Continuous Time Impulses: The Unit Impulse

The continuous time unit impulse is a continuous time signal which is zero for all time , but has the value at time , such that the "spike has an area of one," roughly speaking. More specifically, an approximation to is the signal defined as

where is a parameter. Note that if is small, is simply a a tall spike of height and width centered at , and thus it has an area of one. As approaches zero from above, approaches , in a certain sense.

Given an arbitrary signal , we can form the "staircase" approximation to :

This approximating signal is piecewise constant over intervals of length and agrees with at all times which are multiples of . You may recall that staircase approximations are used in basic integration theory.

As approaches zero, the right side of the equation above becomes an integral and the approximation to becomes exact, so we obtain

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 .) This is actually the formal definition of the unit impulse. Obviously, these two equations are really the same thing.