Tutorials - Digital Signal Processing: An Introduction with MATLAB and Applications - page 269

:

/ ð f Þ¼ \H ð f Þ¼ \ ½ 0 jsgn ð f Þ¼ tan 1 ð 0 Þ¼ tan 1 ð1Þ¼ 2 ;

f 0

tan 1 ð 0 Þ¼ tan 1 ðþ1Þ¼ 2 ;

f \0

Hence /(f)isconstant for physical frequencies f C 0.

Comparison: A constant time-delay T gives a constant (frequency-independent)
time-delay and a frequency-dependent phase shift /(f); HT gives frequency-
independent (constant) phase shift (90) and frequency-dependent time-delay
[t d = (p/2)/x].

| H ( f )| φ ( f )

1 + π /2

0

f f

0

− π /2

Tutorial 34

Q1: Design a BP Chebychev-I 1 dB ripple filter with center frequency
f 0 = 10 kHz, bandwidth BW = 1 kHz, maximum gain G m = 1, and stopband
gain B-10 dB for f B f 1 = 7 kHz and f C f 2 = 13 kHz. True load resistance is
R L ¼ 10 X :

Solution:

x u ¼ x 0 þ x b = 2 ¼ 2p ð 10k Þþ 2p ð 1k Þ= 2 ¼ 21pk rad/s
where x b ¼ BW ¼ x u x l :
x l ¼ x 0 x b = 2 ¼ 2p ð 10k Þ 2p ð 1k Þ= 2 ¼ 19pk rad/s :
x g ¼ ¼

p

p

x l x u ð 21pk Þð 19pk Þ ¼ 19 : 97pk rad/s.

Using LP ? BP transformation (Tables): x N ¼ x 2 x g ; we find the normalized LP

xx b

frequencies that correspond to f 1 and f 2 as follows:
x 1N =-7.25 and x 2N = 5.32.
We check the order n that gives gain B-10 dB for |x 2N | B x N B |x 1N |; or
5.32 B x N B 7.25. From Chebychev-I curves (Tables), n = 1 gives gain B-
10 dB for x N = 7.25, but not for x N = 5.32. Hence, n = 2 is the suitable choice.
From Tables we obtain the normalized transfer function as follows:

a 0
1 : 1 þ 1 : 09s N þ s N

G M
1 þ e 2

1
1 þ 0 : 25

¼ a 0 H ð s N Þ¼ ) G dc ¼ p ¼ p

1 : 1 ) a 0 ¼ 0 : 98 :