Overview

The computational background to digital signal processing (DSP) involves a number of techniques of numerical analysis. Those techniques which are of particular value are:

• solutions to linear systems of equations

• finite difference analysis

• numerical integration

A large number of DSP algorithms can be written in terms of a matrix equation or a set of matrix equations. Hence, computational methods in linear algebra are an important aspect of the subject. Many DSP algorithms can be classified in terms of a digital filter. Two important classes of digital filter are used in DSP, as follows.

Convolution filters are nonrecursive filters. They use linear processes that operate on the data directly.

Fourier filters operate on data obtained by computing the discrete Fourier transform of a signal. This is accomplished using the fast Fourier transform algorithm.

2.1 Digital Filters

Digital filters fall into two main categories:

• real-space filters

• Fourier-space filters

Real-space filters Real-space filters are based on some form of moving window principle. A sample of data from a given element of the signal is processed giving (typically) a single output value. The window is then moved on to the next element of the signal and the process repeated. A common real-space filter is the finite impulse response (FIR) filter. This is a non-recursive filter (in a non-recursive filter the output depends only on the present and previous inputs), involving a discrete convolution operation for sample i of the form