Tutorials - Digital Signal Processing: An Introduction with MATLAB and Applications - page 281

Tutorial 46

Q: Using the bilinear transform, design a 4th-order BP Butterworth digital filter
with center frequency X o ¼ 1 : 5, maximum gain G m = 1, and bandwidth X b ¼ 0 : 4.
Solution: Since the transformation LP ? BP is quadratic (Tables), we need a
2nd-order prototype analog LPF. The transfer function of this filter is given by
(Tables):

a o
b o þ b 1 s N þ s N

H ð s N Þ¼

where b o = 1 and b 1 = 1.41.

G d : c ¼ G m ¼ 1 ¼ a o
b o

1
1 þ 1 : 41s N þ s N

) a o ¼ b o ¼ 1 : ) H ð s N Þ¼ :

= 2 ¼ tan ð 1 : 7 = 2 Þ¼ 1 : 14

X o þ X b
2

x u ¼ tan ð X u = 2 Þ¼ tan

= 2 ¼ tan ð 1 : 3 = 2 Þ¼ 0 : 76

X o X b
2

x l ¼ tan ð X l = 2 Þ¼ tan

x b ¼ x u x l ¼ 0 : 38; x g ¼

p

x l x u ¼ 0 : 93

Using the LP ? BP transformation s N ¼ s 2 þ x g
sx b

(from Tables), we get the transfer

function of the analog BPF as follows:

1
1 þ 1 : 41 s 2 þ x g

H ð s Þ¼

2

sx b þ s 2 þ x g sx b

Using the bilinear transformation s = (z - 1)/(z + 1), we obtain the transfer
function of the required BP digital filter as follows:

1
1 þ 1 : 41 ½ð z 1 Þ=ð z þ 1 Þ 2 þ x g

H ð z Þ¼

2

½ð z 1 Þ=ð z þ 1 Þ x b þ ½ð z 1 Þ=ð z þ 1 Þ 2 þ x g ½ð z 1 Þ=ð z þ 1 Þ x b

Tutorial 47

Q: A radar station received the two signals at two different time intervals. A
correlator is used to analyze these signals using their autocorrelations. The
correlator outputs are as shown below. Comment on the structure of these signals.