What exactly is the amplitude spectrum of the Fourier Transform?

[khkwang]

Homework Statement

Not really a homework question, but related none the less. I'm confused about what exactly the amplitude spectrum is. As well as the power spectrum.

Homework Equations

Not really taking a purely mathematical approach here, I'm using numpy for python. Specifically the fft module to produce:

f(t) = sum of 3 waves; same amp, phase; varying angular frequencies e.g. 3, 8, 5
F(freq) = fft(f): Fourier transform of f
freq = fftfreq(len(f)) dt): Produces frequency bins for F
amp_spec = abs(F)
pow_spec = amp_spec**2

The Attempt at a Solution

From what I gather, it is the absolute value of the Fourier Transform which is somewhat like a histogram of frequencies of the components that the initial f(t) wave oscillates at.

When I plot the graph amp_spec vs. freq, I get something similar to what I expect --peaks at 3, 8 and 5. But there is a mirror of this graph on the negative x-axis which I do not know the reason for but I suspect is just the way the function calculates the values (please correct me if I'm wrong).

The major question for me is what exactly does the value for the amplitude spectrum represent? When I use a dt (for my time array) of 0.01, I get the first graph shown below with ~250 as the max value on the y axis. But when I use 0.001 as my dt, the max value on the y-axis shoots up to ~2500. Incidentally, the number of frequency bins also goes up for reasons I also don't know...

Another question I have is what units would the y-axis be in? That and once I figure out what the amplitude spectrum is, what would the power spectrum be?

I've tried google with varying key terms and I still can't get a definite answer.

 

[revnaknuma]

when you consider the vibration of a violin string, Fourier transform gives the amplitudes and frequencies of the harmonics.

 

[khkwang]

I've already considered that but it doesn't explain why the amplitude of the Fourier transform grows so much larger as one decreases the interval between time steps. It also doesn't really make sense to me that the amplitude of the Fourier transform is ~250 or ~2500 when the amplitude of the original wave f(t) was only 3 max.

 

[klondike]

I've already considered that but it doesn't explain why the amplitude of the Fourier transform grows so much larger as one decreases the interval between time steps. It also doesn't really make sense to me that the amplitude of the Fourier transform is ~250 or ~2500 when the amplitude of the original wave f(t) was only 3 max.

The FT of ideal sinusoid is a pair of Dirac Deltas which has infinitesimal frequency width with infinite magnitude. All energy is "concentrated" at that frequency.

Numerical implementation of FT, i.e. FFT, or discrete-time Fourier Transform, there is no way to make freq step infinitesimal hence the energy of the signal spreads through interval of the FFT bin so the magnitude will not be infinite. The magnitude drops as the bin size increases.

 

[khkwang]

The FT of ideal sinusoid is a pair of Dirac Deltas which has infinitesimal frequency width with infinite magnitude. All energy is "concentrated" at that frequency.

Numerical implementation of FT, i.e. FFT, or discrete-time Fourier Transform, there is no way to make freq step infinitesimal hence the energy of the signal spreads through interval of the FFT bin so the magnitude will not be infinite. The magnitude drops as the bin size increases.

Ahh, I think I get it. But by bin size I assume you mean a single individual bin's size and not the total number of bins right? Otherwise forget what I just said.