A new approach for channel equalization using ... - Semantic Scholar

A New Approach for Channel Equalization Using Wiener Filtering C´assio B. Ribeiro, Marcello L. R. de Campos, and Paulo S. R. Diniz Electrical Engineering Program COPPE/Federal University of Rio de Janeiro P.O. Box 68504, 21945-970, Rio de Janeiro, RJ, Brazil Abstract—In this work we present one efficient structure based on Wiener filter to perform channel equalization in a block communications system. We show that it is possible to derive a structure to estimate a block of transmitted symbols that use the same Wiener filter to estimate all symbols in the block. In terms of bit error rate, experimental results show that the system performs better than Discrete MultiTone (DMT) for a wide range of signalto-noise ratios. A multistage Wiener filter recently introduced was used to implement a reduced-rank version of the equalizer, and the performance of the system was shown to remain almost unchanged for rank values much smaller than the signal autocovariance matrix rank.

I. I NTRODUCTION In modern communications systems it is sometimes desirable to employ block transmissions in order to increase system capacity or provide multiple access. In such systems the equalization is performed independently for each user, like in Orthogonal Frequency Division Multiplex (OFDM), Discrete MultiTone (DMT) and Discrete Wavelet MultiTone (DWMT) [1]. This implies that several filters should be designed in order to recover the transmitted blocks, and thus the receiver can become very complex as the number of users grows. We introduce a very efficient implementation of the Wiener filter for channel equalization in a communications system. The transmitter is designed to eliminate inter-block interference (IBI) between two consecutive transmitted blocks. We show that all elements of the transmitted block can be estimated from the elements of the received block using only one Wiener filter preceded by trivial linear transformations. These transformations circulate the elements of a received block of signals and thus can be implemented very efficiently in digital signal processors by means of circular buffers with minimum added complexity. The bit error rate (BER) of the proposed system was compared to that of DMT for a wide range of signalto-noise ratios over a dispersive channel, and the results show that the proposed system performs better than DMT as the SNR grows. The MultiStage Wiener Filter (MSWF) introduced in

[2] is used as an alternative implementation of the Wiener filter that allows straightforward use of rank-reduction, without the need of eigen-decomposition of the input covariance matrix. In a reduced-rank filter, the input signal is projected in a lower-dimension subspace, and filter optimization occurs within this subspace [2, 3]. This approach has the advantage of reducing the number of filter coefficients that need to be estimated, leading to faster convergence speed. In Section II we describe the communications system. In Section III we present the proposed equalization structure. In Section IV we describe the MSWF and how it can be used to implement reduced-rank equalizer. In Section V we show simulation results comparing the performance of the proposed system with DMT, and evaluating the performance of the reduced-rank equalizer as the filter rank decreases. II. S YSTEM D ESCRIPTION We consider baseband transmissions through a channel modeled as an Lth-order FIR filter. The received signal is given by x(n) =

L X

l

=0

h (l)u(n

l) +  (n)

(1)

where u(n) is the transmitted signal, h(l) is the channel impulse response, and  (n) is zero mean additive Gaussian noise. In order to explore efficiently channel capacity, it is often desirable to employ block transmissions. Let us con(k), where sider the transmission of a length-P vector u P  L. From (1), we realize that the received signal is (k) given by the length-P vector x

x (k) = H0 u (k) + H1 u (k

1) +  (k)

(2)

(k) is a zero mean complex Gaussian noise vecwhere  tor, and the P  P matrices H0 and H1 are defined as [H0 ]i;j = h (i j ) and [H1 ]i;j = h (i j + P ).

