Decoding Error-Correction Codes Utilizing Bit-Error Probability ...

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Chapter 1 Introduction 1.1 Motivation The term “error-control coding” implies a technique by which redundant symbols are attached “intelligently” to information messages by an error correction encoder in the transmitter. These redundancy symbols are used to correct the erroneous data at the error control decoder in the receiver. In other words error-control coding is achieved by restrictions placed on the characteristics of the encoder of the system. These restrictions make it possible for the decoder to correctly extract the original source signal with high reliability and fidelity from the possibly corrupted received or retrieved signals. The purpose of the research conducted in this dissertation is to be able to estimate the individual bit-error probabilities of binary symbols or codewords while they are being received. It turns out that the bit or symbol error probability of a codeword is a function of the received-bit amplitudes A and the channel noise power σ 2 , both of which are assumed to be unknown a-priori at the receiver. In this study coherent detection is implemented with Costas phase-locked loop receiver which facilitates the joint estimation of these two parameters, and as a consequence, the bit-error probabilities. In Chapter 6 it is shown how the individual bit-error probability estimates reduce the decoding complexity of the (23,12,7) Golay code and the (47,24,11) QR code. It is also shown how the bit-error probability estimates facilitate erasure decoding of Reed-Solomon codes over an additive white Gaussian noise (AWGN) channel. 1
The (23,12,7) Golay code is a perfect linear error-correcting code that can correct all patterns of three or fewer errors in 23 bit positions. A simple BCH decoding algorithm, given in [1, pg. 19], can decode the (23,12,7) Golay code provided there are no more than two errors. The shift-search algorithm, developed by Reed [2], sequentially inverts the information bits until the third error is canceled. It then utilizes the BCH decoding algorithm to correct the remaining two errors. However, the computational complexity is very high to decode the third error in terms of CPU time. In this dissertation a simplified decoding algorithm, called the reliability-search algorithm, is proposed. This algorithm uses bit-error probability estimates to cancel the third error, and then uses the BCH decoding algorithm to correct the remaining two errors. Simulation results show that this new algorithm significantly reduces the decoding complexity for correcting the third error while maintaining the same BER performance. Another interesting class of error-control codes are the quadratic residue codes. These codes are near half rate, have very good mathematical structure, and have a high capability to correct random errors in data. Recently, He, Reed, and Truong [3] have developed an algebraic decoding algorithm for the (47,24,11) Quadratic Residue code that can correct up to true minimum distance i.e., five errors. However, the computational complexity is very high to decode the fifth error in terms of CPU time. The reason for this can be explained as follows: The algebraic decoding algorithm needs to calculate the greatest common divisor (gcd) of two polynomials for both the four and five error cases. For the four error case, the decoder must calculate the gcd(f 1 , f 2 ), where deg(f 1 ) = 3 and deg(f 2 ) = 33. For this case, the complexity of gcd(f 1 , f 2 ) is acceptable. However, for the five error case, the decoder must calculate gcd(g 1 , g 2 ), where deg(g 1 ) = 34 and deg(g 2 ) = 258. Clearly, the complexity to compute gcd(g 1 , g 2 ) is very high and considerable more memory is needed. In order to circumvent calculating the greatest common divisor of such high degree polynomials, the reliability-search algorithm can be utilized to decode the fifth error. Simulation results show that the decoding complexity is significantly reduced for correcting the fifth error compared to the algebraic decoding algorithm. 2
Page 1 and 2: COMMUNICATION SCIENCES INSTITUTE D
Page 3 and 4: Dedication First and foremost, this
Page 5 and 6: our course studies and with my rese
Page 7 and 8: 5 Error-Correction Codes: Mathemati
Page 9 and 10: List of Figures 2.1 One and two-sid
Page 11: these symbol erasures to decode the
Page 15 and 16: perfect phase-lock is not achieved.
Page 17 and 18: where E{·} is the statistical expe
Page 19 and 20: and thus, the Fourier transform of
Page 21 and 22: A similar computation for the fourt
Page 23 and 24: To proceed with the proof, first no
Page 25 and 26: Therefore, one can write the comple
Page 27 and 28: for all f, f ′ > 0 or for all f,
Page 29 and 30: where z(t) is the original complex
Page 31 and 32: Consider now two new Gaussian baseb
Page 33 and 34: where σ 2 = E{x 2 (t)} = E{y 2 (t)
Page 35 and 36: signal-to-noise ratio (SNR), or equ
Page 37 and 38: where T b is the pulse width in tim
Page 39 and 40: where ̂θ(t) is the estimated phas
Page 41 and 42: Hence, for perfect lock-on, the out
Page 43 and 44: little distortion due to filtering.
Page 45 and 46: first-order loop filter corresponds
Page 47 and 48: 3.3 The Dominant Noise Term in N(t,
Page 49 and 50: The conditional variances Var{n 1 (
Page 51 and 52: 3.4 The Phase-Error Variance The go
Page 53 and 54: of the closed-loop transfer functio
Page 55 and 56: where S nx (f) and S ny (f) are the
Page 57 and 58: yields a higher phase-error varianc
Page 59 and 60: channel, and the fraction of noise
Page 61 and 62: The power of ˜w(t), assuming the s
Page 63 and 64:
L Q = −13.5 dB. Therefore, one ca
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The signal s(t) represents a binary
Page 67 and 68:
The Cauchy-Schwartz inequality [12]
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In this study the information signa
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The data sequence m(t) is modeled b
Page 73 and 74:
Define σ 2 N 0 /2T b to be the no
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densities, given either pulse signa
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A substitution of (4.26) and (4.24)
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σ 2 . Therefore, it is reasonable
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1.02 Estimate of A: SNR = 6 dB A =
Page 83 and 84:
1 0.9 SNR = 0dB SNR = 4dB SNR = 8dB
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The following properties of a group
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Let R be a ring and I be an ideal o
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of GF(q). If α is a primitive elem
Page 91 and 92:
g(x) is called a primitive polynomi
Page 93 and 94:
Proposition 5.11 The ideal I(x) is
Page 95 and 96:
Let M r (x) be given by M r (x) =
Page 97 and 98:
Proposition 5.13 The mapping betwee
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Λ 0 = 1 v∑ Λ 1 = X i = X 1 + X
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5.5 Reed-Solomon Codes Reed-Solomon
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Unlike the binary case, the syndrom
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The Berlekamp-Massey shift register
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Chapter 6 Decoding Algorithms The p
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To be a (23,12,7) cyclic BCH codewo
Page 111 and 112:
In other words, adding an error vec
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of 13 attempts is the same in both
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Definition 6.2 (Quadratic Residue C
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The frequency distributions for cor
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decoding operation. The first of th
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5. Find the roots of Λ(x) using th
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RS(255,223) 15 Erasures and 2 Error
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4. The phase-error variance is deri
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Reference List [1] E. R. Berlekamp,
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[33] G. Hedin, J. K. Holmes, W. C.
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Without loss of generality, assume
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Thus, a substitution of (A.5) into
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∫ ∞ −∞ The second integral
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Appendix B Proof of Uniform Converg
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Appendix C Convolution of S nx (f)
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Appendix D Power Spectral Density S
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where E{a n a m } = E{a n }E{a m }
Page 145:
When I first moved to Los Angeles,
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Decoding Error-Correction Codes Utilizing Bit-Error Probability ...

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