Information Loss in Deterministic Signal Processing Systems

149 Pages·2.636 MB·English

by  Geiger, Bernhard C., Kubin, Gernot

Understanding Complex Systems
Bernhard C. Geiger
Gernot Kubin
Information Loss
in Deterministic
Signal Processing
Systems
Springer Complexity
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DanBraha,NewEnglandComplexSystemsInstituteandUniversityofMassachusettsDartmouth,USA
Péter Érdi, Center for Complex Systems Studies, Kalamazoo College, USA and Hungarian Academy of
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Understanding Complex Systems
Founding Editor: S. Kelso
Future scientific and technological developments in many fields will necessarily
depend uponcomingtogripswithcomplexsystems.Such systems arecomplex in
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Bernhard C. Geiger Gernot Kubin
(cid:129)
Information Loss
in Deterministic Signal
Processing Systems
123
Bernhard C.Geiger Gernot Kubin
Institute for Communications Engineering Signal ProcessingandSpeech
Technical University of Munich Communication Lab
Munich Graz University of Technology
Germany Graz
Austria
ISSN 1860-0832 ISSN 1860-0840 (electronic)
Understanding ComplexSystems
ISBN978-3-319-59532-0 ISBN978-3-319-59533-7 (eBook)
DOI 10.1007/978-3-319-59533-7
LibraryofCongressControlNumber:2017943204
©SpringerInternationalPublishingAG2018
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to our families
Acknowledgements
Wethankallourco-authorswehadthepleasuretoworkwithduringthelastyears.
In particular, we thank Christian Feldbauer, for developing first results together
withus,andTobiasKoch,UniversidadCarlosIIIdeMadrid,forhelpingusmaking
some of our results mathematically more rigorous. We are indebted to Rana Ali
Amjad,ClemensBlöchl,OnurGünlü,KairenLiu,AndreiNedelcu,andLarsPalzer,
all from Technical University of Munich, for reading and suggesting changes in
early drafts of this book.
This book grew out of the first author’s Ph.D. thesis and of course notes of the
course “Information-Theoretic System Analysis and Design” that he gave at the
Technical University of Munich in 2016. We thus want to thank the students that
attended this course with great interest, asked many smart questions, helped
improving the course notes, and kept the lecturer motivated to invent challenging
end-of-chapter problems. Finally, we thank Gerhard Kramer, Technical University
of Munich, who made this lecture possible.
The book has been written during the first author’s stay at the Institute for
Communications Engineering, Technical University of Munich. The stay has been
funded by the Erwin Schrödinger Fellowship J 3765 of the Austrian Science Fund
and by the German Ministry of Education and Research in the framework of an
Alexander von Humboldt Professorship.
vii
Contents
1 Introduction .... .... .... ..... .... .... .... .... .... ..... .. 1
1.1 Related Work.... .... ..... .... .... .... .... .... ..... .. 6
1.2 Outline and Prerequisites.... .... .... .... .... .... ..... .. 7
1.3 Motivating Example: Analysis of Digital Systems. .... ..... .. 8
1.4 Motivating Example: Quantizer Design. .... .... .... ..... .. 11
Part I Random Variables
2 Piecewise Bijective Functions and Continuous Inputs .... ..... .. 17
2.1 The PDF of Y ¼gðXÞ. ..... .... .... .... .... .... ..... .. 18
2.2 The Differential Entropy of Y ¼gðXÞ .. .... .... .... ..... .. 20
2.3 Information Loss in PBFs ... .... .... .... .... .... ..... .. 21
2.3.1 Elementary Properties. .... .... .... .... .... ..... .. 21
2.3.2 Upper Bounds on the Information Loss ... .... ..... .. 26
2.3.3 Computing Information Loss Numerically . .... ..... .. 29
2.3.4 Application: Polynomials .. .... .... .... .... ..... .. 30
3 General Input Distributions..... .... .... .... .... .... ..... .. 35
