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# 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);
```

### 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:

1. Create an analog Butterworth filter

2. 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.

3. Use Control System Toolbox impulse function to plot the continuous-time unit impulse response of an LTI system.

4. 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.

expand all

### Algorithms

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.

## 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.