# The Nonuniform Discrete Fourier Transform

• S. Bagchi
• S. K. Mitra
Part of the Information Technology: Transmission, Processing, and Storage book series (PSTE)

## Abstract

In many applications, when the representation of a discrete-time signal or a system in the frequency domain is of interest, the Discrete-Time Fourier Transform (DTFT) and the z-transform are often used. In the case of a discrete-time signal of finite length, the most widely used frequency-domain representation is the Discrete Fourier Transform (DFT), which is simply composed of samples of the DTFT of the sequence at equally spaced frequency points, or equivalently, samples of its z-transform at equally spaced points on the unit circle. A generalization of the DFT, introduced in this chapter, is the Nonuniform Discrete Fourier Transform (NDFT), which can be used to obtain frequency domain information of a finite-length signal at arbitrarily chosen frequency points. We provide an introduction to the NDFT and discuss its applications in the design of 1-D and 2-D FIR digital filters. We begin by introducing the problem of computing frequency samples of the z-transform of a finite-length sequence. We develop the basics of the NDFT, including its definition, properties and computational aspects. The NDFT is also extended to two dimensions. We propose NDFT-based nonuniform frequency sampling techniques for designing 1-D and 2-D FIR digital filters, and present design examples. The resulting filters are compared with those designed by other existing methods.

## Keywords

Discrete Fourier Transform Filter Design Goertzel Algorithm Filter Design Method Stopband Edge
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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