$2$ points$)$ Using the appropriate property from Problem $38,$ show that the Fourier transform of the zero$-$state

response y$($t$)$ of the system to an arbitrary input f$($t$)$ is

Y $($$)=$ H$($$)$F $($$),$

where Y $($$),$ H$($$),$ and F $($$)$ are the Fourier transforms of y$($t$),$ h$($t$),$ and f$($t$),$ respectively. The Fourier

transform of the impulse response function h$($t$)$ is identical to the frequency response function H$($$)$ of the

system.

$2.(8$ points$)$ As a specific example, consider a LTIC system with the impulse response function

h$($t$)=$ $2$

n

d

e$$nt sin$($dt$)$u$($t$),$

where n $>0,0$ $<1,$ and

wd $=$ n

$1$ $2.$

By direct integration, determine the frequency response function of the system by computing the Fourier

transform of the impulse response function. Express you answer in the standard form

H$($$)=$Y

$$F $=$ bm $($$)$m $+$ bm$1($$)$m$1+$ b$1($$)+$ b$0$

$($$)$n $+$ an$1($$)$n$1+$ a$1($$)+$ a$0$

$.$

$3.(5$ points$)$ Using the time differentiation property and the results from parts $1$ and $2,$ find the ODE represen$-$

tation of the system. Express your answer in the form

dny

dtn $+$ an$1$

dn$1$y

dtn$1++$ a$1$

dy

dt $+$ aoy$($t$)=$ bm

dmf

dtm $+$ bm$1$

dm$1$f

dtm$1++$ b$1$

df

dt $+$ bof$($t$).$