0-7803-7206-9/01/$17.00 © 2001 IEEE

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We will consider a transmission scheme similar to OFDM [4, 5], where the first L symbols of the transmitted block consist of a repetition of the last L symbols (cyclic prefix). Thus, only the last N = P L symbols of the transmitted block actually contain information. By discarding the first L symbols in the received vector IBI is eliminated at the receiver. The communications system is shown in Figure 1, where Tx introduces the cyclic prefix whereas Rx removes the cyclic prefix. This system can be represented equivalently by

x(k) = Hu(k) + (k)

(3)

where H is an N  N circulant matrix defined as [H]i;j = h ((i j ) mod (L + 1)), where mod denotes the signed remainder after division, and u(k ) is the lengthN vector with the users’ symbols. An example of H for P = 5 and L = 2, is given below: 2 6

h(0) h(2) h(1)

~ = 6 h(1) h(0) h(2) H 4

3 7 7 5

h(2) h(1) h(0)

An N  N circulant matrix is diagonalizable by the matrix that implements the N -point DFT. This implies that H 1 exists if and only if the channel transfer function has no zero in the DFT grid[4]. This fact guarantees that it is possible to invert the channel, and thus perform equalization, for the adopted transmission scheme.

For the Wiener filter to estimate the process ui (k ), i = 1 the corresponding cross-correlation is given

1; : : : ; N

by

p

xui

=

u2 [h(N

i)






h(N

1)

h(0)




h(N

p

xu0

=

Tp c

xu0

=

E [x(k )u0 (k )] = u2 [h(0)




h(N

1)]

T

(4)

(5)

xu1

;

(6)

where Tc circulates the elements of pxu1 , and is defined as  

T

c

I

c

=

(7)

e1

where Ic is an (N 1)  N matrix constructed with the last (N 1) rows of the N  N identity matrix, and e1 is a length-N row vector constructed with the first row of the N  N identity matrix. Based on the relation above, we can derive the relation between the Wiener filter w0 , to estimate u0 (k ) from x(k), and the Wiener filter w1 , to estimate u1 (k) from x(k). The Wiener filter w1 is constructed as

w1 = R 1 p

xu1

x

;

(8)

where Rx 1 is the inverse of the autocorrelation matrix of x(k), given by E [x(k)xH (k)], where (
)H denotes transpose complex conjugate. From (6) and using the fact that Rx 1 is Toeplitz, and Tc is unitary,

w1 = R 1 (T 1 p 0 ) 1 = (R T )p 0 1 = T (R p 0 ) = T w0 x

p

T

1)

From (4) and (5),

III. P ROPOSED E QUALIZATION S TRUCTURE In order to use a conventional Wiener filter to perform the equalization task, we should use N Wiener filters, each filter responsible for the estimation of one symbol ui (k ), i = 0; : : : ; N 1. In many applications, online estimation of N Wiener filters may limit the throughput of the system or otherwise may increase unnecessarily the cost of the receiver. We can avoid the computation of the N filters by exploring the fact that the channel matrix is circulant. In the sequel we will assume that the elements of ui (k ) are independent identically distributed samples of a random process, with zero mean and variance u2 . A necessary condition to build the cyclic prefix is that N  L. Without loss of generality we will consider the transmission of a block with N = L + 1 symbols. The cross-correlation vector between the received vector in (3) and the desired process u0 (k ) is given by

i

x

c

c

xu

H c

xu

(9)

xu

x

c

The relation in (9) can be extended for all ui (k ), i 1. From (4) and (5),

=

0; : : : ; N

(Tc )

i

p

xui

=

p

xu0

(10)

Following the same guidelines of (9) we conclude that

w

i

= (Tc )

i

w0

(11)

Figure 2 shows a block diagram of the proposed equalization structure. It should be noted that the multiplication by (Tc )i does not involve actual multiplications and additions and can be performed very efficiently in digital signal processors.

291

 (n)

u(k)

u  (k )

Tx

? x(n) x (k) - - S/P Rx

u(n)

- h(n)

P/S

x(k)

Fig. 1. Block diagram of the communications system. d0 (k)

- w0H

u^0 (k)

-

 (k )

u(k)

- H~

?

-

d1 (k)

B1

-

- Tc

- w0H

u^1 (k)

?

