# (f) Assume that a source data value 1011 was received at a destination as 1010. Show how Hamming's Error-Correcting code bits will be appended to source data to identified and correct the error of one bit at the destination. You may assume that transmission error occurs only in the source data and not the source parity bits.

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Hamming Code is a type of error-correcting code that adds extra parity bits to the data to detect and correct errors. Let's assume the source data is 4 bits (1011), and we want to use Hamming Code to identify and correct a single-bit error.

Source Data (4 bits):

$D_1D_2D_3D_4 = 1011$

Hamming Code Construction:

We will add three parity bits (P1, P2, P4) to the data to create a 7-bit codeword (H1H2D1H3D2D3D4).

1. Insert Data Bits into Hamming Code:

$H_1H_2D_1H_3D_2D_3D_4$

$1?1?1011$

2. Calculate Parity Bits:

P1 (Parity for positions 1, 2, 4): Calculate the parity for bits in positions 1, 2, 4 (including parity bits themselves).

$P_1 = D_1 \oplus D_2 \oplus D_4 = 1 \oplus 0 \oplus 1 = 0$

- **P2 (Parity for positions 1, 3, 4): Calculate the parity for bits in positions 1, 3, 4 (including parity bits themselves).

$P_2 = D_1 \oplus D_3 \oplus D_4 = 1 \oplus ? \oplus 1 = ?$

- P4 (Parity for positions 2, 3, 4): Calculate the parity for bits in positions 2, 3, 4 (including parity bits themselves).

$P_4 = D_2 \oplus D_3 \oplus D_4 = 0 \oplus ? \oplus 1 = ?$

3. Insert Parity Bits into Codeword:

$1101?1011$

Result:

The Hamming Code for the source data 1011 is 1101011. The question marks represent the calculated parity bits.

Single-Bit Error at Destination:

If an error occurs during transmission, let's say the received codeword is 1100011, indicating an error in bit 3.

Syndrome Calculation:

Calculate the syndrome by comparing received parity bits with recalculated parity bits.

$S_1 = D_1 \oplus D_2 \oplus D_4 \oplus P_1 = 1 \oplus 0 \oplus 1 \oplus 0 = 0$

$S_2 = D_1 \oplus D_3 \oplus D_4 \oplus P_2 = 1 \oplus ? \oplus 1 \oplus ? = ?$

$S_4 = D_2 \oplus D_3 \oplus D_4 \oplus P_4 = 0 \oplus ? \oplus 1 \oplus ? = ?$

Error Correction:

The non-zero syndrome indicates an error. The position of the non-zero syndrome corresponds to the bit with the error. In this case, $$S_1 \neq 0$$, so bit 3 is in error.

Correct the error by flipping the bit at the error position:

$1100011 \rightarrow 1101011$

Now, the corrected codeword matches the original Hamming Code for the source data 1011.