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.

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

\sum_{n=-\infty}^\infty a_ne^{inx} and \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: a_0 real, a_{-n} = \overline{a_n}, b_0 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 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_{-\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.

See also

References

External links

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