Analog Signals and Systems - Digital Signal Processing: An Introduction with MATLAB and Applications - page 50

ω

| H ( ) |

G m

n = 1

n = 2

− ω c ω c

0

Fig. 1.32 The magnitude frequency response of a Butterworth LPF

1.5.2 Butterworth LPF

The magnitude frequency response of a Butterworth LPF (B-LPF) is given by:

G m

j H ð x Þj¼ r ; ð 1 : 45 Þ

1 þ x c 2n

where G m is the maximum gain, n is the filter order, and x c is the cutoff frequency
(see Fig. 1.32 ). The Butterworth LPF's magnitude function is maximally flat at
x = 0, i.e., all its 2n - 1 derivatives are equal to 0 at x = 0. At x = 0 (DC), the
filter gain=G m , and the power gain GP o = |H(0)| 2 = G 2 —this is the maximum

power gain of the filter. At x = x c (cutoff), the filter gain ¼ G m = 2

p

, and the power
gain GP c ¼j H ð x c Þj 2 ¼ G m = 2 ¼ half the maximum power gain GP o . In dB terms:
GP c (dB) ¼ 10 log 10 ð GP o = 2 Þ¼ GP o (dB) 10 log 10 2 ¼ GP o (dB) 3

The cutoff frequency is also called the 3-dB frequency because the power is 3 dB
down from the power at DC.
As n increases, the cutoff becomes sharper, but more circuit components are
needed to build the filter. The Butterworth LPF has a flat response at x = 0. Its
transfer function for different orders can be obtained from Tables. The nth-order
B-LPF has n poles and no zeros in its transfer function.

1.5.3 Chebychev-I LPF

Chebychev-I filters tend to have sharper roll-off than Butterworth filters but have
ripple in the passband, as illustrated in Fig. 1.33 . The magnitude frequency
response of a Chebychev-I LPF (C-LPF) is given by:

G m
1 þ e 2 C n

j H ð x Þj¼ r ;

x
x c