Learn more about Signal Processing Toolbox

Filter Implementation and Analysis

On this page…
Filtering Overview Convolution and Filtering Filters and Transfer Functions Filtering with the filter Function

Filtering Overview

This section describes how to filter discrete signals using the MATLAB^® filter function and other Signal Processing Toolbox™ functions. It also discusses how to use the toolbox functions to analyze filter characteristics, including impulse response, magnitude and phase response, group delay, and zero-pole locations.

Back to Top

Convolution and Filtering

The mathematical foundation of filtering is convolution. The MATLAB conv function performs standard one-dimensional convolution, convolving one vector with another:

conv([1 1 1],[1 1 1])
ans =
     1     2     3     2     1

Note   Convolve rectangular matrices for two-dimensional signal processing using the conv2 function.

A digital filter's output y(k) is related to its input x(k) by convolution with its impulse response h(k).

If a digital filter's impulse response h(k) is finite length, and the input x(k) is also finite length, you can implement the filter using conv. Store x(k) in a vector x, h(k) in a vector h, and convolve the two:

x = randn(5,1);     % A random vector of length 5
h = [1 1 1 1]/4;    % Length 4 averaging filter
y = conv(h,x);

Back to Top

Filters and Transfer Functions

In general, the z-transform Y(z) of a digital filter's output y(n) is related to the z-transform X(z) of the input by

where H(z) is the filter's transfer function. Here, the constants b(i) and a(i) are the filter coefficients and the order of the filter is the maximum of n and m.

Note   The filter coefficients start with subscript 1, rather than 0. This reflects the standard indexing scheme used for MATLAB vectors.

MATLAB filter functions store the coefficients in two vectors, one for the numerator and one for the denominator. By convention, it uses row vectors for filter coefficients.

Filter Coefficients and Filter Names

Many standard names for filters reflect the number of a and b coefficients present:

• When n = 0 (that is, b is a scalar), the filter is an Infinite Impulse Response (IIR), all-pole, recursive, or autoregressive (AR) filter.

• When m = 0 (that is, a is a scalar), the filter is a Finite Impulse Response (FIR), all-zero, nonrecursive, or moving-average (MA) filter.

• If both n and m are greater than zero, the filter is an IIR, pole-zero, recursive, or autoregressive moving-average (ARMA) filter.

The acronyms AR, MA, and ARMA are usually applied to filters associated with filtered stochastic processes.

Back to Top

Filtering with the filter Function

It is simple to work back to a difference equation from the z-transform relation shown earlier. Assume that a(1) = 1. Move the denominator to the left-hand side and take the inverse z-transform.

In terms of current and past inputs, and past outputs, y(n) is

This is the standard time-domain representation of a digital filter, computed starting with y(1) and assuming zero initial conditions. This representation's progression is

A filter in this form is easy to implement with the filter function. For example, a simple single-pole filter (lowpass) is

b = 1;          % Numerator
a = [1 -0.9];   % Denominator

where the vectors b and a represent the coefficients of a filter in transfer function form. To apply this filter to your data, use

y = filter(b,a,x);

filter gives you as many output samples as there are input samples, that is, the length of y is the same as the length of x. If the first element of a is not 1, filter divides the coefficients by a(1) before implementing the difference equation.

Back to Top

Filtering, Linear Systems and Transforms Overview

The filter Function

© 1984-2009- The MathWorks, Inc.    -   Site Help   -   Patents   -   Trademarks   -   Privacy Policy   -   Preventing Piracy   -   RSS