Signal Processing Toolbox™

impinvar

Impulse invariance method for analog-to-digital filter conversion

Syntax

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

Description

[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. Note that the accuracy of the pole values is still limited to the accuracy obtainable by the roots function.

Examples

Example 1

Convert an analog lowpass filter to a digital filter using impinvar with a sampling frequency of 10 Hz:

[b,a] = butter(4,0.3,'s');
[bz,az] = impinvar(b,a,10)
bz =
  1.0e-006 *
   -0.0000    0.1324    0.5192    0.1273         0
az =
    1.0000   -3.9216    5.7679   -3.7709    0.9246

Example 2

Illustrate the relationship between analog and digital impulse responses [2].

Note This example requires the impulse function from Control System Toolbox™ software.

The steps used in this example are:

Create an analog Butterworth filter
Use impinvar with a sampling frequency Fs of 10 Hz to scale the coefficients by 1/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 (Fs) to 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.

Algorithm

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

It finds the partial fraction expansion of the system represented by b and a.
It replaces the poles p by the poles exp(p/fs).
It finds the transfer function coefficients of the system from the residues from step 1 and the poles from step 2.

References

[1] Parks, T.W., and C.S. Burrus, Digital Filter Design, John Wiley & Sons, 1987, pp.206-209.

[2] Antoniou, Andreas, Digital Filters, McGraw Hill, Inc, 1993, pp.221-224.