Question $2(70$ pts$)$

In Matlab, the trellis diagram for the $1$ st row of Table $7\mathrm{.}4($Constraint Length, $K=3$ and rate $\frac{1}{2}$ code$)$

can be stated as follows,

trellis $=$ poly$2$trellis $(3,{\{}^{\text{'}}1+x+x^{{?}^{{?}^{?}}}{2}^{\text{'}},{?}^{\text{'}}1+x^{{?}^{{?}^{?}}}{2}^{\text{'}}\})$

a$)(10$ pts$)$ Plot the simulated BER for the coded BPSK system for rate $\frac{1}{2}$ convolutionally code

with constraint lengths $K=3,K=5$ and $K=7$ respectively. You can generate $5$ Million

information bits, $\frac{E_b}{N_0}$ varies between $2dB$ to $8dB,$ and set the zero mean Gaussian channel

noise variance is $1.$

b$)(20$ pts$)$ For Hard decision decoding $($HDD$),$ plot the coded BER for each value of $K$ and its

BER upperbound. You can use "bercoding" command to compute the coded BER

upperbound. For a given $\frac{E_b}{N_0},$ is it a good idea to use the BER upperbound when the simulated

BER is not available? Comment on your results. To compute the upperbound, you can use

"distspec$($trellis$,10)".$

c$)(10$ pts$)$ Plot execution time of the Viterbi decoder for $5$ million information bits for each

value of $K$ given in Q$2,$ part $($a$).$ Hint: You can use tic, toc commands or see etime function.

d$)(30$ pts$)$ Now you should seek the performance of soft decision decoding $($SDD$)$ instead of

HDD$.$ You should partition the interval between $-1$ and $+1$ into $L$ intervals.

As an example, for $L=4,$ you can set the decision thresholds as $-0\mathrm{.}500\mathrm{.}5$ or for $L=8,$ you

can set the decision thresholds as $[-0\mathrm{.}75-0\mathrm{.}5-0\mathrm{.}2500\mathrm{.}250\mathrm{.}50\mathrm{.}75$ and so on$.$ Let $s_i$ be a soft

decision $2$ bit $(L=4$ level$)$ sample, then $s_i$ is obtained from the received coded BPSK symbol

$r_i$ from,

$s_i=\{\begin{array}{cc}0,& r_i-0\mathrm{.}5\\ 1,& -0\mathrm{.}50\mathrm{.}5\end{array}$

To obtain the quantized soft decisions from the received coded BPSK symbolds, you can use

the function "quantiz". Then, for a rate $\frac{1}{2}$ convolutional code with $K=3,$ obtain the BER and

BER upperbound for SDD$.$ Also compare your results with BER and its upperbound for HDD$.$

Comment on your results.