Digital Signal Processing Reference
In-Depth Information
Inversely, the transform of a product is a convolution sum:
m
()
⎧
()
ˆˆ
xy f
=
x
y f
Fourier transform (continuous time)
⊗
⎪
⎪
m
()
()
ˆˆ
xy v
=
x
y v
Fourier transform (discrete time)
⊗
⎪
⎨
[2.46]
m
()
()
xy
A
=
x
ˆˆ
y
A
Fourier series (period
T
)
⎪
⎪
⊗
m
()
()
ˆˆ
x y
A
=
Nx
y
A
Discrete Fourier transform (period
N
)
⎪
⎩
⊗
2.2.5.
Energy conservation (Parseval's theorem)
The energy of a continuous time signal
x
(
t
)
or a discrete time signal
x
(
k
)
can be
calculated by integrating the square of the Fourier transform modulus
()
x v
or its
transform in standardized frequency
()
x v
:
x x
,
=
x x
ˆˆ
,
[2.47]
The function or the series
2
x
is known as the energy spectrum, or energy
spectral density of a signal
x
, because its integral (or its summation) gives the energy
of the signal
x
.
The power of a periodic signal of continuous period
T
, or discrete period
N
, can
be calculated by the summation of the square of the decomposition modulus in the
Fourier series or discrete Fourier transform. For the Fourier series, this is written as:
x x
,
=
x x
ˆˆ
,
[2.48]
and for the discrete Fourier transform as:
x xNxx
,
=
ˆˆ
,
[2.49]
To sum up, by recalling the definition of the inner product, Parseval's theorem
can be written as:
+∞
+∞
2
2
∫
()
∫
()
- continuous time signals:
=
ˆ
x t t
xf f
−∞
−∞
T
+∞
1
∑
A
2
2
()
()
- T-periodic signals:
xt dt
x
ˆ
A
T
0
=−∞
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