Laplace Transform of x(t)
To find the Laplace Transform of the signal x(t) = e^(-at) Cos(Wo t) u(-t), we first need to understand the components involved.
Understanding the Components
- e^(-at): This is an exponentially decaying function.
- Cos(Wo t): This represents a cosine wave with frequency Wo.
- u(-t): The unit step function which is 1 for t < 0="" and="" 0="" for="" t="" />= 0, indicates that the signal is defined for negative time.
Laplace Transform Definition
The Laplace Transform is defined as:
L{x(t)} = ∫[0 to ∞] e^(-st) x(t) dt
However, due to the presence of u(-t), we need to adjust our limits:
L{x(t)} = ∫[-∞ to 0] e^(-st) e^(-at) Cos(Wo t) dt
Integration by Parts
Using integration techniques, the Laplace Transform can be computed as:
L{x(t)} = ∫[-∞ to 0] e^(-(s+a)t) Cos(Wo t) dt
This integral involves a standard form, which can be evaluated to yield:
L{x(t)} = (s + a) / ((s + a)² + Wo²)
Region of Convergence (ROC)
The ROC for this transform is defined by the condition s + a > 0, which means the real part of s must be greater than -a.
Graphing the ROC
- The ROC is a half-plane to the right of the line Re(s) = -a on the complex plane.
- It indicates stability and convergence of the Laplace transform.
Conclusion
The Laplace Transform of the given signal is (s + a) / ((s + a)² + Wo²), with the ROC being Re(s) > -a. This provides insight into the behavior of the system in the frequency domain.