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, 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.

Examples

Analog and Digital Butterworth Filters

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);

freqz(bz,az,1024,fs)

Analog and Digital Impulse Responses

Convert a third-order analog elliptic filter to a digital filter using impulse invariance. Specify a sample rate $$f_s=10$$ 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);

impz(bz,az,[],fs)

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

$$H(s)=\sum _k\frac{{\displaystyle r_k}}{{\displaystyle s-p_k}}⟺h(t)=\sum _k{r}_k{e}^{p_{k}t}.$$

Overlay the impulse response of the analog filter. Impulse invariance introduces a gain of $$1/f_s$$ 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
plot(t,h)
hold off

Input Arguments

b, a — Analog filter transfer function coefficients

vectors

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

fs — Sample rate

1 Hz (default) | positive scalar

Sample rate, specified as a positive scalar.

Data Types: single | double

tol — Tolerance

0.001 (default) | positive scalar

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

bz, az — Digital filter transfer function coefficients

vectors

Digital filter transfer function coefficients, returned as vectors.

Algorithms

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.

References

[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