FourierTransform

A Fourier transform is a mathematical technique used to transform a function from its original domain (often time or space) into a different domain, typically frequency or wavenumber. This transformation allows us to analyze the function in terms of its constituent frequencies or wavelengths.

The Fourier transform of a continuous, time-domain function $f(t)$ is represented by the frequency-domain function $F(\omega)$, and is defined as follows:

$$ F(\omega) = \int_{-\infty}^{\infty} f(t) e^{-i \omega t} , dt $$

Here, $i$ is the imaginary unit, $\omega$ is the angular frequency (in radians per second), and $t$ is time. The inverse Fourier transform, which converts the frequency-domain function back into the time-domain function, is given by:

$$ f(t) = \frac{1}{2\pi} \int_{-\infty}^{\infty} F(\omega) e^{i \omega t} , d\omega $$

For discrete-time functions, the Discrete Fourier Transform (DFT) is used. Given a sequence of $N$ complex numbers $x[n]$, the DFT $X[k]$ is defined as:

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

And the Inverse Discrete Fourier Transform (IDFT) is:

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

A widely used algorithm to compute the DFT efficiently is the Fast Fourier Transform (FFT), which reduces the computational complexity from $O(N^2)$ to $O(N\log N)$.

Characteristic Functions

See Bochner's Representation Theorem

Conservation of Energy

Parseval and Plancherel's Theorems state a mathematical expression of a conservation law which affirms that the energy of a process remains unchanged whether represented in the time or frequency domain.