Performance of fractionally spaced blind channel shortening

International Journal of Applied Engineering Research, Jan, 2007 by S.A. Elahmar, A. Djebbari, M. Bouziani, J.M. Rouvaen

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.,