I have two discrete-time LTI systems whose transfer functions satisfy $h_1[n] * h_2[n]= \delta[n]$. We also know that system 1 is causal and stable. Does $h_1[n] * h_2[n]= \delta[n]$ tell us anything about $h_2$?
Intuitively, I would say that $h_2[n]$ is stable, but I don't have any idea how to reason about it.
It tells us that the systems are inverses of each other. The DFT of
$$h_1[n]*h_2[n]= \delta[n]$$
is
$$H_1[k] \cdot H_2[k] = 1 $$
so we get
$$H_2[k] = \frac{1}{H_1[k]}, H_1[k] = \frac{1}{H_2[k]}$$
In order for $h_2[n]$ to be causal and stable, $h_1[n]$ has to be minimum phase. Causality and stability of $h_1$ are not sufficient to guarantee causality and stability of the inverse.
A good example of this is an all pass filter. It's perfectly causal and stable but its inverse is anti-causal, i.e. $h_{AP,inv}[n] = h_{AP}[-n]$
Well Hilmar is 100% correct but I want to give a practical application of 2 such systems.
If 2 systems are inverse of each other you can use them to make a noise removing system:
The block with $H_{1}(z)$ is usually a sensor which measures the main frequencies of the input signal sends them to a PID , then the PID using software sets the resonant frequency.
