# How to get some law parameters from its FFT figure?

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 ?

• Is this homework? If so, you should specify it using the homework tag. It seems like that to me, so here are some hints: (1) A is like a DC component. Its frequency is 0. (2) A real signal's FT is symmetric. A complex signals doesn't have to. (3) A FT has both magnitude (which I believe you are plotting) and phase. – Serge Mar 8 '13 at 16:58
• it is not a homework per se. – JCLL Mar 8 '13 at 17:02
• How closely spaced are the $256$ samples? For best results, take the sample spacing to be $(256\nu)^{-1}$ seconds so that the $256$ samples fill up the time interval of one period, e.g. $\[0,\nu^{-1})$. Note that the $257$-th sample will occur at $t = \nu^{-1}$ and is not part of the data set. See, for example, this answer to understand why not. – Dilip Sarwate Mar 8 '13 at 17:19

## 1 Answer

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.