Digital Filters Design for Signal and Image Processing

5.2. Synthesizing FIR filters using frequential specifications

Synthesizing finite impulse response filters is the main step that helps to fix coefficient values of the impulse response. These samples, called filter coefficients, are obtained by trying to approach as closely as possible an ideal frequency response. Many models exist and it is difficult to present an exhaustive list. However, several classes are notable for their simplicity or their performance in terms of approximating an ideal filter.

The first method presented here is the best known for its properties and for its simplicity. Commonly known as the windowing method, it corresponds to a weighting of the truncated impulse response of a filter following directly from specifications with ideal frequency. In the section that follows, we present many weightings allowing for a compromise between attenuation in the stop-band and the rapid decrease of the transition band. In the following sections we will also discuss this part of the influence of truncation on the impulse response of the ideal filter.

The second method, which entails more complex calculations, is an optimal approach in the sense of minimizing a “cost” function expressed by the gap between the impulse response of the ideal filter and that which we are trying to synthesize.

5.2.1. Windows

Usually, we cannot simultaneously process all the samples of a signal; we process them in reduced segments, chosen with an analysis window. By choosing a size of an adapted ...