Eldar Y.C. Sampling theory. Beyond bandlimited systems

Cambridge University Press, 2015. — 835 p.Covering the fundamental mathematical underpinnings together with key principles and applications, this book provides a comprehensive guide to the theory and practice of sampling from an engineering perspective. Beginning with traditional ideas such as uniform sampling in shift-invariant spaces and working through to the more recent fields of compressed sensing and sub-Nyquist sampling, the key concepts are addressed in a unified and coherent way. Emphasis is given to applications in signal processing and communications, as well as hardware considerations, throughout.
The book is divided into three main sections: first is a comprehensive review of linear algebra, Fourier analysis, and prominent signal classes figuring in the context of sampling, followed by coverage of sampling with subspace or smoothness priors, including nonlinear sampling and sample rate conversion. Finally, sampling over union of subspaces is discussed, including a detailed introduction to the field of compressed sensing and the theory and applications of sub-Nyquist sampling.
With 200 worked examples and over 250 end-of-chapter problems, this is an ideal course textbook for senior undergraduate and graduate students. It is also an invaluable reference or self-study guide for engineers and students across industry and academia.List of abbreviations.Standard sampling.
Beyond bandlimited signals.
Outline and outlook.
Signal expansions: some examples.
Subspaces.
Properties of subspaces.
Inner product spaces.
The inner product.
Orthogonality.
Calculus in inner product spaces.
Hilbert spaces.
Linear transformations.
Subspaces associated with a linear transformation.
Invertibility.
Direct-sum decompositions.
The adjoint.
Basis expansions.
Set transformations.
Bases.
Riesz bases.
Riesz basis expansions.
Projection operators.
Orthogonal projection operators.
Oblique projection operators.
Pseudoinverse of a transformation.
Definition and properties.
Matrices.
Frames.
Definition of frames.
Frame expansions.
The canonical dual.
Exercises.
Fourier analysis.
Linearity and time-invariance.
The impulse response.
Causality and stability.
Definition of the CTFT.
Properties of the CTFT.
Examples of the CTFT.
Fubini’s theorem.
Discrete-time impulse response.
Discrete-time Fourier transform.
Properties of the DTFT.
Continuous-discrete representations.
Poisson-sum formula.
Sampled correlation sequences.
Exercises.
Sampling and reconstruction spaces.
Practical sampling theorems.
The Shannon-Nyquist theorem.
Sampling by modulation.
Aliasing.
Orthonormal basis interpretation.
Towards more general sampling spaces.
Shift-invariant spaces.
Spline functions.
Digital communication signals.
Multiple generators.
Refinable functions.
Gabor spaces.
Wavelet expansions.
Union of subspaces.
Signal model.
Union classes.
Stochastic and smoothness priors.
Exercises.
Riesz basis in SI spaces.
Riesz basis condition.
Examples.
Biorthogonal basis.
Expansion coefficients.
Alternative basis expansions.
Partition of unity.
Redundant sampling in SI spaces.
Redundant bandlimited sampling.
Missing samples.
Multiple generators.
Riesz condition.
Biorthogonal basis.
Exercises.
Sampling setups.
Sampling process.
Unconstrained recovery.
Predefined recovery kernel.
Design objectives.
Geometric interpretation.
Equal sampling and prior spaces.
Sampling in general spaces.
The direct-sum condition.
Unique recovery.
Computing the oblique projection operator.
Oblique biorthogonal basis.
Consistent recovery.
Recovery error.
Least squares recovery.
Minimax recovery.
Constrained recovery.
Minimal-error recovery.
Least squares recovery.
Minimax recovery.
Unified formulation of recovery techniques.
Recovery methods.
Papoulis’ generalized sampling.
Exercises.
Smoothness prior.
Least squares solution.
Minimax solution.
Examples.
Multichannel sampling.
Least squares solution.
Minimax-regret solution.
Comparison between least squares and minimax.
Stochastic priors.
The hybrid Wiener filter.
Constrained reconstruction.
Summary of methods.
Unified view.
Sampling with noise.
Constrained reconstruction problem.
Least squares solution.
Minimax MSE filters.
Summary of the different filters.
Bandlimited interpolation.
Unconstrained recovery.
Exercises.
Nonlinear sampling.
Nonlinear model.
Wiener-Hammerstein systems.
Bandlimited signals.
Reproducing kernel Hilbert spaces.
Subspace-preserving nonlinearities.
Equal prior and sampling spaces.
