Continuous-time filters - Continuous and Discrete Time Signals and Systems - page 359

Fig. 7.14. Magnitude spectrum
of the Butterworth bandpass
filter designed in Example 7.11.

1

0.7943

0.6

0.4

0.1

0

0 50 100 150 200 250 300 350 400 450

for 1 ≤ n

≤ 3. Substituting different values of n yields

S = [ − 0 . 7610 + j1 . 3182 − 0 . 7610 − j1 . 3182 − 1 . 5221] .

The transfer function of the lowpass filter is given by

K
( S + 0 . 7610 + j1 . 3182)( S + 0 . 7610 + j1 . 3182)( S + 1 . 5221)

H ( S ) =

or

K
S 3 + 3 . 0442 S 2 + 4 . 6336 S + 3 . 5264 .

H ( S ) =

= 3 . 5364. The

To ensure a dc gain of unity for the lowpass filter, we set K

transfer function of the unity gain lowpass filter is given by

3 . 5264
S 3 + 3 . 0442 S 2 + 4 . 6336 S + 3 . 5264 .

H ( S ) =

To derive the transfer function of the required bandpass filter, we use Eq. (7.69)
with ξ p1 = 100 radians/s and ξ p2 = 200 radians/s. The transformation is given
by

= s 2 + 2 10 4
100 s

S ,

from which the transfer function of the bandpass filter is calculated as follows:

H ( s ) = H ( S )

S = s 2 + 2 10 4
100 s

3 . 5264

= ,

3 + 3 . 0442 2 + 4 . 6336

s 2 + 2 10 4
100 s

s 2 + 2 10 4
100 s

s 2 + 2 10 4
100 s

+ 3 . 5264

which reduces to

3 . 5264 10 6 s 3
s 6 + 3 . 0442 10 2 s 5 + 1 . 0633 10 5 s 4 + 1 . 5703 10 7 s 3 + 2 . 1267 10 9 s 2 + 1 . 2177 10 11 s + 8 10 12 .

H ( s ) =

The magnitude spectrum of the bandpass filter is given in Fig. 7.14, which
confirms that the given specifications for the bandpass filter are satisfied.