Impulse invariance method for analog-to-digital filter conversion


[bz,az] = impinvar(b,a,fs)
[bz,az] = impinvar(b,a,fs,tol)


[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. If you leave out the argument fs, or specify fs as the empty vector [], it takes the default value of 1 Hz.

[bz,az] = impinvar(b,a,fs,tol) uses the tolerance specified by tol to determine whether poles are repeated. A larger tolerance increases the likelihood that impinvar interprets closely located poles as multiplicities (repeated ones). The default is 0.001, or 0.1% of a pole's magnitude. The accuracy of the pole values is still limited to the accuracy obtainable by the roots function.


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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);

hold on
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impinvar performs the impulse-invariant method of analog-to-digital transfer function conversion discussed in reference [1]:

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

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

See Also

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Introduced before R2006a