Filter convolution sum

The response for a non-unit impulse is simply the unit impulse response scaled. So using the linearity principle, we can state that  

As the unit impulse is simply a single sample, a full input sequence r[ ], can be characterised by a sequence of scaled unit impulses  

What are the values of w As the delta function will set everything outside — to 0, it should be clear that w[n = x[n for a given w.Therefore, we can state  

This may seem a rather trivial result, but it shows that a signal can be fully described by a set of appropriately scaled impulse responses. The time invariance property of LTl systems tells us that a shifted input will give a shifted response. For the impulse function, the following is therefore true  

The linear property states that a6[ ] i—ah[n] for any a and so the following is also true  

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