Transforms of Discrete-Time Signals

Abstract

In this chapter, the invertible transforms used to work on discrete-time signals are discussed. Given a complex variable z, the z-transform is defined as an infinite series in the z-plane that exists in the region(s) of the plane where the series exhibits absolute convergence to an analytic function. The corresponding infinite-length signal is required to be absolutely summable. Unit-amplitude z values identify the unit circle, on which the z-transform becomes a continuous function of frequency, called the discrete-time Fourier transform (DTFT). The DTFT representation can also be extended to sequences for which the z-transform does not exist, such as signals that are only square-summable, or periodic signals like sinusoids. If a sequence has finite length, it may be represented in the frequency domain by a finite number of values obtained by properly sampling the DTF, i.e., by the discrete Fourier transform (DFT). The properties of the DFT emerge clearly if this transform is introduced passing through the discrete Fourier series (DFS) of the signal’s periodic extension. The DFT can be efficiently computed via fast Fourier transform (FFT). Each inverse transform represents an expansion of the signal in an orthogonal basis. At the end of the chapter, an appendix provides an overview of the mathematical foundations of analog and discrete-time signal expansions.