NECETRONS: Physical Meaning of Fourier Transform

Physical Meaning of Fourier Transform

The time signal squared

represents how the energy contained in the signal distributes over time

, while its spectrum squared

represents how the energy distributes over frequency (therefore the term power density spectrum). Obviously, the same amount of energy is contained in either time or frequency domain, as indicated by Parseval's formula:

$\begin{displaymath}\int_{-\infty}^\infty \vert x(t) \vert^2 dt =\int_{-\infty}^\infty \vert X(f)\vert^2 df \end{displaymath}$

The complex spectrum

of a time signal

can be written in polar form

$\begin{displaymath}X(f)=X_r(f)+jX_i(f)=\vert X(f)\vert e^{j\angle{X(f)}} \end{displaymath}$

and the inverse transform becomes:

$\begin{displaymath}x(t)=\int_{-\infty}^{\infty} \vert X(f)\vert e^{j\angle{X(f)}... ...fty}^{\infty} \vert X(f)\vert e^{j(2\pi ft+\angle{X(f)})}\; df \end{displaymath}$