$2)$ Determine and sketch an appropriate channel model for two$-$path ionospheric propagation, where the relative time delay between the two received signal paths is $0\mathrm{.}3$ msec and the transmitted signal bandwidth W is $10$ kHz$.(5\backslash \%)$

$3)$ Suppose that a communication channel with a nominal bandwidth allocation of $5$ kHz is to be used for transmitting digital information at a rate of $\backslash (9600\backslash$mathrm$\{$bits$\}/\backslash$mathrm$\{$sec$\}\backslash ).$ The channel multipath spread is $\backslash ($ T$\_\{$m$\}=10\backslash$mathrm$\{$msec$\}\backslash ).$ Specify a modulation method for achieving the desired data rate and indicate whether or not an equalizer is necessary at the receiver for the intersymbol interference. $(5\backslash \%)$

$4)$ A communication system employs dual antenna diversity and binary orthogonal FSK modulation. The received signals at the two antennas are

$\backslash [$

$\backslash$begin$\{$array$\}\{$l$\}$

r$\_\{1\}($t$)=\backslash$alpha$\_\{1\}$ s$($t$)+$n$\_\{1\}($t$)\backslash \backslash$

r$\_\{2\}($t$)=\backslash$alpha$\_\{2\}$ s$($t$)+$n$\_\{2\}($t$)$

$\backslash$end$\{$array$\}$

$\backslash ]$

where $\backslash (\backslash$alpha$\_\{1\}\backslash )$ and $\backslash (\backslash$alpha$\_\{2\}\backslash )$ are statistically i$.$i$.$d$.$ Rayleigh random variables, and $\backslash ($ n$\_\{$l$\}($t$)\backslash )$ and $\backslash ($ n$\_\{2\}($t$)\backslash )$ are statistically independent, zero$-$mean white Gaussian random processes with power$-$spectral density $\backslash ($ N$\_\{0\}/2\backslash$mathrm$\{$~W$\}/\backslash$mathrm$\{$Hz$\}\backslash ).$

The two signals are demodulated, squared, and then combined prior to detection.

a$)$ Sketch the functional block diagram of the entire receiver including the demodulator, the combiner, and the detector. $(5\backslash \%)$

b$)$ Plot the probability of error for the detector and compare the result with the case of no diversity. $(5\backslash \%)$

$5)$ The SNR required at the detector to achieve reliable communication in a DS spread$-$spectrum communication system is $15$ dB $.$ If the interference$-$to$-$signal power at the receiver is $20$ dB $,$ determine the processing gain required to achieve reliable communication. $(5\backslash \%)$

Question III: $[25\backslash \%]$

$1)$ For a signal with Rayleigh$-$distributed amplitude, what is the probability that the received signal power is at least $6$ dB below the mean power. $(5\backslash \%)$

$2)$ Compute the probability that the output power of a selection diversity system is $5$ dB lower than the mean power of each branch, when using $\backslash (\backslash$mathrm$\{$Nr$\}=2\backslash )$ antennas. $(5\backslash \%)$

$3)$ Consider a DS spread$-$spectrum system, with a signal wave and PN code as shown below, determine the spread code. Determine the chip interval if the bit interval is $4$ msec $.(5\backslash \%)$

$4)$ For a Code Division Multiple Access $($CDMA$)$ system, where each user transmits data spread over $4$ chips, and each user utilizes a code sequence consisting of $1\text{'}$s and $-1\text{'}$s$,$ show how a receiver can decode the signals for two users $(\backslash (\backslash$mathrm$\{$U$\}\_\{1\}\backslash )$ and $\backslash (\backslash$mathrm$\{$U$\}\_\{2\}\backslash ))$ using their respective spreading codes. $(10\backslash \%)$

Good Luck!