Signal Space Coding over Rings

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C.7.1 Introduction 339 C.8 Example 340 C.9 Polynomials over rings 341 C.9.1 Definition of a Polynomial over R 341 C.9.2 Addition and multiplication of polynomials over rings 341 C.9.3 Division between polynomials over rings 342 C.9.3.1 The division algorithm in U (x) 342 References 343 xi
Chapter 1: Introduction 1 1 Introduction 1.1 Overview In his well known paper, Claude Shannon [2] presented the problem of the communication in the presence of noise in terms of a geometrical view of transmitted signals, considering them as vectors in an N-dimensional vector space. He stated a theorem about bounds on the transmission of signals over the Additive White Gaussian Noise (AWGN) channel. This theorem states that for an AWGN channel characterised by a given signal-to-noise ratio and a defined bandwidth, it is possible to have error-free transmission of information at a given rate, providing that a sufficiently efficient coding technique is applied. The information to be transmitted is represented by Shannon as belonging to a message space. The set of signals to be transmitted belongs to a signal space. For digital information, the relationship between the message space and the signal space is in general a correspondence between groups of m bits and a signal of the signal space. Therefore there are three main entities involved in a communication system: the message space, the signal space, and the mapping between these spaces. Signals are seen as points in an N-dimensional signal space. Dimensionality of the signal space used in the transmission is a parameter that can be increased to provide an error-free transmission. In this sense, an increase of the dimensionality is another way of approaching error-free transmission. Shannon considered coding as a method for selecting some signal space with a higher dimensionality than that of the corresponding message space. However, coding can be also considered as performed by a particular technique applied over the message space. In view of this, coding can be seen as performed by using a coding machine, either block or convolutional, that modifies the message space, increasing its dimension, and providing it with the dimension of the signal space to be used. From the latter point of view, a communication system designed for optimum performance under the light of the Shannon theorems, can be related now to three main entities: a coding machine that generates symbols or labels as outputs, a signal space, and an efficient mapping procedure that maps each output label into a given
Page 1 and 2: Signal Space Coding over Rings by J
Page 3 and 4: Declaration No portion of the work
Page 5 and 6: Acknowledgements I would like to ex
Page 7 and 8: RI Rotationally Invariant ring-BCM
Page 9 and 10: Dedication To the Memory of my fath
Page 11 and 12: 2.3.5 An example of TCM 21 2.3.6 Pr
Page 13 and 14: 3.12.2 Walsh Functions 84 3.12.3 Ra
Page 15 and 16: 4.7.4 Mapping of elements of Z Q in
Page 17 and 18: 6.7 Cyclic codes over the ring of i
Page 19: C.1.2 Order of a group 331 C.1.3 Ab
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s(k) Chapter 4: Ring-Trellis-Coded
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Chapter 5: Wavelet based ring-TCM s
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Chapter 7: Conclusions and further
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Appendix A: Multi-resolution analys
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Appendix B: Power Spectral Density
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Appendix B: Power Spectral Density
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Appendix C: Groups and rings 331 Ap
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Appendix C: Groups and rings 333 C.
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Appendix C: Groups and rings 339 In
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Appendix C: Groups and rings 341 .
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References 343 References 1. G. D.
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References 345 19. J. G. Proakis an
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References 347 39. V. Acha, and R.
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References 349 57. D. Divsalar, and
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References 351 74. J. L. Massey, an
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References 353 92. R. A. Carrasco,
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References 355 109. G. Ungerboeck,
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Signal Space Coding over Rings

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