I have a problem that I thought was going to be simple, but it has become surprisingly stubborn and I question my method...
I have used the method (described below), but I wanted to confirm that this was the right approach, and see if there was another method I could use.
What I basically have is two wideband signals, (on the order of 100 Mhz BW), in quadrature. So I have the I and Q of those two signals. What I am trying to find is the phase difference between those two signals as a function of frequency.
The method I am currently trying is the following:
- I take my quadrature complex signals, and window each of them with a hamming window 3 times over.
- I take the FFT of the windowed complex signals, at some arbitrary length, say 2048. Call the results $S_1(w)$ and $S_2(w)$.
- I multiply $S_1(w)$ by the conjugate of $S_2(w)$.
- I take the argument out of the complex exponential vector from the above step (inverse tangent of imag over real). This gives me the phase differences between the two signals as a function of frequency.
My questions on this method:
- Is this the right approach for this simple type of problem?
- When I tried this method for a simulation with 1, 2 or a couple tones, I can get good results. I find the index of where my tones exist, (look at the abs of the FFT), I take that index, and look at the $\Delta$phase value from step 4 above. This gives me the right phase.
- However, when I look the $\Delta$phase values for frequency indicies where said those frequencies dont exist, I get wild answers. Why is this? I realize those frequencies dont exist but shouldn't the phases just be 0 - 0 = 0 in that case?
- Based on this (with a couple tones), can I readily extend this to a wideband signal and also expect good results?
- My last question regards the nature of the phase delay here - physically what is happening is that one of the signals (optical) is going through a device that delays different frequencies differently. Of course the delays are manifested as phase offsets between all frequencies... but at the same time there is a time delay associated with this as well. Even if we were to measure the $\Delta$phase between the two signals properly, wouldnt it still be impossible to measure the time delay between them at those frequencies? I can have a time delay of 10 wavelenghts, but a phase offset of 0.
Thanks in advance!
EDIT FROM COMMENT FEEDBACKS:
I think I am clearer now as to what my question really is: For a case where you have well separated tones and high SNR, my method can work.
However, what to do in the wideband case? In this sense, suppose $s_1[n]$ and $s_2[n]$ both contain two tones that are very close to each other in the frequency space. But your FFT can only give you so much frequency resolution. So instead of applying my method to two peaks, you only have one peak to go on. Now what?
- What relation would the $\Delta$phase of that peak have to the actual $\Delta$phase of the 'true' peaks? Is it their average?
- One method I am thinking of is correlating the two signals in the time domain, and working backwards to get the phase offsets per frequency taking advantage of the fourier translation property, although part of me thinks it might not work since there is no new information, just looking at the problem from a different domain.
- Another thought I had for the wideband case would be to put the whole thing through many narrow band filters, and then apply my original method. Is there a name for something like this?