Tutorials - Digital Signal Processing: An Introduction with MATLAB and Applications - page 280

Hence, the transfer function of the digital filter would be:

z
z z 1

z
z z 2

H ð z Þ¼ T s c 1 þ c 2

1
z z 1

1
z z 1

¼ T s zc 1 þ c 1

c 1 ð z z 1 Þþ c 1 ð z z 1 Þ
ð z z 1 Þð z z 1 Þ

¼ T s z

¼ T s z ð c 1 þ c 1 Þ z ð c 1 z 1 þ c 1 z 1 Þ
z 2 ð z 1 þ z 1 Þ z þj z 1 j 2

0 : 15z
z 2 1 : 36z þ 0 : 667 :

¼

Tutorial 45

Q: Using the impulse invariance method, design a digital filter with sampling
frequency 100 Hz and impulse response that matches the response of the following
3rd-order analog Butterworth filter:

1
ð s þ 1 Þð s p Þð s p Þ ;

H ð s Þ¼

where p =-0.5 + 0.866i

Solution: Using Partial Fraction Expansion, we write the transfer function as

follows:

H a ð s Þ¼ 1

s þ 1 þ c s p þ c

s p ;

where c =-0.5 - 0.28i.
The z-domain poles are:
z 1 = exp (p 1 T s ) = exp (- T s ) = 0.99,
z 2 = exp (pT s ) = 0.995 + 0.0086i,
z 3 ¼ z 2 ¼ 0 : 995 0 : 0086i :

z
z 0 : 99 þ c

z
z z 2

z
z z 2

) H ð z Þ¼ T s þ c