Discrete Time Impulses: The Unit Sample

The discrete time unit sample (or unit impulse) is the discrete time signal , where

An arbitrary "discrete time spike" of weight occuring at time is then given by the signal . This signal equals at time , and is zero at all other times. Given a signal and a fixed time , the signal agrees with at time , but is zero at all other times. Thus, if we form the superposition of all such signals as varies this gives us the signal back:

We call the first equation above the representation of as a superposition of impulses. The second equation above is sometimes called the sifting property. ( is "sifted" at time to obtain the value of at time .) Obviously, these two equations are really the same thing.