FIR filter design - Continuous and Discrete Time Signals and Systems - page 688

Fig. 15.14. Desired
specifications of a bandpass
filter.

| H bp ( W )|

1 + d p

1 − d p

stop
band I

pass band stop

band II

d s2
d s1

W

0 W s1 W p1 W p2 W s2 p

Eq. (14.2b) and implement H lp ( Ω ) instead. Based on the frequency character-
istics of the highpass FIR filter illustrated in Fig. 15.12, the specifications of
the H lp ( Ω ) in Eq. (14.2b) are given by

pass band (0 ≤ Ω ≤ Ω s ) ( 1 − δ s ) ≤ H lp ( Ω ) ≤ (1 + δ s );
stop band ( Ω p < Ω ≤ π ) 0 ≤ H lp ( Ω ) ≤δ p .
The impulse response of the lowpass FIR filter h lp [ k ] is then transformed to the
impulse response h hp [ k ] of the highpass FIR filter using the following equation:
h hp [ k ] = δ [ k − m ] − h lp [ k ] .

15.3 Design of bandpass filters using windowing

The design specifications for bandpass filters are specified in Fig. 15.14 and are
given by

stop band I (0 ≤ Ω ≤ Ω s1 ) ≤ H ( Ω ) ≤δ s1 ;

stop band II ( Ω s2 ≤ Ω ≤ π ) ≤ H ( Ω ) ≤δ s2 ;

pass band ( Ω p1 < Ω ≤ Ω p2 ) ( − δ p ) ≤ H ( Ω ) ≤ (1 + δ p ) ,

where we assume that the values of ripples δ s1 and δ s2 allowed in the two stop
bands are different. The algorithm used to design a bandpass FIR filter using
windowing is similar to the design for the highpass filter described in Section
15.2, except that the impulse response of an ideal bandpass filter is used in
step 2.
The transfer function of an ideal bandpass filter was defined in Section 14.1.3,
and is reproduced here for convenience:

1 Ω c1 ≤ Ω ≤ Ω c2
0

H ibp ( Ω ) = (15.31)

Ω < Ω c1

and Ω c2 ≤ Ω ≤ π.