In this lesson, we will discuss linear time-invariant (LTI) systems - these are systems that are both linear and time-invariant. We will see that an LTI system has an input- output relationship described by a convolution.

Using the sifting property, we can write a signal *x*(*t*) as:

which is writing a general signal *x*(*t*) as a function of an
impulse function. This expresses the input *x*(*t*) as an integral (continuum sum) of shifted
impulses that are weighted by weights *x*(*τ*). Another way to put this is that you can
build a CT signal out of impulses.

We can write:

This expresses the input *x*(*t*) as an integral (continuum sum)
of shifted impulses that are weighted by weights *x*(*τ*).

Now take a system and define the impulse response of the system as

*h*(*t*) = *S*[*δ*(*t*)]

and the response of the system to a shifted impulse as:

*h*(*t*,*τ*) =
*S*[*δ*(*t* - *τ*)]

If the system is linear, then

*S*[*α**x*_{1}(*t*) +
*β**x*_{2}(*t*)] =
*α**y*_{1}(*t*) + *β**y*_{2}(*t*)

Let

But what if the system is also Time-Invariant?

Then *S*[*δ*(*t* - *τ*)] =
*h*(*t* , *τ*) = *h*(*t* - *τ*),
since we had *S*[*δ*(*t*)] = *h*(*t*). Therefore,

We have seen that if we have a linear time-invariant system, then the output
is the input convolved with the system's impulse response *h*(*t*). In other words, we can
completely characterize an LTI system by its impulse response.

This is a very important result!

Convolution Integral:

Here, *h*(*τ*) is flipped and shifted across *x*(*τ*).

Convolution is a tough concept to get at first. I have 2 rules that will greatly improve the quality of your life:

- DRAW A PICTURE of
*x*(*τ*) and*h*(*t*-*τ*) - FLIP THE "EASY" FUNCTION

Why can we pick which function to flip?

Because convolution is commutative:

Change variables: *λ* = *t* -
*τ* → *τ* = *t* - *λ*,
*dτ* = -*dλ*.

(minus signs cancel)

Let's examine convolution formula:

- Flip
*h*(*τ*) and shift it to form*h*(*t*-*τ*).

Note:*h*(*t*-*τ*) is a function of*τ*, not*t*!

*t*is the shift parameter. - Fix
*t*and multiply*x*(*τ*) with*h*(*t*-*τ*) for all values of*τ*. - Integrate
*x*(*τ*)*h*(*t*-*τ*) over all*τ*to get*y*(*t*) which is a single value that depends on*t*. Remember that*τ*is the integration variable and that*t*is treated like a constant when doing the integral. - Repeat for all values of
*t*.

Fortunately, it usually falls out that there are only several regions of
interest and the rest of *y*(*t*) is zero.

__Ex.__ Find *y*(*t*) = *x*(*t*)**h*(*t*).

Form *x*(*τ*) and *h*(*t* - *τ*)
(to shift by *h*(- *τ*) by *t*, just add *t* to all points) and continue
from there.

When you finish notice:

- (a) nonzero "width" of
*x*(*t*) = 3

(b) nonzero "width" of*h*(*t*) = 4

(c) nonzero "width" of*y*(*t*) = 7 -
*y*(*t*) is "smoother" than*x*(*t*) or*h*(*t*)