Digital Signal Processing Reference
In-Depth Information
calculated in Section 5.2. The results are listed in Table 5.2, and their magni-
tude and phase spectra of the CTFT are plotted in Table 5.3. In Section 5.3,
we presented the partial fraction method for calculating the inverse CTFT. In
Section 5.4, we covered the following symmetry properties of the CTFT.
(1) The CTFT
X
(
ω
)ofa
real-valued
signal
x
(
t
) is Hermitian symmetrical, i.e.
X
(
ω
)
=
∗
(
−ω
). Due to the Hermitian symmetry property, the magnitude
spectrum
X
(
ω
)
is an even function of
ω
, while the phase spectrum <
X
(
ω
)
is an odd function of
ω
.
(2) The CTFT
X
(
ω
)ofa
real-valued and even
signal
x
(
t
) is also real-valued
and even, i.e. Re
{
X
(
ω
)
=
Re
X
(
−ω
)
}
and Im
{
X
(
ω
)
=
0.
(3) The CTFT
X
(
ω
)ofa
real-valued and odd
signal
x
(
t
) is also pure imaginary
and odd, i.e. Re
{
X
(
ω
)
=
0 and Im
{
X
(
ω
)
=−
Im
{
X
(
−ω
)
}
.
X
Section 5.5 considered the transformation properties of the CTFT, which are
summarized as follows.
(1) The linearity property states that the CTFT of a linear combination of
aperiodic signals is given by the same linear combination of the CTFT of
the individual aperiodic signals.
(2) If an aperiodic signal is
time-scaled
, the CTFT is inversely time-scaled.
(3) A time shift of
t
0
in the aperiodic signal does not affect the magnitude of
the CTFT. However, the phase changes by an additive factor of
ω
t
0
. This
property is referred to as the
time-shifting
property.
(4) A frequency shift of
ω
0
in the aperiodic signal does not affect the magnitude
of the signal in the time domain. However, the phase of the signal in the
time domain changes by an additive factor of
ω
t
0
. This property is referred
to as the
frequency-shifting
property.
(5) The CTFT of a
time-differentiated
periodic signal is obtained by
multiplying
the CTFT of the original signal by a factor of j
ω
.
(6) The CTFT of a
time-integrated
periodic signal is obtained by
dividing
the
CTFT of the original signal by a factor of j
ω
with a scaled impulse function
at
ω =
0
.
(7) The
duality property
states that there is symmetry between the time wave-
form and its frequency-domain representation such that the two functions
in a CTFT pair are dual with respect to each other. Given an arbitrary
time-domain waveform
x
(
t
) and its CTFT waveform
X
(
ω
), for example, a
second CTFT pair exists with the time-domain representation
X
(
t
), having
the same waveform as
X
(
ω
), and the CTFT 2
π
x
(
−ω
) in the frequency
domain.
(8) Convolution in the time domain is equivalent to multiplication of the CTFT
in the frequency domain, and vice versa. The convolution property leads
to an alternative approach for evaluating the output response of an LTIC
system to any arbitrary input.
Search WWH ::
Custom Search