Homework Help: Questions and Answers: Consider a data link layer using a 7-bit Hamming code for error detection and correction. A transmitted codeword is 1101010. After receiving the codeword, the receiver detects that there is an error.
a) What is the purpose of using a Hamming code in the link layer?
b) Identify the position of the error in the received codeword using the Hamming code mechanism.
c) Correct the error and provide the correct codeword.
Answer:
Let’s solve this problem step by step using the Hamming code mechanism.
a) Purpose of using Hamming code in the link layer:
The primary purpose of using Hamming code in the data link layer is to detect and correct errors during data transmission. It helps in ensuring the integrity and reliability of the transmitted data by identifying and correcting single-bit errors.
Hamming codes can detect up to two errors and correct single-bit errors in the received data. The redundancy in the form of parity bits helps in identifying the erroneous bit and allows the receiver to correct it without needing retransmission.
b) Identify the position of the error in the received codeword using the Hamming code mechanism:
Let’s start by understanding how Hamming codes work in this case. A 7-bit Hamming code includes both data bits and parity bits. We can use the following steps to identify the error position:
1. Identify positions of parity bits: In a 7-bit Hamming code, the bits are arranged such that the positions that are powers of 2 are reserved for parity bits.
- Bit positions:
1,2,4 - Data bit positions:
3,5,6,7So the structure is as follows:
Position: 1 2 3 4 5 6 7
Bit: P1 P2 D3 P4 D5 D6 D7
2. Received codeword: 1101010
Let’s map the bits to the positions:
Position: 1 2 3 4 5 6 7
Bit: 1 1 0 1 0 1 0
3. Calculate parity for each parity bit:
P1 (Position 1) covers bits 1, 3, 5, 7.
- Calculating: 1 ⊕ 0 ⊕ 0 ⊕ 0 = 1
- Expected parity: 0 (since parity should make the total even).
- P1 error: yes (because calculated parity is 1, but expected is 0).
P2 (Position 2) covers bits 2, 3, 6, 7.
- Calculating: 1 ⊕ 0 ⊕ 1 ⊕ 0 = 0
- Expected parity: 1.
- P2 error: no.
P4 (Position 4) covers bits 4, 5, 6, 7.
- Calculating: 1 ⊕ 0 ⊕ 1 ⊕ 0 = 0
- Expected parity: 1.
- P4 error: yes.
4. Error Position Calculation: The error position is determined by the binary value formed by the results of the parity checks. If there is an error in P1, P2, or P4, their respective bit values contribute to the position of the error:
- P4 = 1, P2 = 0, P1 = 1 → (101)2 = 5
Therefore, the error is at position 5.
c) Correct the error and provide the correct codeword:
Since the error is at position 5, we need to flip the bit at that position.
- Original codeword:
1101010 - Flip the 5th bit (from 0 to 1):
1101110
Thus, the correct codeword is: 1101110.
Final Answer:
a) The purpose of using Hamming code is for error detection and correction in the data link layer.
b) The error is at position 5.
c) The correct codeword is 1101110.
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