# Phase of a signal with multi delayed windowed (rectangular window) signal with different amplitudes

I have a signal $x(t)$ which is composed of multi delayed linear chirps with different amplitudes and phases but they share the same $w$ and $\alpha$, the signal can be expressed as follows:

$$\DeclareMathOperator{\rect}{\mathrm{rect}} x(t) =\sum_{i} a_i \cos\left(\omega t+\frac{\alpha}{2} t^2 +\phi_i\right) \times \rect\left(\frac{t-T_i}{\Delta T_i}\right)$$

where $a_i$ can be any real number. The signal $x(t)$ differentiated w.r.t time $t$

My goal is not to estimate the different parameters $a_i$,$\phi_i$, $T_i$ or $\Delta T_i$ but to estimate the time varying phase term $\omega t+\frac{\alpha}{2} t^2$.

Any ideas/tips?

• Do you know $\omega$? – Marcus Müller Aug 17 '18 at 12:19
• No, not really. I tried for example to fit a curve to patches of the signal. But curve fitting gives me really unstable estimates of $w$ and $\alpha$ – Ahmad Aug 17 '18 at 14:54
• so, $\omega$ is an unknown, OK, but all the chirps have the same $\omega$. now, do you have any restrictions on $\phi_i$? how many components do you have, roughly? 2? 5? 200? Or do you even maybe know the exact number? – Marcus Müller Aug 17 '18 at 14:59
• all the $\phi_i$ s are bounded between $0$ and $\pi$. components are between 2 and 10 at most. – Ahmad Aug 17 '18 at 15:12
• huh, low numbers of components might make the frequency recovery harder (CLT doesn't kick in), I'm not sure I have an elegant solution. Anyway, this is a superposition of phase-modulated signals, with random initial phase. So, what happens to your formula when you apply $\frac{\mathrm{d}}{\mathrm d\, T}$ to it? – Marcus Müller Aug 17 '18 at 15:23

Assuming they all have the same $\omega$ and $\alpha$, meaning they are all following the same frequency ramp, then you can multiply by a ramp in the opposite direction of a starting guess while measuring the derivative of the phase of each carrier (a simple approach to see this would be to observe the result with an Short-Time Fourier Transform). Point is once you have found (and removed) the frequency slope, you will be left with distinct and constant frequencies, offset in frequency based on the time difference of their starting position and the frequency ramp rate (which also gives you information as you converge toward the final frequency ramp).
For the frequency ramp correction, it would be simplest to do that by multiplying by a complex frequency ramp ($e^{-j(\omega t + \alpha /2 t^2)}$ ) rather than a cosine and then observe just the positive frequencies in the complex output (What you would observe from bins 0 to N/2 in the STFT).