Digital Signal Processing Reference
In-Depth Information
+∞
1
2
1
2
+
∑
2
2
()
∫
()
- discrete time signals:
x k
=
x v
ˆ
dv
−
k
=−∞
N
−
1
N
−
1
1
∑
A
2
2
()
()
-
N
-periodic signals:
xk
x
ˆ
A
N
k
=
0
=
0
Let us demonstrate this property in the case of continuous time signals. Let
x
-
be
the returned signal
x
(for all values of
t, x
-
(
t
)
= x
(
-t
))
.
By combining the formulae
[2.37] and [2.38], we obtain:
m
() ()
*
−
*
= ⎡
⎣ ⎦
ˆ
x
f
x
f
Thus:
n
() ()
m
() () ()
*
2
−
*
−
*
()
xxf
=
xf
ˆ
xf
=
xf
ˆ
⎡ ⎤ ⎡ ⎤
⎣ ⎦ ⎣ ⎦
xf
ˆ
=
xf
ˆ
⊗
thus, by inverse Fourier transform calculated at
t
= 0:
+∞
()
2
−
*
()
=
∫
ˆ
x
x
0
x
f
df
⊗
−∞
Moreover:
+∞
+∞
+∞
()
2
−
*
∫
−
*
∫
*
∫
()
() ( )
() ()
x
x
0
=
xtx
0
−
tdt
=
xtx tdt
=
xt
dt
⊗
−∞
−∞
−∞
which concludes the demonstration.
2.2.6.
Other properties
MODULATION - The product of a continuous time signal
x
(
t
)
by a complex cisoid
of frequency
f
0
offsets the Fourier transform by
f
0
:
j
2
π
f
t
() ()
( ) (
)
yt
=
xte
y f
ˆ
=
x f
ˆ
−
f
[2.50]
0
0
The product of a discrete time signal
x
(
t
) by a complex cisoid of frequency
v
0
offsets the Fourier transform by
v
0
:
j
2
π
vk
() ()
() (
)
yk
=
xke
yv
ˆ
=
xv v
ˆ
−
[2.51]
0
0
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