3.1 Information Loss for Systems with General Inputs .... ..... .. 36
3.2 Systems with Infinite Information Loss . .... .... .... ..... .. 37
3.3 Alternative Proof of Theorem 2.1 . .... .... .... .... ..... .. 40
4 Dimensionality-Reducing Functions... .... .... .... .... ..... .. 43
4.1 Information Dimension ..... .... .... .... .... .... ..... .. 44
4.1.1 Properties and Equivalent Definitions of Information
Dimension .... ..... .... .... .... .... .... ..... .. 45
4.1.2 dðXÞ-dimensional Entropy and Mixture
of Distributions. ..... .... .... .... .... .... ..... .. 49
4.1.3 Operational Characterization of Information
Dimension .... ..... .... .... .... .... .... ..... .. 53
ix
x Contents
4.2 Relative Information Loss ... .... .... .... .... .... ..... .. 54
4.2.1 Elementary Properties. .... .... .... .... .... ..... .. 55
4.2.2 Bounds on the Relative Information Loss.. .... ..... .. 56
4.2.3 Relative Information Loss for System Reducing
the Dimensionality of Continuous Random Variables.. .. 57
4.2.4 Relative Information Loss and Perfect Reconstruction . .. 60
4.2.5 Outlook: Relative Information Loss
for Discrete-Continuous Mixtures.... .... .... ..... .. 62
4.3 Application: Principal Components Analysis. .... .... ..... .. 64
4.3.1 The Energy-Centered Perspective.... .... .... ..... .. 65
4.3.2 PCA with Given Covariance Matrix.. .... .... ..... .. 67
4.3.3 PCA with Sample Covariance Matrix. .... .... ..... .. 68
5 Relevant Information Loss. ..... .... .... .... .... .... ..... .. 73
5.1 Definition and Properties .... .... .... .... .... .... ..... .. 73
5.1.1 Elementary Properties. .... .... .... .... .... ..... .. 75
5.1.2 An Upper Bound on Relevant Information Loss ..... .. 78
5.2 Signal Enhancement and the Information Bottleneck Method . .. 80
5.3 Application: PCA with Signal-and-Noise Models . .... ..... .. 82
Part II Stationary Stochastic Processes
6 Discrete-Valued Processes . ..... .... .... .... .... .... ..... .. 93
6.1 Information Loss Rate for Discrete-Valued Processes .. ..... .. 94
6.2 Information Loss Rate for Markov Chains... .... .... ..... .. 96
6.3 Outlook: Systems with Memory... .... .... .... .... ..... .. 99
6.3.1 Partially Invertible Systems .... .... .... .... ..... .. 100
6.3.2 Application: Fixed-Point Implementation of a Linear
Filter. .... .... ..... .... .... .... .... .... ..... .. 102
7 Piecewise Bijective Functions and Continuous Inputs .... ..... .. 105
7.1 The Differential Entropy Rate of Stationary Processes.. ..... .. 105
7.2 Information Loss Rate in PBFs ... .... .... .... .... ..... .. 106
7.2.1 Elementary Properties. .... .... .... .... .... ..... .. 106
7.2.2 Upper Bounds on the Information Loss Rate... ..... .. 108
7.2.3 Application: AR(1)-Process in a Rectifier.. .... ..... .. 109
7.3 Outlook: Systems with Memory... .... .... .... .... ..... .. 111
8 Dimensionality-Reducing Functions... .... .... .... .... ..... .. 115
8.1 Relative Information Loss Rate ... .... .... .... .... ..... .. 115
8.2 Application: Downsampling.. .... .... .... .... .... ..... .. 118
8.2.1 The Energy-Centered Perspective.... .... .... ..... .. 119
8.2.2 Information Loss in a Downsampling Device... ..... .. 119
8.3 Outlook: Systems with Memory... .... .... .... .... ..... .. 123
Contents xi
9 Relevant Information Loss Rate . .... .... .... .... .... ..... .. 127
9.1 Definition and Properties .... .... .... .... .... .... ..... .. 127
9.1.1 Upper Bounds on the Relevant Information Loss Rate. .. 129
9.2 Application: Anti-aliasing Filter Design for Downsampling... .. 130
9.2.1 Anti-aliasing Filters for Non-Gaussian Processes ..... .. 132
9.2.2 FIR Solutions for Information Maximization ... ..... .. 135
10 Conclusion and Outlook... ..... .... .... .... .... .... ..... .. 137
References.. .... .... .... .... ..... .... .... .... .... .... ..... .. 141