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Since we are considering discrete time signals and systems, an ideal impulse is easy to simulate on a computer or some other digital device. It is simply a signal that is 1 at the point n = 0, and 0 everywhere else.
By the sifting property of impulses, any signal can be decomposed in terms of an infinite sum of shifted, scaled impulses.
The function ${\delta}_{k}\mathrm{[n]}=\delta \mathrm{[n-k]}$ peaks up where $n=k$ .
Since we know the response of the system to an impulse and any signal can be decomposed into impulses, all we need to do to find the response of the system to any signal is to decompose the signal into impulses, calculate the system's output for every impulse and add the outputs back together. This is the process known as Convolution . Since we are in Discrete Time , this is the Discrete Time Convolution Sum .
When a system is "shocked" by a delta function, it produces an output known as its impulse response. For an LTI system, the impulse response completely determines the output of the system given any arbitrary input. The output can be found using discrete time convolution.
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