-

- w0H

. . .

u^2 (k)

-

? -

dN

- TNc

- w0H

1

u^N

1

1

(k ) ? -

MSWF to estimate d0 (k ) from x0 (k ) is shown in Figure 3, where the length-(N i + 1) vector ri is a normalized vector in the direction of the cross-correlation vector between the observed and desired processes in the previous stage, i.e.,

-

(k)

-

r

i

Fig. 2. Block diagram of the proposed channel equalization structure.

Finally, from (3) and (11), the estimate of u(k ) is given by

6

^ 1=6 H 6

3

w0 w0 T H

H

c

.. .

4

w0 T H

N c

1

7 7 7 5

(13)

IV. M ULTISTAGE W IENER F ILTER The multistage Wiener filter was introduced in [2] by Goldstein et. al. It is a representation of the N th-order Wiener filter achieved by a multistage decomposition and can be naturally divided into an analysis filterbank and an error-synthesis filterbank. In [3, 6] this decomposition is employed for interference suppression in DS-CDMA systems, and adaptation algorithms are proposed in [3]. The analysis filterbank performs a change in the coordinate system prior to Wiener filtering. A four-stage (N = 4) example of the analysis filterbank for the

= q (E [xi

1 (k)d 1 (k)] 1 (k)d 1 (k)]) E [x 1 (k)d 1 (k)] E [xi

i

H

i

i

i

(14) and the (N i)(N i+1) matrix Bi spans the nullspace of ri , i.e., Bi ri = 0. The analysis filterbank can be represented by [2]

^(k) = H ^ 1 Hu(k) + H ^ 1 (k) (12) d ^ 1 is the estimated inverse of H ^ , calculated as where H 2

x3 (k) = d4(k )

Fig. 3. Analysis filter bank for a four-stage MSWF.

. . .

. . .

x2(k)

B2

d2 (k)

- T2c

B3

d2 (k)

r2

x1(k)

d1 (k)

? x(k)

r3

d3 (k) -

r1

x0 ( k )

"

L

=

N

r1 B1 r2 H

Y2

!

N






=1

B

H i

Y1

#H

N

r

N

1

i

=1

B

H i

i

(15) The error-synthesis filterbank operates on the output of the analysis filterbank, and is constructed as a nested chain of scalar Wiener filters [2], as shown in Figure 4 for N = 4. Let the length-N vector d(k ) =  T represent the output of d1 (k ) d2 (k )


dN (k ) the analysis filterbank, and define the length-N vector

w

H z

=



w1

w1 w2 : : :

(

1)N +1

QN

=1 w

i





i

(16) to represent the error-synthesis filterbank. From (15) and (16), the mapping from the MSWF to the equivalent Wiener filter is given by

w0

H

=

w L H z

N

(17)

The MSWF provides us a natural way to implement rank-reduction [2]. In order to design a MSWF with rank

292

()

- 0 (k-) +  -6

d0 k

()

+  -6

d2 k

() ()

()

2 k

+  -6

d3 k

d4 k

()

+  -6

1 k

()

3 k

-

-

6

-

6

-

6

0 -5

6

Magnitude Response (dB)

()

d1 k

w1

w2

w3

w4

-10 -15 -20 -25 -30 -35

Fig. 4. Synthesis filter bank for a four-stage MSWF.

-40 0

0.2

0.4

0.6

D , it suffices to proceed with the multistage decomposition until D -stages are obtained. In this case, the analysis filterbank in (15) is calculated as

D

r1 B1 r2 H

=








1 =1 B

D

H i

i

r

iH

(18)

D

and the error-synthesis filterbank is given by

w

H z

=



w1

w1 w2 : : :

(

1)

D

+1 Q

D

=1

i

Wiener DMT

0.1 Average Bit-Error Rate

L

Q

w

1

Fig. 5. Channel magnitude response. 1

h

0.8

Normalized frequency



i

(19)

0.01 0.001 0.0001 1e-05 1e-06 1e-07

V. E XPERIMENTAL R ESULTS

0

5

10

15

20

25

30

Signal-to-Noise Ratio (dB)

