FIR filter design - Continuous and Discrete Time Signals and Systems - page 691

Fig. 15.15. Magnitude response
of the bandpass FIR filter
designed in Example 15.6.

20 log 10 | H ( W )|

0

−20

−40

−60

W

0

0.25 p 0.5 p 0.75 p p

The impulse response of the bandpass FIR filter is given by

h bp [ k ] = h ibp [ k ] w kaiser [ k ] .

where h ibp [ k ] is specified in Eq. (15.35) with m = 23 and w kaiser [ k ] is specified

in Eq. (15.36).
The magnitude spectrum of the bandpass FIR filter is plotted in Fig. 15.15.
It is observed that the bandpass filter satisfies the design specifications.
In Example 15.6, we designed a bandpass FIR filter directly. As for the
highpass FIR filter, an alternative procedure to design a bandpass FIR filter
is to exploit Eq. (14.4e) and implement two lowpass FIR filters with impulse
responses H lp1 ( k ) and H lp2 ( k ). The specifications for the two lowpass filters
should be carefully derived such that the pass- and stop-band ripples of the
combined system are limited to values allowed in the original bandpass filter's
specifications.

15.4 Design of a bandstop filter using windowing

As illustrated in Fig. 15.16, the design specifications for a bandstop filter are
given by

pass band I (0 ≤ Ω ≤ Ω p1 ) ( − δ p1 ) ≤ H bs ( Ω ) ≤ (1 + δ p1 );
pass band II ( Ω p2 ≤ Ω ≤ π ) 1 − δ p2 ) ≤ H bs ( Ω ) ≤ (1 + δ p2 );
stop band ( Ω s1 < Ω ≤ Ω s2 ) ≤ H bs ( Ω ) ≤δ s .

The steps involved in the design of a bandpass FIR filter using windowing are
similar to those specified for the bandpass filter in Section 15.3.
The transfer function of an ideal bandstop filter was defined in Section 14.1.4,
and is reproduced here for convenience:

0

Ω c1 ≤ Ω ≤ Ω c2

H ibs ( Ω ) = (15.37)

1 Ω < Ω c1 and Ω c2 < Ω ≤π,