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FURTHER PROPERTIES OF TRANSFORMS
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But since f(t) = 0 for t < 0, the lower limit of this integral may be replaced by zero. Since (t - to) is now the dummy variable of integration, the integral may be recognized as the Laplace transform of f(t); thus, the theorem is proved. This result is also useful in inverting transforms. It follows that, if f(t) is the inverse transform of f(s), then the inverse transform of POf (s) is the function
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Example 4.5. Find the Laplace transform of f(t) = P_ i 0
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This function is pictured in Fig. 4.2. It is clear that f(t) may be represented by the difference of two functions, f(t) = ;[u(t) - u(t - h)] where u(t - h) is the unit-step function translated h units to the right. We may now use the linearity of the transform and the previous theorem to write immediately f(s) = I l - e-hs h s This result is of considerable value in establishing the transform of the unit-impulse function, as will be described in the next section.
t----t
FIGURE 4-2
Pulse function of Example 4.5.
T H E LAPLACE
TRANSFORM
lkansform of the Unit-impulse Function
Consider again the function of Example 4.5. If we allow h to shrink to zero, we obtain a new function which is zero everywhere except at the origin, where it is infinite. However, it is important to note that the area under this function always remains equal to unity. We call this new function 8(t), and the fact that its area is unity means that m G(t)& = 1 I --m The graph of s(t) appears as a line of infinite height at the origin, as indicated in Table 2.1. The function 8(t) is called the unit-impulse function or, alternatively, the delta function. It is mentioned here that, in the strict mathematical sense of a limit, the function f(t) does not possess a limit as h goes to zero. Hence, the function 8(t) does not fit the strict mathematical definition of a function. To assign a mathematically precise meaning to the unit-impulse function requires use of the theory of distributions, which is clearly beyond the scope of this text. However, for our work in automatic control, we shall be able to obtain useful results by formal manipulation of the delta function, and hence we ignore these mathematical difficulties. We have derived in Example 4.5 the Laplace transform of f(t) as
Formally, then, the Laplace transform of 8(t) can be obtained by letting h go to zero in LCf(t)}. Applying L HBpital s rule,
h-0 S
(4.1)
This verifies the entry in Table 2.1. It is interesting to note that, since we rewrote f(t) in Example 4.5 as f(t) = \$u(t) then 8(t) can be written as 6(t) = /imo + u(r) - u(t - h)
- u(t - h)]
In this form, the delta function appears as the derivative of the unit-step function. The reader may find it interesting to ponder this statement in relation to the graphs of s(t) and u(t) and in relation to the integral of 8(t) discussed previously. The unit-impulse function finds use as an idealized disturbance in control systems analysis and design.
FURTHJZR
PROPERTIES OF TRANSFORMS
lkansform of an Integral
If LCf(t)} = f(s), then
L 110 lf(t)dt 1 = y
This important theorem is closely related to the theorem on differentiation. Since the operations of differentiation and integration are inverses of each other when applied to the time functions, i.e.,
-\$/-otf(t)dt = j-ot %dt = f(t)
sf(s) s fsf(s) = f(s)
(4.2)
it is to be expected that these operations when applied to the transforms will also be inverses. Thus assuming the theorem to be valid, Eq. (4.2) in the transformed variable s becomes
In other words, multiplication of f(s) by s corresponds to differentiation of f(t) with respect to t, and division off(s) by s corresponds to integration off(t) with respect to t. The proof follows from a straightforward integration by parts.