Fourier Transforms

I'm having a terrible time trying to understand Fourier transforms. I'm very visual so leaving the $X,Y,Z,t$ domain is not working form me :)

I'm trying to figure out the basics at the moment. Like, taking a Sine wave (they're odd right?) and converting it into its real and imaginary numbers. I'm pretty sure I got that working, but to make sure, what should the plotted data look like?

Also, how do I find the power spectrum of a transform? How do I use FT to identify the $n$ most significant frequencies in a signal? That last question shows how lost I am!

What I have:

I know how to get the real and imaginary numbers from a signal. I know how to get the phase and the magnitude. What I need to get is the power spectrum and the most significant frequencies. Also any dumbed down explanation of what's going on would be very helpful!

Thanks!

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I'll try and give some intuition from an Electrical engineering perspective. It is useful to think of a Fourier transform as giving you the frequency domain picture of a given signal, i.e., it gives you a picture of both the amplitude and the phase of the different frequency components that make up the signal.

If you are familiar with Fourier series representation of periodic signals, the Fourier transform can be viewed as an extension to non periodic signals. Of course, not all continuous functions have a well defined Fourier transform, but if you look at typical engineering applications, that is not an issue.

When dealing with deterministic signals, the power spectrum is given by the squared magnitude of the Fourier transform. If $f(t)$ is the signal in the (continuous) time domain and $F(\omega)$ is the frequency domain representation, then the Power spectrum is $|F(\omega)|^2$.

When dealing with random signals, the power spectral density is the Fourier transform of the autocorrelation function of the process (Provided the process is wide sense stationary).

The above explanations carry over to the discrete domain as well. So, if you are working with DFTs of signal samples, things are not significantly different.

As for the N most significant frequencies, I think you need to take the DFT of your signal and then look at the frequencies in increasing order of amplitudes.

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