Antihomomorphism

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, or Rayleigh's Identity, 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 square integrable (with respect to the Lebesgue measure), complex-valued functions on R of period 2π with Fourier series

$A(x)=\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}{a}_n{e}^{inx}$

and

$B(x)=\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}{b}_n{e}^{inx}$

respectively. Then

$\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}{a}_n\overline{b_n}=\frac{1}{2\pi }\underset{-\pi }{\overset{\pi }{\int }}A(x)\overline{B(x)}dx,$

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

More generally, given an abelian topological 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 R, G^ is also 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-time Fourier transform in applied contexts.

Notation used in engineering and physics

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

$\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}|x(t){|}^{2}dt=\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}|X(f){|}^{2}df$

where $X(f)=\mathcal{ℱ}\{x(t)\}$ 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:

$\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}|x[n]{|}^2=\frac{1}{2\pi }\underset{-\pi }{\overset{\pi }{\int }}|X(e^{i\varphi }){|}^{2}d\varphi$

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:

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

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

See also

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

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

Notes

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References

• 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.

External links

• Parseval's Theorem on Mathworld