FIR filter design - Continuous and Discrete Time Signals and Systems - page 692

Fig. 15.16. Desired
specifications of a bandstop
filter.

| H bs ( W )|

1 + d p1

1 + d p2
1 − d p2

1 − d p1

pass
band I

stop
band

pass
band II

d s

W

0 W p1 W s1 W s2 W p2 p

As shown in Table 14.1, the impulse response of the ideal bandstop filter with
normalized cut-off frequencies of Ω n1 , Ω n2 ( Ω n2 > Ω n )isgivenby

h ibs [ k ] = δ [ k ] −

Ω n2 sinc[ Ω n2 k ] + Ω n1 sinc[ Ω n1 k ] .

(15.38)

By applying a delay m , the modified impulse response of an ideal bandpass
filter is obtained:

h ibs [ k ] = δ [ k − m ] −

Ω n2 sinc[ Ω n2 ( k − m )] + Ω n1 sinc[ Ω n1 ( k − m )] .

(15.39)

In the following example, we illustrate the steps involved in designing a practical
bandstop filter using the windowing method.

Example 15.7
Design a bandstop FIR filter, using a Kaiser window, with the following speci-
fications:

(i) pass-band edge frequencies, Ω p1 = 0 . 25 π and Ω p2 = 0 . 625 π radians/s;
(ii) stop-band edge frequencies, Ω s1 = 0 . 375 π and Ω s2 = 0 . 5 π radians/s;
(iii) stop-band attenuations, δ s1 > 50 dB and δ s2 > 50 dB.

Solution
The cut-off frequencies of the bandpass filter are given by

= 0 . 5(0 . 25 π + 0 . 375 π ) = 0 . 3125 π

Ω c1

and

= 0 . 5(0 . 5 π + 0 . 625 π ) = 0 . 5625 π.

Ω c2

The normalized cut-off frequencies are given by Ω n1 = 0 . 3125 and Ω n2 =

0 . 5625. The impulse response of an ideal bandpass filter is given by

h ibs [ k ] = δ [ k − m ] − 0 . 5625 sinc[0 . 5625( k − m )]
+ 0 . 3125 sinc[0 . 3125( k − m )] . (15.40)