Digital Signal Processing Reference
In-Depth Information
This property is used in telecommunication systems by modulating the
amplitude,
f
0
being the frequency of the carrier sine wave.
TIME SHIFTING -
Shifting
a continuous time signal x
(
t
)
by a time t
0
(if t
0
>
0,
it
consists of a lag; if t
0
<
0,
it consists of a
lead) means multiplying the Fourier
transform by the cisoid of “frequency” t
0
:
() (
)
()
j
2
π
f t
()
yt
=−
xt t
y f
ˆ
=
e
ˆ
x f
[2.52]
0
0
Shifting
the discrete time signal x
(
k
)
by a time k
o
∈
Z means multiplying the
Fourier transform b the cisoid of “frequency” k
0
:
() (
)
()
j
2
π
vk
()
yk
=−
xk k
yv
ˆ
=
e
xv
ˆ
[2.53]
0
0
DERIVATIVE -
Let x
(
t
)
be a continuous time signal of derivative
() ()
dx
dt
x t
=
t
.
Then, for all values of f:
()
ˆ
()
x
f
=
j
2
π
f x
ˆ
f
[2.54]
For a T-periodic signal, whose derivative is also T-periodic, the decomposition
into Fourier series of the derivative signal can be written as:
A
ˆ
()
()
A
x
=
j
2
π
x
ˆ
A
[2.55]
T
TIME SCALING -
The time dependent expansion by a factor a
∈ ℜ
of a continuous
time signal leads to a frequency contraction by a factor
1
a
:
( )
1
f
⎛⎞
() ( )
yt
=
xat
ˆ
y f
=
aa
ˆ
⎜
⎝⎠
[2.56]
For a discrete time signal, the equivalent property must be used with greater
care. Let a
∈ Z,
and let
x be a zero discrete time signal for all the non-multiple
instants of a. Consider the signal y obtained from the signal x by the time scaling by
the factor a; then, the Fourier transform of y is obtained by the contraction of x:
v
⎛⎞
() ( )
()
yk
=
xak
ˆ
yv
=
⎜
⎝⎠
x
a
ˆ
[2.57]
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