1-D digital filter
y = filter(b,a,X)
[y,zf] = filter(b,a,X)
[y,zf] = filter(b,a,X,zi)
y = filter(b,a,X,zi,dim)
[...] = filter(b,a,X,,dim)
The filter function filters a data sequence using a digital filter which works for both real and complex inputs. The filter is a direct form II transposed implementation of the standard difference equation (see "Algorithm").
y = filter(b,a,X) filters the data in vector X with the filter described by numerator coefficient vector b and denominator coefficient vector a. If a(1) is not equal to 1, filter normalizes the filter coefficients by a(1). If a(1) equals 0, filter returns an error.
If X is a matrix, filter operates on the columns of X. If X is a multidimensional array, filter operates on the first nonsingleton dimension.
[y,zf] = filter(b,a,X) returns the final conditions, zf, of the filter delays. If X is a row or column vector, output zf is a column vector of max(length(a),length(b))-1. If X is a matrix, zf is an array of such vectors, one for each column of X, and similarly for multidimensional arrays.
[y,zf] = filter(b,a,X,zi) accepts initial conditions, zi, and returns the final conditions, zf, of the filter delays. Input zi is a vector of length max(length(a),length(b))-1, or an array with the leading dimension of size max(length(a),length(b))-1 and with remaining dimensions matching those of X.
You can use filter to find a running average without using a for loop. This example finds the running average of a 16-element vector, using a window size of 5.
data = [1:0.2:4]'; windowSize = 5; filter(ones(1,windowSize)/windowSize,1,data) ans = 0.2000 0.4400 0.7200 1.0400 1.4000 1.6000 1.8000 2.0000 2.2000 2.4000 2.6000 2.8000 3.0000 3.2000 3.4000 3.6000
If you have the Signal Processing Toolbox™, you can design a filter, d, using designfilt. Then you can use Y = filter(d,X) to filter your data.
The filter function is implemented as a direct form II transposed structure,
a(1)*y(n) = b(1)*x(n) + b(2)*x(n-1) + ... + b(nb+1)*x(n-nb) - a(2)*y(n-1) - ... - a(na+1)*y(n-na)
where n-1 is the filter order, which handles both FIR and IIR filters , na is the feedback filter order, and nb is the feedforward filter order. Due to normalization, assume a(1) = 1.
The operation of filter at sample m is given by the time domain difference equations
The input-output description of this filtering operation in the z-transform domain is a rational transfer function,
 Oppenheim, A. V. and R.W. Schafer. Discrete-Time Signal Processing, Englewood Cliffs, NJ: Prentice-Hall, 1989, pp. 311-312.