i got a couple of questions regardion fourier transformations. I'm not that interested
Lets take a box/rectangular function as an example.
The "Frequency-Spectrum" of that function is would look somewhat like this:
_o_________
_o__o______
_o__o__o___
_o__o__o__o
I need a lot of different frequencies with lower amplitude, the higher the frequency gets. I kind of get that.
1) The fourier transform looks like this, however:
So i get that it's supposed to be mirrored at the y-axis. Where is the "wave-form" coming from, though? And why are the negative values? If i am only interested in the frequency band components of the original signals, would it enough to look at the absolute values? I suppose thats called a continous fourier transform?!
2) Lets take the sin(x) function. It should only have one frequency. How would the fourier transform look like here Intuitively i would assume its only non-zero for one specific frequency (probably mirrored at the y-axis as well). Is that right? Or would i get some sort of "wave"-form as well?
Best Answer
1) The positive frequencies correspond to anticlockwise rotations of the complex exponential. ($e^{i\phi}$) Similarly, the negative frequencies correspond to the clockwise rotations. When the inverse Fourier transform is performed on the box, the imaginary parts of the positive/negative frequencies cancel out, resulting in a real signal.
2) Periodic functions get transformed to Dirac deltas.