Impulse invariance method for analog-to-digital filter conversion
[bz,az] creates a digital
filter with numerator and denominator coefficients
respectively, whose impulse response is equal to the impulse response
of the analog filter with coefficients
1/fs. If you leave out the argument
fs as the empty vector
it takes the default value of 1 Hz.
[bz,az] uses the tolerance
tol to determine whether poles are
repeated. A larger tolerance increases the likelihood that
closely located poles as multiplicities (repeated ones). The default
is 0.001, or 0.1% of a pole's magnitude. Note that the accuracy of
the pole values is still limited to the accuracy obtainable by the
Convert an analog lowpass filter to a digital filter using
a sampling frequency of 10 Hz:
[b,a] = butter(4,0.3,'s'); [bz,az] = impinvar(b,a,10);
Illustrate the relationship between analog and digital impulse responses .
This example requires the
The steps used in this example are:
Create an analog Butterworth filter
a sampling frequency
Fs of 10 Hz to scale the coefficients
Fs. This compensates for the gain that will
be introduced in Step 4 below.
Use Control System Toolbox
impulse function to plot the continuous-time
unit impulse response of an LTI system.
Plot the digital impulse response,
multiplying the numerator by a constant (
compensate for the 1/
Fs gain introduced in the
impulse response of the derived digital filter.
[b,a] = butter(4,0.3,'s'); [bz,az] = impinvar(b,a,10); sys = tf(b,a); impulse(sys); hold on; impz(10*bz,az,,10);
Zooming the resulting plot shows that the analog and digital impulse responses are the same.
impinvar performs the impulse-invariant method
of analog-to-digital transfer function conversion discussed in reference :
It finds the partial fraction expansion
of the system represented by
It replaces the poles
It finds the transfer function coefficients of the system from the residues from step 1 and the poles from step 2.
 Parks, T.W., and C.S. Burrus, Digital Filter Design, John Wiley & Sons, 1987, pp.206-209.
 Antoniou, Andreas, Digital Filters, McGraw Hill, Inc, 1993, pp.221-224.