Parseval's theorem

In mathematics, Parseval's theorem ^[1] usually refers to the result that the Fourier transform is unitary; loosely, that the sum (or integral) of the square of a function is equal to the sum (or integral) of the square of its transform. It originates from a 1799 theorem about series by Marc-Antoine Parseval, which was later applied to the Fourier series. It is also known as Rayleigh's energy theorem, or Rayleigh's identity, after 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 ${\displaystyle A(x)}$ and ${\displaystyle B(x)}$ are two complex-valued functions on ${\displaystyle \mathbb {R} }$ of period ${\displaystyle 2\pi }$ that are square integrable (with respect to the Lebesgue measure) over intervals of period length, with Fourier series

${\displaystyle A(x)=\sum _{n=-\infty }^{\infty }a_{n}e^{inx}}$

and

${\displaystyle B(x)=\sum _{n=-\infty }^{\infty }b_{n}e^{inx}}$

respectively. Then

${\displaystyle \sum _{n=-\infty }^{\infty }a_{n}{\overline {b_{n}}}={\frac {1}{2\pi }}\int _{-\pi }^{\pi }A(x){\overline {B(x)}}\,\mathrm {d} x,}$

(Eq.1)

where ${\displaystyle i}$ is the imaginary unit and horizontal bars indicate complex conjugation. Substituting ${\displaystyle A(x)}$ and ${\displaystyle {\overline {B(x)}}}$:

${\displaystyle {\begin{aligned}\sum _{n=-\infty }^{\infty }a_{n}{\overline {b_{n}}}&={\frac {1}{2\pi }}\int _{-\pi }^{\pi }\left(\sum _{n=-\infty }^{\infty }a_{n}e^{inx}\right)\left(\sum _{n=-\infty }^{\infty }{\overline {b_{n}}}e^{-inx}\right)\,\mathrm {d} x\\[6pt]&={\frac {1}{2\pi }}\int _{-\pi }^{\pi }\left(a_{1}e^{i1x}+a_{2}e^{i2x}+\cdots \right)\left({\overline {b_{1}}}e^{-i1x}+{\overline {b_{2}}}e^{-i2x}+\cdots \right)\mathrm {d} x\\[6pt]&={\frac {1}{2\pi }}\int _{-\pi }^{\pi }\left(a_{1}e^{i1x}{\overline {b_{1}}}e^{-i1x}+a_{1}e^{i1x}{\overline {b_{2}}}e^{-i2x}+a_{2}e^{i2x}{\overline {b_{1}}}e^{-i1x}+a_{2}e^{i2x}{\overline {b_{2}}}e^{-i2x}+\cdots \right)\mathrm {d} x\\[6pt]&={\frac {1}{2\pi }}\int _{-\pi }^{\pi }\left(a_{1}{\overline {b_{1}}}+a_{1}{\overline {b_{2}}}e^{-ix}+a_{2}{\overline {b_{1}}}e^{ix}+a_{2}{\overline {b_{2}}}+\cdots \right)\mathrm {d} x\end{aligned}}}$

As is the case with the middle terms in this example, many terms will integrate to ${\displaystyle 0}$ over a full period of length ${\displaystyle 2\pi }$ (see harmonics):

${\displaystyle {\begin{aligned}\sum _{n=-\infty }^{\infty }a_{n}{\overline {b_{n}}}&={\frac {1}{2\pi }}\left[a_{1}{\overline {b_{1}}}x+ia_{1}{\overline {b_{2}}}e^{-ix}-ia_{2}{\overline {b_{1}}}e^{ix}+a_{2}{\overline {b_{2}}}x+\cdots \right]_{-\pi }^{+\pi }\\[6pt]&={\frac {1}{2\pi }}\left(2\pi a_{1}{\overline {b_{1}}}+0+0+2\pi a_{2}{\overline {b_{2}}}+\cdots \right)\\[6pt]&=a_{1}{\overline {b_{1}}}+a_{2}{\overline {b_{2}}}+\cdots \\[6pt]\end{aligned}}}$

More generally, if ${\displaystyle A(x)}$ and ${\displaystyle B(x)}$ are instead two complex-valued functions on ${\displaystyle \mathbb {R} }$ of period ${\displaystyle P}$ that are square integrable (with respect to the Lebesgue measure) over intervals of period length, with Fourier series

${\displaystyle A(x)=\sum _{n=-\infty }^{\infty }a_{n}e^{2\pi ni\left({\frac {x}{P}}\right)}}$

and

${\displaystyle B(x)=\sum _{n=-\infty }^{\infty }b_{n}e^{2\pi ni\left({\frac {x}{P}}\right)}}$

respectively. Then

${\displaystyle \sum _{n=-\infty }^{\infty }a_{n}{\overline {b_{n}}}={\frac {1}{P}}\int _{-P/2}^{P/2}A(x){\overline {B(x)}}\,\mathrm {d} x,}$

