Problem I $(15$pts$).$

For the signal $e^{-3t}u(t),$ determine the bandwidth of an anti$-$aliasing filter if the essential bandwidth of the signal contains $99\%$ of the signal energy.

Hint: use Parseval's theorem

Problem II $(25$ pts$).$

A$.$ Using only the fact that ${}^{k}u[k]\frac{z}{z-}$ and properties of the $z-$transform, find the $z-$ transform of:

a$.x[n]=n^2{}^{n}u[n]$

b$.,x[n]=(\frac{1}{2}{)}^{n}u[n]+(\frac{1}{3}{)}^{n}u[n]$

B$.$ Derive the amplitude and phase response of $H(z)$ of the digital filters shown in the figure.

Hint: $|F()|=\sqrt[\phantom{2}]{F()F(-)},|F(){|}^2=F()F^{**}()$

Problem III $(20$ pts$).$

Consider the following LTID system:

a$.$ Determine the system's frequency response function, $H().$

b$.$ Determine the unit$-$impulse response, $h[n].$

c$.$ Sketch the magnitude response and the angle response, and determine the filter type.

d$.$ Solve for the $3-dB$ cutoff frequency, $3dB.$

Problem IV $(20$pts$).$

Design a second order digital low$-$pass Butterworth filter with a $3$ dB cut$-$off frequency of $2$ kHz and minimum attenuation of $30$ dB at $4\mathrm{.}25$ kHz for a sampling rate of $10$ kHz $.$

Problem V $(20$pts$).$

A discrete$-$time LTI system is specified by the difference equation

$[k+1]-0\mathrm{.}5y[k]=f[k+1]+0\mathrm{.}8f[k]$

a$.$ Derive the transfer function in the z$-$domain.

b$.$ Find the amplitude and phase response of the system.

c$.$ Find the system response $y[k]$ for the input $f[k]=cos(0\mathrm{.}5k-\frac{}{3})$