Question $1$

$($a$)$ Given an even$-$parity $(4,3)$ block code:

explain how to find the minimum Hamming distance and minimum Hamming

weight, and what the values are of the minimum Hamming distance and minimum

Hamming weight for this block code.

$($b$)$ Given the $(6,5)$ single$-$parity$-$check code with odd parity, find the probability of

successfully detecting erroneously received words using binomial distribution,

when the binary symmetric channel error probability is $f=0\mathrm{.}3.$ Justify your

reasoning.

$($c$)$ If a packet can accommodate $252$ bits, there are two coding schemes:

Scheme A encodes a sequence of bits using a $(63,57)$ Hamming code.

Scheme B segments a sequence of bits into blocks and encodes each block

using a $(7,4)$ Hamming code.

Compute the throughput of information bits and the coding rate for both schemes.

Justify the advantages and disadvantages between these two schemes.

$($d$)$ If four$-$level Pulse Amplitude Modulation $(4-$PAM$)$ is used to group bits in pairs

and map each pair to one of four possible levels or symbols and $+3$

Volts$),$ complete the following task:

$($i$)$ Draw an abstract graph showing the four symbols of $4-$PAM, the probability

density function associated to each symbol and the optimal decision

thresholds at the receiver, when the signal is transmitted over a AWGN

channel with zero$-$mean and the variance ${}^2=1.$

$($ii$)$ Obtain a mathematical expression for the symbol error rate $($SER$)$ of $4-$PAM

when ${}^2$ is the variance of AWGN. Assume that the four symbols are

equiprobable. Justify each step of your work.