Problem 30 - Advanced Engineering Mathematics

Question

(a) In the Hamming \((8,4)\) code a word $$ \mathbf{W}=\left(w_{1} w_{2} w_{3} w_{4}\right) $$ of length four is transformed into a code word of length eight: $$ \mathbf{C}=\left(c_{1} c_{2} c_{3} w_{1} c_{4} w_{2} w_{3} w_{4}\right), $$ where the parity check equations are $$ \begin{aligned} &c_{4}+w_{2}+w_{3}+w_{4}=0 \\ &c_{3}+w_{1}+w_{3}+w_{4}=0 \\ &c_{2}+w_{1}+w_{2}+w_{4}=0 \\ &c_{1}+c_{2}+c_{3}+w_{1}+c_{4}+w_{2}+w_{3}+w_{4}=0. \end{aligned} $$ Encode the word \(\left(\begin{array}{lll}0 & 1 & 1 & 0\end{array}\right)\). (b) From the system in part (a), determine the parity check matrix \(\mathbf{H}\). (c) Using the matrix \(\mathbf{H}\) from part (b), compute the syndrome S of the received message $$ \mathbf{R}=\left(\begin{array}{lllllll} 0 & 0 & 1 & 1 & 1 & 1 & 0 & 0 \end{array}\right) . $$

Step-by-Step Solution

Answer

(a) Encoded word: (0, 1, 1, 0, 0, 1, 1, 0). (b) H matrix determined. (c) Syndrome: (1, 1, 0, 0).

Step 1: Encode the Word (0, 1, 1, 0)

The word \(\mathbf{W} = (0, 1, 1, 0)\) needs to be encoded into an 8-bit code word. First, apply the parity check equations to find the values of \(c_4, c_3, c_2,\) and \(c_1\).1. \(c_4 + 1 + 1 + 0 = 0 \Rightarrow c_4 = 0\)2. \(c_3 + 0 + 1 + 0 = 0 \Rightarrow c_3 = 1\)3. \(c_2 + 0 + 1 + 0 = 0 \Rightarrow c_2 = 1\)4. \(c_1 + 1 + 1 + 0 + 0 + 1 + 1 + 0 = 0 \Rightarrow c_1 = 0\)Thus, the encoded word \(\mathbf{C}\) is \((0, 1, 1, 0, 0, 1, 1, 0)\).

Step 2: Determine the Parity Check Matrix H

The parity check matrix \(\mathbf{H}\) can be derived from the parity check equations given for the Hamming code:\[\mathbf{H} = \begin{bmatrix}1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 \1 & 0 & 0 & 1 & 0 & 0 & 1 & 1 \0 & 1 & 0 & 1 & 0 & 1 & 0 & 1 \0 & 0 & 1 & 1 & 1 & 0 & 0 & 1\end{bmatrix}\]

Step 3: Compute the Syndrome S from Received Message

To find the syndrome \(\mathbf{S}\) for a received message \(\mathbf{R} = (0, 0, 1, 1, 1, 1, 0, 0)\), multiply \(\mathbf{R}\) by \(\mathbf{H}^T\) (the transpose of \(\mathbf{H}\)):1. First row: \(0 + 0 + 1 = 1\)2. Second row: \(0 + 1 + 1 + 1 = 3 = 1 \mod 2\)3. Third row: \(0 + 0 + 1 + 1 = 2 = 0 \mod 2\)4. Fourth row: \(1 + 1 + 0 = 2 = 0 \mod 2\)The syndrome \(\mathbf{S} = (1, 1, 0, 0)\).

Key Concepts

Parity Check Matrix

In Hamming codes, such as the \(8,4\) Hamming code, the Parity Check Matrix (denoted as \(\mathbf{H}\)) is crucial. \(\mathbf{H}\) helps ensure that the code can detect and correct errors. This matrix is derived from the parity check equations, which are specific rules used to compose the Hamming code. In our exercise, given the parity check equations:

• \(c_{4} + w_{2} + w_{3} + w_{4} = 0\)
• \(c_{3} + w_{1} + w_{3} + w_{4} = 0\)
• \(c_{2} + w_{1} + w_{2} + w_{4} = 0\)
• \(c_{1} + c_{2} + c_{3} + w_{1} + c_{4} + w_{2} + w_{3} + w_{4} = 0\)

\(\mathbf{H}\) was formed by identifying the dependencies of each bit. Each row in \(\mathbf{H}\) corresponds to one equation, where \(1\) indicates the bit's presence. For example, the first row of \(\mathbf{H}\) \[ \begin{bmatrix} 1 \ 1 \ 1 \ 0 \ 0 \ 0 \ 0 \ 0 \end{bmatrix} \] indicates that the first parity condition involves the first three bits. Understanding this matrix is vital, not only for encoding but also for decoding and error correction.

Syndrome Analysis

Syndrome analysis is essential in error detection and correction. Once a message is received, syndrome analysis helps determine if any errors occurred during transmission. It involves multiplying the received message \(\mathbf{R}\) by the transpose of the parity check matrix \(\mathbf{H}^T\). This operation checks for inconsistencies that could indicate errors. For the received message \(\mathbf{R} = (0, 0, 1, 1, 1, 1, 0, 0)\), this is performed as follows:

• Compute each row by taking the dot product of \(\mathbf{H}^T\) and \(\mathbf{R}\). If there's a discrepancy, it will appear here.
• If the syndrome vector \(\mathbf{S}\) is non-zero, an error is signaled, and its value helps identify the specific error location.

In our case, the resulting syndrome \(\mathbf{S} = (1, 1, 0, 0)\) indicates the presence of a single bit error. Identifying such syndromes ensures that we catch errors and maintain integrity.

Error Detection and Correction

Hamming codes are renowned for their ability to detect and correct errors swiftly and efficiently. This is achieved by using the syndrome derived from the parity check matrix. When the syndrome is zero, we know the message is likely error-free. However, a non-zero syndrome indicates an error, and its specific pattern points to the location of the error.Hamming codes typically detect single-bit errors and can correct them because of this unique mapping system. Let’s break it down:

• The parity check equations allow us to identify which bit in the code word might be incorrect.
• With the matrix, any alteration to a single bit causes the syndrome to reflect those changes, thus enabling correction.
• Interpreting the syndrome pattern, like \(\mathbf{S} = (1, 1, 0, 0)\), can directly locate the bit needing correction.

This method is fundamental in digital communications, where maintaining message accuracy is crucial. The instant detection and correction ability make Hamming codes a practical choice for error management.