Problem $39$: $(15$ points$)$

A key goal of EE $350$ is to insure that you have a solid understanding of the relationship between the ODE, impulse response, and frequency response function representation of a LTI system. Consider a linear time$-$invariant causal $($LTIC$)$ system with input $f(t),$ impulse response function representation $h(t),$ and zero$-$state response $y(t).$

$(2$ 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

$h(t)=\frac{{}_n^2}{{}_d}{e}^{-{}_{n}t}sin({}_{d}t)u(t)$

where and

$wd={}_n\sqrt[\phantom{2}]{1-{}^2}.$

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

$H()=\frac{(tilde(Y))}{(tilde(F))}=\frac{b_m({)}^m+b_{m-1}({)}^{m-1}+cdots{b}_1()+b_0}{({)}^n+a_{n-1}({)}^{n-1}+cdots{a}_1()+a_0}.$

$(5$ points$)$ Using the time differentiation property and the results from parts $1$ and $2,$ find the ODE representation of the system. Express your answer in the form

$\frac{d^{n}y}{d{t}^n}+a_{n-1}\frac{d^{n-1}y}{d{t}^{n-1}}+$cdots$+a_1\frac{dy}{dt}+a_{o}y(t)=b_m\frac{d^{m}f}{d{t}^m}+b_{m-1}\frac{d^{m-1}f}{d{t}^{m-1}}+$cdots$+b_1\frac{df}{dt}+b_{o}f(t).$