parseval's theorem

Table of Contents

1. Discrete time

For two sequences, \(x\) and \(y\) with discrete fourier transform sequences \(X\) and \(Y\):

\begin{equation} \sum_{n = 0}^{N-1} |x[n]|^2 = \frac{1}{N}\sum_{n = 0}^{N-1} |X[n]|^2 \end{equation}

implying that the total energy is conserved by the dft, or that the energy in the time and frequency domain is equal. The same property can be observed from the fact that the fourier matrix is orthonormal.

2. Continuous time

If \(x\) and \(y\) are continuous time functions and \(X\) and \(Y\) are continuous fourier transforms (so, complex valued functions with period \(2\pi\))

\begin{equation} \int_{-\infty}^{\infty} x(t)y^{*}(t)dt = \frac{1}{2\pi}\int_{-\infty}^{\infty} X(\omega)Y^{**}(\omega) d\omega \end{equation}

which is making a statement about energy conservation when multiplying two continuous signals

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