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

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Reed-Solomon codes have many communication and memory applications and are very powerful in dealing with burst errors. Reed-Solomon codes also allow for a very efficient means of erasure decoding. An erasure is a symbol error with its location known to the decoder. However, in current or traditional decoding procedures, the locations of the error symbols are not known a-priori and must be estimated. Knowledge of the bit-error probabilities enable the decoder to determine, with considerable confidence, the symbols that have the highest likelihood of being in error. Once the locations of the most likely symbol errors are known, the decoder only needs to calculate the amplitudes of these symbol erasures to decode the codeword. Simulations were conducted on a (255,223) Reed- Solomon code over an additive white Gaussian noise (AWGN) channel to determine the accuracy of the symbol erasure estimate. <strong>Error</strong>/erasure decoding was performed using a combination of 15 errors and 2 erasures and compared to error only decoding for 16 errors using the Berlekamp-Massey algorithm. The performance of the error/erasure decoder yielded a slightly lower symbol error probability compared to the error only decoder. 1.2 Organization of the Dissertation This dissertation consists of seven chapters with five appendices and is organized as follows: Chapter 1 is the introduction and the motivation for conducting the research in this dissertation. Chapter 2 establishes a rigorous mathematical development for the spectral representation of a real wide-sense stationary stochastic noise process, and develops the fundamental results of narrow-band stochastic processes. These concepts are fundamental to understanding the estimation methods used in the study. The first purpose of Chapter 3 is to analyze the Costas phase-locked loop under the strict requirement of perfect phase lock, that is, no phase error. It is shown that the correlation of the received signal with a coherent reference signal yields the following baseband representation: Signal corrupted with Gaussian noise at the output of the inphase (I) channel, and Gaussian noise only with no signal at the output of the quadrature (Q) channel. The second purpose of this chapter is to analyze the Costas circuit when 3
perfect phase-lock is not achieved. In Section 3.4, a new analytical expression for the phaseerror variance is derived when there is a mismatch between the modulation bandwidth and the RF filter bandwidth in the receiver. The last section uses the phase-error variance to ascertain the fraction of signal and noise power that leaks from the in-phase channel into the quadrature channel, and the fraction of noise power that leaks from the quadrature channel into the in-phase channel. The main purpose of Chapter 4 is 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 probability. Chapter 5 is devoted to the mathematical preliminaries and the general properties of error-control coding. The basic concepts of groups, rings, fields, primitive elements, cyclic codes, Reed-Solomon codes, syndrome equations, and the Berlekamp-Massey (BM) algorithm are discussed. 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) Quadratic Residue 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. Finally, the main results of this dissertation are summarized in Chapter 7, and some future areas of research are proposed. The appendices give the detailed proofs of the facts stated in the dissertation. 4
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 and 12: these symbol erasures to decode the
Page 13: The (23,12,7) Golay code is a perfe
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
Page 65 and 66:
The signal s(t) represents a binary
Page 67 and 68:
The Cauchy-Schwartz inequality [12]
Page 69 and 70:
In this study the information signa
Page 71 and 72:
The data sequence m(t) is modeled b
Page 73 and 74:
Define σ 2 N 0 /2T b to be the no
Page 75 and 76:
densities, given either pulse signa
Page 77 and 78:
A substitution of (4.26) and (4.24)
Page 79 and 80:
σ 2 . Therefore, it is reasonable
Page 81 and 82:
1.02 Estimate of A: SNR = 6 dB A =
Page 83 and 84:
1 0.9 SNR = 0dB SNR = 4dB SNR = 8dB
Page 85 and 86:
The following properties of a group
Page 87 and 88:
Let R be a ring and I be an ideal o
Page 89 and 90:
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
Page 99 and 100:
Λ 0 = 1 v∑ Λ 1 = X i = X 1 + X
Page 101 and 102:
5.5 Reed-Solomon Codes Reed-Solomon
Page 103 and 104:
Unlike the binary case, the syndrom
Page 105 and 106:
The Berlekamp-Massey shift register
Page 107 and 108:
Chapter 6 Decoding Algorithms The p
Page 109 and 110:
To be a (23,12,7) cyclic BCH codewo
Page 111 and 112:
In other words, adding an error vec
Page 113 and 114:
of 13 attempts is the same in both
Page 115 and 116:
Definition 6.2 (Quadratic Residue C
Page 117 and 118:
The frequency distributions for cor
Page 119 and 120:
decoding operation. The first of th
Page 121 and 122:
5. Find the roots of Λ(x) using th
Page 123 and 124:
RS(255,223) 15 Erasures and 2 Error
Page 125 and 126:
4. The phase-error variance is deri
Page 127 and 128:
Reference List [1] E. R. Berlekamp,
Page 129 and 130:
[33] G. Hedin, J. K. Holmes, W. C.
Page 131 and 132:
Without loss of generality, assume
Page 133 and 134:
Thus, a substitution of (A.5) into
Page 135 and 136:
∫ ∞ −∞ The second integral
Page 137 and 138:
Appendix B Proof of Uniform Converg
Page 139 and 140:
Appendix C Convolution of S nx (f)
Page 141 and 142:
Appendix D Power Spectral Density S
Page 143 and 144:
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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