How to get some law parameters from its FFT figure?

Question

Given an hidden law $$I(t)=\frac{1}{2}[A+B.cos(2\pi\nu_0t+\phi_0)]$$ I observe its signal with 256 samples and compute its FFT over this interval, and represent the FFT result from 0 to 1 (normalised frequency).

I get a figure whose pic is at 0.83.

How can I retreive A,B and $\phi_0$ from this graphics ?

A second question is : is it normal that this figure is not symmetric ?

A final question is : doing an analytical computation on classical Fourier transform gives a result. How much does what I observe from FFT and the analytical result differ ?

Answer 1

If you feed an FFT a finite set of samples from a function of theoretically infinite range, you have implicitly windowed your function with a rectangular window before the FFT. Thus, the FFT result will be the convolution of the spectrum of your function and the transform of a rectangular window (a periodic Sinc). This convolution will be an identity operation only if the samples happen to span exactly an integer multiple of periods of a purely periodic function. So how much your FFT result will change or differ will depend on the offset and length of your FFT aperture/window with respect to some reference point on and the scale of your function.

In the case of a single sinusoid in zero noise, I reference 2 papers by Clay Turner on solving for the 3 unknowns using only 3 or 4 non-aliased samples, somewhere on my web page: http://www.nicholson.com/rhn/dsp.html No FFT needed.