Question

A 12-bit Hamming code whose hexadecimal value is \(0 \mathrm{xE} 4 \mathrm{~F}\) arrives at a receiver. What was the original value in hexadecimal? Assume that not more than 1 bit is in error.

Short answer

The original value was 0xBD.

Step-by-step solution

Understand Hamming Code

Hamming code is a method for error detection and correction using redundant bits, called parity bits, inserted into data. With a 12-bit code, some bits are parity bits, and the rest are data bits.

Convert Hexadecimal to Binary

Convert the hexadecimal value 0xE4F to binary. Hex E is 1110, 4 is 0100, and F is 1111. Therefore, the 12-bit binary representation of 0xE4F is 1110 0100 1111.

Identify Parity Bits

In a 12-bit Hamming code configuration, identify positions for parity bits, commonly at positions that are powers of two: 1, 2, 4, and 8. The layout will be as follows: _ _ data _ data data data _ data data data data, where underscores (_) are parity bits.

Check Parity Bits

Calculate the parity for each parity bit position. For Hamming code, parity should cover specific bits: - Parity at position 1 (covers bits 1, 3, 5, 7, 9, 11): 1 XOR 1 XOR 0 XOR 0 XOR 1 = 1 - Parity at position 2 (covers bits 2, 3, 6, 7, 10, 11): 1 XOR 0 XOR 0 XOR 0 XOR 1 = 0 - Parity at position 4 (covers bits 4, 5, 6, 7, 12): 0 XOR 1 XOR 0 XOR 0 XOR 1 = 1 - Parity at position 8 (covers bits 8 to 12): 1 XOR 1 XOR 1 XOR 0 = 1 Add these binary parities to find the syndrome, which is 1011 (binary).

Analyze the Syndrome

The syndrome indicates which bit is incorrect. If no error exists, the syndrome is 0000; if a bit error exists, the syndrome is non-zero. The syndrome 1011 (binary) equals 11 (decimal), pointing to bit position 11 as incorrect.

Correct the Error

With bit 11 identified as incorrect, flip it from 1 to 0. The corrected 12-bit binary becomes 1110 0100 1101.

Extract Data Bits

Remove the parity bits from the corrected binary 1110 0100 1101, leaving only the data bits 1011 1101.

Convert Binary Data to Hexadecimal

Convert the extracted binary data 1011 1101 to hexadecimal. 1011 is B in hexadecimal, and 1101 is D. Therefore, the original hexadecimal value was 0xBD.

Key concepts

Error Detection

Error detection is the process of identifying errors within a data sequence. This is crucial when data is transmitted from one point to another, as errors can occur due to noise within the communication channel.

In communication systems, certain algorithms or codes are used to detect errors. Hamming Code is one such method, which integrates parity bits into data to help identify and correct errors. If an error is detected, the system can either request a retransmission or, in some cases like Hamming Code allows, attempt to correct the error itself.

The efficiency of error detection greatly affects the reliability and stability of data communication, ensuring that data integrity is preserved across transmissions. Hamming Codes are particularly useful for their ability to not only detect a single error within the data but also provide a way to pinpoint its location for correction.

Error Correction

Error correction is a continuation of error detection that not only identifies the presence of errors but also locates and corrects them. Hamming Code takes advantage of its constructed redundancy to perform this correction without needing a retransmission.

During the error correction process in a Hamming Code, a calculation of syndrome bits occurs, which helps determine the exact position of the erroneous bit. This process uses the parity bits that were added to the data.

For example, if a single bit error occurs, the Hamming Code creates a non-zero syndrome. By analyzing this syndrome, you can figure out which bit is incorrect and flip it to the correct value. This correction capability enhances the robustness of data transmission, reducing the need for multiple data transmissions just to handle every minor glitch or error.

Parity Bits

Parity bits are fundamental components of Hamming Code. These are special bits added to a data sequence to help with both error detection and correction.

Each parity bit's value is determined by a specific set of positions in the data, calculated according to the rule of exclusive OR (XOR). This means that the parity bit ensures an even number of 1s across the specific positions covered. In the Hamming Code, **parity bits** are strategically placed in positions that are powers of two, such as 1, 2, 4, 8, etc.

The magic of the parity bits lies in their ability to cover multiple data bits, creating a logical system where each bit is part of an equation (or parity check) involving at least one parity bit. This systematic coverage enables the detection of errors in a very methodical way. When a single error occurs, the parity checks will help to produce a unique syndrome, pinpointing the faulty bit which can then be corrected. Parity bits thus play a crucial role in maintaining the accuracy of data transmission.