Filter in MATLAB^®

Moving-Average Filter

A moving-average filter is a common method used for smoothing noisy data. This example uses the filter function to compute averages along a vector of data.

Create a 1-by-100 row vector of sinusoidal data that is corrupted by random noise.

t = linspace(-pi,pi,100);
rng default  %initialize random number generator
x = sin(t) + 0.25*rand(size(t));

A moving-average filter slides a window of length windowSize along the data, computing averages of the data contained in each window. The following difference equation defines a moving-average filter of a vector x:

For a window size of 5, compute the numerator and denominator coefficients for the rational transfer function.

t = linspace(-pi,pi,100);
rng default  %initialize random number generator
x = sin(t) + 0.25*rand(size(t));

windowSize = 5; 
b = (1/windowSize)*ones(1,windowSize);
a = 1;

Find the moving average of the data and plot it against the original data.

t = linspace(-pi,pi,100);
rng default  %initialize random number generator
x = sin(t) + 0.25*rand(size(t));

windowSize = 5; 
b = (1/windowSize)*ones(1,windowSize);
a = 1;

y = filter(b,a,x);

plot(t,x)
hold on
plot(t,y)
legend('Input Data','5-day Moving Average')

fig2plotly(gcf);

Filter Matrix Rows

This example filters a matrix of data with the following rational transfer function.

Create a 2-by-15 matrix of random input data.

rng default  %initialize random number generator
x = rand(2,15);

Define the numerator and denominator coefficients for the rational transfer function.

rng default  %initialize random number generator
x = rand(2,15);

b = 1;
a = [1 -0.2];

Apply the transfer function along the second dimension of x and return the 1-D digital filter of each row. Plot the first row of original data against the filtered data.

rng default  %initialize random number generator
x = rand(2,15);

b = 1;
a = [1 -0.2];

y = filter(b,a,x,[],2);

t = 0:length(x)-1;  %index vector

plot(t,x(1,:))
hold on
plot(t,y(1,:))
legend('Input Data','Filtered Data')
title('First Row')

fig2plotly(gcf);

Plot the second row of input data against the filtered data.

rng default  %initialize random number generator
x = rand(2,15);

b = 1;
a = [1 -0.2];

y = filter(b,a,x,[],2);

t = 0:length(x)-1;  %index vector

plot(t,x(1,:))
hold on
plot(t,y(1,:))
legend('Input Data','Filtered Data')
title('First Row')

figure
plot(t,x(2,:))
hold on
plot(t,y(2,:))
legend('Input Data','Filtered Data')
title('Second Row')

fig2plotly(gcf);

Filter Data in Sections

Use initial and final conditions for filter delays to filter data in sections, especially if memory limitations are a consideration.

Generate a large random data sequence and split it into two segments, x1 and x2.

x = randn(10000,1);

x1 = x(1:5000);
x2 = x(5001:end);

The whole sequence, x, is the vertical concatenation of x1 and x2.

Define the numerator and denominator coefficients for the rational transfer function,

x = randn(10000,1);

x1 = x(1:5000);
x2 = x(5001:end);

b = [2,3];
a = [1,0.2];

Filter the subsequences x1 and x2 one at a time. Output the final conditions from filtering x1 to store the internal status of the filter at the end of the first segment.

x = randn(10000,1);

x1 = x(1:5000);
x2 = x(5001:end);

b = [2,3];
a = [1,0.2];

[y1,zf] = filter(b,a,x1);

Use the final conditions from filtering x1 as initial conditions to filter the second segment, x2.

x = randn(10000,1);

x1 = x(1:5000);
x2 = x(5001:end);

b = [2,3];
a = [1,0.2];

[y1,zf] = filter(b,a,x1);

y2 = filter(b,a,x2,zf);

y1 is the filtered data from x1, and y2 is the filtered data from x2. The entire filtered sequence is the vertical concatenation of y1 and y2.

Filter the entire sequence simultaneously for comparison.

x = randn(10000,1);

x1 = x(1:5000);
x2 = x(5001:end);

b = [2,3];
a = [1,0.2];

[y1,zf] = filter(b,a,x1);

y2 = filter(b,a,x2,zf);

y = filter(b,a,x);

isequal(y,[y1;y2])

ans =

  logical

   1