# The graph of Fourier Transform

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.

• 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
• See the graph – abcd Mar 18 '17 at 7:45

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.

• 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!).

You can refer the following link. Here you can find intuitive explanation to Fourier Transform.

The Frequency Domain values after Fourier Transform represent the contribution of that each Frequency in the signal