Module $7$ Homework Assignment. $($Please Help With This Assignment$)$

What to Submit

On a piece of paper, solve all the problems below one by one using clear and unambiguous handwriting. Circle your final answer to each problem. Take a picture of your paper, name it $$crc$-$

homework$,$ save it as a PDF$,$ and submit it for the Module $7$ Homework Assignment. Problems $($Reference Section $6\mathrm{.}2\mathrm{.}3$ in the textbook$)$

Problem $1.$

Suppose we want to transmit the message $1011001001001011$ and protect it from errors using the CRC$-8$ generator $100000111.$

$1\mathrm{.}1.$ Determine the message that is transmitted.

$1\mathrm{.}2.$ Suppose the leftmost bit of the message is inverted due to noise on the transmission link. What is the result of the receiver$$s CRC calculation? How does the receiver know that an error has occurred?

Problem $2.$

The CRC algorithm as presented in this chapter, requires lots of bit manipulations. It is$,$ however, possible to do them by taking multiple bits at a time, via a table$-$driven method, that enables efficient software implementations of CRC$.$ I outline the strategy here for long division $3$ bits at a time $($See table below$)$; in practice, we would divide $8$ bits at a time, and the table would have $256$ entries.

Let the generator G $=1101.$ To build the table for G$,$ we take each $3-$bit sequence p$,$ append three trailing $0$s$,$ and then find the quotient q $=$ p$000$ G$,$ ignoring the remainder. The third column is the product of G $$ q $($note: do exclusive$-$or when adding bits up$),$ and the first $3$ bits should be equal p$.$

$2\mathrm{.}2$ Use the table you developed above to divide $101001011001100$ by G $=1101.$

IMPORTANT WARNING:

For each problem above, you MUST show your calculations like those shown in Figure $6\mathrm{.}7$ in the textbook to justify your answer. Significant points will be deducted otherwise. $2\mathrm{.}1.$ Fill in the missing entries in the following table $($I did two rows for you already.$)$

$\backslash$begin$\{$tabular$\}\{|$c$|$l$|$c$|\}$

$\backslash$hline$\backslash ($ p $\backslash )$ & $\backslash ($ q$=$p $000\backslash$div G $\backslash )$ & $\backslash ($ G $\backslash$times q $\backslash )\backslash \backslash$

$\backslash$hline $000$ & $000$ & $000000\backslash \backslash$

$\backslash$hline $001$ & & $\backslash \backslash$

$\backslash$hline $010$ & & $\backslash \backslash$

$\backslash$hline $011$ & & $101110\backslash \backslash$

$\backslash$hline $100$ & & $\backslash \backslash$

$\backslash$hline $101$ & $110$ & $\backslash \backslash$

$\backslash$hline $110$ & & $\backslash \backslash$

$\backslash$hline $111$ & & $\backslash \backslash$

$\backslash$hline

$\backslash$end$\{$tabular$\}$

$2\mathrm{.}2$ Use the table you developed above to divide $101001011001100$ by $\backslash ($ G$=1101\backslash ).$

Hint: The first $3$ bits of the dividend are $\backslash ($ n$=101\backslash ).$ so from the table. the correspondina first $3$ bits of the