In this section we show experimental results obtained through computer simulations. The performance of the proposed system is measured as the average bit error rate (BER) for 100 transmissions of 128000 QPSKmodulated symbols for signal-to-noise ratios (SNR) varying from 0 to 30 dB. We will consider two channel models, both FIR filters with 64 coefficients. The cyclic prefix length is 64 and N = 128. The Wiener filter w0 was estimated using the recursive least squares algorithm with forgetting factor  = 0:999. A training sequence was used to train the adaptive equalizer, and after convergence the filter coefficients were frozen. Figure 5 shows the magnitude response of the channel model considered for the first set of simulations. Figure 6 shows the resulting average BER. For high SNR, the proposed system performs better than DMT, while for low SNR the proposed system performs comparably to DMT. In order to explain this behavior, one has to pay attention at the main difference between the two systems: in the proposed system, all symbols are transmitted in the same channel, while in DMT the channel is divided in subchannels that carry one symbol each [4]. As SNR increases, for the proposed system the accuracy of the estimate of all symbols increases, while in DMT the accuracy

Fig. 6. Average BER for first set of simulations.

of the estimate of the symbols transmitted in subchannels that have low SNR remain poor. After the SNR increases beyond a certain level, both systems are able to have an accurate estimate of all symbols, leading to zero BER, but this level is different for the two systems. As a side result of the discussion above, for channels with notch characteristics in certain frequencies, the proposed system should have even better performance than observed for the first set of simulations, when compared to DMT. This would be the case of a channel that has a zero close to the unit circle, as shown in Figure 7. This channel model was used for the second set of simulations, and the resulting average BER is shown in Figure 8. As expected, we can observe that for this channel model, the performance of the proposed system increases faster with SNR than in the first set of simulations. For the third set of simulations, the performance of the reduced-rank MSWF equalizer is evaluated for a fixed SNR. Figure 9 shows the resulting average BER for rank varying from 4 to 128 using the channel model shown in Figure 5 and SNR=15 dB. The performance of the system remains near optimum for filter rank > 20.

293

0

0.2

Average Bit-Error Rate

Magnitude Response (dB)

-5 -10 -15 -20 -25 -30

0.15

0.1

0.05

-35 -40

0 0

0.2

0.4

0.6

0.8

1

0

Normalized frequency

60

80

100

120

Fig. 9. Average BER for rank varying from 4 to 128 and SNR=15 dB.

Wiener DMT

0.1 Average Bit-Error Rate

40

Filter Rank

Fig. 7. Channel magnitude response. 1

20

to values much smaller than the signal rank, while still obtaining performance comparable to the optimum filter. For the conventional Wiener or MSWF implementations, since the same filter is used to estimate all the transmitted symbols, several implementation alternatives are available, allowing the use of more complex and robust adaptive algorithms, leading to faster convergence speed.

0.01 0.001 0.0001 1e-05 1e-06 1e-07 0

5

10

15

20

25

30

R EFERENCES

Signal-to-Noise Ratio (dB)

Fig. 8. Average BER for first set of simulations.

VI. C ONCLUSIONS In this work an equalization structure that uses a sequence of linear transformations followed by a single Wiener filter was proposed to perform channel equalization in a communications system with block transmissions. In fact, the linear transformations can be implemented by means of shifts in circular buffers available in modern digital signal processors, and do not require either additions or multiplications. In case the transmitter has access to channel information, pre-equalization can also be implemented by moving the equalizer from the receiver to the transmitter. The performance of the proposed system, measured as the average BER, is compared to DMT. It was found that the proposed system performs better than DMT for the evaluated channel models over a wide range of SNR. This behavior was analyzed qualitatively, and it was found that it is closely related to presence of zeros near the unit circle. The MSWF was considered to implement the equalizer, allowing a straightforward implementation of rank reduction. It was found that the filter rank can be reduced

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