filter

1-D digital filter

Syntax

y = filter(b,a,x)

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

y = filter(b,a,x,zi,dim)

[y,zf] = filter(___)

Description

y = filter(b,a,x) filters the input data x using a rational transfer function defined by the numerator and denominator coefficients b and a.

If a(1) is not equal to 1, then filter normalizes the filter coefficients by a(1). Therefore, a(1) must be nonzero.

• If x is a vector, then filter returns the filtered data as a vector of the same size as x.

• If x is a matrix, then filter acts along the first dimension and returns the filtered data for each column.

• If x is a multidimensional array, then filter acts along the first array dimension whose size does not equal 1.

y = filter(b,a,x,zi) uses initial conditions zi for the filter delays. The length of zi must equal max(length(a),length(b))-1.

y = filter(b,a,x,zi,dim) acts along dimension dim. For example, if x is a matrix, then filter(b,a,x,zi,2) returns the filtered data for each row.

[y,zf] = filter(___) also returns the final conditions zf of the filter delays, using any of the previous syntaxes.

Examples

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 $$x$$:

$$y(n)=\frac{1}{windowSize}(x(n)+x(n-1)+...+x(n-(windowSize-1))).$$

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')

Filter Matrix Rows

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

$$H(z)=\frac{b(1)}{a(1)+a(2)z^{-1}}=\frac{1}{1-0.2z^{-1}}$$

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')

Specify Initial Conditions

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.

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,

$$H(z)=\frac{b(1)+b(2)z^{-1}}{a(1)+a(2)z^{-1}}=\frac{2+3z^{-1}}{1+0.2z^{-1}}.$$

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

Input Arguments

b — Numerator coefficients of rational transfer function

vector

Numerator coefficients of the rational transfer function, specified as a vector.

Data Types: double | single | int8 | int16 | int32 | int64 | uint8 | uint16 | uint32 | uint64 | logical
Complex Number Support: Yes

a — Denominator coefficients of rational transfer function

vector

Denominator coefficients of the rational transfer function, specified as a vector.

Data Types: double | single | int8 | int16 | int32 | int64 | uint8 | uint16 | uint32 | uint64 | logical
Complex Number Support: Yes

x — Input data

vector | matrix | multidimensional array

Input data, specified as a vector, matrix, or multidimensional array.

Data Types: double | single | int8 | int16 | int32 | int64 | uint8 | uint16 | uint32 | uint64 | logical
Complex Number Support: Yes

zi — Initial conditions for filter delays

[] (default) | vector | matrix | multidimensional array

Initial conditions for filter delays, specified as a vector, matrix, or multidimensional array.

• If zi is a vector, then its length must be max(length(a),length(b))-1.

• If zi is a matrix or multidimensional array, then the size of the leading dimension must be max(length(a),length(b))-1. The size of each remaining dimension must match the size of the corresponding dimension of x. For example, consider using filter along the second dimension (dim = 2) of a 3-by-4-by-5 array x. The array zi must have size [max(length(a),length(b))-1]-by-3-by-5.

The default value, specified by [], initializes all filter delays to zero.

Data Types: double | single | int8 | int16 | int32 | int64 | uint8 | uint16 | uint32 | uint64 | logical
Complex Number Support: Yes

dim — Dimension to operate along

positive integer scalar

Dimension to operate along, specified as a positive integer scalar. If you do not specify the dimension, then the default is the first array dimension whose size does not equal 1.

Consider a two-dimensional input array, x.

• If dim = 1, then filter(b,a,x,zi,1) operates along the columns of x and returns the filter applied to each column.

  

• If dim = 2, then filter(b,a,x,zi,2) operates along the rows of x and returns the filter applied to each row.

  

If dim is greater than ndims(x), then filter considers x as if it has additional dimensions up to dim with sizes of 1. For example, if x is a matrix with a size of 2-by-3 and dim = 3, then filter operates along the third dimension of x as if it has the size of 2-by-3-by-1.

Data Types: double | single | int8 | int16 | int32 | int64 | uint8 | uint16 | uint32 | uint64 | logical

Output Arguments

y — Filtered data

vector | matrix | multidimensional array

Filtered data, returned as a vector, matrix, or multidimensional array of the same size as the input data, x.

If x is of type single, then filter natively computes in single precision, and y is also of type single. Otherwise, y is returned as type double.

Data Types: double | single

zf — Final conditions for filter delays

vector | matrix | multidimensional array

Final conditions for filter delays, returned as a vector, matrix, or multidimensional array.

• If x is a vector, then zf is a column vector of length max(length(a),length(b))-1.

• If x is a matrix or multidimensional array, then zf is an array of column vectors of length max(length(a),length(b))-1, such that the number of columns in zf is equivalent to the number of columns in x. For example, consider using filter along the second dimension (dim = 2) of a 3-by-4-by-5 array x. The array zf has size [max(length(a),length(b))-1]-by-3-by-5.

