On Frequency-Domain Implementation of Digital FIR Filters Using Overlap-Add and Overlap-Save Techniques

TL;DR

This paper provides new insights into frequency-domain digital FIR filter implementations using overlap-add and overlap-save, analyzing system representations, effects of quantization, and computational complexity to optimize efficiency.

Contribution

It introduces two novel system representations for finite-wordlength implementations and derives formulas for optimal DFT lengths, demonstrating improved efficiency over time-domain methods.

Findings

Frequency-domain methods are more efficient for shorter filters.

Two system representations help analyze quantization and aliasing effects.

Optimal DFT length formulas improve computational efficiency.

Abstract

In this paper, new insights in frequency-domain implementations of digital finite-length impulse response filtering (linear convolution) using overlap-add and overlap-save techniques are provided. It is shown that, in practical finite-wordlength implementations, the overall system corresponds to a time-varying system that can be represented in essentially two different ways. One way is to represent the system with a distortion function and aliasing functions, which in this paper is derived from multirate filter bank representations. The other way is to use a periodically time-varying impulse-response representation or, equivalently, a set of time-invariant impulse responses and the corresponding frequency responses. The paper provides systematic derivations and analyses of these representations along with filter impulse response properties and design examples. The representations are particularly useful when analyzing the effect of coefficient quantizations as well as the use of shorter DFT lengths than theoretically required. A comprehensive computational-complexity analysis is also provided, and accurate formulas for estimating the optimal DFT lengths for given filter lengths are derived. Using optimal DFT lengths, it is shown that the frequency-domain implementations have lower computational complexities (multiplication rates) than the corresponding time-domain implementations for filter lengths that are shorter than those reported earlier in the literature. In particular, for general (unsymmetric) filters, the frequency-domain implementations are shown to be more efficient for all filter lengths. This opens up for new considerations when comparing complexities of different filter implementations.