# If the convolution of two signals is a unit impulse, what does this tell us?

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.