Data Types: double | single

More About

Rational Transfer Function

The input-output description of the filter operation on a vector in the Z-transform domain is a rational transfer function. A rational transfer function is of the form

$$Y(z)=\frac{b(1)+b(2)z^{-1}+\mathrm{...}+b(n_b+1)z^{-n_b}}{1+a(2)z^{-1}+\mathrm{...}+a(n_a+1)z^{-n_a}}X(z),$$

which handles both finite impulse response (FIR) and infinite impulse response (IIR) filters [1]. Here, X(z) is the Z-transform of the input signal x, Y(z) is the Z-transform of the output signal y, n_a is the feedback filter order, and n_b is the feedforward filter order. Due to normalization, assume a(1) = 1.

For a discrete signal with L elements, you can also express the rational transfer function as the difference equation

$$\begin{array}{c}a(1)y(L)=b(1)x(L)+b(2)x(L-1)+\mathrm{...}+b(n_b+1)x(L-n_b)\\ -a(2)y(L-1)-\mathrm{...}-a(n_a+1)y(L-n_a).\end{array}$$

Furthermore, you can represent the rational transfer function using its direct-form II transposed implementation, as in the following diagram of an IIR digital filter. In the diagram, n_a = n_b = n–1. If your feedback and feedforward filter orders are different, or n_a ≠ n_b, then you can treat the higher-order terms as 0. For example, for a filter with a = [1,2] and b = [2,3,2,4], you can assume a = [1,2,0,0].

The operation of filter at a sample point m is given by the time-domain difference equations

$$\begin{array}{l}y(m)=b(1)x(m)+w_1(m-1)\\ w_1(m)=b(2)x(m)+w_2(m-1)-a(2)y(m)\\ ⋮=⋮⋮\\ w_{n-2}(m)=b(n-1)x(m)+w_{n-1}(m-1)-a(n-1)y(m)\\ w_{n-1}(m)=b(n)x(m)-a(n)y(m).\end{array}$$

By default, the filter function initializes the filter delays as zero, where w_k(0) = 0. This initialization assumes both past inputs and outputs to be zero. To include nonzero past inputs in the present data, specify the initial conditions of the present data as the filter delays. You can consider the filter delays to be the final conditions that are obtained from applying the same transfer function to the past inputs (and past outputs). You can specify the fourth input argument zi when using filter to set the filter delays, where w_k(0) = zi(k). You can also specify the second output argument zf when using filter to access the final conditions, where w_k(L) = zf(k).

Tips

• To use the filter function with the b coefficients from an FIR filter, use y = filter(b,1,x).

• If you have Signal Processing Toolbox™, use y = filter(d,x) to filter an input signal x with a digitalFilter (Signal Processing Toolbox) object d. To generate d based on frequency-response specifications, use designfilt (Signal Processing Toolbox).

• See Digital Filtering (Signal Processing Toolbox) for more on filtering functions.

References

[1] Oppenheim, Alan V., Ronald W. Schafer, and John R. Buck. Discrete-Time Signal Processing. Upper Saddle River, NJ: Prentice-Hall, 1999.

Extended Capabilities

Tall Arrays

Calculate with arrays that have more rows than fit in memory.

The filter function supports tall arrays with the following usage notes and limitations:

The two-output syntax [y,zf] = filter(___) is not supported when dim > 1.

For more information, see Tall Arrays.

C/C++ Code Generation

Generate C and C++ code using MATLAB® Coder™.

Usage notes and limitations:

• If supplied, dim must be a constant.

• See Variable-Sizing Restrictions for Code Generation of Toolbox Functions (MATLAB Coder).

GPU Code Generation

Generate CUDA® code for NVIDIA® GPUs using GPU Coder™.

Usage notes and limitations:

• If supplied, dim must be a constant.

• See Variable-Sizing Restrictions for Code Generation of Toolbox Functions (MATLAB Coder).

Thread-Based Environment

Run code in the background using MATLAB® backgroundPool or accelerate code with Parallel Computing Toolbox™ ThreadPool.

This function fully supports thread-based environments. For more information, see Run MATLAB Functions in Thread-Based Environment.

GPU Arrays

Accelerate code by running on a graphics processing unit (GPU) using Parallel Computing Toolbox™.

Usage notes:

• y = filter(d,x) is supported for digitalFilter object d and gpuArray input x. For IIR filters, digitalFilter objects store filter coefficients as a matrix of second-order sections (SOS). For a gpuArray input, it is not necessary to convert the SOS matrix to a gpuArray.

For more information, see Run MATLAB Functions on a GPU (Parallel Computing Toolbox).

Distributed Arrays

Partition large arrays across the combined memory of your cluster using Parallel Computing Toolbox™.

This function fully supports distributed arrays. For more information, see Run MATLAB Functions with Distributed Arrays (Parallel Computing Toolbox).

Version History

Introduced before R2006a