Definitions

# Parseval's theorem

In mathematics, Parseval's theorem 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, after John William Strutt, Lord Rayleigh.

Although the term "Parseval's theorem" is often used to describe the unitarity of any Fourier transform, especially in physics and engineering, the most general form of this property is more properly called the Plancherel theorem.

## Statement of Parseval's theorem

Suppose that A(x) and B(x) are two Riemann integrable, complex-valued functions on R of period 2π with (formal) Fourier series

$A\left(x\right)=sum_\left\{n=-infty\right\}^infty a_ne^\left\{inx\right\}$ and $B\left(x\right)=sum_\left\{n=-infty\right\}^infty b_ne^\left\{inx\right\}$

respectively. Then

$sum_\left\{n=-infty\right\}^infty a_noverline\left\{b_n\right\} = frac\left\{1\right\}\left\{2pi\right\} int_\left\{-pi\right\}^pi A\left(x\right)overline\left\{B\left(x\right)\right\} dx,$

where i is the imaginary unit and horizontal bars indicate complex conjugation.

Parseval, who apparently had confined himself to real-valued functions, actually presented the theorem without proof, considering it to be self-evident. There are various important special cases of the theorem. First, if A = B one immediately obtains:

$sum_\left\{n=-infty\right\}^infty |a_n|^2 = frac\left\{1\right\}\left\{2pi\right\} int_\left\{-pi\right\}^pi |A\left(x\right)|^2 dx,$

from which the unitarity of the Fourier series follows.

Second, one often considers only the Fourier series for real-valued functions A and B, which corresponds to the special case: $a_0$ real, $a_\left\{-n\right\} = overline\left\{a_n\right\}$, $b_0$ real, and $b_\left\{-n\right\} =overline\left\{b_n\right\}$. In this case:

$a_0 b_0 + 2 Re sum_\left\{n=1\right\}^infty a_noverline\left\{b_n\right\} = frac\left\{1\right\}\left\{2pi\right\} int_\left\{-pi\right\}^pi A\left(x\right) B\left(x\right)dx,$

where $Re$ denotes the real part. (In the notation of the Fourier series article, replace $a_n$ and $b_n$ by $a_n / 2 - i b_n / 2$.)

## Applications

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

$int_\left\{-infty\right\}^\left\{infty\right\} | x\left(t\right) |^2 dt = int_\left\{-infty\right\}^\left\{infty\right\} | X\left(f\right) |^2 df$

where $X\left(f\right) = mathcal\left\{F\right\} \left\{ x\left(t\right) \right\}$ represents the continuous Fourier transform (in normalized, unitary form) of x(t) and f represents the frequency component (not angular frequency) of x.

The interpretation of this form of the theorem is that the total energy contained in a waveform x(t) summed across all of time t is equal to the total energy of the waveform's Fourier Transform X(f) summed across all of its frequency components f.

For discrete time signals, the theorem becomes:

$sum_\left\{n=-infty\right\}^\left\{infty\right\} | x\left[n\right] |^2 = frac\left\{1\right\}\left\{2pi\right\} int_\left\{-pi\right\}^\left\{pi\right\} | X\left(e^\left\{jphi\right\}\right) |^2 dphi$

where X is the discrete-time Fourier transform (DTFT) of x and φ represents the angular frequency (in radians per sample) of x.

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

$sum_\left\{n=0\right\}^\left\{N-1\right\} | x\left[n\right] |^2 = frac\left\{1\right\}\left\{N\right\} sum_\left\{k=0\right\}^\left\{N-1\right\} | X\left[k\right] |^2$

where X[k] is the DFT of x[n], both of length N.

## Equivalence of the norm and inner product forms

We shall refer to

$int_\left\{-infty\right\}^infty x\left(t\right)overline\left\{y\right\}\left(t\right) dt = int_\left\{-infty\right\}^infty X\left(f\right)overline\left\{Y\right\}\left(f\right) df$
as the inner product form, and to
$int_\left\{-infty\right\}^infty |x\left(t\right)|^2 dt = int_\left\{-infty\right\}^infty |X\left(f\right)|^2 df$
as the norm form. It is not difficult to show that they are (pointwise) equivalent. One can use the polarization identity
$aoverline\left\{b\right\} = frac\left\{1\right\}\left\{4\right\}\left(|a+b|^2 + i|a+ib|^2 + i^2|a+i^2b|^2 + i^3|a+i^3b|^2\right),$
which is true for all complex numbers a and b, and the linearity of both integration and the Fourier transform.

## References

• Parseval, MacTutor History of Mathematics archive.
• George B. Arfken and Hans J. Weber, Mathematical Methods for Physicists (Harcourt: San Diego, 2001).
• Hubert Kennedy, (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.