$7.$ A transfer function, $\backslash (\backslash$mathrm$\{$H$\}(\backslash$mathrm$\{$z$\})\backslash ),$ has one zero at the origin of the complex z$-$plane, one zero at $0\mathrm{.}72$ on the real axis, and a pair of complex conjugate poles at $\backslash (0\mathrm{.}72\backslash$pm $\backslash$mathrm$\{$j$\}0\mathrm{.}96\backslash ).$

a$.$ Draw a pole$-$zero plot of the transfer function.

b$.$ Sketch $($without MATLAB$)$ the magnitude vs$.$ frequency plot of the system,

c$.$ Write the transfer function, $\backslash ($ H$($z$)\backslash ).($See Hint.$)$

d$.$ Use a table of z $-$transforms to write the discrete$-$time function $\backslash (\backslash$mathrm$\{$h$\}[\backslash$mathrm$\{$n$\}]\backslash ).$

e$.$ Write the difference equation for the impulse response of the system.

f$.$ Given a sampling frequency of $16$ kHz $,$ calculate the frequency of oscillation of the system.

Maanitude