Discrete Time Convolution

Let the given signal x[n] be

Let the Impulse Response be

Now we break the signal in its components i.e. expressed as a sum of unit impulses scaled and delayed or advanced appropriately. Simultaneously we show the output as sum of responses of unit impulses function scaled by the same multiplying factor and appropriately delayed or advanced.

Summing the left and the right hand sides of the above figures we get the input x[n] and the total response on the left and the right sides respectively. Thus we see the graphical analog the above formula.

The total response referred to as the Convolution sum need not always be found graphically. The formula can directly be applied if the input and the impulse response are some mathematical functions. We show this by a example.