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Digital Image Processing (2000)

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Filter Design

9.1 Introduction

One of the most widely used signal/image processing operations is linear filtering. Filtering is used to pass certain frequency components of a signal through the system called the filter, and to block other frequency components. The range of frequencies being passed through the filter is called the passband, and the range of frequencies that are blocked is the stopband. In all realizable filters, the two bands are separated by a finite-width frequency region known as the transition band. The magnitude spectrum of a prototypical lowpass filter is shown in Figure 9.1.1.

Figure 9.1.1. Frequency response of a lowpass filter showing the passband (red), 0≤f≤0.2, transition band (green), 0.2<f<0.25, and stopband (blue), 0.25≤f≤ 0.5.

In this chapter we follow the common convention of defining the fundamental frequency interval in units of Hertz (Hz) as
The ability to design digital filters to a given specification is essential in implementing a broad class of linear, shift-invariant image processing operations, including noise filtering, image smoothing or sharpening, edge detection, template matching, and more. Recognizing this need, the Digital Image Processing package includes a number of popular FIR filter design functions. This chapter serves as an introduction to the theory and practice of digital filter design. For reasons of practicality and brevity, the discussion is restricted to a class of filters known as linear-phase FIR filters. Section 9.2 describes the basic features of FIR filters. Section 9.3 introduces three commonly used design algorithms: frequency sampling, the window method, and equiripple filter method. These methods may be used to design arbitrary 1D FIR filters. Extensions to 2D filters are presented in Section 9.4. Section 9.5 presents an example design of an important class of FIR filters, the so-called Nyquist or Lth band filter. These filters may be used in interpolation/decimation operations.