(Eq.2)

Even more generally, given an abelian locally compact group G with Pontryagin dual G^, Parseval's theorem says the Pontryagin–Fourier transform is a unitary operator between Hilbert spaces L²(G) and L²(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 ${\displaystyle \mathbb {R} }$, G^ is also ${\displaystyle \mathbb {R} }$ 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 ${\displaystyle f(x)}$ is a square-integrable function over ${\displaystyle [-\pi ,\pi ]}$ (i.e., ${\displaystyle f(x)}$ and ${\displaystyle f^{2}(x)}$ are integrable on that interval), with the Fourier series

${\displaystyle f(x)\simeq {\frac {a_{0}}{2}}+\sum _{n=1}^{\infty }(a_{n}\cos(nx)+b_{n}\sin(nx)).}$

Then ^{[4] [5] [6]}

${\displaystyle {\frac {1}{\pi }}\int _{-\pi }^{\pi }f^{2}(x)\,\mathrm {d} x={\frac {a_{0}^{2}}{2}}+\sum _{n=1}^{\infty }\left(a_{n}^{2}+b_{n}^{2}\right).}$

Notation used in engineering

In electrical engineering, Parseval's theorem is often written as:

${\displaystyle \int _{-\infty }^{\infty }|x(t)|^{2}\,\mathrm {d} t={\frac {1}{2\pi }}\int _{-\infty }^{\infty }|X(\omega )|^{2}\,\mathrm {d} \omega =\int _{-\infty }^{\infty }|X(2\pi f)|^{2}\,\mathrm {d} f}$

where ${\displaystyle X(\omega )={\mathcal {F}}_{\omega }\{x(t)\}}$ represents the continuous Fourier transform (in normalized, unitary form) of ${\displaystyle x(t)}$, and ${\displaystyle \omega =2\pi f}$ 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 discrete time signals, the theorem becomes:

${\displaystyle \sum _{n=-\infty }^{\infty }|x[n]|^{2}={\frac {1}{2\pi }}\int _{-\pi }^{\pi }|X_{2\pi }({\phi })|^{2}\mathrm {d} \phi }$

where ${\displaystyle X_{2\pi }}$ is the discrete-time Fourier transform (DTFT) of ${\displaystyle x}$ and ${\displaystyle \phi }$ represents the angular frequency (in radians per sample) of ${\displaystyle x}$.

Alternatively, for the discrete Fourier transform (DFT), the relation becomes:

${\displaystyle \sum _{n=0}^{N-1}|x[n]|^{2}={\frac {1}{N}}\sum _{k=0}^{N-1}|X[k]|^{2}}$

where ${\displaystyle X[k]}$ is the DFT of ${\displaystyle x[n]}$, both of length ${\displaystyle N}$.

We show the DFT case below. For the other cases, the proof is similar. By using the definition of inverse DFT of ${\displaystyle X[k]}$, we can derive

${\displaystyle {\begin{aligned}{\frac {1}{N}}\sum _{k=0}^{N-1}|X[k]|^{2}&={\frac {1}{N}}\sum _{k=0}^{N-1}X[k]\cdot X^{*}[k]={\frac {1}{N}}\sum _{k=0}^{N-1}\left[\sum _{n=0}^{N-1}x[n]\,\exp \left(-j{\frac {2\pi }{N}}k\,n\right)\right]\,X^{*}[k]\\[5mu]&={\frac {1}{N}}\sum _{n=0}^{N-1}x[n]\left[\sum _{k=0}^{N-1}X^{*}[k]\,\exp \left(-j{\frac {2\pi }{N}}k\,n\right)\right]={\frac {1}{N}}\sum _{n=0}^{N-1}x[n](N\cdot x^{*}[n])\\[5mu]&=\sum _{n=0}^{N-1}|x[n]|^{2},\end{aligned}}}$

where ${\displaystyle *}$ represents complex conjugate.

See also

Parseval's theorem is closely related to other mathematical results involving unitary transformations:

• Parseval's identity
• Plancherel's theorem
• Wiener–Khinchin theorem
• Bessel's inequality

Notes

1. 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).
2. 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.
3. 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.
4. Arthur E. Danese (1965). Advanced Calculus. Vol. 1. Boston, MA: Allyn and Bacon, Inc. p. 439.
5. Wilfred Kaplan (1991). Advanced Calculus (4th ed.). Reading, MA: Addison Wesley. p.  519. ISBN   0-201-57888-3.
6. Georgi P. Tolstov (1962). Fourier Series . Translated by Silverman, Richard. Englewood Cliffs, NJ: Prentice-Hall, Inc. p.  119.

External links

• Parseval's Theorem on Mathworld