Chapter 10 Problems

Assume that the probability that a bit in a data unit is corrupted during transmission is p. Find the probability that x number of bits are corrupted in an nbit data unit for each of the following cases. a.) n = 8, x = 1, p = .2 b.) n = 16, x = 3, p = .3 c.) n = 32, x = 10, p = .4

(n!/x!(n-x)!)(p^x )(1 - p)^n - x a.) .336 b.) .146 c.) .084

Prove that the code represented by the following codewords is not linear. You need to find only one case that violates the linearity.

10111 XOR 11111 does not give another codeword therefore the codewords are not linear

Traditional checksum calculation needs to be done in one's complement arithmetic. Computers and calculators today are designed to do calculations in two's complement arithmetic. One way to calculate the traditional checksum is to add the numbers in two's complement arithmetic, find the quotient and remainder of dividing the result by 2^16, and add the quotient and the remainder to get the sum in one's complement. The checksum can be found by subtracting the sum from 2^16 − 1. Use the above method to find the checksum of the following four numbers: 43,689, 64,463, 45,112, and 59,683.

43689 + 64463 + 45112 + 59683 = 212947 Divide by 2^16 until remainder becomes less than 2^16. Result is hex C029

Referring to the CRC-32 polynomial in Table 10.4, answer the following questions: a.) Does it detect a single error? Defend your answer. b.) Does it detect a burst error of size 16? Defend your answer. c.) What is the probability of detecting a burst error of size 33? d.) What is the probability of detecting a burst error of size 55?

CRC-32: x^32 + x^26 + x^23 + x^22 + x^16 + x^12 + x^11 + x^10 + x^8 + x^7 + x^5 + x^4 + x^2 + x + 1 a.) Yes, since it has multiple terms and a non zero coefficient for x^0 b.) Yes, since R = 32 > 16 c.) 1 - (1/2)^(r-1) = 99.99% chance 465 out of 1012 are not undected d.) 1 - (1/2)^r = 99.99% chance 233 out of 1012 are not undetected

What is the maximum effect of a 2-ms burst of noise on data transmitted at the following rates? a.) 1500 bps b.) 12 Kbps c.) 100 Kbps d.) 100 Mbps

Effect Bits = data rate * burst length a.) 1500(2*10^-3) = 3 bits effected b.) 12x10^3(2x10^-3) = 24 bits effected c.) 200 bits effected d.) 200000 bits effected

Manually simulate the Fletcher algorithm (Figure 10.18) to calculate the checksum of the following bytes: (2B)16, (3F)16, (6A)16, and (AF)16. Also show that the result is a weighted checksum.

In notebook, final CRC is 69D4

Manually simulate the Adler algorithm (Figure 10.19) to calculate the checksum of the following words: (FBFF)16 and (EFAA)16. Also show that the result is a weighted checksum.

In notebook, final CRC is E7C8EBB9

Referring to the CRC-8 polynomial in Table 10.7, answer the following questions:

Same as P10-9

Assume a packet is made only of four 16-bit words (A7A2)16, (CABF)16, (903A)16, and (A123)16. Manually simulate the algorithm in Figure 10.17 to find the checksum.

Step 1: Add all the values together to get 0002A3BE Step 2: Extract the leftmost digits and add them to the right most. 0002 + A3BE = A3C0. Stop here because no carry is generated Step 3: Complement the result to get the checksum (5C3F)

In Table 10.1, the sender sends dataword 10. A 3-bit burst error corrupts the codeword. Can the receiver detect the error? Defend your answer.

The answer depends heavily on what the codeword is corrupted to. If the codeword is corrupted to 010 for example the error will be detected because that codeword is invalid. If the codeword is corrupted to 000 however, the error will not be detected.

This problem shows a special case in checksum handling. A sender has two data items to send: (4567)16 and (BA98)16. What is the value of the checksum?

The value of the checksum is the complement of 4567 + BA98 which equals FFFF. Therefore the checksum value would be 0000.

Which of the following CRC generators guarantee the detection of a single bit error? a.) x^3 + x + 1 b.) x^4 + x^2 c.) 1 d.) x^2 + 1

To detect single bit errors, a CRC generator must have at least two terms and the coefficient of x^0 must be nonzero Generators a and d meet these criteria

Assuming even parity, find the parity bit for each of the following data units. a.) 1001011 b.) 0001100 c.) 1000000 d.) 1110111

a.) 0 b.) 0 c.) 1 d.) 0

Exclusive-OR (XOR) is one of the most used operations in the calculation of codewords. Apply the exclusive-OR operation on the following pairs of patterns. Interpret the results

a.) 00000 b.) 11100 c.) 01100

Although it can be formally proved that the code in Table 10.3 is both linear and cyclic, use only two tests to partially prove the fact: a.) Test the cyclic property on codeword 0101100. b.) Test the linear property on codewords 0010110 and 1111111.

a.) 0101100 left shifted = 1011000 which is a valid codeword in 10.3, therefore cyclic b.) 0010110 XOR 1111111 = 1101001 which is a valid codeword in 10.3, therefore linear

What is the Hamming distance for each of the following codewords? a.) d (10000, 00000) b.) d (10101, 10000) c.) d (00000, 11111) d.) d (00000, 00000)

a.) 1 b.) 2 c.) 5 d.) 0

Using the code in Table 10.2, what is the dataword if each of the following codewords is received?

a.) ERROR b.) ERROR c.) 0000 d.) 1101

Show how the following errors can be detected: a.) An error at (R3, C3). b.) Two errors at (R3, C4) and (R3, C6). c.) Three errors at (R2, C4), (R2, C5), and (R3, C4). d.) Four errors at (R1, C2), (R1, C6), (R3, C2), and (R3, C6).

a.) If the parity bits for the row and column don't match an error is detected. b.) Parity for columns 4 and 6 won't match allowing the error to be detected c.) Parity for rows 2 and 3 will corrupt d.) There is no way to detect these errors as they do not impact the parity

Referring to the CRC-8 in Table 5.4, answer the following questions: a.) Does it detect a single error? Defend your answer. b.) Does it detect a burst error of size 6? Defend your answer. c.) What is the probability of detecting a burst error of size 9? d.) What is the probability of detecting a burst error of size 15?

a.) Yes, it has more than 1 coefficient and the coefficient for x^0 is 1. b.) Yes, since r = 8 c.) 1-(1/2)^(r - 1) = 99.2% chance d.) 1-(1/2)^r = 99.6% chance

Apply the following operations on the corresponding polynomials: a.) (x^3 + x^2 + x + 1) + (x^4 + x^2 + x + 1) b.) (x^3 + x^2 + x + 1) − (x^4 + x^2 + x + 1) c.) (x^3 + x^2) × (x^4 + x^2 + x + 1) d.) (x^3 + x^2 + x + 1) / (x^2 + 1)

a.) x^4 + x^3 + 1 b.) x^4 + x^3 + 1 c.) X^7 + x^5 + x^4 + x^3 + x^6 + x^4 + x^3 + x^2 = x^7 + x^6 + x^5 + x^2 d.) ...

a.) What is the polynomial representation of 101110? b.) What is the result of shifting 101110 three bits to the left? c.) Repeat part b using polynomials. d.) What is the result of shifting 101110 four bits to the right? e.) Repeat part d using polynomials.

a.) x^5 + x^3 + x^2 + x b.) 101110000 c.) x^8 + x^6 + x^5 + x^4 d.) 10 e.) x

Given the dataword 101001111 and the divisor 10111, show the generation of the CRC codeword at the sender site (using binary division).

in notebook (remainder = 0110)