Convolution - Can someone explain this solution?

[Lomion]

Convolution & Inverses

Given an impulse response h[n] to a system, and the impulse response g[n] of the inverse system, why is $$h[n] * g[n] = \delta[n]$$? Where the * sign is used to denote convolution.

 

[xanthym]

Given an impulse response h[n] to a system, and the impulse response g[n] of the inverse system, why is $$h[n] * g[n] = \delta[n]$$? Where the * sign is used to denote convolution.

This result follows from the properties of Laplace Transforms.
First, define the Laplace Transforms of the Impulse Responses h(n) and g(n):
H(s) = L{h(n)}
G(s) = L{g(n)}
Next, find the Laplace Transform of the given Convolution, remembering that the Laplace Transform of a Convolution is the product of the Laplace Transforms of the convolved functions:
L{h(n)*g(n)} = L{h(n)}L{g(n)} = H(s)G(s)
However, since both h(n) and g(n) are Impulse Response functions, their Laplace Transforms are their system Transfer Functions. Moreover, since we are given that h(n) represents the Inverse system to g(n), their TRANSFER FUNCTIONS must be be reciprocal to each other:
H(s)G(s) = 1
-----> L{h(n)*g(n)} = 1
-----> h(n)*g(n) = DIRAC-DELTA(n)
where we used the result that L^(-1)(1)=DIRAC-DELTA(n).
~

 

[wais]

Convolution is a mathematical operation that involves two functions, in this case h[n] and g[n], and produces a third function that represents the output of the first function when it is passed through the second function. In other words, convolution is a way to combine two functions in order to understand how the output of one affects the other.

In the context of systems, convolution is often used to analyze the behavior of a system when it is given an input signal. The impulse response of a system, h[n], is a function that describes how the system responds to a brief input signal, or an impulse. Similarly, the impulse response of the inverse system, g[n], describes how the inverse system responds to a brief input signal.

When we convolve these two impulse responses, h[n] * g[n], we are essentially passing the impulse response of the original system through the impulse response of the inverse system. This results in the output of the original system being "undone" by the inverse system, leaving us with the original input signal, which is represented by the delta function, δ[n].

In other words, h[n] * g[n] = δ[n] means that when we pass the impulse response of the original system through the impulse response of the inverse system, we get back the original input signal. This is why convolution is often used in signal processing and system analysis, as it allows us to understand the relationship between different systems and their inputs.