# FFT phase result interpretation?

• A
• Gauzi
In summary, you can see that the phase of the complex signal changes depending on the definition of fft.f

#### Gauzi

I have a complex signal eg: cos(wt + phase1) + i*cos(wt + phase2)
the frequency of both the waves is same. When i have a look at the phase spectrum of the above signal, i am not able to interpret the phase values. They are making no sense. I tried to determine phase shift for real signals and then for complex signal with different frequencies (eg: cos(w1t+ph1) + i*cos(w2t + ph2) ) and in these cases it was possible for me to interpret correct phase values. Could some one please guide or hint me int this direction. i am trying hard to understand it!

When i have a look at the phase spectrum of the above signal, i am not able to interpret the phase values. They are making no sense
Well, at least you have something to look at. Without a picture or a clearer description of what you see, it is difficult to assist here !
I take it you are familiar with the euler formula and have enough trigonometry available to unravel cos(wt + phase1) + i*cos(wt + phase2) into a complex frequency spectrum ?

What is it you expect and what is it you see ?

I expect the phase result to be somehow related to input signal. I will give an example which I could understand:

Input --> x = cos(2*pi*10*t - 60°) + i*cos( 2*pi*20*t + 90°)
Process --> fft(x)
Output --> I get peaks at 10 and 20 hz with amplitudes symmetric with -10 and -20(this is as expected)
--> when i have a look at phase values at 10hz and 20 hz i get -60°and 90° (in degrees) respectively

But when x --> cos(2*pi*10*t - 60°) + i*cos( 2*pi*10*t + 90°)
Process --> fft(x)
Output --> i get symmetrical peak at 10hz but value of phase is -78°
why is the resultant phase -78° and how does it relate to input wave? may be i am missing some very simple concept here

why is the resultant phase -78°
The input is ##x(t) = \cos(20\pi t - 60^o) + i\cos(20\pi t + 90^o)##. Performing Fourier transform on this yields
$$x(\nu) = FT[x(t)] = \frac{1}{2} \left(\delta(\nu-10)e^{-i\pi/3} + \delta(\nu+10)e^{i\pi/3} \right) + \frac{e^{i\pi/2}}{2} \left(\delta(\nu-10)e^{i\pi/2} + \delta(\nu+10)e^{-i\pi/2} \right)$$
You are interested only in the positive frequency part, namely
$$x_+(\nu) = \frac{1}{2} \left(\delta(\nu-10)e^{-i\pi/3} + e^{i\pi/2} \delta(\nu-10)e^{i\pi/2} \right) = \frac{1}{2}\delta(\nu-10) \left( e^{-i\pi/3}+ e^{i\pi} \right)$$
I leave it to you to calculate the phase of ##\left( e^{-i\pi/3}+ e^{i\pi} \right)## part.
By the way, if I calculate the phase mathematically using the above way, I don't get the same answer of ##78^o## as you did.

BvU
Gauzi: Funny, I should expect a phase of +60°and -90° for 10 and 20 Hz respectively, but it may depend on the fft definition ?

Gauzi: Funny, I should expect a phase of +60°and -90° for 10 and 20 Hz respectively, but it may depend on the fft definition ?
can you give a simple example on how it would depend on definition of fft?
It will be very useful for me to understand fft! :)