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

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List of Tables 4.1 Estimation of received bit amplitude and noise power . . . . . . . . . . . . . 69 5.1 Finding the error locator polynomial . . . . . . . . . . . . . . . . . . . . . . 95 vii
List of Figures 2.1 One and two-sided power spectral density for an arbitrary process . . . . . 11 2.2 Power spectral densities of the real and complex process . . . . . . . . . . . 15 2.3 Quadrature demodulator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 3.1 Costas phase-locked loop . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 3.2 Linearized baseband model of the Costas loop . . . . . . . . . . . . . . . . . 40 3.3 Phase-error variance with filter mismatch . . . . . . . . . . . . . . . . . . . 46 4.1 Costas phase-locked loop receiver functional diagram . . . . . . . . . . . . . 59 4.2 Estimated value of the received amplitude A . . . . . . . . . . . . . . . . . 70 4.3 Probability of p k ≥ 1/2 − ε . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 6.1 Reliability-search frequency distribution . . . . . . . . . . . . . . . . . . . . 102 6.2 Shift-search frequency distribution . . . . . . . . . . . . . . . . . . . . . . . 103 6.3 Probability mass functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 6.4 Error/Erasure decoding of the RS(255,223) code . . . . . . . . . . . . . . . 112 viii
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: 5 Error-Correction Codes: Mathemati
Page 11 and 12: these symbol erasures to decode the
Page 13 and 14: The (23,12,7) Golay code is a perfe
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
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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Decoding Error-Correction Codes Utilizing Bit-Error Probability ...

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