A discrete time signal consists of a sequence of numbers denoted by xn, x(n), or x(nT). Sequences of particular importance have been given special names.
Figure 1   A discrete time filter with
a unit pulse input and and a response h(n).
The response of a system to a unit pulse is h(n).
Since the unit pulse is zero except when its argument, n, is zero, a sequence x(n) can be written as the weighted sum of a sequence of unit pulses.
where is zero except when k=n. Since the response to a unit pulse delayed by k units is h(n-k), the response to the weighted sum of delayed unit pulses is
This is the convolution summation. By changing variables you can show that the convolution summation can also be written as,
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