There are many applications of Laplace transforms in which it is reasonable to assume that the signals involved are causal, that is, zero for times t<0. For example, if we apply an input that is zero for time, t<0, to a causal system, the output will also be zero for t < 0. Also, the choice of time origin is arbitrary in many problems. Therefore, time t=0 was often chosen as the time at which an input is presented to the system, and the behavior of the system for time t >= 0 is of interest to us. In such problems, it is advantageous to define the unilateral or one-sided Laplace transform, which is based only on the nonnegative time, that is, t>=0 portions of a signal.
Causal signals help to remove the ambiguity inherent in the bilateral transform and therefore it does not consider the region of convergence. To analyze the behavior of a causal system, we can use the differentiation property for the unilateral Laplace transform.
We represent the unilateral Laplace transform of a signal x(t) as follows:
X(s) = ∫x(t) e^st dt.
We do include discontinuities and impulses that occur at t= 0 in the integral, it is made clear by implying the lower limit of 0. Hence, X(x) depends on x(t) for t >= 0. The inverse unilateral Laplace transform is also given by the same equation as the inverse Laplace transform depends only on X(s). The relationship between X(s) and x(t) can be denoted as:
x(t) ←(Lu)→ X(s)
In the given relationship, Lu denotes the unilateral transform. It is already known, that for signals that are zero for times t<0, the unilateral and bilateral Laplace transforms are equivalent.
Properties of the lateral Laplace Transform
The properties of the Laplace transform are similar to those of the Fourier transform hence, we simply state many of them. The unilateral and bilateral transforms have many properties in common, although there are also some important differences. In the properties discussed next, we assume that:
x(t) ←(Lu) —> X(s)And y(t) ← (Lu) —> Y(s)
1. Linearity
ax(t) + by(t) ←(Lu) —> aX(s) + bY(s)
2. Scaling
x(at) ←(Lu) —> 1/a X(s/a), for a>0. The inverse scaling in s is introduced by scaling in time.
3. Time shift
x(t-T) ←(Lu) —> e^-st X(s)
The complex exponential e^-st helps in corresponding the shift of T in time to the multiplication of the Laplace transform. As the unilateral transform is defined solely in terms of the nonnegative-time portions of the signal, there is a restriction on the shift. Hence, this property applies only if the shift does not move a nonzero t >= 0 component of the signal to t < 0, or does not move a nonzero t < 0 part of the signal t>=0. The time-shift property is most commonly applied to causal signals x(t) with shifts T > 0, in which case the shift restriction is always satisfied.
Inversion of unilateral Laplace Transform
Direct inversion of the Laplace transform using the equation mentioned requires an understanding of contour integration. Instead, we shall determine inverse Laplace transforms using the one-to-one relationship between a signal unilateral Laplace transform. Given knowledge of several basic transform pairs and Laplace transform properties, we are able to invert a very large class of Laplace transform in this manner.
In the study of LTI systems described by integrodifferential equations, we fully encounter Laplace transforms that are a ratio of polynomials in s. In this case, the transform is obtained by expressing X(s) as a sum of terms for which we already know the time function, using a partial-fraction expansion. Suppose:
X(s)= B(s)/A(s)
If X(s) is an improper rational function that is M>=N, then we can express X(s) in the following form:
X(s) = Σcs + X’(s).
Where, X’(s) = B’(s)/A(s).
The numerator polynomial B’(s) now has the order one less than that of the denominator and the partial fraction expansion method is used to determine the inverse transform of X’(s). Given that the impulse and the derivatives a
The partial-fraction expansion procedure is applicable to either real or complex poles. A complex pole usually results in complex-valued expansion coefficients and a complex exponential function of time. If the coefficients in the denominator polynomial are real, then all the complex poles occur in complex-conjugate pairs. In cases where X(s) has real-valued coefficients and thus corresponds to a real-valued time signal, we may simplify the algebraic by combining complex-conjugate poles in the partial-fraction expansion in such a way as to ensure real-valued expansion coefficients and a real-valued inverse transform. This is accomplished by combining all pairs of complex-conjugate poles into quadratic terms with real coefficients. The inverse Laplace transforms of these quadratic terms are exponentially damped sinusoids.
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