OVERVIEW

The aim of this chapter is to show how the filters presented in this book were designed, thereby providing a basis for understanding the limitations of the design. The filters described here were designed using the well-known windowing technique. In this technique, the desired frequency response H ( F), is first created in the frequency domain and then its inverse Fourier transform (IFT) is found. This yields the impulse response function H ( k), which is then discretized, truncated, and windowed to form h _k in the spatial or time domain k. Note that in the following discussion both H ( k) and H ( F) should be treated as continuous functions, whereas H _k and H _F are their respective discretized forms. The windowing and truncation of H _k yield the finite impulse response (FIR) digital filter, h _k.

In principle, the domain of the impulse response function H ( F) is infinite, but to be able to use it in a digital context it is truncated to some reasonable length. If an FFT is performed on the truncated form H _k, ringing is observed on the response H _F in the frequency domain. Ringing is characterized by a damped oscillatory behavior at the transition edges of the filter; this is a consequence of the well-known Gibbs phenomenon ^[1]. Moreover, truncation of the impulse response function...