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Performance of fractionally spaced blind channel shortening.

Abstract

In this paper we derive the Multicarrier Equalization by Restoration of RedundancyY (MERRY) algorithm for a fractionally-spaced channel shortening (or time domain equalizer) (FSTEQ and show that the FSTEQ MERRY algorithm converges significantly faster than the non-fractional TEQ MERRY algorithm. The main reason is that a fractionally-spaced blind adaptive TEQ admits infinitely many realizations of perfect channel shortening for a specific delay whereas a non-fractionally-spaced TEQ admits only one realization. Computer simulation demonstrates the performance improvement provided by the blind adaptive fractionally-spaced TEQ using MERRY algorithm for multicarrier systems.

Keywords: blind, fractionally-spaced time-domain equalizer (FSTEQ, multicarrier modulation, orthogonal frequency division multiplexing (OFDM).

Introduction

Multicarrier (MC) modulation, such as orthogonal frequency division multiplexing (OFDM) and discrete multitone (DMT) can easily combat channel dispersion when the channel delay spread is not great than the length of the cyclic prefix (CP) [1]. However, when the CP is not long enough, the orthogonality of the subcarriers is lost, causing intercarrier and intersymbol interference (ICI and ISI), and a prefilter is needed at the receiver to shorten the effective channel to appropriate length. This prefilter is called a time-domain equalizer (TEQ [2-7].

Channel shortening is a generalization of equalization, since equalization amounts to shortening the channel to length one. Most channel shortening (or TEQs) schemes in the literature have been designed in the context of ADSL, which runs over twisted pair telephone lines [3-6]. As a consequence, most of the TEQ designs in the literature are trained and nonadaptive, have high complexity.

Recently, blind and adaptive TEQ design has received increasing attention. The Multicarrier Equalization by Restoration of RedundancY (MERRY) algorithm [7], induces channel shortening by restoring the redundancy in the received data that is due to the CP. The algorithm is low-complexity and converges to the minimum MSE solution (for a white input).

A fractionally-spaced TEQ is a transversal filter whose tap spacing is less than the symbol interval [T.sub.s]. It was shown in [8-11] that when the tap spacing is less than the reciprocal of twice the highest frequency of the analog channel, a fractionally-spaced equalizer can realize any analog filter, including the best linear receiver; and its steady state performance--mean square error (MSE)--is insensitive to the timing phase. In this paper, we apply the MERRY algorithm to the blind adaptive fractionally-spaced TEQ and we will examine its converging speed. The paper is organized as follows. In Sec. 2, we formulate the fractionally-spaced TEQ problem for a multicarrier system (OFDM). In Sec. 3, the MERRY algorithm is derived for a blind fractionally-spaced setting. In Sec. 4, we first derive the number of realizations of perfect channel shortening for both the Ts-spaced and fractionally-spaced TEQs and then relate the number of realizations to the convergence speed of blind channel shortening algorithms. In Sec. 5, we show by computer simulations that the FSTEQ-MERRY algorithm converges significantly faster than the non-fractionally-spaced TEQ MERRY algorithm. This is because that a FSTEQ admits infinitely many realizations of perfect equalization for a specific delay.

[FIGURE 1 OMITTED]

where (I)FFT denotes (inverse) fast Fourier transform, P/S: parallel to serial, S/P: serial to parallel, CP: add cyclic prefix, and xCP: remove cyclic prefix.

Fractionally-Spaced Teq for Multi-Carrier Systems

The baseband discrete-time model of multicarrier system with a fractionally-spaced TEQ is depicted in Figure 1, and Table I gives the FSTEQ notation. For simplicity, we will consider the Ts/2-spaced TEQ, where Ts denotes the symbol period. For notation convenience, the index k is reserved for Ts-spaced quantities and index n for Ts/2-spaced quantities. Each of the N frequency bins is modulated with a Quadrature Amplitude Modulated (QAM) signal. Modulation is performed via an inverse fast Fourier transform (IFFT), which converts the frequency-domain data into a time-domain signal, and FFT block is used at the receiver for demodulation. A (possibly complex-valued) Ts/2-spaced symbol sequence {[bar.x](n)} is transmitted through a pulse shaping filter. The received signal [bar.r](n) is also corrupted by additive channel noise. A fractionally-spaced time-domain equalizer (FSTEQ of length 2[L.sub.w] (where [L.sub.w] is the length of Ts-spaced TEQ is employed to shorten the channel, and the FSTEQ output {[bar.y](n)} is decimated by a factor of two to create the [T.sub.s]-spaced output sequence {y(k)} . Decimation is accomplished by disregarding alternate samples thus producing the Ts-spaced y(k). Finally, the resulting shortened combined channel is equalized by a bank of one-tap frequency-domain equalizers (FEQs).

