Speech and Audio Signal Processing: Processing and Perception of Speech and Music, Second Edition

CHAPTER 7

DIGITAL FILTERS AND DISCRETE FOURIER TRANSFORM

7.1 INTRODUCTION

If we define a filter as a device that discriminates among incoming frequencies, then filters can be implemented, for example, by mechanical, acoustical, pneumatic, or electrical elements. In the past few decades, discrete-time algorithms employing computational elements have played an increasing role in filter applications. Most commonly, these elements also incorporate discrete numerical values (as opposed to continuous circuit variables¹) to represent signals. For simplicity we refer to these devices as digital filters. In this chapter, we deal with some of the issues that arise in the design of digital filters. Many of the ideas of digital signal processing pertain directly to digital filter design. However, in addition, many of the filtering concepts developed for the design of analog electrical filters [11] apply equally well to digital filter design. Section 7.2 is a review of these concepts. In Section 7.3 we show how simple mathematical transformations bridge the gap between the analog and digital world. The remainder of this chapter focuses on methods that have been primarily applied to discrete-time linear systems. Of particular interest is the application of the discrete Fourier transform (DFT) to filter theory and applications. Both filtering and the DFT are widely used for speech-processing ...