Fourier analysis to retrieve components of individual spectra

Question

I have a basic, simple question, I am a physics student, and searching internet gives me a lot of signal processing theory but couldn't find this basic answer, which I plan to implement in my speech detection algorithm (if the answer to this question is yes) so :

Imagine I have two sine waves :

G1 = A1*sin(t)

G2 = A2*sin(t+dt)

We dont know about this two waves however. What we know is a signal superposition of this two waves measured in terms of amplitude over time.

So we can have a measurement data like :

time Amplitude
 t1. a1
 t2. a2
 .   . 
 .   .
 tn  an    

Now, We know using Fourier transformation, we con obtain the frequency of this individual components, but I have another question, is it possible to retrieve the amplitudes and the phase shift A1, A2, dt of the individual input components ? So we can reconstruct the individual input signals fully ?

Answer 1

As I understand, you observe $G_t = A_1\sin(t) + A_2\sin(t+dt)$. Assume first that you know $dt$, the time delay of the second signal. If you have observations over a stretch of time $T$ you could multiply $G$ by $\sin(t)$ and $\sin(t+dt)$ and add over time to obtain: $$ \sum_tG_t\sin(t) = A_1\sum_t\sin(t)^2 + A_2\sum_t\sin(t)\sin(t+dt)$$ $$ \sum_tG_t\sin(t+dt) = A_1\sum_t\sin(t+dt)\sin(t) + A_2\sum_t\sin(t+dt)^2$$ This yields a system of two equations which you can solve for $A_1$ and $A_2$. It is clear that if $dt$ is small, the system above is very nearly singular, and your chances of singling out both signals are slim.

If you do not know the time delay, you could repeat the procedure above for different tentative values of $dt$, solve for $A_1$, $A_2$ in each case, and retain the solution which gives the best fit to $G_t$, i.e., the value of $dt$ for which $\sum_t(G_t - A_1\sin(t) - A_2\sin(t+dt))^2$ is minimum.

The answer above has been suggested to me by the consideration of a method in signal processing known as complex demodulation. If you do a Google search for this terms you will likely find many references that help you with details I am not addressing here.