Fourier Transform

The Fourier transform is a mathematical tool that decomposes a function (typically a signal defined in time or space) into its frequency components. Named after the French mathematician Jean-Baptiste Joseph Fourier, this transform is one of the fundamental analytical methods in fields ranging from engineering to statistics. Mathematically:

$F(\omega )={\int }_{-\infty }^{\infty }f(t)\cdot e^{-i\omega t}dt$

Where:

• $f(t)$: the signal in the time domain
• $F(\omega )$: the representation in the frequency domain
• $\omega$: angular frequency (rad/s)

Signals in the time domain often appear complex. However, the Fourier transform decomposes this signal into simple waves (sine and cosine) at different frequencies.

_{Comparison of Time and Frequency Domains (Generated by Artificial Intelligence)}

The energy of the signal in the time domain equals its energy in the frequency domain:

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

The convolution of two functions in the time domain becomes multiplication in the frequency domain:

$f(t)\ast g(t)\stackrel{\mathcal{F}}{\leftrightarrow }F(\omega )\cdot G(\omega )$

This property is used in the analysis of system responses.

A periodic square wave in the time domain contains harmonic components in the frequency domain (approximated by a Fourier series).

_{Square Wave Time Domain – Generated by Artificial Intelligence.}

_{Square Wave Frequency Domain (Generated by Artificial Intelligence)}

This spectrum shows that only odd harmonics are present in the signal.

$f(t)=e^{-t^2}$

The Fourier transform is again a Gaussian function:

$F(\omega )=\sqrt{\pi }\cdot e^{-\frac{{\omega }^2}{4}}$

This property makes the Gaussian function “ideal” in both time and frequency domains.

In probability theory, the statistical counterpart of the Fourier transform is the characteristic function:

${\varphi }_X(t)=\mathbb{E}\left[e^{itX}\right]$

If $X\sim N(𝜇,{𝜎}^2)$, then:

${\varphi }_X(t)=\exp \left(i\mu t-\frac{1}{2}{\sigma }^2{t}^2\right)$

This transform carries information about the moments of the distribution. It is particularly used in proving results such as the Lévy Continuity Theorem and the Central Limit Theorem.

To apply the Fourier transform on a computer, the signal is sampled into a discrete version. In this case:

$X_k={\sum }_{n=0}^{N-1}{x}_n\cdot e^{-2\pi i\cdot \frac{kn}{N}},k=0,1,...,N-1$

The FFT is an algorithm that accelerates the computation of the DFT, reducing the time complexity from $O(N^2)$ to $O(NlogN)$.

Example applications:

• FFT analysis of audio signals reveals which frequencies are dominant.
• In image processing, spectral analysis identifies patterns in images.