PRACTICAL MATLAB® FOR ENGINEERS PRACTICAL MATLAB

586 Practical MATLAB® Applications for Engineers

R.6.99 The function [pz, zz, kz] = bilinear (pa, za, ka, fs, fc) returns the digital poles, zeros, and
gain given by pz, zz, and kz of its discrete transfer function H(z), where pa, za, and
ka are the analog poles, zeros, and gain of H(s), where fs = 1 /Ts and fc = 1 /( 2 π).
The bilinear transformation with the optional frequency prewarping is given by
[pz, zz, kz] = bilinear (pa, za, ka, fs) returns the s-domain analog transfer function
specifi ed by pa, za, ka, fs into the z discrete equivalent transfer function given by
the following substitutions:

Hz Hs

sfszz

() () ()

()

 2 ∗ ^11

where the column vectors za and pa are the analog zeros and poles of H(s), and fs

represents its sample frequency in hertz.

R.6.100 The impulse invariant response method converts the impulse response of the desired
analog fi lter h(t) into an equivalent digital FIR h(n), by properly sampling h(t), with
the hope that the analog response would closely follow its digital response. Hence,
the digital fi lter preserves the impulse response of the original analog fi lter.

R.6.101 Observe that the bilinear transformation is a process done entirely in the frequency
domain, whereas the impulse response method is done in the time domain.

R.6.102 The impulse invariant IIR fi lter is realized by the following steps:

a. h(t) = £−^1 [H(s)]

b. Substitute t by nT

c. Obtain the digital transfer function H(z) = {Z[h(nT)]} * T

Observe that the analog fi lter impulse response would then have the same val-

ues as the sample digital fi lter impulse response at the sampling instances nT.

Recall that when the inverse Laplace transform of H(s) cannot be obtained

directly, then a partial fraction expansion of H(s) may be required for obtaining in

this way a parallel structure implementation.

R.6.103 The impulse invariant method suffers from aliasing, and this is the main reason
that it is not widely used. The bilinear transformation does not suffer from alias-
ing and is by far more popular than the impulse invariance method.

R.6.104 The step invariant technique is an alternate way in the implementation of an IIR
fi lter by working with the step response instead of the impulse response. The step
response of the analog fi lter would then have the same values at the sampling
instances as the digital fi lter step response.

R.6.105 Mathematically, the impulse invariance response transforms H(s) into H(z) [H(s) →
H(z)] by making the following substitution:
11
sa 1 ze^1 aT

⇒

(transformation used for simple poles of H(s)).

sa

sasab

ze bT

ze bT ze

aT

aT aT











222  

1

2 122

1

12

⇒

cos( )

cos( )

b

sasab

ze bT

ze bT ze

aT

222 aT aT

1

2 122

1

 12







⇒  

sin( )

cos( )