# 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

example

````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. ```

example

````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`.```

example

````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.```

example

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

## Examples

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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 $x$:

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

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.

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

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')```

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,

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

```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

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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

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

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

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

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 of size greater than 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

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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`

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`

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### 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

`$Y\left(z\right)=\frac{b\left(1\right)+b\left(2\right){z}^{-1}+...+b\left({n}_{b}+1\right){z}^{-{n}_{b}}}{1+a\left(2\right){z}^{-1}+...+a\left({n}_{a}+1\right){z}^{-{n}_{a}}}X\left(z\right),$`

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, na is the feedback filter order, and nb 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

`$\begin{array}{c}a\left(1\right)y\left(L\right)=b\left(1\right)x\left(L\right)+b\left(2\right)x\left(L-1\right)+...+b\left({n}_{b}+1\right)x\left(L-{n}_{b}\right)\\ -a\left(2\right)y\left(L-1\right)-...-a\left({n}_{a}+1\right)y\left(L-{n}_{a}\right).\end{array}$`

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, na = nb = n–1. If your feedback and feedforward filter orders are different, or na ≠ nb, 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

By default, the `filter` function initializes the filter delays as zero, where wk(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 wk(0) = `zi(k)`. You can also specify the second output argument `zf` when using `filter` to access the final conditions, where wk(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.

## Version History

Introduced before R2006a