2.9e Parseval's Theorem

Parseval's Theorem

Parseval’s Theorem states that the total energy of any $p$-periodic discrete signal $x(n)$ in the time domain can be expressed as a linear combination of its period with its Fourier coefficients, such that
${\sum }_{n=0}^{p-1}|x(n){|}^2=p{\sum }_{n=0}^{p-1}|X_n{|}^2,$
where the right hand side represents the total energy of $x(n)$ in the frequency domain.

Intuition

For discrete-time, we can take the vector equivalent of $x(n)$ to be $\vec{x}\in {\mathbb{C}}^p$. We can then express $|x(n){|}^2$ as the squared norm of vector $\vec{x}$, such that

||\vec{x}||^2 &= \langle \vec{x}, \vec{x} \rangle \\ &= \vec{x}^T\vec{x}\textasteriskcentered \\ \end{align}$$ Applying the equation $\vec{x} = \Psi X$, we get $$||\vec{x}||^2=(X^T\Psi^T)(\Psi\textasteriskcentered\\X\textasteriskcentered).$$ Since we already know that $\Psi^T\Psi\textasteriskcentered=pI$, $$\begin{align} ||\vec{x}||^2 &= X^T(pI)X\textasteriskcentered \\ &= pX^TX\textasteriskcentered \\ &=p\langle X, X \rangle \\ &= p||X||^2. \end{align}$$ Which is equivalent to $\sum^{p-1}_{n=0}|x(n)|^2 = p\sum^{p-1}_{n=0}|X_n|^2$ in the signal domain.