The signal x(t)=e^(-|t| ) is defined for all values of t. a. Plot the signal x(t

Question

The signal x(t)=e^(-|t| ) is defined for all values of t.
a. Plot the signal x(t) and determine if this signal is finite energy. That is, compute the integral

?_(-?)^? |x(t)|^2 dt
and determine if it is finite.
The work I currently have:

x(t)=?_(-?)^0 e^t )^2 +?_0^? (e^(-t) )^2 =
?_(-?)^0 e^2t +?_0^? e^(-2t) =
[e^(3t/3) ]_(-?)^0+[e^(3t/3) ]_0^?=
(0-1/3)+(-1/3+0)=-2/3=finite

b. If you determine that x(t) is absolutely integrable or that the integral
?_(-?)^? |x(t)|^2 dt
is finite, could you say that x(t) has finite energy? Explain why or why not.
Hint: Plot |x(t)| and |x(t)|^2.
c. From your results above, is it true the energy signal
y(t)=e^(-t) cos (2*pi*t)u(t)
Is less than half the energy of x(t)? Explain. To verify your result, use symbolic MATALB to plot y(t) and to compute its energy.
d. To discharge a capacitor of 1mF charged with a voltage of 1 volt we connect it, at time t=0, with a resistor of R?. When we measure the voltage in the resistor we find it to be v_R (t)=e^(-t) u(t). Determine the resistance R. If the capacitor has a capacitance of 1

Explanation / Answer

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