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