Intuition behind Fourier transforms of complex signals

[mirceapricop]

I'm working on a project involving applying Fourier transforms on a two-dimensional signal. When I learned Fourier transforms, they were presented to me as being capable of transforming a real-valued input signal into frequency space. So naturally, my first impulse was to separate the components and run analysis on each of them individually.

After a bit of reading, I came across the fact that Fourier transforms can actually take a complex-valued signal as input, and someone was using this to transform 2d-signals by having the first dimension show up in the real part and the other one in the imaginary part of the signal values. I tried duplicating this technique and it seems to give great results.

I understand the intuition behind transforming a real-valued signal, but why does this technique work for the two-dimensional case? A short explanation or pointing me to the right article would mean a lot.

[PM 2Ring]

There's a good discussion of this and related topics in the FFTW tutorial.

[Yakk]

A Fourier transform takes a function f:A->X, and produces another function g:B->X, such that g describes the "coefficients" of a family of sinusoidal that, when added together, produce f.

g(b) is the "coefficient" in front of the sinusoidal h(x) := e^{2 pi i b x}. (Yes, e^x is a sinusoidal -- in particular, e^{2 pi i x} is).

How we add them up depends on our B. Basically, we define a measure on our B, and then integrate with respect to that measure. (a measure is a way of weighting the elements. A counting measure says each element has weight 1. A measure on a continuum (like the real line, or the interval [0,1]) tends to say any one element has weight 0, while intervals have non-zero weight.)

The wonder of the Fourier Transform is the symmetry between extracting these coefficients, and reconstructing the original signal. How that works is going to be beyond what I could put in a post. (anyone else want to try?) But the basic idea remains: you are building a map from the space of sinusoidal (in this case, sinusoidals with a complex index!) to coefficients on said sinusoidals. You then add up each of those (integrate them together, because adding up an uncountable number of functions doesn't work well without integrating them) to generate your original function.

As another example, FFT uses a discrete family of sinusoidal indexed by a discrete set B. This changes how we add them up, and restricts what set of functions we can represent. In this case, the "sum of sinusodials" is more obvious, because instead of this strange looking integral, we get an actual sum of functions.

[mirceapricop]

I think I got it, thank you.

So as a consequence of how the maths work, we actually get the ability to transform complex signals into a sum of sines with complex coefficients, but most of the time it's being only used with a real-valued input signal.

So if I "multiplex" the 2 real signals in the complex one, the resulting frequency coefficients should still describe or represent both original signals at the same time.

One more question, would a power spectrum be constructed for the two signals in the same way as before, by taking the magnitudes of each coefficient? If so, how can one single spectrum represent both signals?

[Yakk]

First, you seem to be doing a FFT, not a general Fourier Transform.

Second, your amplitudes are going to be complex values.

By multiplex, I assume you mean construct a complex signal f from signals g and h:
f(x+iy) = g(x) * h(y) for x y real?
or do you mean
f(x) = g(x) + i h(x) for x real?

I'm guessing the second.

If the Fourier transform maps frequency to a complex amplitude, seeing how g and h are represented is pretty trivial, isn't it?

[mirceapricop]

You are absolutely right, I am doing a FFT, and constructing the complex signal using the second form.

So from what I read, when doing a real FFT of N points, you only get N/2 "significant" frequency magnitudes, because the negative harmonics are equal to the positive ones so we can ignore them. This means only magnitudes 0 - N/2 are enough to get a representation of the signal.

For my complex signal, these negative harmonics also contain information about the original signal, so the full spectrum from 0 - N is needed to represent the original complex signal. I've also seen you can obtain the individual separated signals out of it through odd even decomposition, but I don't need that.

I am using the spectra to compare different signals, with the Euclidean distance between them as a distance measure for the original signals. Then I can control the comparison "strict-ness" by only considering a certain number of harmonics. This would mean, I can use the resulting full spectrum to compare two 2d-signals directly, instead of first splitting them and comparing individual component frequencies. Right?

EDIT: I just found out there are much better distance measures for spectra, don't burn me on that