Digital Signal Processing Reference
In-Depth Information
and we split the last integral to obtain
1
2
π
1
2
π
X
(
e
j
(
ω
−
σ
)
)
e
j
(
ω
−
σ
)
n
d
ω
Y
(
e
j
σ
)
e
j
σ
n
d
σ
l
g
r
,
y
i
d
.
,
©
,
L
s
π
π
−
π
−
π
=
x
[
n
]
y
[
n
]
These fundamental results are summarized in Table 5.1.
Tabl e 5 .1
The convolution and modulation theorems.
Time Domain
Frequency Domain
[
]
∗
[
]
(
e
j
ω
)
(
e
j
ω
)
x
n
y
n
X
Y
x
[
n
]
y
[
n
]
X
(
e
j
ω
)
∗
Y
(
e
j
ω
)
5.4.3
Properties of the Frequency Response
Since an LTI system is completely characterized by its impulse response, it
is also uniquely characterized by its frequency response. The frequency re-
sponse provides us with a different perspective on the properties of a given
filter, which are embedded in the magnitude and the phase of the response.
Just as the impulse response completely characterizes a filter in the
discrete-time domain, its Fourier transform, called the filter's
frequency re-
sponse
, completely characterizes the filter in the frequency domain. The
properties of LTI systems are described in terms of their DTFTs magnitude
and phase, each of which controls different features of the system's behav-
ior.
Magnitude.
The most powerful intuition arising from the convolution
theorem is obtained by considering the magnitude of the spectra involved
in a filtering operation. Recall that a Fourier spectrum represents the energy
distribution of a signal in frequency; by appropriately “shaping” the magni-
tude spectrum of a filter's impulse response we can easily boost, attenuate,
and even completely eliminate, a given part of the frequency content in the
filtered input sequence. According to the way the magnitude spectrum is
affected by the filter, we can classify filters into three broad categories (here
as before we assume that the impulse response is real, and therefore the as-
sociated magnitude spectrum is symmetric; in addition, the 2
π
periodicity
of the spectrum is implicitly understood):
