Problem $4[15$ marks$]$ Consider a communication problem where we wish to transmit the word "SMILE"

using the Pulse Amplitude Modulation $($PAM$)$ scheme and then decode it at the receiver $.$ This is done as

follows:

The individual text letters are ASCII encoded to get a corresponding binary word. For example, the

ASCII encoded $8-$bit binary word corresponding to letter $"S"$ is $"01010011".$

The bits in the binary word are grouped together to form symbols. Each symbol contains $lo{g}_{2}M$ bits,

where $M$ is a power of $2.$ For instance, if $M=4,$ the binary word $"01010011"$ will result in $4$ symbols $-$

$01,01,00,11.$

Each symbol is mapped to a different amplitude level. For example, if we choose the mapping $00$

$A_0,01A_1,10A_2$ and are amplitude levels$),$ then letter $"S"$ corresponds

to the amplitude level sequence $A_1,A_1,A_0,A_3.$

The amplitude levels are transmitted over a noisy channel.

The receiver's goal is to decode the noisy amplitude levels back to letters.

The above scheme can be modeled as the following $M-$ary hypothesis testing for detecting each of the

transmitted amplitude levels corrupted in noise,

$H_i$:$y=A_i+w,i=0,1,$dots,$M-1$

where $wN(0,{}^2),$ and the amplitude levels $A_1,A_2,$cdots,$A_m$ are given. We wish to distinguish between

hypothesis $H_0,H_1,$cdots,$H_{M-1}$ using the observation $y.$ Assume uniform costs and equal priors.

$($a$)$ Compute the minimum probability of error $(P_e)$ detector and show that it corresponds to the minimum

distance detector.

$($b$)$ Compute the minimum $P_e$ when $M=2.$

$($c$)$ Use the detector computed in $($a$)$ to decode the transmitted message "SMILE" from the measurement

data provided in Problem$4.$mat file. The file contains two sets of measurement data:Question $4.[15$ marks$]$ Consider a random DNA sequence of nucleotides stored in FASTA format in the file randomDNA.fasta. Write a Matlab script file to $-$ read this file, $-$ convert the $\text{'}\backslash ($ a $\backslash )\text{'}$ and $\text{'}\backslash ($ g $\backslash )\text{'}$ in the sequence to $\backslash (-1\backslash )$ and the $\text{'}\backslash ($ c $\backslash )\text{'}$ and $\text{'}\backslash ($ t $\backslash )\text{'}$ to $\backslash (+1\backslash ),-$ construct a DNA random walk by performing the cumulative sum of the $\backslash (-1\backslash )$ and $\backslash (+1\backslash ),$ and $-$ plot the DNA walk.