Table Of ContentSelect Topics in
1
Signal Analysis
Select Topics in
Signal Analysis
Selected Topics in Signal Analysis
Harish Parthasarathy
ECE division, NSUT
Harish Parthasarathy
December 12, 2020
Professor
Electronics & Communication Engineering
Netaji Subhas Institute of Technology (NSIT)
New Delhi, Delhi-110078
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Preface
Preface
This book developed from a course given by the author to undergraduate
and post-graduate students at the NSUT over a period of two semesters. The
first chapter on matrix theory discusses in reasonable depth the theory of Lie
algebras leading upto Cartan’s classification theory. It also discusses some ba-
sic elements of functional analysis and operator theory in infinite dimensional
Banach and Hilbert spaces. The theory of Lie groups and Lie algebras play a
fundamentalroleinroboticdynamicswith3-Dlinksaswellasinimageprocess-
ing. We discuss here in full details the construction of Cartan subalgebras of a
semisimple Lie algebra leading to the root space decomposition and consequent
classification of the simple Lie algebras. In functional analysis, we discuss some
issuesrelatedtographsofoperators,theadjointmap,thebasictheoremsofBa-
nachspacetheoryandsomeaspectsofoperatorsinHilbertspaces. Thesetopics
areimportantmainlybecauseoftheapplicationstheyfindinquantummechan-
icsandmoderninfinitedimensionalcontroltheory. Thesecondchapterisabout
basic probability theory and the topics discussed in this book find applications
tostochasticfilteringtheoryfordifferentialequationsdrivenbywhiteGaussian
noise. Thethirdchapterisonantennatheorywithafocusonmodernquantum
antenna theory in which we discuss quantum electrodynamics within a cavity
resonatorinwhichtheMaxwellphotonfieldinteractswiththeDiracfieldofelec-
trons and positrons to radiate out a quantum electromagnetic field into space
whose statistics in any state of the cavity like say a joint coherent state of the
electrons, positrons and photons can be calculated and controlled by insertion
ofclassicalphotonprobesusingalaserandclassicalcurrentprobesusingawire.
The last chapter of Part I is based upon many interesting and useful discussion
IhavehadoverthepastfewmonthswithDr.Steven A.Langford, ageophysicist
who has made a variety of experiments on the refractive index of materials and
liquids. Dr.Langfordhassuggestedtomemanywaysofanalysingandmodeling
the refractive index of materials using classical and quantum statistics always
stressing on the fact that the problem of modeling the refractive index is in-
timately connected with resonance with dark matter, the cosmic expansion of
the universe, the cosmic microwave background radiation and the phenomenon
of reflection and refraction of De-Broglie matter waves at a boundary between
two matrix valued potentials which take into account the spin of the quantum
mechanicalparticle. MydebttoDr.Langfordistooheavytoberepaidinwords.
ThematerialinPartIIdealsprimarilywithapplicationsoflargedeviationthe-
ory to problems in engineering and physics. The main idea is that when weak
noise enters into a system, then we can use large deviation theory to approxi-
matelycomputetheprobabilityofthesystemstatedeviatingfromthenoiseless
system state by an amount greater than a prescribed threshold in terms of the
system control parameters. Then, we adpat the control parameters so that
this probability is minimized. We have adapted this principle to a variety of
problems including fluid dynamics, plasma physics, hydrodynamics in curved
space-time and even to quantum filtering theory. It should be noted that large
deviations cannot be directly applied to quantum mechanics because the evolv-
ing observables at different times do not commute and hence by the Heisenberg
3 4
uncertainty principle they do not have a joint probability distribution. How- 1.7.1 Controllability of the Yang-Mills non-Abelian field equations
ever, if we filter the state using Belavkin’s non-demoltion measurement based 1.7.2 Appendix
quantum filtering theory, then these filtered states form an Abelian family and 1.8 Controllability of supersymmetric field theoretic problems
hence have a joint probability density to which large deviation theory can be 1.9 Large deviations and control theory
applied. The final chapter in this book deals with quantum field theoretic as- 1.10 Approximate controllability of the Maxwell equations
pectsofthehumaneyemodeledasanantenna. Itakethisopportunitytothank 1.11 Controllability problems in quantum scattering theory
my colleague Prof.Rajveer Yaduvanshi for posing the problem of performing a 1.12 Controllability in the context of representations of Lie groups
quantum mechanical analysis of the eye cavity as a quantum antenna and for 1.13 Irreducible representations and maximal ideals
some nice suggestions. We can formulate large deviations problems even this 1.14 Controllability of the Maxwell-Dirac equations using external
context, for example, by including classical external random gauge fields and classical current and field sources
currents interacting with the quantum Dirac and Maxwell fields and then ask- 1.15 Application of the representation theory of SL(2,C) as an
