Sifting property

Consider the product . The delta function is non zero only at the origin so it follows the signal is the same as .

 More generally 

It is important to understand the above expression. It means the product of a given signal x[n] with the shifted Unit Impulse Function is equal to the time shifted Unit Impulse Function multiplied by x[k]. Thus the signal is 0 at time not equal to k and at time k the amplitude is x[k]. So we see that the unit impulse sequence can be used to obtain the value of the signal at any time k. This is called the Sampling Property of the Unit Impulse Function. This property will be used in the discussion of LTI systems. For example consider the product . It gives .

Likewise, the product x[n] u[n] i.e. the product of the signal u[n] with x[n] truncates the signal for n < 0 since u[n] = 0 for n <0

Similarly, the product x[n] u[n-1] will truncate the signal for n < 1.

Now we move to the Continuous Time domain. We now introduce the Continuous Time Unit Impulse Function and Unit Step Function.