Assume that a source data value 1001 was received at a destination as 1101. Show how Hamming’s Error-Correcting code will be appended to source data so this error of one bit is identified and corrected at the destination. You may assume that the transmission error occurs in the source data, not the error correction code

Hamming’s Error-Correcting Code (ECC) can correct single-bit errors in transmitted data.

The basic idea is to add extra bits (parity bits) to the original data in such a way that the resulting code can detect and correct errors.

Let’s use Hamming(7,4) code as an example, which adds 3 parity bits to a 4-bit data word, resulting in a 7-bit codeword. The positions of these parity bits are chosen to cover certain combinations of data bits.

Steps to Append Hamming’s ECC:

1. Original Data (4 bits): 1001
2. Calculate Parity Bits: The parity bits are usually calculated based on the positions of the data bits. In Hamming(7,4) code, the positions of the parity bits are powers of 2 (1, 2, 4). Let’s denote the original data bits as D1, D2, D3, D4 (left to right).

• Parity bit P1 covers D1, D2, D4.
• Parity bit P2 covers D1, D3, D4.
• Parity bit P4 covers D2, D3, D4. To calculate each parity bit, we count the number of 1s in the positions it covers and set the parity bit to 1 if the count is odd, and 0 otherwise. For example:
• P1 = D1 ⊕ D2 ⊕ D4 = 1 ⊕ 0 ⊕ 1 = 0 (Even parity)
• P2 = D1 ⊕ D3 ⊕ D4 = 1 ⊕ 0 ⊕ 1 = 0 (Even parity)
• P4 = D2 ⊕ D3 ⊕ D4 = 0 ⊕ 0 ⊕ 1 = 1 (Odd parity)

1. Append Parity Bits to Data: The parity bits are inserted at their respective positions in the codeword. So, the resulting 7-bit codeword is: 0111001

Transmission and Error Detection/Correction:

Now, let’s assume that during transmission, one bit is flipped. The received data becomes 1101001.

1. Check Parity Bits: The receiver recalculates the parity bits based on the received data.

• P1′ = D1′ ⊕ D2′ ⊕ D4′ = 1 ⊕ 1 ⊕ 1 = 1 (Odd parity)
• P2′ = D1′ ⊕ D3′ ⊕ D4′ = 1 ⊕ 0 ⊕ 1 = 0 (Even parity)
• P4′ = D2′ ⊕ D3′ ⊕ D4′ = 0 ⊕ 0 ⊕ 1 = 1 (Odd parity)

1. Identify the Error: By comparing the calculated parity bits with the received parity bits, the receiver can identify the position of the error. In this case, the error occurs in position 1 (P1), as the calculated parity bit P1′ is different from the received parity bit.
2. Correct the Error: The receiver corrects the error by flipping the bit in the detected position. So, the corrected data becomes: 1001

Thus, using Hamming’s Error-Correcting Code, the receiver can identify and correct the error that occurred during transmission.