On the Compensation of Delay in the Discrete Frequency Domain
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On the Compensation of Delay in the Discrete Frequency Domain Gareth Parker Defence Science and Technology Organisation, P.O. Box 1500, Edinburgh, South Australia 5111, Australia Email: [email protected] Received 31 October 2003; Revised 19 February 2004; Recommended for Publication by Ulrich Heute The ability of a DFT filterbank frequency domain filter to effect time domain delay is examined. This is achieved by comparing the quality of equalisation using a DFT filterbank frequency domain filter with that possible using an FIR implementation. The actual performance of each filter architecture depends on the particular signal and transmission channel, so an exact general analysis is not practical. However, as a benchmark, we derive expressions for the performance for the particular case of an allpass channel response with a delay that is a linear function of frequency. It is shown that a DFT filterbank frequency domain filter requires considerably more degrees of freedom than an FIR filter to effect such a pure delay function. However, it is asserted that for the more general problem that additionally involves frequency response magnitude modifications, the frequency domain filter and FIR filters require a more similar number of degrees of freedom. This assertion is supported by simulation results for a physical example channel. Keywords and phrases: frequency domain, FDAF, transmultiplexer, equaliser, delay.
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INTRODUCTION
The term “frequency domain adaptive filter” (FDAF) [1] is often applied to any adaptive digital filter that incorporates a degree of frequency domain processing. Some time domain adaptive filtering algorithms, such as the least mean square (LMS), may be well approximated using such “frequency domain” processing, by employing fast Fourier transform (FFT) algorithms to perform the necessary convolutions [1]. The computational complexity of such implementations of these adaptive filters can be, for a large number of taps, considerably less than the explicit time domain forms. It is this computational advantage that is often the main motivation for using these architectures. Other advantages also exist, such as the ability to achieve a uniform rate for all convergence modes (see, e.g., [1]). Architectures that could be more deservedly labelled “frequency domain” can be achieved by transforming the time domain input signal into a form in which individual frequency components can be directly modified. This process can be approximated using a filterbank “analyser” [2], shown in Figure 1, which channels the input x(n) into relatively narrow, partially overlapping subbands, or “bins.” For clarity of illustration, the complex oscillator inputs to the multipliers in the analyser, e− j2πn fk / fs , are denoted simply by − fk , k = 0 · · · K − 1. In the synthesiser, the conjugate oscillators e j2πn fk / fs are similarly denoted by fk .
With a sampling frequency fs Hz, the output of a K bin filterbank analyser with decimation M at time mM/ fs is a vector of bins X(m) = [X(m, f0 ), . . . , X(m
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