Let the discrete$-$time end$-$to$-$end impulse response of a linearly modulated system sampled

at three times the symbol rate is

$[(4/3),...,(4/3)][(1)/2,(1)/4,12,$ f T f T j j j j j j $-=-+-+-+-1/2,0,/4,1/4,(12)/4,(3)/2]$

Assume the noise $()$ k at the output of the sampler is discrete$-$time AWGN.

a$)$ Let r $=-+[(34)....(34)]$ r k r k T and $=-+[(34)....(34)]$ k k T $.$ Find matrix U and

vector b such that r Ub $=+$ $.$

b$)$ Find a length $9$ ZF equalizer cZF where the desired data is b$[$k$].$

c$)$ Compute the noise enhancement factor.

d$)$ Express the channel as three parallel symbol rate channels $\{(),1,2,3\}$ F z i i $=$ and the

equalizer $\{(),1,2,3\}$ C z i i $=.$ Verify $3$i i i $=1$C z F z z $()()=-$d $,$ and specify what is d$.$

e$)$ Express the SNR at the output of the ZF equalizer, assuming $,$ and

$[|[]|]$ b k E $2=$ s $[|[]|]$ k N $2=$ o $.$ Find the symbol error probability at E N s o $/15$dB $=$ if $16$QAM is the

modulation.