Theorem in mathematics
In mathematics, Parseval's theorem John William Strutt, Lord Rayleigh.
[2]
Although the term "Parseval's theorem" is often used to describe the unitarity of any Fourier transform, especially in physics , the most general form of this property is more properly called the Plancherel theorem .[3]
Statement of Parseval's theorem
Suppose that
A
(
x
)
{\displaystyle A(x)}
and
B
(
x
)
{\displaystyle B(x)}
are two complex-valued functions on
R
{\displaystyle \mathbb {R} }
of period
2
π
{\displaystyle 2\pi }
that are
A
(
x
)
=
∑
n
=
−
∞
∞
a
n
e
i
n
x
{\displaystyle A(x)=\sum _{n=-\infty }^{\infty }a_{n}e^{inx}}
and
B
(
x
)
=
∑
n
=
−
∞
∞
b
n
e
i
n
x
{\displaystyle B(x)=\sum _{n=-\infty }^{\infty }b_{n}e^{inx}}
respectively. Then
∑
n
=
−
∞
∞
a
n
b
n
¯
=
1
2
π
∫
−
π
π
A
(
x
)
B
(
x
)
¯
d
x
,
{\displaystyle \sum _{n=-\infty }^{\infty }a_{n}{\overline {b_{n}}}={\frac {1}{2\pi }}\int _{-\pi }^{\pi }A(x){\overline {B(x)}}\,\mathrm {d} x,}
(Eq.1 )
where
i
{\displaystyle i}
is the
complex conjugation
. Substituting
A
(
x
)
{\displaystyle A(x)}
and
B
(
x
)
¯
{\displaystyle {\overline {B(x)}}}
:
∑
n
=
−
∞
∞
a
n
b
n
¯
=
1
2
π
∫
−
π
π
(
∑
n
=
−
∞
∞
a
n
e
i
n
x
)
(
∑
n
=
−
∞
∞
b
n
¯
e
−
i
n
x
)
d
x
=
1
2
π
∫
−
π
π
(
a
1
e
i
1
x
+
a
2
e
i
2
x
+
⋯
)
(
b
1
¯
e
−
i
1
x
+
b
2
¯
e
−
i
2
x
+
⋯
)
d
x
=
1
2
π
∫
−
π
π
(
a
1
e
i
1
x
b
1
¯
e
−
i
1
x
+
a
1
e
i
1
x
b
2
¯
e
−
i
2
x
+
a
2
e
i
2
x
b
1
¯
e
−
i
1
x
+
a
2
e
i
2
x
b
2
¯
e
−
i
2
x
+
⋯
)
d
x
=
1
2
π
∫
−
π
π
(
a
1
b
1
¯
+
a
1
b
2
¯
e
−
i
x
+
a
2
b
1
¯
e
i
x
+
a
2
b
2
¯
+
⋯
)
d
x
{\displaystyle {\begin{aligned}\sum _{n=-\infty }^{\infty }a_{n}{\overline {b_{n}}}&={\frac {1}{2\pi }}\int _{-\pi }^{\pi }\left(\sum _{n=-\infty }^{\infty }a_{n}e^{inx}\right)\left(\sum _{n=-\infty }^{\infty }{\overline {b_{n}}}e^{-inx}\right)\,\mathrm {d} x\\[6pt]&={\frac {1}{2\pi }}\int _{-\pi }^{\pi }\left(a_{1}e^{i1x}+a_{2}e^{i2x}+\cdots \right)\left({\overline {b_{1}}}e^{-i1x}+{\overline {b_{2}}}e^{-i2x}+\cdots \right)\mathrm {d} x\\[6pt]&={\frac {1}{2\pi }}\int _{-\pi }^{\pi }\left(a_{1}e^{i1x}{\overline {b_{1}}}e^{-i1x}+a_{1}e^{i1x}{\overline {b_{2}}}e^{-i2x}+a_{2}e^{i2x}{\overline {b_{1}}}e^{-i1x}+a_{2}e^{i2x}{\overline {b_{2}}}e^{-i2x}+\cdots \right)\mathrm {d} x\\[6pt]&={\frac {1}{2\pi }}\int _{-\pi }^{\pi }\left(a_{1}{\overline {b_{1}}}+a_{1}{\overline {b_{2}}}e^{-ix}+a_{2}{\overline {b_{1}}}e^{ix}+a_{2}{\overline {b_{2}}}+\cdots \right)\mathrm {d} x\end{aligned}}}
As is the case with the middle terms in this example, many terms will integrate to
0
{\displaystyle 0}
over a full period of length
2
π
{\displaystyle 2\pi }
(see harmonics ):
∑
n
=
−
∞
∞
a
n
b
n
¯
=
1
2
π
[
a
1
b
1
¯
x
+
i
a
1
b
2
¯
e
−
i
x
−
i
a
2
b
1
¯
e
i
x
+
a
2
b
2
¯
x
+
⋯
]
−
π
+
π
=
1
2
π
(
2
π
a
1
b
1
¯
+
0
+
0
+
2
π
a
2
b
2
¯
+
⋯
)
=
a
1
b
1
¯
+
a
2
b
2
¯
+
⋯
