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Impulse invariance method for analog-to-digital filter conversion



[bz,az] = impinvar(b,a,fs) creates a digital filter with numerator and denominator coefficients bz and az, respectively, whose impulse response is equal to the impulse response of the analog filter with coefficients b and a, scaled by 1/fs, where fs is the sample rate.

[bz,az] = impinvar(b,a,fs,tol) uses the tolerance specified by tol to determine whether poles are repeated.


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Convert a sixth-order analog Butterworth lowpass filter to a digital filter using impulse invariance. Specify a sample rate of 10 Hz and a cutoff frequency of 2 Hz. Display the frequency response of the filter.

f = 2;
fs = 10;

[b,a] = butter(6,2*pi*f,'s');
[bz,az] = impinvar(b,a,fs);


Convert a third-order analog elliptic filter to a digital filter using impulse invariance. Specify a sample rate fs=10 Hz, a passband edge frequency of 2.5 Hz, a passband ripple of 1 dB, and a stopband attenuation of 60 dB. Display the impulse response of the digital filter.

fs = 10;

[b,a] = ellip(3,1,60,2*pi*2.5,'s');
[bz,az] = impinvar(b,a,fs);


Derive the impulse response of the analog filter by finding the residues, rk, and poles, pk, of the transfer function and inverting the Laplace transform explicitly using


Overlay the impulse response of the analog filter. Impulse invariance introduces a gain of 1/fs to the digital filter. Multiply the analog impulse response by this gain to enable meaningful comparison.

[r,p] = residue(b,a);
t = linspace(0,4,1000);
h = real(r.'*exp(p.*t)/fs);

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Input Arguments

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Analog filter transfer function coefficients, specified as vectors.

Example: [b,a] = butter(6,2*pi*10,'s') specifies a 6th-order Butterworth filter with a cutoff frequency of 10 Hz.

Data Types: single | double

Sample rate, specified as a positive scalar.

Data Types: single | double

Tolerance, specified as a positive scalar. The tolerance determines whether poles are repeated. A larger tolerance increases the likelihood that impinvar interprets closely located poles as multiplicities (repeated ones). The default tolerance corresponds to 0.1% of a pole magnitude. The accuracy of the pole values is still limited to the accuracy obtainable by the roots function.

Data Types: single | double

Output Arguments

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Digital filter transfer function coefficients, returned as vectors.


impinvar performs the impulse-invariant method of analog-to-digital transfer function conversion discussed in reference [2]:

  1. It finds the partial fraction expansion of the system represented by b and a.

  2. It replaces the poles p by the poles exp(p/fs).

  3. It finds the transfer function coefficients of the system from the residues from step 1 and the poles from step 2.


[1] Antoniou, Andreas. Digital Filters. New York: McGraw-Hill, Inc., 1993.

[2] Parks, Thomas W., and C. Sidney Burrus. Digital Filter Design. New York: John Wiley & Sons, 1987.

Version History

Introduced before R2006a

See Also

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