precoded Transmissions with Blind Channel Estimation Capabilities, IEEE Trans. Signal Processing, vol. 49, pp , January [2] D. Williamson, R. A. Kennedy and G. W. Pulford, Block Decision Feedback Equalization, IEEE Trans. Commun., vol. 40, pp , February [3] N. Al-Dhahir and J. M. Cioffi, MMSE Decision-Feedback Equalizers: Finite Length Results, IEEE Trans. Inform. Theory, vol. 41, pp , July [4] I. Barhumi, G. Leus and M. Moonen, Time Varynig FIR Equalization of Doubly-Selective Channels, in Int. Conf. on Comm., (Anchorage, Alaska USA), May accepted. [5] G. Leus, I. Barhumi and M. Moonen, MMSE-Time Varying FIR Equalization of Doubly-Selective Channels, in Int. Conf. on ASSP, (Hong Kong), April accepted. [6] I. Barhumi, G. Leus and M. Moonen, Time Varynig FIR Equalization of Doubly-Selective Channels, IEEE Trans. Wireless Commun., Dec submitted. [7] G. B. Giannakis and C. Tepedelenlioğlu, Basis Expansion Models and Diversity Techniques for Blind Identification and Equalization of Time Varying Channels, Proc. IEEE, vol. 86, no. 10, pp , Oct., [8] A. M. Sayeed and B. Aazhang, Joint Multipath-Doppler Diversity in Mobile Wireless Communications, IEEE Trans. Commun., vol. 47, pp , January [9] X. Ma, and G. B. Giannakis, Maximum-Diversity Transmissions over Doubly-Selective Wireless Channels, IEEE Trans. Inform. Theory, revised, February see also Proc. of WCNC., vol. 1, pp , Orlando, FL, March 17-21, [10] G. Leus, S. Zhou and G. B. Giannakis, Orthogonal Multiple Access over Time- and Frequency-Selective Fading, IEEE Trans. Inform. Theory, revised, June, see also Proc. of the ACSSC, Pacific Grove, California, November 4-7, 2001.