A continuous$-$time signal $x(t)=e^{-at}u(t)$ is given for $a>0.$ Which of the following is equal to the continuous$-$time Fourier Transform $x(\frac{?}{W})$ of this signal?

A$)\frac{1}{a^2\frac{+}{w^2}}$

B$)\frac{1}{a-jw}$

C$)\frac{1}{2a+jw}$

D$)\frac{1}{a+2jw}$

E$)\frac{1}{a+jw}$

The input and impulse response of a continuous$-$time LTI system are given below.

$x(t)=u(t-7)$ and $h(t)=e^{-3t}u(t).$ Accordingly, which of the following is equal to the result of the convolution operation $\frac{d}{dt}x(t)**h(t)?$

A$)y(t)=e^{-3(t-7)}u(t+7)$

B$)y(t)=-e^{-3(t-7)}u(t-7)$

C$)y(t)=e^{-3(t-7)}u(t-7)$

D$)y(t)=2e^{-3(t-7)}u(t-7)$

E$)y(t)=e^{-3(t+7)}u(t-7)$

The input and impulse response of a continuous$-$time LTI system are given below.

$x(t)=u(t-1)$ and $h(t)=e^{-3t}u(t).$ Accordingly, which of the following is equal to the result of the convolution operation $\frac{d}{dt}x(t)**h(t)?$

A$)y(t)=2e^{-3(t-1)}u(t-1)$

B$)y(t)=-e^{-3(t-1)}u(t-1)$

C$)y(t)=e^{-3(t-1)}u(t+1)$

D$)y(t)=e^{-3(t-1)}u(t-1)$

E$)y(t)=e^{-3(t+1)}u(t-1)$