5.10 Final-value theorem
The final-value theorem allows us to find the final value x (∞) directly from its Laplace transform x(s).If x(t) is a causal signal,then
This proves the final value theorem.
Example 12:-EFind the final value of
Solution
This verifies the result obtained from final-value theorem.
5.11 Time periodicity
Let us consider a function f(t) that is periodic as shown in Fig.8. The function f(t) can berepresented as the sum of time-shifted functions as shown in Fig.9.
Figure 8A periodic function
Figure 9 Decomposition of periodic function
(11)
Wherex1(t) is the waveform described over the first period of x(t). That is, x1(t) is the same asthe function x(t) gated_ over the interval 0 <t<T.
Taking the Laplace transform on both sides of equation (11) with the time-shift propertyapplied, we get
(12)
In equation (12), X1(t) is the Laplace transform of x (t) defined over first period only.Hence, we have shown that the Laplace transform of a periodic function is the Laplace transformevaluated over its first period divided by 1-e-Ts.
Example 13:-Find the Laplace transform of the periodic signal x (t) shown in Fig.10.
Figure 10
SolutionFrom Fig1, we find that T= 2 Seconds.
The signal x (t) considered over one period is denoted as x1(t) and shown in Fig. (a).
Figure1. (a) Figure 1.(b) Figure 1.(c)
The signal x1(t) may be viewed as the multiplication ofxA(t) and &g(t).
Taking Laplace Transform, we get
6. Inverse Laplace transform
The inverse Laplace transform of X(s) is defined by an integral operation with respect to variable s as follows:
(13)
Since s is complex, the solution requires knowledge of complex variables. In other words,the evaluation of integral in equation (13) requires the use of contour integration in the complexplane, which is very difficult. Hence, we will avoid using equation (13) to compute inverseLaplace transform.
In many situations, the Laplace transform can be expressed in the form
(14)
The function x(s) as defined by equation (14) is said to be rational function of s, sinceit is a ratio of two polynomials. The denominator Q(s) can be factored into linear factors.A partial fraction expansion allows a strictly proper rational function ((?)to be expressed as afactor of terms whose numerators are constants and whose denominator corresponds to linear ora combination of linear and repeated factors. This in turn allows us to relate such terms to theircorresponding inverse transform.
For performing partial fraction technique on X(s) the function X(s) has to meet the followingconditions:
(i) X(s) must be a proper fraction. That is,) m<n. When X(s) is improper, we can use longdivision to reduce it to proper fraction.
(ii) Q(s) should be in the factored form.