This is a detailed chapter on digital filter design. Specific digital filters such as notch and antinotch filters, and sharp-cutoff lowpass filters such as Butterworth filters are discussed in detail. Also discussed are allpass filters and some of their applications, including the implementation of notch and antinotch filters. Computational graphs (structures) for allpass filters are presented. It is explained how continuous-time filters can be transformed into discrete time by using the bilinear transformation. A simple method for the design of linear-phase FIR filters, called the window-based method, is also presented. Examples include the Kaiser window and the Hamming window. A comparative discussion of FIR and IIR filters is given. It is demonstrated how nonlinear-phase filters can create visible phase distortion in images. Towards the end, a detailed discussion of steady-state and transient components of filter outputs is given. The dependence of transient duration on pole position is explained. The chapter concludes with a discussion of spectral factorization.

The objective of this chapter is to discuss digital filters. We start from a review of theory of Fourier Transform for continuous functions. The continuous Fourier Transform is then discretized. The discretized Fourier Transform and inverse Fourier Transform, however, are not approximate equations – they are exact. Using the shifting theorem, a filter can easily be expressed in the frequency domain. A Finite Impulse Response (FIR) filter is then defined. By adding another implicit convolution to the original convolution for FIR filter, the filtered data depends on not only the input (the original time series) but also the output (the filtered data). This is an iterative relation that forms the Infinite Impulse Response (IIR) filter. These filters are examples of so-called linear systems that have an input and output. The gain is defined by the filter, which is the ratio between the input and output in the frequency domain. Several FIR and IIR filter functions in MATLAB are discussed.