Digital Signal Processing Reference
In-Depth Information
At the outset let us note that if the variable
t
is consistent in relation to time,
then the variable
f
is consistent in relation to a frequency. We will accept that the
Fourier transform of a signal is defined (i.e. the above integral converges) if the
signal is of finite energy.
The Fourier transform does not lead to any loss of information. In fact, knowing
()
x
f
, we can recreate
x
(
t
)
from the following inversion formula; for all
t
:
+∞
()
=
∫
( )
2
j
π
f t
ˆ
[2.26]
x t
x
f
e
df
−∞
We demonstrate this theorem by directly calculating the above integral and using
the formulae [2.10] and [2.13]:
+∞
+∞ +∞
∫
()
j
2
π
f t
∫ ∫
()
j
2
π
f u
j
2
π
f t
ˆ
x
f
e
df
=
x u
e
du e
df
−∞
−∞ −∞
+∞
+∞
⎡
⎤
( )
j
2
π
f
t
−
u
∫ ∫
()
=
x u
e
df
du
⎢
⎥
−∞
−∞
⎣
⎦
+∞
∫
()( )
=
xu
δ
t u du
−
−∞
()
=
xt
which concludes the demonstration.
For a periodic continuous time signal of period
T
,
the Fourier transform
()
x
is
f
1
T
a series of pulses with
spacing whose weight can be calculated using its Fourier
series decomposition
()
x
A
:
+∞
A
⎛
⎞
A
()
()
ˆ
=
ˆ
A
δ
−
[2.27]
xf
x
f
⎜
⎟
T
⎝
⎠
=−∞
with:
T
A
1
−
j
2
π
t
=
∫
()
()
x
ˆ
A
xte
T
dt
[2.28]
T
0
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