Digital Signal Processing Reference
In-Depth Information
H
(
W
)
16
<
H
(
W
)
3
0.245
p
.
W
W
−
p
−
p
/2
−
p
−
p
/2
0
p
/2
p
0
p
/2
p
−0.245
p
(a)
(b)
value of -0.245
π
radians at
Ω
=
0
.
37
π
radians/s. From
Ω
=
0
.
37
π
radians/s to
Fig. 11.18. (a) Magnitude
spectrum and (b) phase
spectrum of the LTID system
considered in Example 11.19. The
responses are shown in the
frequency range
Ω
= [−π, π].
= π
radians/s.
For negative frequencies, the phase increases to its maximum value of
0.245
π
radians at
=−
0.37
π
radians/s, after which the phase decreases and
approaches zero at
Ω
= π
radians/s, the phase increases and approaches zero at
Ω
Ω
=−π
radians/s.
It is also observed that the transfer function
H
(
Ω
) satisfies the Hermitian
symmetry property stated in Eq. (11.39a). Since the impulse response
h
[
k
]isa
real-valued function, the magnitude spectrum
H
(
Ω
)
is an even function of
Ω
and is therefore symmetric about the
y
-axis in Fig. 11.18(a). On the other hand,
the phase spectrum
<
H
(
Ω
) is an odd function of
Ω
and is therefore symmetric
about the origin in Fig. 11.18(b). In cases where the impulse response
h
[
k
]isa
real-valued function, the plots in the range
Ω
=
[0
,π
] are sufficient to represent
the frequency response completely. The frequency response within the range
Ω
=
[
−π,
0] can then be obtained using the Hermitian symmetry property.
Example 11.20
Derive and plot the frequency responses of the LTID systems with the following
impulse responses:
(i)
h
[
k
]
=
sin(
π
k
/
6)
π
k
;
(11.61)
(ii)
g
[
k
]
= δ
[
k
]
−
sin(
π
k
/
6)
π
k
.
(11.62)
Solution
(i) We express
h
[
k
] as a sinc function as
h
[
k
]
=
1
6
sin (
π
k
/
6)
π
k
/
6
=
1
6
sinc(
k
/
6).
Using Table 11.2, the transfer function is given by
1
≤π/
6
Ω
H
(
Ω
)
=
(11.63)
0
π/
6
<
≤π.
Ω
The impulse response
h
[
k
] and its magnitude spectrum
H
(
Ω
)
are plotted in
Figs. 11.19(a) and (b) within the frequency range
Ω
=
[0
,π
]. Since the transfer
function
H
(
Ω
) is real-valued, the phase spectrum is zero. The transfer function,
Eq. (11.63), or equivalently the impulse response, Eq. (11.61), represents an
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