$(1)($a$)$ A second order IIR filter has the following transfer function parameters:

Poles are given by: $p_{1,2}=\frac{0\mathrm{.}9}{{?}_{+}}-70\mathrm{.}528.$

Zeros are given by: $z_{1,2}=\frac{1}{{?}_{+}}-70\mathrm{.}528.$

A sampling frequency $f_{\text{s}\text{a}\text{m}\text{p}}=2kHz.$

A DC magnitude response equivalent to $18\mathrm{.}0618$ dB $.$

$($a$)$ Construct the transfer function of the filter, namely $H(z).$

$($b$)$ Determine the frequency response of the filter $H(e^j).$ Hence, plot the magnitude response. What kind of filter is it$?$

$($c$)$ Determine the frequencies in Hz $,$ at which the magnitude responses are $1$ and $0.$

$($d$)$ Write out the difference equation of the filter. Hence, draw the block diagram of the diff. equation, then calculate the min. number of multiplications$/$sec$.$

$($e$)$ Derive an expression for the step$-$response of the filter. Hence, determine and plot the steady$-$state and transient responses.

$($f$)$ Calculate the transient duration in msec.

$($g$)$ Given a discrete$-$time signal $x(n)$ with the z$-$transform

$x(z)=20\mathrm{.}0855+1\mathrm{.}0043z^{-1}+0\mathrm{.}8034z^{-2}-9\mathrm{.}6411z^{-3}.$

Determine and plot $x(n).$ What is the length of the signal?

Consider the signal $y(n)$ with $Y(z)=log(x(z)),$ where $y(n)$ is the inverse ztransform of $Y(z).$

Expand $Y(z)$ in a power series and then determine and plot $y(n).$ Hint:. Retain only the most significant terms in the series.

$($h$)$ Consider the discrete$-$time signal $x(n),$ which is absolutely summable and has the following z $-$transform:

$x(z)=\frac{-1\mathrm{.}5z^4-3z^3-0\mathrm{.}5z^2-0\mathrm{.}5z}{z^4+3z^3+6\mathrm{.}25z^2+4\mathrm{.}5z+1}$

Find the poles and zeros. Hence, determine the ROC.

Find $x(n)$ and specify the causal and no$-$causal parts of $x(n).$ Hence plot it$.$ item Determine the energy of the signal.

Plot the magnitude and phase spectra of the signal.