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tamtam402
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Homework Statement
Let x(t) = e-100tu(t)
u(t) = 0 for t < 0
u(t) = 1 for t > 0
Evaluate the following integral (from -∞ to ∞):
X(ω) = ∫ x(t)e-iωtdt
Homework Equations
See below.
The Attempt at a Solution
I tried to evaluate the integral by splitting it in two parts, since x(t) takes two different values. Keep in mind I'm replacing ∞ with a to evaluate the limit later.
from -∞ to 0:
X1(ω) = ∫ e-iωtdt
X1(ω) = (1/-iω) e-iωt (from -∞ to 0)
X1(ω) = (1/-iω)(1 - eiωa)
from 0 to ∞:
X2(ω) = e-100∫ e-iωtdt
X2(ω) = e-100 (1/-iω) e-iωt (from 0 to ∞)
X2(ω) = e-100 (1/-iω)(e-iωa - 1)
X(ω) = X1(ω) + X2(ω) = (1/-iω)(1 - eiωa) + e-100 (1/-iω)(e-iωa - 1)
= (1/-iω) (1 - eiωa + e-100e-iωa - e-100)
This is where I'm stuck. The answer is supposed to be X(ω) = 1 / (100 + iω), and I have absolutely no idea how I'm supposed to get there. Would someone mind helping me out?
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