We already defined the unit step function *u*(*t*) as:

Ex: Find and plot
*u*(*t* - *t*_{0}) and *u*(*t* - *t*_{1})

Ex. Define a block function (window) as

Then |

Example: Plot |

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.

1. Scaling

*Kδ*(*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*-*t _{0}*)

So we get an impulse with weight equal to the value of *x*(*t*)
where the impulse is located:

*y*(*t*) = *x*(*t _{0}*)

__Example__ What is *Kx*(*t*)*δ*(*t*-*t _{0}*)?

__Example__ What is 3*u*(*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*-*t _{0}*) is defined by the integral
(as long as the function

__Ex.__

ALSO: