ParsevalAndPlancherelsTheorems

Parseval's Theorem and Plancherel's Theorem both deal with the transformation of processes from time domain to frequency domain using Fourier series and Fourier transforms, respectively.

1. Parseval's Theorem (for Fourier Series): This theorem states that the total energy of a process is preserved when transformed from the time domain to the frequency domain. In other words, the sum (or integral) of the square of a function's absolute value over its domain is equal to the sum (or integral) of the square of the absolute value of its Fourier series coefficients. Mathematically, it is expressed as:

   $$\int_{-\infty}^{\infty} |f(t)|^2 dt = \frac{1}{2\pi} \int_{-\infty}^{\infty} |\hat{f}(\omega)|^2 d\omega$$

   where $\hat{f}(\omega)$ denotes the Fourier transform of $f(t)$.

2. Plancherel's Theorem (for Fourier Transforms): Plancherel's Theorem is an extension of Parseval's Theorem to the Fourier Transform, and it states a similar property: the total energy of a process is preserved when transforming from time domain to frequency domain using Fourier Transform. That is, the integral of the square of the absolute value of a function is equal to the integral of the square of the absolute value of its Fourier transform. It is formally expressed as:

   $$\int_{-\infty}^{\infty} |f(t)|^2 dt = \int_{-\infty}^{\infty} |\hat{f}(\omega)|^2 d\omega$$