Composition of LTI systems

 Suppose that we have two LTI systems with impulses responses h1(n) and h2(n), and we connect them in a cascade structure as shown in the following figure:

We can combine the two convolutions

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To get

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Exchanging the two summations and changing variables we can show that

where

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This tells us that the cascade system is equivalent to a single LTI system where the impulse response of the single LTI system is the convolution of the impulses responses of the two component systems. Of course, we also know that the frequency responses compose in a very simple way, namely

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