Task $2.$ Let's compare the $4$G LTE and $5$G NR systems from a Doppler$-$shift perspective and study the significance of a scalable subcarrier spacing in $5$G NR$.$ In LTE, the subcarrier spacing of an OFDM transmission is fixed to $15$ kHz $,$ while in $5$G NR the subcarrier spacing is scalable: in a basic situation one can choose values $\backslash (15\backslash$mathrm$\{$kHz$\},30\backslash$mathrm$\{$kHz$\}\backslash )$ ja $\backslash (\backslash$mathbf$\{60$ ~ k H z$\}\backslash )$ for carrier frequencies below $6$ GHz $,$ and values $\backslash (\backslash$mathbf$\{60$ ~ k H z$\},120\backslash$mathrm$\{$kHz$\}\backslash )$ ja $240$ kHz for higher frequencies. Moreover, let's consider that the LTE works at the $800$ MHz band and $5$ G NR at $3\mathrm{.}5$ GHz band and $28$ GHz $($millimeter wave$)$ band.

a$)$ Compute the maximum Doppler$-$shift $($in Hz$)$ for LTE ja $5$G NR bands by assuming four different mobile speeds: $\backslash (4\backslash$mathrm$\{$~km$\}/\backslash$mathrm$\{$h$\}\backslash )($pedestrian$),\backslash (50\backslash$mathrm$\{$~km$\}/\backslash$mathrm$\{$h$\}\backslash )($urban vehicular$),120\backslash (\backslash$mathrm$\{$km$\}/\backslash$mathrm$\{$h$\}\backslash )($highway vehicular$)$ ja $\backslash (300\backslash$mathrm$\{$~km$\}/\backslash$mathrm$\{$h$\}\backslash )($high$-$speed train$)$

b$)$ A fundamental requirement for using OFDM is that the channel remains $($practically$)$ constant during one OFDM symbol. Compute the channel coherence time $(50\backslash \%$ correlation$)$ for different speeds and frequencies.

c$)$ Compute the duration of an OFDM symbol $($without guard times$)$ for different subcarrier spacings $($both LTE and $5$G NR$)$ and compare the symbol duration with the channel coherence time for different speeds$/$frequencies$.$ Is the duration of OFDM symbols always smaller than the channel coherence time with different speeds$/$frequencies$?$ For example, would $5$G NR at $28$ GHz work properly with $15\backslash (\backslash$mathbf$\{$k m$\}/\backslash$mathbf$\{$h$\}\backslash )?$