The 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 

Since the unit impulse is simply a single sample, a lull input sequence x[n] can be characterised ly a sequence of scaled unit impulses  

What are the values of w Since the delta function will set ever5dhing outside n — kXo 0, it should be clear that w[n = x[n] for a given n. Therefore, we can state that 

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 LTI systems tells us that a shifted input will give a shifted response. For the impulse function, the following is therefore true  

We can use these two properties on Equation (10.59) to find the output of the filter 

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