Applications of Gray Codes in Various Fields
Gray codes are utilized for error correction in digital communications, genetic algorithms, Karnaugh maps, and solving the Towers of Hanoi problem. They play a crucial role in improving system robustness, simplifying Boolean algebra expressions, guiding genetic algorithm mutations, and tracking disk movements in the Towers of Hanoi puzzle. The applications demonstrate the versatility and importance of Gray codes across diverse domains.
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Presentation Transcript
Uses Error correction For more information: Genetic algorithms Karnaugh maps Towers of Hanoi 2/7
Error correction In modern digital communications, Gray codes play an important role in error correction. For example, in a digital modulation scheme such as QAM where data is typically transmitted in symbols of 4 bits or more, the signal's constellation diagram is arranged so that the bit patterns conveyed by adjacent constellation points differ by only one bit. By combining this with forward error correction capable of correcting single-bit errors, it is possible for a receiver to correct any transmission errors that cause a constellation point to deviate into the area of an adjacent point. This makes the transmission system less susceptible to noise. 3/7
Genetic algorithms Due to the Hamming distance properties of Gray codes, they are sometimes used in genetic algorithms. They are very useful in this field, since mutations in the code allow for mostly incremental changes, but occasionally a single bit-change can cause a big leap and lead to new properties. 4/7
Karnaugh maps Gray codes are also used in labeling the axes of Karnaugh maps. The Karnaugh map, also known as the K-map, is a method to simplify boolean algebra expressions. 5/7
Towers of Hanoi The binary-reflected Gray code can also be used to serve as a solution guide for the Towers of Hanoi problem. If one counts in Gray code of a bit size equal to the number of disks in a particular Tower of Hanoi, begins at zero, and counts up, then the bit changed each move corresponds to the disk to move, where the least-significant-bit is the smallest disk and the most-significant-bit is the largest. 6/7
Microprocessor Systems (MIPS) Professor: dr Veljko M. Milutinovi Student: Petar Risti 32/2012 Software engineering School of Electrical Engineering University of Belgrade 7/7
