Filtering practice — Digital filtering. Recursive filters

We present two methods for synthesizing recursive (or infinite impulse response) filters:

• by derivation equivalence ;

• by equivalence of integration, this is the bilinear transformation.

These filters are mainly used to transpose cells with well-known analog transfer functions, such as Butterworth or Chebychev filters. They use a reduced number of coefficients, so lend themselves well to rapid calculation, but they are very sensitive to coefficient errors and can become unstable.

We'll also look at the case of fast algorithm filters, which are not very efficient, but remain interesting because they require very little computation and can run at high frequencies on modest machines.

In recent years, the sensitivity of digital filters to coefficient errors has been the subject of much research, and less sensitive structures have been proposed. Here are a few examples of ladder and lattice filters.

Finally, it's becoming less and less expensive to have a Fast Fourier Transform (FFT), which can be used to filter a signal. Unfortunately, the Fourier transform operates on the signal as a whole. The FFT algorithm, on the other hand, works on signal sections of limited duration, and splitting the signal into pieces introduces spurious transients that cannot always be eliminated. There is, however, a method that can be used for finite impulse response filters, which will be described in this article.

This article is part of a series devoted to the practice of filtering:

• Filtering practice. Introduction [R 1 100] ;

• Practical filtering. Analog filtering [R 1 102] ;

• Practical filtering. Digital filtering. Transverse filters [R 1 105] ;

• Practical filtering. Digital filtering. Recursive filters [R 1 106].