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.

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

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

y = filter(b,a,x);

plot(t,x)
hold on
plot(t,y)
legend('Input Data','Filtered Data')

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.

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.

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')

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

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

Define a moving-average filter with a window size of 3.

windowSize = 3;
b = (1/windowSize)*ones(1,windowSize);
a = 1;

Find the 3-point moving average of a 1-by-6 row vector of data.

x = [2 1 6 2 4 3];
y = filter(b,a,x)

y = 1×6

    0.6667    1.0000    3.0000    3.0000    4.0000    3.0000

By default, the filter function initializes the filter delays as zero, assuming that both past inputs and outputs are zero. In this case, the first two elements of y are the 3-point moving average of the first element and the first two elements of x, respectively. In other words, the first element 0.6667 is the 3-point average of 2, and the second element 1 is the 3-point average of 2 and 1.

To include additional past inputs and outputs in your data, specify the initial conditions as the filter delays. These initial conditions for the present inputs are the final conditions that are obtained from applying the same transfer function to the past inputs (and past outputs). For example, include past inputs of [1 3]. Without filter delays, the past outputs are (0+0+1)/3 and (0+1+3)/3.

x_past = [1 3];
y_past = filter(b,a,x_past)

y_past = 1×2

    0.3333    1.3333

However, you can continue applying the same transfer function to generate further nonzero outputs, assuming that the tails of these past inputs are zero. These further outputs are (1+3+0)/3 and (3+0+0)/3, which represent the final conditions obtained from the past inputs. To compute these final conditions, specify the second output argument of the filter function.

[y_past,zf] = filter(b,a,x_past)

y_past = 1×2

    0.3333    1.3333

zf = 2×1

    1.3333
    1.0000

To include the past inputs in the present data, specify the filter delays by using the fourth input argument of the filter function. Use the final conditions from the past data as the initial conditions for the present data.

y = filter(b,a,x,zf)

y = 1×6

    2.0000    2.0000    3.0000    3.0000    4.0000    3.0000

In this case, the first element of y becomes the 3-point moving average of 1, 3, and 2, which is 2, and the second element of y becomes the moving average of 3, 2, and 1, which is 2.

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,

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.

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

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

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.

y = filter(b,a,x);

isequal(y,[y1;y2])

ans = logical
   1