Coupled system instability Suppose there are two linear devices (containing some multi-physics but the details don't matter here), and these devices are given to us as two black boxes that can be studied experimentally. Black box #1 takes signal X as input and produces signal Y as output. Black box #2 takes signal Y as input and produces signal X as output. It is conceptually similar to a microphone converting an acoustic signal to an electric signal, and a loudspeaker converting an electric signal to an acoustic signal. If these two black boxes get connected to each other then an instability can probably arise. The question is what measurements on the two black boxes would produce all necessary and sufficient information to assess whether the coupled system would go unstable, and at what frequency. Would the right approach be to study the response of each black box to a sinusoidal input? Or the response to a delta-pulse input? What is the criterion for instability for the coupled system, and what measurements and calculations would be needed for assessing it?
Assuming both black boxes are LTI systems, that's reasonably easy to do.
Let's call the transfer function of the first box $H_A(\omega)$ and the second one $H_B(\omega)$. The open look transfer function is simply the product of the, i.e. $H_O(\omega) = H_A(\omega) \cdot H_B(\omega)$
A sufficient stability condition is $|H_O(\omega)| &lt; 1$. For an actual acoustic feedback loop that's really all there is to it since there is always plenty of group delay in the system and the phase is spinning so fast that any type of phase criteria is pointless.
So measure the transfer functions (fully calibrated !), multiply them and check whether the max gain is smaller than unity.