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# impzlength

Impulse response length

## Syntax

``len = impzlength(b,a)``
``len = impzlength(sos)``
``len = impzlength(d)``
``len = impzlength(hs)``
``len = impzlength(___,tol)``

## Description

example

````len = impzlength(b,a)` returns the impulse response length for the causal discrete-time filter with the rational system function specified by the numerator, `b`, and denominator, `a`, polynomials in z–1. For stable IIR filters, `len` is the effective impulse response sequence length. Terms in the IIR filter’s impulse response after the `len`-th term are essentially zero. ```

example

````len = impzlength(sos)` returns the effective impulse response length for the IIR filter specified by the second order sections matrix, `sos`. `sos` is a K-by-6 matrix, where the number of sections, K, must be greater than or equal to 2. If the number of sections is less than 2, `impzlength` considers the input to be the numerator vector, `b`. Each row of `sos` corresponds to the coefficients of a second order (biquad) filter. The ith row of the `sos` matrix corresponds to ```[bi(1) bi(2) bi(3) ai(1) ai(2) ai(3)]```.```

example

````len = impzlength(d)` returns the impulse response length for the digital filter, `d`. Use `designfilt` to generate `d` based on frequency-response specifications.```

example

````len = impzlength(hs)` returns the impulse response length for the filter System object™, `hs`. You must have the DSP System Toolbox™ software to use `impzlength` with a filter System object.```
````len = impzlength(___,tol)` specifies a tolerance for estimating the effective length of an IIR filter’s impulse response. By default, `tol` is `5e-5`. Increasing the value of `tol` estimates a shorter effective length for an IIR filter’s impulse response. Decreasing the value of `tol` produces a longer effective length for an IIR filter’s impulse response.```

## Examples

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Create a lowpass allpole IIR filter with a pole at 0.9. Calculate the effective impulse response length. Obtain the impulse response. Plot the result.

```b = 1; a = [1 -0.9]; len = impzlength(b,a)```
```len = 93 ```
```[h,t] = impz(b,a); stem(t,h)```

`h(len)`
```ans = 6.1704e-05 ```

Design a 4th-order lowpass elliptic filter with a cutoff frequency of 0.4π rad/sample. Specify 1 dB of passband ripple and 60 dB of stopband attenuation. Design the filter in pole-zero-gain form and obtain the second-order section matrix using `zp2sos`. Determine the effective impulse response sequence length from the second-order section matrix.

```[z,p,k] = ellip(4,1,60,.4); [sos,g] = zp2sos(z,p,k); len = impzlength(sos)```
```len = 80 ```

Use `designfilt` to design a 4th-order lowpass elliptic filter with normalized passband frequency 0.4π rad/sample. Specify 1 dB of passband ripple and 60 dB of stopband attenuation. Determine the effective impulse response sequence length and visualize it.

```d = designfilt('lowpassiir','FilterOrder',4,'PassbandFrequency',0.4, ... 'PassbandRipple',1,'StopbandAttenuation',60, ... 'DesignMethod','ellip'); len = impzlength(d)```
```len = 80 ```
`impz(d)`

This example requires DSP System Toolbox™ software.

Design a 4th-order lowpass elliptic filter with a cutoff frequency of 0.4*pi rad/sample. Specify 1 dB of passband ripple and 60 dB of stopband attenuation. Design the filter in pole-zero-gain form and obtain the second order section matrix using `zp2sos`. Create a biquad filter System object™ and input the System object to `impzlength`.

```[z,p,k] = ellip(4,1,60,.4); [sos,g] = zp2sos(z,p,k); hBqdFilt = dsp.BiquadFilter('Structure','Direct form I',... 'SOSMatrix', sos,... 'ScaleValues',g); len = impzlength(hBqdFilt)```
```len = 80 ```

Design IIR Butterworth and FIR equiripple filters for data sampled at 1 kHz. The passband frequency is 100 Hz and the stopband frequency is 150 Hz. The passband ripple is 0.5 dB and there is 60 dB of stopband attenuation. Obtain System objects for the filters and compare the filter impulse response sequence lengths.

```d = fdesign.lowpass('Fp,Fst,Ap,Ast',100,150,0.5,60,1000); Hd1 = design(d,'butter','SystemObject',true); Hd2 = design(d,'equiripple','SystemObject',true); len = [impzlength(Hd1) impzlength(Hd2)]```
```len = 183 49 ```

## Input Arguments

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Numerator coefficients, specified as a scalar (allpole filter) or a vector.

Example: `b = fir1(20,0.25)`

Data Types: `single` | `double`
Complex Number Support: Yes

Denominator coefficients, specified as a scalar (FIR filter) or vector.

Data Types: `single` | `double`
Complex Number Support: Yes

Matrix of second order sections, specified as a K-by-6 matrix. The system function of the K-th biquad filter has the rational Z-transform

`${H}_{k}\left(z\right)=\frac{{B}_{k}\left(1\right)+{B}_{k}\left(2\right){z}^{-1}+{B}_{k}\left(3\right){z}^{-2}}{{A}_{k}\left(1\right)+{A}_{k}\left(2\right){z}^{-1}+{A}_{k}\left(3\right){z}^{-2}}.$`
The coefficients in the Kth row of the matrix, `sos`, are ordered as follows.
`$\left[\begin{array}{cccccc}{B}_{k}\left(1\right)\text{ }& {B}_{k}\left(2\right)\text{ }& {B}_{k}\left(3\right)& {A}_{k}\left(1\right)\text{ }& {A}_{k}\left(2\right)& {A}_{k}\left(3\right)\end{array}\right]$`
The frequency response of the filter is the system function evaluated on the unit circle with
`$z={e}^{j2\pi f}.$`

Digital filter, specified as a `digitalFilter` object. Use `designfilt` to generate a digital filter based on frequency-response specifications.

Example: `d = designfilt('lowpassiir','FilterOrder',3,'HalfPowerFrequency',0.5)` specifies a third-order Butterworth filter with normalized 3-dB frequency 0.5π rad/sample.

Filter System object, specified as one of the following:

The following Filter System objects are supported by this analysis function:

Using `impzlength` with a filter System object requires the DSP System Toolbox software.

Tolerance for IIR filter effective impulse response length, specified as a positive number. The tolerance determines the term in the absolutely summable sequence after which subsequent terms are considered to be 0. The default tolerance is `5e-5`. Increasing the tolerance returns a shorter effective impulse response sequence length. Decreasing the tolerance returns a longer effective impulse response sequence length.

## Output Arguments

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Length of the impulse response, specified as a positive integer. For stable IIR filters with absolutely summable impulse responses, `impzlength` returns an effective length for the impulse response beyond which the coefficients are essentially zero. You can control this cutoff point by specifying the optional `tol` input argument.

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