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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?

Danh Thanh Nguyen
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  • Hi, thanks for the suggestion. I looked at those two already. The tutorial seems to refer to Fourier Series as it does not mention in which way a frequency contributes to the original signal. – Danh Thanh Nguyen Dec 23 '15 at 14:32
  • One book among many which explains the Fourier transform is Rudin Real and Complex Analysis ch. 9. Mathematically it takes a complex-valued function in $L^1$ on the real line and converts it to another complex-valued function on the real line. The transform can be extended to $L^2$ and also to distributions. This is all math. Now, if you use the Fourier transform to model something in the real world, the function being transformed can in some cases represent a behavior as a function of time, in which case the argument of the transformed function is often thought of as being frequency. – ForgotALot Dec 23 '15 at 15:56
  • An enjoyable example of how we use Fourier transforms in the real world is light. The light energy emitted by a source (e.g., the sun, a fire on earth) can be imagined to be eternal and then we take the Fourier transform and we get energy as a function of frequency. We reason about that energy as a function of frequency, for example to see what % of it is in the visible range, and what color humans with normal vision see it as. In this case, the usual unit of "amplitude" is energy. In other cases, it could be something else, e.g., the amplitude of the E and B vectors of the field. – ForgotALot Dec 23 '15 at 16:04
  • Given a function $g:\mathbb R \to \mathbb C$, we might like to express $g$ as a "linear combination" of the functions $f_\omega$, $\omega \in \mathbb R$, where $f_\omega(x) = e^{2 \pi i \omega x}$. Each function $f_\omega$ is a "wave" with frequency $\omega$. By "linear combination", I actually mean that we want $g(x) = \int_{-\infty}^{\infty} c(\omega) f_\omega(x) ,d\omega$. This is like a continuous version of a linear combination. The function $c(\omega)$ is called the Fourier transform of $g$. You give $c$ a frequency, and $c$ tells you the coefficient for that frequency. – littleO Dec 23 '15 at 16:07

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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.)

r.e.s.
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