Parseval’s theorem – derivation

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The Parseval’s theorem (a.k.a Plancherel theorem) expresses the energy of a signal in time-domain in terms of the average energy in its frequency components.

Suppose if the x[n] is a discrete-time sequence of complex numbers of length N : xn={x0,x1,…,xN-1}, its N-point discrete Fourier transform (DFT)[1] : Xk={X0,X1,…,XN-1} is given by

\[X[k] = \sum_{n=0}^{N-1} x[n] e^{-j\frac{2 \pi}{N} k n} \]

The inverse discrete Fourier transform is given by

\[\tilde{x}[n] = \frac{1}{N} \sum_{k=0}^{N-1} X[k] e^{j \frac{2 \pi}{N} kn}\]

Suppose if x[n] and y[n] are two such sequences that follows the above definitions, the Parseval’s theorem is written as

\[\boxed{ \sum_{n=0}^{N-1} x[n] y^{\ast}[n] = \frac{1}{N} \sum_{k=0}^{N-1} X[k] Y^{\ast}[k]} \]

where, * indicates conjugate operation.

Deriving Parseval’s theorem

\[\begin{aligned} \sum_{n=0}^{N-1} x[n] y^{\ast}[n] &= \sum_{n=0}^{N-1} x[n] \left(\frac{1}{N} \sum_{k=0}^{N-1} Y[k] e^{j\frac{2 \pi}{N} k n} \right )^\ast \\ &= \frac{1}{N}\sum_{n=0}^{N-1} x[n] \sum_{k=0}^{N-1} Y^\ast[k] e^{-j\frac{2 \pi}{N} k n} \\ &= \frac{1}{N} \sum_{k=0}^{N-1} Y^\ast[k] \cdot \sum_{n=0}^{N-1} x[n] e^{-j\frac{2 \pi}{N} k n} \\ &= \frac{1}{N} \sum_{k=0}^{N-1} X[k] Y^\ast[k] \end{aligned}\]

If x[n] = y[n], then

\[ \begin{aligned} \sum_{n=0}^{N-1} x[n] x^{\ast}[n] &= \frac{1}{N} \sum_{k=0}^{N-1} X[k] X^\ast[k] \\ \Rightarrow \sum_{n=0}^{N-1} |x[n]|^2 &= \frac{1}{N} \sum_{k=0}^{N-1} |X[k]|^2\end{aligned} \]

Time-domain and frequency domain representations are equivalent manifestations of the same signal. Therefore, the energy of the signal computed from time domain samples must be equal to the total energy computed from frequency domain representation.

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Reference

[1] Bertrand Delgutte and Julie Greenberg , “The discrete Fourier Transform”, Biomedical Signal and Image Processing Spring 2005, MIT web

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