{\displaystyle {\begin{aligned}\sum _{n=-\infty }^{\infty }a_{n}{\overline {b_{n}}}&={\frac {1}{2\pi }}\left[a_{1}{\overline {b_{1}}}x+ia_{1}{\overline {b_{2}}}e^{-ix}-ia_{2}{\overline {b_{1}}}e^{ix}+a_{2}{\overline {b_{2}}}x+\cdots \right]_{-\pi }^{+\pi }\\[6pt]&={\frac {1}{2\pi }}\left(2\pi a_{1}{\overline {b_{1}}}+0+0+2\pi a_{2}{\overline {b_{2}}}+\cdots \right)\\[6pt]&=a_{1}{\overline {b_{1}}}+a_{2}{\overline {b_{2}}}+\cdots \\[6pt]\end{aligned}}}
More generally, if
A
(
x
)
{\displaystyle A(x)}
and
B
(
x
)
{\displaystyle B(x)}
are instead two complex-valued functions on
R
{\displaystyle \mathbb {R} }
of period
P
{\displaystyle P}
that are
A
(
x
)
=
∑
n
=
−
∞
∞
a
n
e
2
π
n
i
(
x
P
)
{\displaystyle A(x)=\sum _{n=-\infty }^{\infty }a_{n}e^{2\pi ni\left({\frac {x}{P}}\right)}}
and
B
(
x
)
=
∑
n
=
−
∞
∞
b
n
e
2
π
n
i
(
x
P
)
{\displaystyle B(x)=\sum _{n=-\infty }^{\infty }b_{n}e^{2\pi ni\left({\frac {x}{P}}\right)}}
respectively. Then
∑
n
=
−
∞
∞
a
n
b
n
¯
=
1
P
∫
−
P
/
2
P
/
2
A
(
x
)
B
(
x
)
¯
d
x
,
{\displaystyle \sum _{n=-\infty }^{\infty }a_{n}{\overline {b_{n}}}={\frac {1}{P}}\int _{-P/2}^{P/2}A(x){\overline {B(x)}}\,\mathrm {d} x,}
(Eq.2 )
Even more generally, given an abelian
Hilbert spaces
L 2 (
G ) and
L 2 (
G^ ) (with integration being against the appropriately scaled
Haar measures on the two groups.) When
G is the
unit circle T ,
G^ is the integers and this is the case discussed above. When
G is the real line
R
{\displaystyle \mathbb {R} }
,
G^ is also
R
{\displaystyle \mathbb {R} }
and the unitary transform is the
Fourier transform on the real line. When
G is the
cyclic group Z n , again it is self-dual and the Pontryagin–Fourier transform is what is called
discrete Fourier transform in applied contexts.
Parseval's theorem can also be expressed as follows:
Suppose
f
(
x
)
{\displaystyle f(x)}
is a square-integrable function over
[
−
π
,
π
]
{\displaystyle [-\pi ,\pi ]}
(i.e.,
f
(
x
)
{\displaystyle f(x)}
and
f
2
(
x
)
{\displaystyle f^{2}(x)}
are integrable on that interval), with the Fourier series
f
(
x
)
≃
a
0
2
+
∑
n
=
1
∞
(
a
n
cos
(
n
x
)
+
b
n
sin
(
n
x
)
)
.
{\displaystyle f(x)\simeq {\frac {a_{0}}{2}}+\sum _{n=1}^{\infty }(a_{n}\cos(nx)+b_{n}\sin(nx)).}
Then[4] [5] [6]
1
π
∫
−
π
π
f
2
(
x
)
d
x
=
a
0
2
2
+
∑
n
=
1
∞
(
a
n
2
+
b
n
2
)
.
{\displaystyle {\frac {1}{\pi }}\int _{-\pi }^{\pi }f^{2}(x)\,\mathrm {d} x={\frac {a_{0}^{2}}{2}}+\sum _{n=1}^{\infty }\left(a_{n}^{2}+b_{n}^{2}\right).}
Notation used in engineering
In electrical engineering , Parseval's theorem is often written as:
∫
−
∞
∞
|
x
(
t
)
|
2
d
t
=
1
2
π
∫
−
∞
∞
|
X
(
ω
)
|
2
d
ω
=
∫
−
∞
∞
|
X
(
2
π
f
)
|
2
d
f
{\displaystyle \int _{-\infty }^{\infty }|x(t)|^{2}\,\mathrm {d} t={\frac {1}{2\pi }}\int _{-\infty }^{\infty }|X(\omega )|^{2}\,\mathrm {d} \omega =\int _{-\infty }^{\infty }|X(2\pi f)|^{2}\,\mathrm {d} f}
where
X
(
ω
)
=
F
ω
{
x
(
t
)
}
{\displaystyle X(\omega )={\mathcal {F}}_{\omega }\{x(t)\}}
represents the
continuous Fourier transform
(in normalized, unitary form) of
x
(
t
)
{\displaystyle x(t)}
, and
ω
=
2
π
f
{\displaystyle \omega =2\pi f}
is frequency in radians per second.
