EE3101_7 Chapter 4-1 Error Correction Coding_r0.pptx - EE3101 Communication Engineering Chapter 4 Error Correction Coding Course Intended Learning

EE3101_7 Chapter 4-1 Error Correction Coding_r0.pptx -...

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EE3101 Communication Engineering Chapter 4, Error Correction Coding
Course Intended Learning Outcomes Explain the basic concepts of error detection/correction coding and perform error analysis 4.1 Block Codes 4.2 Cyclic Codes
4.1.1 Parity-check code 1) Single-Parity-check-code Redundant bit = 1 bit even or odd parity Code rate k/(k+1) Can only detect only odd number of bit errors, cannot correct errors Probability of undetected error P ud (even number of bits are inverted) p is the bit error probability
Message Parity Codewor d 000 0 0 000 100 1 1 100 010 1 1 010 110 0 0 110 001 1 1 001 101 0 0 101 011 0 0 011 111 1 1 111 Example: Even-Parity Code A (4, 3) even parity, error-detection code Probability of channel symbol error is; Can detect single or triple error patterns. Probability of undetected error is equal to the probability that two or four errors occur anywhere in a codeword.
2) Rectangular Code (Product Code) Can be considered as a parallel data transmission. Data bloc; M rows and N columns Append a horizontal parity check to each row and a vertical parity check to each column Coded block; (M+1) rows and (N+1) columns Can correct single bit error Serial transmission; 1 1 1 0 0 1 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1 0 0 1 1 0 1 1 0 Vertical parity check Horizontal parity check 1 110010 000111 100001 11100110110
4.1.2 Linear Block Codes Belong to a class of parity check codes characterized by (n, k) notation Encoder transforms a block of k message digits (a message vector) into a longer block of n codeword digits (a code vector) constructed from a given alphabet of elements 2 k k-tuples (sequence of k digits) messages are linearly, uniquely mapped to 2 k n- tuples codewords This mapping is accomplished via a look-up table.
4.1.2.1 Vector Spaces Set of all binary n-tuples V n is called a vector space over the binary field of two elements (0 and 1). The field has two operations, addition (XOR) and multiplication (AND). Addition Multiplication 4.1.2.2 Vector Subspaces Subset S of the vector space V n