ing the question, what is the rate function for the quantum average of a field alternative way of characterizing Lorentz transformations to control
observable like the quantum current field or quantum electromagnetic field (or problems
even the space-time moments for the far field radiation pattern generated by 1.16 Construction of irreducible representations
thequantumcurrentswithinthecavitybodyandonitssurface)inagivenstate 1.17MoreonrootspacedecompositionofasemisimpleLiealgebra
of electrons, positrons and photons and hence deduce the probability for these 1.18 A problem in Lie group theory
quantum averages to deviate from the corresponding noiseless (ie, zero classical 1.19 More problems in linear algebra and functional analysis
noise)valuesbyanamountgreaterthanagiventhreshold. Thiswouldenableus 1.19.1 Riesz’ representation theorem
to design control algorithms by introducing non-random classical control fields 1.19.2 Lie’s theorem on solvable Lie algebras
and current sources so that this deviation probability is a minimum. 1.19.3 Engel’s theorem on nil-representations of a Lie algebra
1.20 Spectral theory in Banach Algebras
Table of Contents Author 1.21 Atiyah-Singer index theorem:A supersymmetric proof
Preface 1.22 Jordan decomposition on a semisimple Lie algebra
Chapter 1:Matrix Theory 1.23 Construction of a Cartan subalgebra
1.1 Course Outline 1.24 A criterion for regularity of an element in a Lie algebra
1.2 Prerequisites of linear algebra 1.25 Cartan subalgebras of a semisimple Lie algebra are maximal
1.2.1 Fields Abelian
1.2.2 Rings 1.26 Maximal nilpotency of a Cartan subalgebra
1.2.3 Simultaneous triangulability and diagonability 1.27LecturePlan,MatrixTheory,M.Tech,Instructor:HarishParthasarathy
1.2.4 The minimax variational principle for calculating all the 1.28 Some other remarks on Lie algebras
eigenvalues of a Hermitian matrix 1.28.1 Root space decomposition of a semisimple Lie algebra and
1.2.5 The Primary Decomposition Theorem some of its properties
1.3 Cartan Subalgebras of a Semisimple Lie algebra 1.29 Question paper on matrix theory
1.4 Exercises in Matrix Theory: 1.30 Root space decompositions of the complex classical Lie alge-
1.4.1 The Polar decomposition bras
1.4.2 The singular value decomposition 1.31 More on the root space decomposition of a semisimple Lie
1.4.3 The Riesz representation theorem algebra
1.4.4 Transpose of a linear operator 1.32 Cartan’s classification of the complex simple Lie algebras
1.4.5 Differentiation of infinite dimensional vector valued functions 1.33 The cornerstone theorems of functional analysis
1.4.6 Invariant subspaces and primary decomposition 1.34 Spectrum of a Compact operator
1.4.7 Normalizer of a Lie subalgebra 1.35 Hahn-Banach theorem
1.4.8 Finite dimensional Irreducible representations of SL(2,C) 1.36 Polar decomposition in infinite dimensional Hilbert spaces
1.5 Conjugacy classes of Cartan sub-algebras 1.37 Von-Neumann’s theorem
1.6 Exercises 1.38 An iterative method for constructing the positive square root
1.7Appendix:Someapplicationsofmatrixtheorytocontroltheory of a bounded positive operator in a Hilbert space
problems 1.39 Hilbert-Schmidt operators
Table of Contents
1. Matrix Theory 1–104
2. Antenna Theory 105–160
3. ProbabilityTheory 161–188
4. Models for the Refractive Index of Materials
and Liquids 189–250
5. Statistics of Refractive Index and Fundamental
Laws of Nature 251–272
6. Miscellaneous Remarks on the Content of
The Previous Chapters 273–284
7. Applications of Large Deviation Theory to
Engineering Problems 285–306
8. Large Deviations for Filtering in a Mixture of
Boson-Fermion noise 307–316
9. Large Deviations for Classical and Quantum Stochastic
Filtering Problems in General Relativity 317–328
10. Quantum Mechanics of The Eye 329–388
Chapter 1
Matrix Theory
1.1 Course Outline
[0] Prerequisites of linear algebra.
Fields, rings, vector spaces over a field, modules over a ring, algebras, ideals
in a ring and an algebra, bases for vector spaces, linear transformations in a
vector space, basis for a vector space, matrix of a linear transformation relative
toabasis, innerproductspaces, unitary, Hermitianandnormaloperatorsinan
inner product space, spectral theorem for normal operators.
[1] Quotient of a vector space by another space.
[2] Simultaneous triangulability of commuting matrices relative to an onb.
[3] Simultaneous diagonability of commuting normal matrices relative to an
onb.
[4] Tensor products of vector spaces.
[5] Variational principles for calculating the eigenvalues of a Hermitian ma-
trix.
[6] Positive definite matrices.
[7] The basic decomposition theorems of matrix theory.
[a] Row reduced Echelon form.
[b] Spectral theorem for normal matrices.
[c] Polar decomposition.
[d] Singular value decomposition.
[e] QR decomposition based on the Gram-Schmidt orthonormalization pro-
cess.
[f] LDU decomposition of positive definite matrices.
[8] Applications of matrix theory to finite state quantum systems.
[a]SchrodingerandHeisenbergevolutioninfinitedimensionalHilbertspaces.
[b] Different kinds of unitary gates for finite state quantum computation:
CNOT, Swap, Fredkin, Toffoli, phase gate, Quantum Fourier transform gate.
[c] Perturbation theory for quantum systems in finite dimensional state
space.
9
