$4.$ Radome Design $[30$pts$].$ A radar dome aka. radome, is a protective dielectric enclosure for microwave and mm$-$wave antenna systems. Microwave landing systems $($MLS$),$ help airplanes land at all weathers and the MLS antenna is enclosed in a radome. The MLS center operating frequency is $5$ GHz and the radome is fabricated by a thermoplastic PEM material which can be assumed lossless, non$-$magnetic with $\backslash (\backslash$epsilon$\_\{$r$\}\backslash$approx $3\backslash ).$

$($a$)$ Assuming that the radome is flat and planar, design for the minimum thickness d of the radome that will produce no reflection at normal incidence at $5$ GHz $.$

$($b$)$ What is the frequency bandwidth $\backslash ($ B W$=$f$\_\{$H$\}-$f$\_\{$L$\}\backslash ),$ where $\backslash ($ f$\_\{$H$\}\backslash )$ and $\backslash ($ f$\_\{$L$\}\backslash )$ are the highest and lowest frequencies that the radome permits $\backslash (90\backslash \%\backslash )$ of the normality incident TEM wave power to be transmitted?

$($c$)$ Write a Matlab scrip that plots the magnitude of the normal incidence reflection and transmission coefficient versus frequency, for a frequency range larger from $\backslash ($ f $\backslash$in$\backslash$left$[0\mathrm{.}5$ f$\_\{$L$\},2$ f$\_\{$H$\}\backslash$right$]\backslash ).$

$($d$)$ If the incident wave is a linear polarized TEM wave at $5$ GHz with oblique incidence, what are the range of incident angles that the radome permits $\backslash (90\backslash \%\backslash )$ of incident power to be transmitted at the plane of incidence, perpendicular to the plane of incidence and $45$deg from the plane of incidence?

$($e$)$ Write a Matlab scrip that plots the magnitude of the total reflection and transmission coefficient versus incident angle at $5$ GHz $.$ You could use a surf or contourf plot in Matlab, where the $\backslash ($ x $\backslash )$ and $\backslash ($ y $\backslash )$ axes correspond to the indecent angles $\backslash (\backslash$theta $\backslash$in$\backslash$left$[0^\{\backslash$circ$\},60^\{\backslash$circ$\}\backslash$right$]\backslash )$ at the plane of incidence and orthogonal to the plane of incidence, respectively; and the z$-$ axis $($or contourf colors$)$ represent the magnitude of the reflection coefficient. Use a separate plot to plot the magnitude of the transmission coefficient. Verify that the power balance holds at all angles.

$($f$)$ If you goal was to re$-$design the radome for increased frequency bandwidth, and wider angle operation, while maintaining the same design specification of $\backslash (90\backslash \%\backslash )$ transmittance, suggest a possible design alternative. For this part you do not need to do a qualitative analysis and design, an intuitive$/$qualitatively justification and reasoning would suffice.