$4\mathrm{.}31.($a$)$ Show that the three LTI systems with impulse responses

h$1($t$)=$ u$($t$),$

h$2($t$)=-28($t$)+5$e$-2$tu$($t$),$

and

all have the same response to x$($t$)=$ cost$.$

$($b$)$ Find the impulse response of another LTI system with the same response to

cost$.$

This problem illustrates the fact that the response to cos t cannot be used to

specify an LTI system uniquely.