# Fourier analysis to retrieve components of individual spectra

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 ?

## 1 Answer

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.

• Thanks for this idea. Quite interesting. I am looking in to it's details. Some basic questions, I think tis can be applied to N terms (here only 2 is shown for ease of understanding), second, the contributing function could be sin or cosine or for that sake, any trigonometric ( cyclic function), so how do I know what function to multiply ? – Ayan Mitra Apr 19 '20 at 20:05
• In a realistic situation, I think you would use both sine and cosine, and you would be able to deal with signals which do not "start" exactly at time zero. Yes, I think you can generalize to more than two terms. – F. Tusell Apr 20 '20 at 6:24