After the cyclic prefix (CP) add, the last v samples are identical to the first v samples in the symbol, i.e.,

x(Mk+i) = x(Mk+i+N), i [member of] {1, ....., v} (1)

where M=N+v is the total symbol duration, and k is the symbol index. The received Ts/2-spaced data sample is

[bar.r](n) = [2[L.sub.h]-1.summation over (i=0)] [[bar.h].sub.i] x [bar.x] (n - i) + [bar.e](n) (2)

where the Ts/2-spaced sequence {[bar.x](n)} is a zero-filled version of the transmitted symbol sequence {x(k)} defined by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3)

The channel is specified by Ts/2-spaced complex valued channel impulse response (CIR) given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4)

with [L.sub.h] corresponding to the Ts-spaced CIR length, and Ts/2-spaced sample [bar.e](n) = [[bar.e].sub.R] (n) + j[[bar.e].sub.I], (n) is an complex Gaussian white noise with, E [[[bar.e].sup.2.sub.R] (n)] = E [[[bar.e].sup.2.sub.I] (n)] = [[sigma].sup.2.sub.e] where E[x] denotes the expectation operator.

To shorten the channel length, the Ts/2-spaced TEQ data [bar.y](n) is similarly obtained from [bar.r](n) by

[bar.y](n) = [2[L.sub.w]-1.summation over (i=0)] [[bar.w].sub.i] x [bar.r](n - i) = [[bar.w].sup.T] [bar.r](n) (5)

where 2[L.sub.w] is the order of the Ts/2-spaced TEQ,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6)

is the FSTEQ complex-valued weight vector, and

[bar.r] = [[[bar.r](n) [bar.r](n-1) .......... [bar.r](n-2[L.sub.w] + 1)].sup.T] (7)

is the TEQ input vector. The FSTEQ output [bar.y](n) is decimated by a factor of 2 to create the Ts-spaced output y(k).

It can easily be shown [8] that the system model of Figure 1 is equivalent to the model depicted in Figure 2 by defining

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (8)

and

[e.sup.e](k) = [bar.e](2n), [e.sup.o](k) = [bar.e](2n + 1), [r.sup.e](k) = [bar.r](2n), [r.sup.o](k) = [bar.r](2n + 1) (9)

Further define

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (10)

and

r(k) = [[r(k) r(k-1) ... r(k-2[L.sub.w] + 1)].sup.T] = [[[([r.sup.e](k)).sup.T] [([r.sup.o](k)).sup.T]].sup.T] (11)

with [r.sup.e](k) = [[[r.sup.e](k) [r.sup.e](k - 1).... [r.sup.e] (k - [L.sub.w] + 1)].sup.T]

and [r.sup.o](k) = [[[r.sup.o](k) [r.sup.o] (k-1).... [r.sup.o] (k-[L.sub.w] + 1)].sup.T]. Then the Ts-spaced TEQ output y(k) is given by

y(k) = [2[L.sub.w-1].summation over (i=0)] [w.sub.i] x r(k-i) = [w.sup.T] r(k) (12)

The channel shortening model equation (12) forms the basis for discussion of the blind adaptive FSTEQ-MERRY algorithm.

[FIGURE 2 OMITTED]

Fractionally-Spaced Merry Algorithm

Throughout this section, we assume that the IFFT input is zero-mean, white, and wide sense stationary (implying that the output bins of each IFFT are uncorrelated)--the length of the effective channel is no larger than the FFT size ([L.sub.c] + 1 [less than or equal to] N)--the noise autocorrelation function [R.sub.e]([DELTA])= 0 for [DELTA] [greater than or equal to] N - [L.sub.w]--and the noise is uncorrelated with the data.

The idea behind MERRY [7] is that if the channel length [L.sub.h] + 1 [less than or equal to] v, then the last sample in the CP should match the last sample in the symbol. The MERRY cost function reflects this goal:

[J.sub.[delta]] =E[[[absolute value of y(Mk + v + [DELTA]) - y(Mk + v + N + [DELTA])].sup.2]] (13)

where v [member of] {0, ..., M-1} and the symbol synchronization parameter [DELTA] represents the desired delay. Minimizing (13) minimizes the energy of the channel outside of a length-v window.

