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. Obtain the Laplace transform of x (t)= e^(-at)cos (wot) u (-t) and indicate its ROC?
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. Obtain the Laplace transform of x (t)= e^(-at)cos (wot) u (-t) and i...
Laplace Transform of x(t) = e^(-at)cos(wot)u(-t)
To find the Laplace transform of the function x(t) = e^(-at)cos(wot)u(-t), we need to consider the properties of the unit step function and the nature of the function itself.
Step 1: Define the Function
- x(t) = e^(-at)cos(wot) for t < 0="" (due="" to="" />
- x(t) = 0 for t ≥ 0
Step 2: Apply the Laplace Transform
The Laplace transform is given by:
L{x(t)} = ∫[0 to ∞] e^(-st)x(t)dt
Since x(t) is defined for t < 0,="" we="" modify="" the="" />
L{x(t)} = ∫[-∞ to 0] e^(-st)e^(-at)cos(wot)dt
This can be rewritten as:
L{x(t)} = ∫[-∞ to 0] e^(-(s+a)t)cos(wot)dt
Using the known Laplace transform of e^(αt)cos(βt), where α = -(s + a) and β = wo:
L{e^(αt)cos(βt)} = (s + a) / [(s + a)^2 + wo^2]
Thus, we have:
L{x(t)} = (s + a) / [(s + a)^2 + wo^2], for s + a < />
Step 3: Region of Convergence (ROC)
- The ROC for this Laplace transform is determined by the condition: Re(s) < />
- This indicates that the transform converges for values of s that are to the left of the vertical line in the complex plane at s = -a.
Conclusion
The Laplace transform of x(t) = e^(-at)cos(wot)u(-t) is:
L{x(t)} = (s + a) / [(s + a)^2 + wo^2] with ROC: Re(s) < -a.="" />
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. Obtain the Laplace transform of x (t)= e^(-at)cos (wot) u (-t) and indicate its ROC?
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