Digital filters - Continuous and Discrete Time Signals and Systems - page 624

h ihp [ k ] = δ [ k ] − h ilp [ k ] = δ [ k ] − Ω c
π

k Ω c
π

sinc . (14.3b)

14.1.3 Ideal bandpass filter

The transfer function H ibp ( Ω ) of an ideal bandpass filter, with cut-off frequencies
of Ω c1 and Ω c2 ,isgivenby

1 Ω c1 ≤ Ω ≤ Ω c2
0 Ω c1 < Ω

H ibp ( Ω ) = (14.4a)

and Ω c2 < Ω ≤ π,

which has a pass band of Ω c1 ≤ Ω ≤ Ω c2 and a stop band of Ω ≤ Ω c1 and
Ω c2 ≤ Ω ≤π . The magnitude response of the ideal bandpass filter is shown
in Fig. 14.2(c).
The transfer function H ibp ( Ω ) is expressed in terms of the transfer functions
of two ideal lowpass filters:

H ibp ( Ω ) = H ilp1 ( Ω ) − H ilp2 ( Ω )

cut - off freq = Ω c1 .

(14.4b)

cut - off freq = Ω c2

Calculating the inverse DTFT of Eq. (14.4a), the impulse response h ibp [ k ]of
the ideal bandpass filter can be expressed as follows:

h ibp [ k ] = h ilp1 [ k ] − h ilp2 [ k ]

Ω c = Ω c1 .

(14.4c)

Ω c = Ω c2

Substituting the expression for h ilp [ k ] given in Eq. (14.1b) into the above equa-
tion, the impulse response h ibp [ k ] of the ideal bandpass filter can be expressed
as follows:
h ibp [ k ] = Ω c2
π

k Ω c2
π

k Ω c1
π

− Ω c1
π

sinc sinc . (14.4d)

Equation. (14.4b) shows that a bandpass filter can be formed by a parallel
configuration of two lowpass filters. The first lowpass filter in the parallel con-
figuration should have a cut-off frequency of Ω c2 , while the second lowpass
filter has a cut-off frequency of Ω c1 . Other configurations of bandpass filters
are also possible, such as a series combination of a lowpass and a highpass
filter.

14.1.4 Ideal bandstop filter

The transfer function H ibs ( Ω ) of an ideal bandstop filter, with cut-off frequencies
Ω c1 and Ω c2 ,isgivenby

0 Ω c1 ≤ Ω ≤ Ω c2
1

H ibs ( Ω ) =

(14.5a)

Ω < Ω c1 and Ω c2 < Ω ≤π,

which has a pass band of Ω < Ω c1 and Ω c2 < Ω ≤π and a stop band of
Ω c1 ≤ Ω ≤ Ω c2 . The magnitude response of the ideal bandstop filter is shown
in Fig. 14.2(d).