Question

1. Suppose we want to transmit the message 10011010 and protect it from errors using the CRC polynomial x²+1.
   1. (8 points) Encode the data bit sequence using the generator polynomial and give the code word.

2. (8 points) Using this polynomial, can all single-bit errors be detected? If not, give an example scenario of errors that goes undetected.

Answer 1

Solution:

i)Given polynomial is x^2 + 1

hence,key is: 101

the above key is the coefficient of polynomials ,for example " x^2 +x+1"  for this key is "111" i.e. 1.x^2 + 1.x+1

Data to be sent is: 10011010

Now let us see CRC solving,

1) First we need to append the k no. of zeroes to he given data

where, k=the degree of the given polynomial, in the above question we need to add 2 zeroes as degree is "2".

so,now the modified data to send is : 1001101000

2)Now,XOR the given modified data and key such that we can get the "code word"

Hence the code word is "11".

ii)

Forouzan states following:

A single-bit error is e(x)=x^i, where i is the position of the bit. If a single-bit error is caught, then e(x)=x^i is not divisible by g(x). (Note that when we say not divisible, we mean that there is a remainder.) If g(x) has at least two terms and the coefficient of x^0 is not zero (the rightmost bit is 1), then e(x) cannot be divided by g(x) and all single bits errors can be caught.

Since in the above polynomial,it has two terms an coefficient of x^0 is not zero instead it is "1", hence we can say that all single bit errors can be detected.

I hope ,i had given the answers u need with my knowledge,so if it is useful then it will be good.