Fourier Transformation of Functions
The Fourier transformation enables the acquisition of a frequency representation (spectral representation) for deterministic, continuous, and non-periodic signals. It articulates the frequency distribution of amplitude, phase, and energy (or power) within the considered signals.
Definition
Consider x(t) to be a non-periodic deterministic signal; its Fourier transform is:
\(X(w)=TF{x(t)}\)
\(X\left(\omega\right)=\int_{-\infty}^{+\infty}{x\left(t\right)e^{-j\omega t}dt}\)
Note :
Noticed :
The Fourier transform of a real signal x(t) is a complex function \(X(f)\) given by :
\(X\left(\omega\right)=X_{re}\left(\omega\right)+j.X_{im}\left(\omega\right)=\left|X(\omega)\right|.e^{j\varphi(\omega)}\)
The module \(\left|X(\omega)\right|\\) is an even function, and the phase \(\varphi(\omega)\) is an odd function.
The modulus, which represents the amplitude of the spectrum, is given by :
\(\left|X(\omega)\right|=\sqrt{{X_{re}\left(\omega\right)}^2+{X_{im}\left(\omega\right)}^2}\)
The argument is given by :\( \varphi\left(\omega\right)=\arg{\left(x\left(\omega\right)\right)}=Arctg(\frac{X_{im}\left(\omega\right)}{X_{re}\left(\omega\right)})\)