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 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. This expresses the input x(t) as an integral (continuum sum) of shifted impulses that are weighted by weights x(τ).
Again, we write:
Now take a system and define the impulse response h(t) of the system as the response of the system to an unit impulse input:
h(t) = S[δ(t)]
Next, define the response of the system to a shifted unit impulse as:
h(t,τ) = S[δ (t - τ)]
If the system is linear, then
S[αx_{1}(t) + βx_{2}(t)] = αy_{1}(t) + βy_{2}(t)
where of course S[x_{1}(t)] = y_{1}(t) and S[x_{2}(t)] = y_{2}(t).Now,
or,
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,
and
We have seen that for a linear time-invariant system, 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!
Again, we write the convolution integral as:
The notation h(t-τ) means that the function h(τ) is flipped and shifted across the function x(τ).
Convolution is a tough concept to get at first. Here are two rules that will greatly simplify doing convolutions:
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:
Fortunately, it usually falls out that there are only several regions of interest and the rest of y(t) is zero.
Example 1 Find y(t) = x(t)*h(t).
Form x(τ) and h(t - τ) (to shift the time-reversed function h(- τ) by t, just add t to all points).
When you finish notice: