Laplace Transform of x(t) = e^t * cos(ωt) * u(t)
To find the Laplace Transform of the signal x(t) = e^t * cos(ωt) * u(t), where u(t) is the unit step function, we can use the formula for the Laplace Transform of a product of an exponential and cosine function.
Step-by-Step Derivation
- The Laplace Transform is defined as:
L{x(t)} = ∫[0 to ∞] e^(-st) * x(t) dt
- Substituting x(t):
L{x(t)} = ∫[0 to ∞] e^(-st) * e^t * cos(ωt) dt
- This simplifies to:
L{x(t)} = ∫[0 to ∞] e^(-(s - 1)t) * cos(ωt) dt
- The Laplace Transform of e^(at) * cos(ωt) is given by:
L{e^(at) * cos(ωt)} = (s - a) / ((s - a)² + ω²)
- For our signal, a = 1:
L{x(t)} = (s - 1) / ((s - 1)² + ω²)
Region of Convergence (ROC)
- The ROC for this transform is:
- Re(s) > 1
- This means that the Laplace Transform converges in the right half of the complex plane, ensuring stability for the system described by the signal x(t).
Summary
- The Laplace Transform of x(t) = e^t * cos(ωt) * u(t) is:
L{x(t)} = (s - 1) / ((s - 1)² + ω²)
- The ROC is:
Re(s) > 1
Understanding the Laplace Transform and its ROC is crucial for analyzing linear time-invariant systems in electronics and communication engineering.