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I am trying to grasp Fourier transform, I read few websites about it, and I think I don't understand it very good. I know how I can transform simple functions but there is few things that are puzzling to me.

Fourier transform takes a function from time domain to a frequency domain, so now I have $\widehat{f(\nu)}$, this is complex-valued function, so as I understand for every frequency I get an imaginary number.

  1. What does this number represent, what is an interpretation of real and imaginary part of $\widehat{f(\nu)}$?

  2. How can I graph $\widehat{f(\nu)}$? As I understand if function is not odd-function, $\widehat{f(\nu)}$ will have complex values and imaginary part will be different then 0. Do I need to plot it in 3d or do I just plot $|\widehat{f(\nu)}|$?. I am asking about plotting, because for example on wikipedia there is a plot of sinc function, which is fourier transform for square function. It is nice, because it is an odd-function in their case. And I am wondering about other functions.

I would be also very grateful for any useful links that can shed some light on the idea of fourier transform and some light theory behind it, preferably done step-by step.

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To be pick<: You should better write $\hat{f}(\nu)$ instead of $\widehat{f(\nu)}$ because it is $f$ which is transformed, not $f(\nu)$. – Dirk Apr 13 '12 at 21:42

The values of a frequency domain function represent how much of that frequency is "in" the function. For example, if you would take the fourier transform of a sine wave, you would get a delta function in the frequency domain: there's a lot of some specific frequency in that function.

Now, this is quite a simple way of saying it; we can deduct quite a bit more from the value, such as the "phase" of that frequency component. But it's also a bit more difficult, since for many signals the fourier transform is not composed of delta functions but a continuous function.

Don't worry too much about it, it takes a while getting used to the idea of fourier transforms. It's only now in my third year in Electrical Engineering that they really feel natural, I must say.

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Thanks for answer, I can 'get it' for simple function, but as I said, what about the functions for which transform values has real and imaginary parts different than 0? – Andna Apr 13 '12 at 21:49
I don't quite get your question; what do you want to know? – akkkk Apr 13 '12 at 22:31
@Andna, If the imaginary part is different from 0, then there is a phase shift, as it happens when an input voltage is applied to a RC-, RL- or RLC circuit. The output voltage is shifted with respect to the input voltage. We say that there is a reactive power associated to this phase shift. If the phase shift is 0, as in a R-circuit, there is only active power which corresponds to the Joule effect (the heat generated by the electric current through the resistor R). – Américo Tavares Apr 13 '12 at 23:13

Not exactly an answer, but may provide some perspective...

Engineers often deal with models in which the system output is the convolution of some kernel $f$ with an input. The fourier transform has the nice property of transforming convolution into point-wise multiplication. It is much easier to comprehend the effect of point-wise multiplication than it is to understand the effect of convolutions. Hence the popularity of fourier transform in engineering.

The fourier transform of the convolution kernel ($\hat f$) can be interpreted in terms of the system response to an input of the form $t \mapsto e^{i \omega t}$ (or $\sin$, $\cos$, etc.). The steady-state response (ie, after transients have 'died away') output of the system is given by $t \mapsto \hat f(i \omega) e^{i \omega t}$. So the behavior of the system can be understood by looking at $\hat f$. Engineers typically look at plots of $\omega \mapsto |\hat f(i\omega)|$, and $\omega \mapsto \arg(\hat f(i\omega))$ (with a $\log$ axis for frequency $\omega$, and a $\log$ axis for the modulus). The fourier transform of the kernel is called the system 'transfer function'.

The value of $|\hat f(i\omega)|$ shows how much a signal at frequency $\omega$ is amplified ($|\hat f(i\omega)|>1$) or attenuated ($|\hat f(i\omega)|<1$). The angle $\arg(\hat f(i\omega))$ indicates the phase shift (interpret loosely as a time delay) between the input signal at frequency $\omega$ and the output.

Stability of systems is also of concern to engineers (and indirectly to the general public). A test (Nyquist stability criterion) based on applying the Argument Principle to $\hat f$ is used to evaluate stability of the system (most frequently for homework problems!).

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