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

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If one equates the first and last expressions for R(τ), given in (2.20), then the two-sided spectral density is related to the one-sided spectral density as follows: ⎧ ⎪⎨ G(f) = ⎪⎩ 1 2 G 1(f), if f > 0 1 2 G 1(−f), if f < 0, (2.21) where G 1 (f) and G(f) are the original one-sided and modern two-sided definition of power spectral density, respectively. The graphical interpretation of the one and two-sided power spectral density for an ½´µ arbitrary stochastic process is illustrated ´µ in Figure 2.1. ¾ Figure 2.1: One and two-sided power spectral density for an arbitrary process 2.3 The Complex Process z(t), Associated with the Real Process x(t) The complex WSS stochastic process z(t), associated with the real process x(t), is defined by the one-sided spectral representation, z(t) = 2 ∫ ∞ 0 e i2πft dX(f). (2.22) For this definition to be consistent with the real process, it is demonstrated next that x(t) = Re{z(t)} = z(t) + z∗ (t) . (2.23) 2 11
To proceed with the proof, first note that the complex conjugate of z(t) is given by z ∗ (t) = 2 ∫ ∞ 0 e −i2πft dX ∗ (f). (2.24) The differential element dX ∗ (f) is calculated from the conjugate of the X(f) in equation (2.8) as follows: (∫ ∞ X ∗ (f) = = = −∞ ∫ ∞ −∞ ∫ ∞ −∞ e −i2πft ) − 1 ∗ x(t)dt −2πit e i2πft − 1 x(t)dt 2πit e −i2π(−f)t − 1 x(t)dt 2πit = −X(−f). (2.25) By (2.25), a substitution of the differential dX ∗ (f) = −dX(−f) into (2.24) yields z ∗ (t) = 2 = 2 ∫ ∞ 0 ∫ 0 −∞ e −i2πft [−dX(−f)] e i2πft dX(f). Finally, a substitution of z(t) and z ∗ (t) into equation (2.23) demonstrates that x(t) = ∫ ∞ −∞ e i2πft dX(f). (2.26) 12
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 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: A similar computation for the fourt
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
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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
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∫ ∞ −∞ 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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