Parseval's theorem

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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, 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 and engineering, the most general form of this property is more properly called the Plancherel theorem.[3]

Contents

[edit] 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(x)=\sum_{n=-\infty}^\infty a_ne^{inx} and B(x)=\sum_{n=-\infty}^\infty b_ne^{inx}

respectively. Then

\sum_{n=-\infty}^\infty a_n\overline{b_n} = \frac{1}{2\pi} \int_{-\pi}^\pi A(x)\overline{B(x)} 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_{n=-\infty}^\infty |a_n|^2 = \frac{1}{2\pi} \int_{-\pi}^\pi |A(x)|^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: a0 real, a_{-n} = \overline{a_n}, b0 real, and b_{-n} =\overline{b_n}. In this case:

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

where \Re denotes the real part. (In the notation of the Fourier series article, replace an and bn by an / 2 − ibn / 2.)

[edit] Applications

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

\int_{-\infty}^{\infty} | x(t) |^2 dt = \int_{-\infty}^{\infty} | 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=-\infty}^{\infty} | x[n] |^2 = \frac{1}{2\pi} \int_{-\pi}^{\pi} | X(e^{j\phi}) |^2 d\phi
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.

[edit] Equivalence of the norm and inner product forms

We shall refer to

\int_{-\infty}^\infty x(t)\overline{y}(t) dt = \int_{-\infty}^\infty X(f)\overline{Y}(f) df

as the inner product form, and to

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

as the norm form. It is not difficult to show that they are (pointwise) equivalent. One can use the polarization identity

a\overline{b} = \frac{1}{4}(|a+b|^2 + i|a+ib|^2 + i^2|a+i^2b|^2 + i^3|a+i^3b|^2),

which is true for all complex numbers a and b, and the linearity of both integration and the Fourier transform.

[edit] See also

[edit] 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.

[edit] Notes

  1. ^ Parseval des Chênes, Marc-Antoine "Mémoire sur les séries et sur l'intégration complète d'une équation aux differences partielle linéaire du second ordre, à coefficiens constans" 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 savans, et lus dans ses assemblées. Sciences, mathématiques et physiques. (Savans 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.
  3. ^ Plancherel, Michel (1910) "Contribution a l'etude de la representation d'une fonction arbitraire par les integrales définies," Rendiconti del Circolo Matematico di Palermo, vol. 30, pages 298-335.

[edit] External links

  • Parseval's Theorem on Mathworld
  • In the movie Good Will Hunting, the theorem that Professor Lambeau finishes writing on the classroom chalkboard just after we first see him is Parseval's theorem. [1]