Parseval's theorem

The set of Parseval 's a statement from functional analysis in the field of Fourier analysis. He states that the norm of a Fourier series coincides with the norm of its Fourier coefficients. The statement was created in 1799 from a set of mathematical series by Marc -Antoine Parseval, which was later extended to the Fourier series. Parseval, which actually focused only on real-valued functions, published his theorem without proof, because he held his accuracy of obvious. A similar statement for the Fourier transform makes the set of Plancherel. Often these two records are not kept apart, but also called the set of Plancherel according to Parseval.

Statements of the Parseval theorem

Let and be two riemann integrable complex-valued functions with period and the Fourier series decomposition

Then we have

The imaginary unit, and * denotes complex conjugate.

There are several special cases of the theorem. If, for example,, One obtains

From which the unitarity of the Fourier series follows.

In addition, often only the Fourier series for real-valued functions A and B meant, which corresponds to the following special case:

In this case,

Where denotes the real part.

Applications

In physics and engineering sciences is expressed by the Parseval's theorem 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:

The Fourier transform of being omitted with prefactor and denotes the frequency of the signal.

For discrete-time signals, the equation becomes

Where X [ k] is the discrete Fourier transform (DFT) of x [ n], both with the interval length N.