Assignment #3

ECE 487 (Data Communications Networks)

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Your Student ID: ________________________

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1. For the following data bits organized in two rows and three columns, please add twodimensional parity-check bits, and give the corresponding codeword. Over the transmission

medium, the second bit in the second row is changed from bit ‘0’ to bit ‘1’. Please describe how

the receiver will process the received codeword. (4 points)

0 1 1

1 0 1

2. For the Hamming Code discussed in our Lecture 3, show how error correction is performed at

the receiver for the following cases. It is required your error detection and correction should be

based on the syndrome bits. (6 points)

i) Dataword: 0111; error pattern: 0100000

ii) Dataword: 0100; error pattern: 1000001

iii) Dataword: 0011; error pattern: 1101000

3. Consider a code (with 4 bits in a dataword and 7 bits in a codeword) using the following three

redundant bits:

r2 = a2 + a1 +a0 modulo-2

r1 = a3 + a1 +a0 modulo-2

r0 = a3 + a2 + a1 modulo-2

Note that the code could also be called a Hamming Code (not the same as the Hamming Code

used in our Lecture 3).

(a) How does the receiver calculate the three syndrome bits? (2 points)

(b) The receiver assumes there is at most one bit error in the received codeword. The

three-bit syndrome creates eight different bit patterns (“000” to “111”). For each bit

pattern, please indicate which bit (among the seven bits in the received codeword) the

receiver considers corrupted. (2 points)

Syndrome

S2S1S0

000 001 010 011 100 101 110 111

Corrupted

bit

(Question 4 is on the next page)

4. For the CRC code system with divisor “1011” discussed in Lecture 3, Denote the 4-bit

dataword at the sender as a3a2a1a0, and the 7-bit codeword at the sender as a3a2a1a0r2r1r0.

(i) Give the codeword for dataword “1101”. You are required to use division to get the

codeword. (2 points)

(ii) For the codeword in (i), if bits a2, a1, and a0 are corrupted during the transmission, give the

syndrome bits at the receiver. Will the receiver accept the received codeword? (2 points)

(iii) Repeat question (ii) if bits a1, r2 and r1 are corrupted during the transmission. (2 points)

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