# How to account for phase and shift when composing a new signal based upon 2 or more recordings?

I have 2 signals that were independently recorded, and want to produce a new signal that would represent what would be recorded if both signal sources were emitting at the same time together. Specifically my case is that I have a recording of dolphins, and a recording of beluga whales and although both recordings were produced independently I would like a sound file with both biologics in the signal.

This post on a MATLAB forum link recommends to multiply the time ordered values together $$x_1(t) \times x_2(t)$$ to produce a new signal as a result of 'mixing'. This blog post link describes the addition of noise as an additive process between the signals. A question on this forum relevant question asks about the composition from multiple signals where the addition is prescribed. But these approaches do not account for phase and I would assume that this would be necessary. It also does not account for the fact that the time of the recording for the onset of each signal greatly affects the alignment of the signals where there is irregular activity. There is quite a large amount of variation in the acoustic activity where the start time changes whether these animal sounds overlap or not.

The approach I currently use, is to take the FFT of each $$x_i(t)$$ separately and then generate signals using iFFT from each FFT output. This produces separate signals that I then add together their values at every time point.

1. Is there a better way to do this?

2. In order to create a more realistic environment would I have to segment the different states of the signals (biologic activity and non-active states) in order to simulate the randomness in their activity. Would that be an approach seen elsewhere? In general a more simulation like approach to account for the stochastic changes in the state of the system where phase then is taken into account?