Properties of bilateral Laplace transform

The analysis formula for the bilateral Laplace transform is as follows:

X(s) = integration(x(t)) e^-st dt. 

The region of convergence is the X(s) defined for the regions in s. It is important to specify the region of convergence in the case of bilateral Laplace transform, this is one of the major differences between unilateral Laplace transform and bilateral Laplace transform. 

Some of the features of the region of convergence are as follows:

1. If the time function is right-sided, that is, if x(t)= 0, t<t0, where t0 is a constant, the ROC will be a right half-plane. 
2. Similarly, a left-sided function will give the ROC in the left half-plane.
3. For a two-sided time function, the ROC will either be a strip or will not exist. 
4. If the ROC contains the jw-axis, then the system would be BIBO stable. And, if the boundary of the ROC is the jw-axis then the system is BIBO unstable. 

The unilateral Laplace transform is restricted to the causal time functions and takes the initial conditions into accounts in a systematic and automated manner in both the cases of differential equations and in the analysis of the systems. 

The bilateral Laplace transform can represent both the causal and non-causal time functions. Initial conditions are accounted for by including additional inputs. 

The properties of the Laplace transform are as follows:

1. Linearity property

If x(t) ←–> X(s), and y(t) ←→ Y(s)

Then the linearity property states that:

ax(t) + by(t) ←→ aX(s) + bY(s)

2. Time-shifting property 

If x(t) ←→ X(s)

Then time-shifting property states that x(t-t0) ←→ e^-st0 X(s). 

3. Frequency shifting property

If x(t) ←→ X(s)

Then the frequency shifting property states that e^s0t.x(t) ←→ X(s-s0) 

4. Time reversal property

If x(t) ←→ X(s)

x(-t) ←→ X(-s)

5. Time scaling property

If x(t) ←→ X(s)

Then time scaling property states that

x(at) ←→ 1/ |a| X(s/a) 

6. Differentiation and integration properties

The differentiation property states that:

If x(t) ←→ X(s)

dx(t)/dt ←→ s. X(s) - s. X(0)

The integration property states that:

           ∫ x(t) dt ←→ 1/sX(s) 

The bilateral Laplace transform involves the values of the signal x(t) for both t = 0 and x = 0 and is given by

x(t) ←→ X(s) = ∫x(t) e^-st dt. 

Therefore, the bilateral Laplace transform is mostly used to solve problems involving noncausal signals and systems, and other applications.

In the case of the unilateral Laplace transform, and bilateral Laplace transforms the properties of linearity, scaling, convolution, and differentiation are identical, though the operations associated with the above-mentioned properties may change the region of convergence or ROC. 

We can consider the linearity property to illustrate the change in ROC that may occur. If x(t) ← Lu→ X(s) with ROC Rx and y(t) ← L → Y(s) with ROC Ry, then ax(t) + by(t) ←L→ aX(s) + bY(s) with ROC that is at least Rx intersection Ry. 

The ROC for a sum of signals is just the intersection of the individual ROCs, the ROC may be larger than the intersection of the individual ROCs if a pole and a zero cancel in the sum aX (s) + bY (s). 

Inversion of the bilateral Laplace Transform 

As in the unilateral case, we consider the inversion of bilateral Laplace transform that is expressed in the ratios of polynomials in s. The primary difference between the inversions of bilateral and unilateral Laplace transforms is that we must use the ROC to determine a unique inverse transform in the bilateral case. 

Suppose we need to invert the ratio of polynomials in s given by:

X(s) = B(s) / A(s) 
= bMs^M + bM-1s^M-1 + ….. + b0/ s^N + aN-1 S^N-1 + … + a1s + a0

As in the unilateral case, if M>= N, then we use long division to express,

X(s) = Σ ck s^kn + X(s) 

The region of convergence associated with X(s) determines if the left-sided or the right-sided inverse transform will be chosen.  A right-sided exponential signal will have its ROC in the right of the pole, while the ROC of a left-sided exponential signal lies to the left of the pole. 

The linearity property states that the region of convergence of a given function is the intersection of the region of convergences or ROCs of the individual terms in the partial-fraction expansion. To find the inverse transform of each term, we need to infer the ROC of each term from the given ROC of X(s).  

Reference

Properties of bilateral Laplace transform