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We have already used the unit step function which
can be used to act as a switch for a single or as a switch, double-switch, or delay.
Using it as a switch, we turn a function on at some time by using
multiplication; for example, Figure 1 shows a function
Figure 1. The function .
If , we can create a double switch which turns the function
on at time
and off again at time
; for example, Figure 2 shows a function
Figure 2. The function .
To delay a signal, both the function and the unit step function must be delayed; for
example, Figure 3 shows a function and
Figure 3. The function .
Because these functions arise so often in engineering, we require their Laplace transforms:
Finally, the second shift theorem states that if , it follows
The proofs are straight forward and are given in AMEM.
It is possible to use the unit step functions to define periodic functions other than the usual trigonometric
functions. For example, Figure 4 shows the function
We may now take the Laplace transform of this infinite series simply by applying it term-by-term and each term is simply a shifted
unit step function. Therefore, we have
If we factor out , we can write this sum as
This is a geometric series where in this case
and therefore, we have that
This is a special case of the following more general theorem:
If is defined for all and it is a periodic function
with period , then
Give the example we previously looked at, with
We must therefore calculate
and therefore the Laplace transform is
Further examples are in the assignments.