Digital Signal Processing Reference
In-Depth Information
Table 5.4. Symmetry and transformation properties of the CTFT
Time domain
Frequency domain
∞
∞
Transformation
1
2
π
X
(
ω
)e
j
ω
t
d
ω
x
(
t
)e
−
j
ω
t
d
t
properties
x
(
t
)
=
X
(
ω
)
=
Comments
−∞
−∞
Linearity
a
1
x
1
(
t
)
+
a
2
x
2
(
t
)
a
1
X
1
(
ω
)
+
a
2
X
2
(
ω
)
a
1
,
a
2
∈
C
1
a
X
ω
a
Scaling
x
(
at
)
a
∈ℜ
, real-valued
Time shifting
x
(
t
−
t
0
)
−
j
ω
t
0
X
(
ω
)
t
0
∈ℜ
, real-valued
Frequency shifting
e
j
ω
0
t
x
(
t
)
X
(
ω − ω
0
)
ω
0
∈ℜ
, real-valued
d
n
x
d
t
n
(j
ω
)
n
X
(
ω
)
Time differentiation
provided d
x
/d
t
exists
t
t
X
(
ω
)
j
ω
Time integration
x
(
τ
)d
τ
+ π
X
(0)
δ
(
ω
)
provided
x
(
τ
)d
τ
−∞
−∞
exists
( j )
n
d
n
X
d
ω
n
t
n
x
(
t
)
Frequency differentiation
provided d
X
/d
ω
exists
CTFT
←−−→
2
π
x
(
−ω
)
X
(
ω
)
Duality
X
(
t
)
i f
x
(
t
)
Time convolution
x
1
(
t
)
∗
x
2
(
t
)
X
1
(
ω
)
X
2
(
ω
)
convolution in time
domain
1
2
π
[
X
1
(
ω
)
∗
X
2
(
ω
)]
Frequency convolution
x
1
(
t
)
x
2
(
t
)
multiplication in time
domain
∞
∞
=
1
2
π
x
(
t
)
2
d
t
X
(
ω
)
2
d
ω
Parseval's relationship
E
x
=
energy in a signal
−∞
−∞
Symmetry properties
X
∗
(
ω
)
CTFT:
X
(
−ω
)
=
real and imaginary components
real component is even;
imaginary component
is odd
Re
X
(
ω
)
=
Re
X
(
−ω
)
Im
X
(
ω
)
=−
Im
X
(
−ω
)
x
(
t
) is a real-valued
function
Hermitian property
magnitude and phase spectra
magnitude spectrum is
even; phase spectrum
is odd
X
(
−ω
)
=
X
(
ω
)
<
X
(
−ω
)
=−<
X
(
ω
)
∞
Even function
x
(
t
)iseven
X
(
ω
)
=
2
x
(
t
) cos(
ω
t
)d
t
simplified CTFT
expression for even
signals
0
∞
Odd function
x
(
t
)isodd
X
(
ω
)
=−
j2
x
(
t
) sin(
ω
t
)d
t
simplified CTFT
expression for odd
signals
0
Re
{
X
(
ω
)
=
Re
X
(
−ω
)
}
Im
{
X
(
ω
)
=
0
Real-valued and even
function
x
(
t
) is even and real-valued
CTFT is real-valued and
even
Real-valued and odd
function
x
(
t
) is odd and real-valued
Re
{
X
(
ω
)
=
0
Im
{
X
(
ω
)
=−
Im
{
X
(
−ω
)
}
CTFT is imaginary and
odd
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