PRACTICAL MATLAB® FOR ENGINEERS PRACTICAL MATLAB

Analog and Digital Filters 587

(transformations used for the case of complex poles of H(s)).

R.6.106 For example, let the analog transfer function of a second-order Butterworth low

pass-band fi lter be given by

Hs

ss

()

.



100

(^2) 14 14 (^100)
Use the impulse invariant method to transform the analog fi lter into a digital
one if the sampling rate is given by T = 0.0314 s.
ANALYTICAL Solution
Rearranging the terms of H(s) into the format given in R.6.105 results in
Hs
ss

()

(. ) (. ) (. )



100

(^222) 2 14 14 2 14 14 2 100 14 14 2
Hs
ss

()

(. ) (.) (.)



100

(^222) 21414 2 (^707100707)
Hs
ss

() *

(. ) () ()



250

(^2) 2 14 14 2 (^5010050)
Hs
ss

() *

()()

(. ) () ()



250 50

2 14 14 2 50 50

12 12

2

Hs

ss

() *()

()

(. ) (.) (.)







250

50

21414 2 707 707

12

12

222

Hs

ss

() *()

.

(. ) (.) (.)



250 ^707

21414 2 707 707

12

222

Then, let a = 7. 0 7, b = 7. 0 7 0 , a n d T = 0.0314, and using the substitutions of R.6.105

H(s) → H(z) the following discrete system is obtained: The numerator of [H(z)] =

[ (^2) ( 50 )1/2]) (^) [z–^1 e−(7. 0 7)(0.0314) (^) sin(7. 0 7 0.0314)] and the denominator of [H(z)] =
[ 1 − 2 z−^1 e−(7. 0 7)(0.0314) (^) cos(7. 0 7 0.0314) + z−^2 e−^2 7. 0 7 0.0314].
Then, after performing the corresponding algebraic manipulations the follow-
ing simplifi ed expression is obtained:
Hz
Yz
Xz
z
zz

()

()

()

.

..





(^250) 

0 0882

1 1 5626 0 6414

1

* 12

As a consequence of the sampling process (Nyquist theorem), the fi lter mag-

nitude is multiplied by 1/T. Therefore, after the transformation process from the