Please help-understand impulse response function?

Question

Hi. say I have two systems

i(t) --> [sys1] --> u(t) --> [sys2]--> o(t)

Where a pulse signal is generated i(t) and goes through the systems. But by the time it reaches u(t) and o(t) it is no longer a (delta) impulse pulse but a wider pulse.

I am then told that the signal I would have gotten at o(t) if u(t) was a delta pulse can be obtained by deconvolving o(t) with u(t). Please can you explain why this is?

There is a delta pulse generated at i(t)

Answer 1

If h(t) is the impulse response of a system in the time domain, then the output of the system to any input i(t) is the convolution integral o(t) = i(t)*h(t) = ∫i(t-x)h(x)dx. In the frequency domain, this is just the product of the Laplace transforms:

o(s) = i(s)H(s)

Therefore, you can always derive the input signal from the output signal by inverse convolution or division of the Laplace transforms.