Spectral Transformations

Abstract

Spectral transformations are key to many speech-processing algorithms. The purpose of a spectral transformation is to represent a signal in a domain where certain signal properties are better accessible or a specific processing task is more efficiently accomplished. In this chapter, the authors summarize the definitions and properties of the Fourier transform (FT) for continuous and discrete time signals as well as the discrete Fourier transform (DFT), its fast realizations, and the z-transform. The FT provides an analysis of signals in terms of its spectral components. A system that is linear and shift invariant is called a linear shift invariant system. The DFT coefficients represent the spectrum of the input signal at equally spaced points on the frequency axis. However, when the support of the signal is larger than the transformation length, the DFT coefficients are not identical to the FT of the complete signal at these frequencies.

Authors 2

1. Peter Vary Aachen Institute of Communication Systems

   RWTH Aachen University

   Affiliation as printed

   RWTH Aachen University, Institute of Communication Systems, Aachen, Germany

2. Rainer Martin

   Ruhr University Bochum

   Affiliation as printed

   Ruhr-Universität Bochum, Institute of Communication Acoustics, Bochum, Germany