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
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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
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Error correction or channel coding has built in redundancies to enable reliable transmission that can withstand effects due to noise, interference and multipath fading. 4.1 Block Codes Data is segmented into blocks of k data bits, i.e. each block has 2 k distinct messages. An encoder transforms each block into a large one of n-bits (code bits or channel symbol) referred to as (n, k) code. n-k are called redundant bits, parity bits or check bits, used for error detection and/or correction. Redundancy = (n-k)/k Code rate = k/n
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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
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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.
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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
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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.
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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).
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