# Documentation

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# impzlength

Impulse response length

## Syntax

• len = impzlength(b,a)
example
• len = impzlength(sos)
example
• len = impzlength(d)
example
• len = impzlength(hs)
example
• len = impzlength(hd)
example
• 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.

example

len = impzlength(hd) returns the impulse response length for the dfilt filter object, hd. You can also input an array of filter objects. If hd is an array of filter objects, each column of len is the impulse response length of the corresponding filter 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) [h,t] = impz(b,a); stem(t,h) h(len) 
len = 93 ans = 6.1704e-05 

Design a 4th-order lowpass elliptic filter with a cutoff frequency of 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 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) impz(d) 
len = 80 

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 dfilt 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'); Hd2 = design(d,'equiripple'); len = impzlength([Hd1 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.

Filter object, specified as a dfilt object.

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.