Parseval's theorem
The Parseval's theorem is a statement from the functional analysis in the field of Fourier analysis . It says that the -norm of a Fourier series agrees with the -norm of its Fourier coefficients . The statement arose in 1799 from a sentence about mathematical series by Marc-Antoine Parseval , which was later extended to the Fourier series. Parseval, who actually only concentrated on real-valued functions, published his theorem without proof, since he considered its correctness to be obvious. Plancherel's theorem makes a similar statement for the Fourier transform . Often these two sentences are not kept apart, but Plancherel's sentence is named after Parseval.
Statements of Parseval's theorem
Let and two Riemann integrable complex-valued functions over with period and the Fourier series decomposition
- and .
Then applies
where is the imaginary unit and denotes the complex conjugation .
There are many different special cases of the theorem. Is z. B. , one obtains
from which the unitarity of the Fourier series follows.
Furthermore, often only the Fourier series for real-valued functions and mean what corresponds to the following special case:
- real, ,
- real, .
In this case it is
where denotes the real part.
Applications
In physics and engineering , Parseval's theorem is used to express that the energy of a signal in the time domain is equal to its energy in the frequency domain. This is expressed in the following equation:
where is the Fourier transform of with the prefactor omitted and denotes the frequency of the signal.
For discrete-time signals the equation becomes
where is the Discrete Fourier Transform (DFT) of , both of interval length .
See also
credentials
- 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.