Understanding the Basics of Fourier Transforms

[spaghetti3451]

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]

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?

 

[7thSon]

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]

I'm not sure exactly what you are asking, but there are two different things that may be confusing you.

The first one is the difference between the Fourier transform or a periodic and a non-periodic signal. The FT of a non-periodic signal is a continuous frequency spectrum. The FT of a periodic signal has an infinite number of frequecy components at multiples of the periodic frequency of the signal (often called "harmonics").

The second one is the difference between the continuous Fourier transform (defined in terms of integrals) and the discrete Fourier transform (DFT) calculated from a sampled signal. Because of aliasing and the Nyquist frequency, for all practical purposes you can consider the sampled signal to be bandwidth limited. Also, for any practical DFT calculation you can only use a finite number of samples. The consequence of those two facts is that the output from a DFT actually looks the same as the continuous FT of a periodic signal (with period equal to the sampling length) that is bandwidth limited (by the Nyquist frequency) and therefore it consists of a finite number of discrete harmonics.

Hope that helps (though you may have to read it slowly several times...)