Iterative recovery.
Linearization approach.
Conditions for invertibility.
Newton algorithm.
Comparison between algorithms.
Recovery algorithms.
Uniqueness conditions.
Algorithm convergence.
Examples.
Exercises.
Resampling.
Bandlimited sampling rate conversion.
Interpolation by an integer factor I.
Decimation by an integer factor D.
Rate conversion by a rational factor I/D.
Rate conversion by arbitrary factors.
Interpolation formula.
Comparison with bandlimited interpolation.
Dense-grid interpolation.
Subspace prior.
Smoothness prior.
Stochastic prior.
Projection-based resampling.
Orthogonal projection resampling.
Oblique projection resampling.
Computational aspects.
Exercises.
Union of subspaces.
Multiband sampling.
Time-delay estimation.
Definition and properties.
Classes of unions.
Unique and stable sampling.
Rate requirements.
Xampling: compressed sampling methods.
Exercises.
Motivation for compressed sensing.
Sparsity models.
Normed vector spaces.
Sparse signal models.
Sensing matrices.
Null space conditions.
The restricted isometry property.
Coherence.
Uncertainty relations.
Sensing matrix constructions.
Recovery algorithms.
^recovery.
Greedy algorithms.
Combinatorial algorithms.
Analysis versus synthesis methods.
Recovery guarantees.
t\ recovery: RIP-based results.
i\ recovery: coherence-based results.
Instance-optimal guarantees.
The cross-polytope and phase transitions.
Guarantees on greedy methods.
Signal model.
Recovery algorithms.
Performance guarantees.
Infinite measurement vectors.
Summary and extensions.
Exercises.
Signal model.
Problem formulation.
Connection with block sparsity.
Uniqueness and stability.
Block RIP.
Block coherence and subcoherence.
Exponential recovery algorithm.
Convex recovery algorithm.
Greedy algorithms.
Block basis pursuit recovery.
Random matrices and block RIP.
Recovery conditions.
Extensions.
Proofs of theorems.
Dictionary and subspace learning.
Dictionary learning.
Subspace learning.
BCS problem formulation.
BCS with a constrained dictionary.
BCS with multiple measurement matrices.
Exercises.
Sparse union of SI subspaces.
Sub-Nyquist sampling.
Union of discrete sequences.
Reduced-rate sampling.
Application to detection.
Matched-filter receiver.
Maximum-likelihood detector.
Compressed-sensing receiver.
Multiuser detection.
Conventional multiuser detectors.
Reduced-dimension MUD (RD-MUD).
Performance of RD-MUD.
Exercises.
Sampling of multiband signals.
I/Q demodulation.
Landau rate.
Direct undersampling of bandpass signals.
Bandpass sampling.
Multiband sampling.
Universal sampling patterns.
Hardware considerations.
Modulated wideband converter.
MWC operation.
MWC signal recovery.
Collapsing channels.
Sign-alternating sequences.
Blind sampling of multiband signals.
Minimal sampling rate.
Blind recovery.
Multicoset sampling and the sparse SI framework.
Sub-Nyquist baseband processing.
Noise folding.
Hardware prototype of sub-Nyquist multiband sensing.
MWC designs.
Sign-alternating sequences.
Effect of CTF length.
Parameter limits.
Exercises.
Finite rate of innovation signals.
Shift-invariant spaces.
Channel sounding.
Other examples.
Periodic pulse streams.
Time-domain formulation.
Frequency-domain formulation.
Prony’s method.
Noisy samples.
Matrix pencil.
Subspace methods.
Covariance-based methods.
Compressed sensing formulation.
Sub-Nyquist sampling.
Coset sampling.
Sum-of-sincs filter.
Noise effects.
Finite and infinite pulse streams.
Multichannel sampling.
Modulation-based multichannel systems.
Filterbank sampling.
Noisy FRI recovery.
MSE bounds.
Periodic versus semiperiodic FRI signals.
General FRI sampling.
Sampling method.
Minimal sampling rate.
Least squares recovery.
Iterative recovery.
Sub-Nyquist radar.
Time-varying system identification.
Ultrasound imaging.
Exercises.
Matrix operations.
Matrix properties.
Special classes of matrices.
Eigenvalues and eigenvectors.
Singular value decomposition.
Linear equations.
Matrix norms.
Induced norms.
Schatten norms.
Probability density function.
Jointly random variables.
Continuous-time random processes.
Discrete-time random processes.
Sampling of bandlimited processes.