# Fourier Transform

Hey guys, I have a quick question about fourier transforms.

I have been told that the fourier transform of a function tells us the minimum components required to support that function and that a real pulse may have extra frequencies, but not too few frequencies.

I don't understand why fourier transform only gives the min possible frequencies. Actually, I haven't seen the math. So I'd be glad some kind person could please give an intuition for what actually goes in a fourier transform in relation to what I have been told.

Just one other quick question:
A fourier transform can contain negative amplitude terms. What is that supposed to mean in a physical sense? It can also contain -ve frequency components. Is that physical?

HallsofIvy
Homework Helper
You seem to be talking about the "Fast Fourier Transform" (FFT) which is a numerical method for finding the fourier transform from the value of the function at a finite number of points (typically a power of 2 for simplicity) rather than the true Fourier Transform. Is that the case?

Well I'm no expert on Fourier Transform but if you haven't heard this before, it is a good start to think of FT as a decomposition of a signal onto a basis of functions. The basis set for the Fourier Transform is the complex exponentials (positive and negative), i.e. $$e^{i\times\omega}$$ . The amplitude for each basis (i.e. a single complex exponential) has the interpretation as the coefficient for each basis function, which is a complex exponential.

Now it's been awhile so you might have to check this, but if you have a purely real signal, the FT of that signal will have hermitian symmetry, i.e. there is a complex conjugate relationship between the coefficients of a $$e^{3\times i}$$ and $$e^{-3\times i}$$. This makes sense because if you have a pure sine wave, its Euler representation has a representation of two equally-weighted exponentials with opposite sign. A sine wave looks the same "in reverse" from the point of view of the basis set, which physically may explain why you can have negative frequencies.

A negative amplitude is possible because remember these are projections onto a set of basis functions. The best way to interpret a negative amplitude might be to think of it as interfering with other signals with positive amplitude to best represent the signal.

I think as Halls suggested the minimum frequency questions would have to relate to discrete Fourier Transform because it is only when it is discretized do you run into issues of sampling, etc. and that is something I know little about.

Hopefully someone else has a more precise and less naive way of looking at these questions.

AlephZero