I'm new to Fourier Transform. I need to get a bit of understanding on it for my CompSci dissertation.
I've looked at several tutorials online. Most of them explain the Fourier Series very well. However, when it comes to Fourier Transform, I could not figure out what it produces since most tutorials state it quite ambiguously as in Wikipedia:
The Fourier transform of a function of time itself is a complex-valued
function of frequency, whose absolute value represents the amount of
that frequency present in the original function, and whose complex
argument is the phase offset of the basic sinusoid in that frequency
What does it mean by
the amount of that frequency present in the original function
Some tutorials as in here describes a frequency-amplitude representation as a result of Fourier Transforming a time domain function. What could possibly be the unit of that 'amplitude' and how does it relate to the amount of a frequency present in the original time domain function?
Best Answer
By Parseval's theorem, if a square-integrable time-domain "waveform" $x(t)$ has Fourier transform $$\hat{x}(f)=\int\limits_{-\infty}^\infty e^{-2\pi ift}x(t) dt$$ then
$$ \int\limits_{-\infty}^\infty |\hat{x}(f)|^2\, df = \int\limits_{-\infty}^\infty |x(t)|^2\, dt$$
where $f$ is frequency in Hz (i.e., cycles per second). The integral on the right-hand side may be interpreted as the total "energy" of the signal, so the integrand $$|\hat{x}(f)|^2$$ on the left-hand side can be interpreted as a density function (called the spectral density) describing the signal energy per unit frequency at the frequency $f$.
(Analogous results hold in the case of discrete signals via the discrete Fourier transform.)