Final answer:
The Fourier transform of the signal x(t) = 2e^-2t u(t) is 2/(−2+jw). To sketch |X(w)|, plot the magnitude of X(w) against w.
Step-by-step explanation:
The Fourier transform of the signal x(t) = 2e-2tu(t) can be calculated using the formula:
X(ω) = ∫[x(t)e-jωt]dt
In this case, x(t) = 2e-2tu(t), where u(t) is the unit step function. The Fourier transform of x(t) is:
X(ω) = 2 ∫[e(-2+jω)t]dt
To evaluate this integral, you can use the formula for the Fourier transform of eatu(t):
F{eatu(t)} = 1/(a+jω)
Substituting a = -2+jω, we get:
X(ω) = 2/(−2+jω)
To sketch |X(ω)|, you can plot the magnitude of X(ω) against ω. The magnitude of a complex number can be calculated using the formula:
|a+jb| = √(a2+b2)