In Ts-spaced TEQ setting, by defining the TEQ coefficient vector and the received signal vector as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (14)

the MERRY adapts [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] according to

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (15)

where [mu] is a small positive adaptive gain and [r.sup.*](x) is the complex conjugate of r(x). Similarly, in a fractionally-spaced ([T.sub.s]/2-spaced) TEQ, by using (12) and, defining the FSTEQ coefficient vector (10) and the received signal vector (11) as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

we have the FSTEQ coefficient updating formulas

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (16)

Perfect Fractionally- Spaced Blind Channel Shortening

In this section, we first derive the number of realizations of perfect channel shortening for both the Ts-spaced and fractionally-spaced TEQs and then relate the number of realizations to the convergence speed of blind channel shortening algorithms.

Denoting with [H.sub.k](z) and [W.sub.k](z) (k = 1, 2) the z-domain transfer function of the k-th channel and filter, respectively, perfect ISI/ICl cancellation is achieved if the combined channel-TEQ transfer function, corresponding to the system of Figure 2, satisfies the relationship

C(z) = [H.sub.1](z) [W.sub.1](z) + [H.sub.2](z) [W.sub.2](z) = [z.sup.[DELTA]] (17)

where [DELTA] is a possible delay.

In this paper we assume that the average power of the transmitted data is known. Thus, perfect channel shortening corresponds to y(k) = x(k-[DELTA]). In the Ts-spaced TEQ, corresponding to k = 1, then (17) is equivalent to

W(z) = 1/H(z) * [z.sup.-[DELTA]] (18)

that is, the transfer function of the perfect TEQ is the reciprocal of the transfer function of the sampled channel scaled by a linear phase factor resulting from the delay. Therefore, there is only one Ts-spaced realization of perfect TEQ for a specific delay.

In general, for a fractionally-spaced TEQ with K >1, however, the perfect channel shortening corresponds to

C(z) = [K.summation over (k=1)] [C.sub.k](z) = [K.summation over (k=1)] [H.sub.k](z) [W.sub.k](z) = [z.sup.-[DELTA]] (19)

since all these [C.sub.k](z) result in C(z) = [z.sup.-[DELTA], we conclude that there exists infinitely many fractionally-spaced perfect TEQ for a specific delay.

Now, we establish the relations of the number of realizations of perfect channel shortening to the convergence speed of the blind TEQ MERRY algorithm.

The ultimate objective of a blind TEQ algorithm is to reach perfect channel shortening. It does not matter, however, which realization of perfect channel shortening the TEQ finally converges to. For a Ts-spaced case, TEQ has to converge to the one and only realization of perfect channel shortening (for a specific delay. However, for a fractionally-spaced TEQ, there are infinitely many realizations of perfect channel shortening (for a specific delay) it can converge to. Since a gradient method is used, the fractionally-spaced equalizer will most probably converge to the realization closest to its initial setting [9].

Simulation Results

We present simulations results of FSTEQ MERRY algorithm for standard DSL test channel (CSA loop 1) [5]. The channel output is contaminated by additive white Gaussian noise with a signal-to-noise ratio (SNR) of 40dB. The FFT size is 512 and the CP length is 32. The step-size [mu] is selected so that the algorithm can achieve the fastest convergence without causing instability. The TEQ is initialized by setting the central tap to 1 and the other taps to 0. In the FSTEQ, 13 taps are selected in each of the two filters ([w.sub.1] and [w.sub.2]), whereas the length of the Ts-spaced TEQ is chosen to be 16.

In Figures 3, 4 and 5, we compare the performance of the MERRY-FSTEQ with conventional MERRY-TEQ in terms of cost function and bit rate for a fixed probability of error [3, 5-7] (we assume a 6-dB margin and 4.2-dB coding gain).

In Figures 3 and 4, it is clearly seen that MERRY algorithm, with fractionally-spaced TEQ, converges faster than the non-fractionally MERRY algorithm and can rapidly provide a solution approaching more the optimal MERRY. In order to perform bit allocation at the end of the initialization period, the Ts-spaced MERRY algorithm needs approximately 16000 iterations, while the FSTEQ MERRY algorithm needs 8000 iterations.

[FIGURE 3 OMITTED]

[FIGURE 4 OMITTED]

In Figure 5 we report the average bit rate versus SNR, obtained after 5000 iterations of the steep-est descent method. As we can see, the performance of the adaptive blind fractionally-spaced MERRY algorithm approaches much more the maximum shortening SNR (SSNR) solution [4]. Therefore, the fractionally-spaced TEQ for multicarrier system allows higher data rates to be achieved due to its more powerful channel shortening capability.

We would like to point out that the simulation results are due to the interplay of two factors. One is that a fractionally-spaced TEQ admits infinitely many realizations of perfect channel shortening for a specific delay. The other is that, the fractionally-spaced TEQ has twice as many taps as the Ts-spaced TEQ does and should result in slower convergence speed.

