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

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

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

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):

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

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]

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

In physics and 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 ${\displaystyle N}$.

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

• Parseval, MacTutor History of Mathematics archive.
• George B. Arfken and Hans J. Weber, Mathematical Methods for Physicists (Harcourt: San Diego, 2001).
• Hubert Kennedy, Eight Mathematical Biographies (Peremptory Publications: San Francisco, 2002).
• Alan V. Oppenheim and Ronald W. Schafer, Discrete-Time Signal Processing 2nd Edition (Prentice Hall: Upper Saddle River, NJ, 1999) p 60.
• William McC. Siebert, Circuits, Signals, and Systems (MIT Press: Cambridge, MA, 1986), pp. 410–411.
• David W. Kammler, A First Course in Fourier Analysis (Prentice–Hall, Inc., Upper Saddle River, NJ, 2000) p. 74.

• Parseval's Theorem on Mathworld