The Fourier Transform of the signal $$x(t) = {e^{ - 3{t^2}}}$$ is of the following form, where A and B are constants:

A linear time invariant system has an impulse response $${e^{2t}},\,\,t\, > \,0.$$ If the initial conditions are zero and the input is $${e^{3t}}$$, the output for $$t\, > \,0$$ is

Let u(t) be the unit step function. Which of the waveforms in Fig.(a) -(d) corresponds to the convolution of $$\left[ {u\left( t \right)\, - \,u\left( {t\, - \,1} \right)} \right]$$ with $$\left[ {u\left( t \right)\, - \,u\left( {t\, - \,2} \right)} \right]$$ ?

The Hilbert transform of $$\left[ {\cos \,{\omega _1}t + \,\sin {\omega _2}t\,} \right]$$ is