[FIGURE 5 OMITTED]

Conclusion

Fractionally-spaced blind channel shortening MERRY algorithm are formulated and shown to converge significantly faster than the corresponding non-fractionally-spaced ones. This is because a fractionally-spaced TEQ admits infinitely many realizations of perfect channel shortening for a specific delay whereas a non-fractionally-spaced TEQ admit only one realization.

References

[1] A. N. Akansu, P. Duhamel, X. Lin, and M. de Courville, "Orthogonal transmultiplexers in communication: A review," IEEE Trans. Signal Process., vol. 46, pp. 979-995, Apr. 1998.

[2] J. S. Chow, J. M. Cioffi, and J. A. C. Ingham, "Equalizer training algorithms for multicarrier modulation systems," in Proc. IEEE Int. Conf Commun., May 1993, pp. 761-765.

[3] N. Al-Dhahir and J. M. Cioffi, "Optimum finite-length equalization for multicarrier transceivers," IEEE Trans. Commun., vol.44, no. 1, pp. 56-64, Jan. 1996

[4] P. J. W. Melsa, R. C. Younce, and C. E. Rohrs, "Impulse response shortening for discrete multitone transceivers," IEEE Trans. Commun., vol. 44, no. 12, pp. 1662-1672, Dec. 1996.

[5] B. Farhang-Boroujeny and M. Ding, "Design methods for time-domain equalizers in DMT transceivers," IEEE Trans. Commun., vol. 49, no. 3, pp. 554-562, Mar. 2001.

[6] G. Arslan, B. L. Evans, and S. Kiaei, "Equalization for discrete multitone receivers to maximize bit rate," IEEE Trans. Signal Processing, vol. 49, no. 12, pp. 3123-3135, Dec. 2001.

[7] R. K. Martin, J. Balakrishnan, W. A. Sethares, and C. R. Johnson, Jr., "A Blind, Adaptive TEQ for multicarrier Systems," IEEE Signal Processing Letters, vol. 9, no. 11, pp. 341-343, Nov. 2002.

[8] R. Johnson Jr., P. Schniter, T.J. Endres, J.D. Behm, D.R. Brown, R.A. Casas, "Blind equalization using the constant modulus criterion: a review," Proc. IEEE 86, pp. 1927-1950. Oct. 1998.

[9] Y. Chen, C. Nikias, "Fractionally-spaced blind equalization with CRIMNO algorithm," In. Proc. IEEE MILCOM, vol. 1, San Diego, CA, pp. 221-225, 11-14 Oct. 1992.

[10] R.D. Gitlin, and S.B. Weinstein, "Fractionally-spaced equalization: an improved digital transversal equalizer," The Bell System Technical Journal, vol. 60, pp. 275-296, Feb. 1981.

[11] PP. Vaidyanathan and B. Vrcelj, "Theory of fractionally spaced cyclic-. prefix equalizers," in Proc. ICASSP'02, vol. 2, 2002, pp. 1277-1280.

S. A. Elahmar (1), A. Djebbari (1), M. Bouziani (1) and J. M. Rouvaen (2)

(1) Telecommunications and Signal Processing Laboratory The University of Djillali Liabes, BP 89 Sidi Bel Abbes, 22 000, Algeria.

(2) Department of Electronic Acoustic Optical The University of Valenciennes, Le Mont Houy Valenciennes, France. Email: silahmar@yahoo.fr, adjebbari@univ-sba.dz, mbouziani@univ-sba.dz and rouvaen@univ-valenciennes.fr
Table I: Fractionally-Spaced Channel Shortening Notation

Notation Definition

x(k) transmitted signal with Ts-spaced
[bar.x](n) transmitted signal with Ts/2-spaced
[bar.e](n) additive noise
[bar.r](n) received data with Ts/2-spaced
[bar.y](n) output of FSTEQ
N, v, M sizes of FFT, CP, and symbol
[DELTA] transmission delay
[bar.h] channel impulse response
[bar.w] FSTEQ impulse response
[bar.c] effective channel [bar.c]=
 [bar.h] * [bar.w]
[L.sub.h], [L.sub.w], [L.sub.c] order of h, w, or c
[R.sub.e]([DELTA]) autocorrelation function of noise
[A.sup.*], [A.sup.T] conjugate and transpose
bps bit per second
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Author:Elahmar, S.A.; Djebbari, A.; Bouziani, M.; Rouvaen, J.M.
Publication:International Journal of Applied Engineering Research
Article Type:Report
Geographic Code:6ALGE
Date:Jan 1, 2007
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