- #1

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QUESTION:

QUESTION:

Compute the convolution of [itex] x(t) [/itex] and [itex] h(t) [/itex] where:

**1:**

[tex] x(t) = u(t) [/tex]

[tex] h(t) = e^{-at}u(t), \,\,\, a>0 [/tex]

**2.**

[tex] x(t) = u(t) - u(t-T_1) [/tex]

[tex] h(t) = u(t) - u(t-T_2), \,\,\, T_1 > T_2 [/tex]

**"ANSWER":**

Let [itex] y(t) [/itex] be the convolution of x(t), h(t)

**(1):**

[tex] t < 0 [/tex]:

y(t) =0

[tex] t \geq 0 [/tex]:

[tex] y(t) = \int_{-\infty}^t e^{-a(t-\tau)}d\tau = \frac{1}{a}[/tex]

**(2):**

[tex] t < 0 [/tex] or [tex] t-T_2 > 0 [/tex]:

y(t) =0

[tex] t \geq 0 [/tex] and [tex] t-T_2 < 0 [/tex]:

[tex] y(t) = \int_0^t d\tau = t [/tex]

[tex] t-T_2 > 0 [/tex] and [tex] t<T_1 [/tex]:

[tex] y(t) = \int_{t-T_2}^t d\tau = T_2 [/tex]

If I need to show more steps, please let me know. Thanks!