Understanding Fourier Transforms for Sound Spectroscopy

[Ana09]

Homework Statement

Hi, I am trying to draw a sound spectrum using the discrete Fourier transform, but there are still a few things I do not understand:
- what does the "imaginary part" really represent? Does it exist physically in the sound? Is it a part of the sound signal that we don't hear or something? Or is it just part of the math?
- How do I graph the imaginary part?

Homework Equations

The DFT's equation.

The Attempt at a Solution

Maybe using argan's diagram? But all the spectra I have seen are plotted in regular cartesian axes...and I have also read that the imaginary part is not represented, as it is just the same as the real part..

 

[Dick]

It depends on the conventions of your DFT. If have something like your signal is equal to the sum of terms like A_n*exp(2*pi*i*n*k/N), if the signal is real, you will have another term like A_{-n}*exp(2*pi*i*(-n)*k/N). The two are complex conjugates (since you will find A_n and A_{-n) are complex conjugates) and sum the give you a real number (in your notation it may be N-n instead of -n. Same thing.) If you expand exp(i*t)=cos(t)+i*sin(t) and combine the two terms you will see that that the real parts correspond to the 'cos' part of the signal and the imaginary parts to the 'sin' part. So there's nothing mystical about it, there's no i's left in the end. When you are graphing a power spectrum what you are graphing is Re(A_n)^2+Im(A_n)^2. So you don't throw away the imaginary parts. You can throw away A_{-n}, since you can reconstruct it knowing A_n.