The interpretation of this form of the theorem is that the total energy of a signal can be calculated by summing power-per-sample across time or spectral power across frequency.
For
signals
, the theorem becomes:
∑
n
=
−
∞
∞
|
x
[
n
]
|
2
=
1
2
π
∫
−
π
π
|
X
2
π
(
ϕ
)
|
2
d
ϕ
{\displaystyle \sum _{n=-\infty }^{\infty }|x[n]|^{2}={\frac {1}{2\pi }}\int _{-\pi }^{\pi }|X_{2\pi }({\phi })|^{2}\mathrm {d} \phi }
where
X
2
π
{\displaystyle X_{2\pi }}
is the discrete-time Fourier transform (DTFT) of
x
{\displaystyle x}
and
ϕ
{\displaystyle \phi }
represents the angular frequency (in radians per sample) of
x
{\displaystyle x}
.
Alternatively, for the discrete Fourier transform (DFT), the relation becomes:
∑
n
=
0
N
−
1
|
x
[
n
]
|
2
=
1
N
∑
k
=
0
N
−
1
|
X
[
k
]
|
2
{\displaystyle \sum _{n=0}^{N-1}|x[n]|^{2}={\frac {1}{N}}\sum _{k=0}^{N-1}|X[k]|^{2}}
where
X
[
k
]
{\displaystyle X[k]}
is the DFT of
x
[
n
]
{\displaystyle x[n]}
, both of length
N
{\displaystyle N}
.
We show the DFT case below. For the other cases, the proof is similar. By using the definition of inverse DFT of
X
[
k
]
{\displaystyle X[k]}
, we can derive
1
N
∑
k
=
0
N
−
1
|
X
[
k
]
|
2
=
1
N
∑
k
=
0
N
−
1
X
[
k
]
⋅
X
∗
[
k
]
=
1
N
∑
k
=
0
N
−
1
[
∑
n
=
0
N
−
1
x
[
n
]
exp
(
−
j
2
π
N
k
n
)
]
X
∗
[
k
]
=
1
N
∑
n
=
0
N
−
1
x
[
n
]
[
∑
k
=
0
N
−
1
X
∗
[
k
]
exp
(
−
j
2
π
N
k
n
)
]
=
1
N
∑
n
=
0
N
−
1
x
[
n
]
(
N
⋅
x
∗
[
n
]
)
=
∑
n
=
0
N
−
1
|
x
[
n
]
|
2
,
{\displaystyle {\begin{aligned}{\frac {1}{N}}\sum _{k=0}^{N-1}|X[k]|^{2}&={\frac {1}{N}}\sum _{k=0}^{N-1}X[k]\cdot X^{*}[k]={\frac {1}{N}}\sum _{k=0}^{N-1}\left[\sum _{n=0}^{N-1}x[n]\,\exp \left(-j{\frac {2\pi }{N}}k\,n\right)\right]\,X^{*}[k]\\[5mu]&={\frac {1}{N}}\sum _{n=0}^{N-1}x[n]\left[\sum _{k=0}^{N-1}X^{*}[k]\,\exp \left(-j{\frac {2\pi }{N}}k\,n\right)\right]={\frac {1}{N}}\sum _{n=0}^{N-1}x[n](N\cdot x^{*}[n])\\[5mu]&=\sum _{n=0}^{N-1}|x[n]|^{2},\end{aligned}}}
where
∗
{\displaystyle *}
represents complex conjugate.
See also
Parseval's theorem is closely related to other mathematical results involving unitary transformations:
Notes
^ Parseval des Chênes, Marc-Antoine Mémoire sur les séries et sur l'intégration complète d'une équation aux différences partielles linéaire du second ordre, à coefficients constants" presented before the Académie des Sciences (Paris) on 5 April 1799. This article was published in Mémoires présentés à l’Institut des Sciences, Lettres et Arts, par divers savants, et lus dans ses assemblées. Sciences, mathématiques et physiques. (Savants étrangers.) , vol. 1, pages 638–648 (1806).
^ Rayleigh, J.W.S. (1889) "On the character of the complete radiation at a given temperature," Philosophical Magazine , vol. 27, pages 460–469. Available on-line here .
^ Plancherel, Michel (1910) "Contribution à l'etude de la representation d'une fonction arbitraire par les integrales définies," Rendiconti del Circolo Matematico di Palermo , vol. 30, pages 298–335.
^ Arthur E. Danese (1965). Advanced Calculus . Vol. 1. Boston, MA: Allyn and Bacon, Inc. p. 439.
.
^ Georgi P. Tolstov (1962). Fourier Series . Translated by Silverman, Richard. Englewood Cliffs, NJ: Prentice-Hall, Inc. p. 119 .
External links