# Correct the phase of several damped sines

I have a signal $s(t)$ which consists of 3 damped sines it can be written as: $$s(t) = A_1\sin(\omega_1t + \phi_1)\exp(-t/t_1) + A_2\sin(\omega_2 t + \phi_2)\exp(-t/t_2) + \\A_3\sin(\omega_3 t + \phi_3)\exp(-t/t_3) + n(t),$$ where $n(t)$ is noise.

1. How to make the $\phi_1 = \phi_2 = \phi_3 = 0$; Any filter can fulfill the task?
2. It is also desirable to extract $A_1$, $A_2$ and $A_3$. But some times $A$s can be negative.
• First: Ai being negative is the same as a phase shift of $\pi$/$180^{\circ}$. Do you know wi and phii? – Oscar Mar 3 '15 at 16:39
• Wi and Phii are unknown. All I have is a time sequence. You are right, negative A can be incorporated into phase term. But we can restrict the phii to 0-pi/2; – PhySics Mar 3 '15 at 16:43
• Just to straighten things out: do you really need all $\Phi_i=0$ or just the same? Is the restriction on $\Phi_i$, i.e., $A$ may be negative important or is it OK to use $A_i > 0$ as restriction? Should it be done in real-time or will you process a batch of samples? – Oscar Mar 3 '15 at 16:59
• Hi Oscar, I need Φi=0. A may be negative is important because it is related to some quantity has physical meaning. There is no restriction on processing time it can be offline. – PhySics Mar 3 '15 at 17:03
• OK. Clarifying those things may at least be helpful for someone else with more detailed knowledge in the subject. My sort of hand waving suggestion is to go with an FFT to estimate $\omega_i$ and (relative) $\Phi_i$. Then it might be possible to get some idea of $t_i$ by looking at the FFT of different parts of the sequence. Finally, it may be possible to wrap it all up to determine $A_i$ (and refine the other parameters) by some model based filter. Once you have the $\Phi_i$s, "just" design an all-pass filter with the correct phase shift for the correct $\omega_i$. – Oscar Mar 3 '15 at 17:20