$($a$)$ If $x(t)$ is a process with zero mean and autocorrelation $f(t_1)f(t_2)(t_1-t_2),$ show that the process

$y(t)=x\frac{t}{f}(t)$ is WSS with autocorrelation $R_{yy}(t_1-t_2)=R_{yy}()=().$

$($b$)$ If $x(t)$ is white noise with autocorrelation $q(t_1)(t_1-t_2),$ show that the process $z(t)=x\frac{t}{\sqrt[\phantom{2}]{?}}q(t)$

is WSS white noise with autocorrelation $q(t_1)().$

$($c$)$ If $y(t)=x(t+a)-x(t-a),$ show that the corresponding autocorrelation is

$R_y()=2R_x()-R_x(+2)-R_x(-2a)$

And show that the corresponding power spectrum density is

$S_y()=4S_x()si{n}^{2}a.$