Signal Definition
The given signal X(t) is defined as:
- X(t)e [u(t) - u(t - 5)]
Here, u(t) is the unit step function, which indicates that the signal is active from t = 0 to t = 5.
Understanding the Signal
- The term u(t) activates the signal at t = 0.
- The term u(t - 5) deactivates the signal at t = 5.
- Therefore, the signal X(t) is non-zero only in the interval [0, 5].
Laplace Transform Calculation
The Laplace transform of a signal defined over a finite interval can be expressed as:
- L{X(t)e [u(t) - u(t - 5)]} = ∫ from 0 to 5 X(t)e^(-st) dt
This integral computes the Laplace transform for the active part of the signal.
Region of Convergence (ROC)
- The ROC for the Laplace transform of a signal defined within a finite interval is typically the entire s-plane except for the poles within that interval.
- For this case, assuming X(t) is a proper function, the ROC will depend on the nature of X(t) and not include the points that would cause divergence.
Conclusion
- The Laplace transform of the signal X(t)e [u(t) - u(t - 5)] can be calculated using the specified integral from 0 to 5.
- The ROC will be determined based on the characteristics of the function X(t) and will generally exclude any poles within the bounds of integration.