# Singularity Functions

### The Unit Step Function

We already defined the unit step function u(t) as: Ex: Find and plot u(t - t0) and u(t - t1)

Ex. Define a block function (window) as Then Example:    Plot ### The Unit Impulse Function

 Now let's look at a signal:  What is its derivative? Define it as:  which has unit area.

 Now, So what if we take ?

The pulse height gets higher and higher and its width goes to 0, but its area is still 1!

So define δ(t) as the unit impulse: AND or equivalently, AND Also, δ(t) can be considered to be the derivative of u(t) but only in a restricted sense since u(t) is a discontinuous function.

Note that the impulse function is not a true function since it is not defined for all values of t. It's a "generalized function." But its idealization will allow us to derive many interesting results.

### Unit Impulse Properties

1.    Scaling

(t) is an impulse with weight or area K: 2.    Multiplication of a function x(t) (that is continuous at 0) by an impulse δ(t):

We get an impulse with area or weight x(0).

3.    Time Shift of an impulse

y(t) = x(t)δ(t-t0) So we get an impulse with weight equal to the value of x(t) where the impulse is located:

y(t) = x(t0)δ(t-t0)

Example What is Kx(t)δ(t-t0)?

Example What is 3u(t-1)δ(t) ?

***SIFTING PROPERTY***

What if we multiply a function by an impulse and then integrate? We integrate out the time variable so the integral is just equal to a number (or later on, a function). We'll see this many times.In this case, the impulse δ(t-t0) is defined by the integral (as long as the function x(t) is continuous at t0).

Ex.

ALSO: 