RademacherCao_ReedSolomon_Final

RademacherCao_ReedSolomon_Final - Reed-Solomon Codes Paul...

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Click to edit Master subtitle style 10/2/11 Reed-Solomon Codes Paul Radmacher Yixin Cao
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Click to edit Master subtitle style 10/2/11 Overview Background Fundamental theories and code construction Decoding Reliability and error correcting performance Applications (Deep space communications, CD)
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10/2/11 Background Developed by Irving Reed and Gustave Solomon in 1960 Reed discovered a hidden structure in the binary codes developed by David Muller: his codes were multinomials of a Galois field. Reed began investigating the possibility that Galois theory could allow the use of non-binary finite field symbols for bytes, rather than just operating at the bit-level. Reed recruited the help of Solomon, and together they published their paper “Polynomial Codes over Certain Finite Fields” in Journal of the Society for Industrial and
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10/2/11 Code Construction There are three ways that the construction of Reed- Solomon codes can be performed: the “original” finite field arithmetic approach the generator polynomial approach the Galois FFT approach
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10/2/11 Original Approach - Construction
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10/2/11 Original Approach - Construction
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10/2/11 Original Approach - Construction Below shows how the non-zero elements of GF(8) can be represented as polynomials by using the primitive polynomial p(x)=x^3+x+1:
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10/2/11 Original Approach - Construction
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10/2/11 Original Approach - Construction
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10/2/11 Original Approach - Construction Following our expression for a Reed-Solomon code, we can generalize with linear algebra. A length “q” code word can be formulated for a message of length “k”. Below, we choose k expressions to form a solvable system:
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10/2/11 Original Approach – Error Correction
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10/2/11 Original Approach – Error Correction
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10/2/11 Original Approach – Error Correction To illustrate how the Reed-Solomon code error correction performs, we wrote a MATLAB function RSdecodeSim.m , which takes (n,k) as input and returns the number of errors after decoding for various numbers of received codeword errors. Here is the plot for (63,21): 0 10 20 30 40 50 60 70 0 5 10 15 20 25 # errors before decoding # errors after decoding
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10/2/11 Original Approach – Error Correction Another possibility at the demodulator: erasures Up to q – k code word coordinates can be erased (only need “k” expressions to solve). Modified Result: Given “v” erasures and “t” errors, correction is possible if and only if 2t + v < q – k + 1
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Polynomial Approach – Construction Most common approach today, used not just for Reed- Solomon codes but for all cyclic codes Any (n,k) cyclic code can be defined by a generator polynomial g(x) of order n-k. Consider code words as
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This note was uploaded on 10/02/2011 for the course ECE 5670 taught by Professor Scaglione during the Spring '11 term at Cornell.

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RademacherCao_ReedSolomon_Final - Reed-Solomon Codes Paul...

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