$(20$ points$)$ Consider the following Consider the following excitation $x(t)$ :

$x(t)=cos(45000t)cos(50000t)$

Also, consider a two$-$stage filter with an impulse response

$h(t)=[(t)-\frac{1}{{}_1}{e}^{-\frac{t}{{}_1}}u(t)]**\frac{1}{{}_2}{e}^{-\frac{t}{{}_2}}u(t)$

The response of the filter $h(t)$ to the excitation $x(t)$ is

$y(t)=x(t)**h(t)$

${}_1$ : Time constant associated with filter cutoff frequency of $1$ kHz

${}_2$ : Time constant associated with filter cutoff frequency of $50$ kHz Using CTFT$,$ find the observed magnitude and phase, in radians, of the response, $y(t),$ of the two$-$stage filter, to the excitation $x(t),$ for only the positive frequencies. Show all mathematical steps, including the full doublesided CTFT of the response, which is Y$($f$),$ for full credit. All complex amplitude scaling factors, in $Y(f),$ must be simplified to the form $A+-jB.$ For all numbers go up to $5$ decimal places.