Compressed Sensing & Sparse Filtering (Signals and Communication Technology)
This book is aimed at presenting concepts, methods and algorithms ableto cope with undersampled and limited data. One such trend that recently gained popularity and to some extent revolutionised signal processing is compressed sensing. Compressed sensing builds upon the observation that many signals in nature are nearly sparse (or compressible, as they are normally referred to) in some domain, and consequently they can be reconstructed to within high accuracy from far fewer observations than traditionally held to be necessary.
Apart from compressed sensing this book contains other related approaches. Each methodology has its own formalities for dealing with such problems. As an example, in the Bayesian approach, sparseness promoting priors such as Laplace and Cauchy are normally used for penalising improbable model variables, thus promoting low complexity solutions. Compressed sensing techniques and homotopy-type solutions, such as the LASSO, utilise l1-norm penalties for obtaining sparse solutions using fewer observations than conventionally needed. The book emphasizes on the role of sparsity as a machinery for promoting low complexity representations and likewise its connections to variable selection and dimensionality reduction in various engineering problems.
This book is intended for researchers, academics and practitioners with interest in various aspects and applications of sparse signal processing.
Why Read This Book
You will learn how modern compressed sensing and sparse-filtering techniques let you recover signals from far fewer measurements than classical sampling theory suggests, with concrete algorithms and practical design guidance. The book emphasizes both theory and application, so you can move from understanding RIP, incoherence and sparse priors to implementing L1 solvers, greedy methods and Bayesian estimators for audio, radar and communications problems.
Who Will Benefit
Engineers and advanced students working in DSP, audio/speech, radar or communications who need to design or implement sparse recovery and compressed-sensing solutions for real systems.
Level: Advanced — Prerequisites: Linear algebra (vector spaces, eigenvalues), probability and statistics, basic digital signal processing (filtering, FFT, spectral analysis), and familiarity with numerical computing (MATLAB/Python).
Key Takeaways
- Understand the theoretical foundations of compressed sensing (sparsity, incoherence, Restricted Isometry Property) and when undersampling is possible
- Apply and compare sparse recovery algorithms such as Basis Pursuit (L1 minimization), greedy methods (OMP, CoSaMP) and iterative shrinkage (ISTA/FISTA)
- Implement Bayesian sparse estimators using sparseness-promoting priors (Laplace, Cauchy, Student-t) and EM-style inference for noisy measurements
- Design compressed-sensing measurement schemes and reconstruction pipelines for audio, speech, radar and communications use cases
- Use convex optimization and numerical tools (ADMM, CVX/SPGL1) to solve large-scale sparse recovery problems efficiently
- Evaluate reconstruction performance with spectral analysis, error metrics, and robustness tests for noise, quantization, and model mismatch
Topics Covered
- 1. Introduction: Undersampling and the Promise of Sparsity
- 2. Sparse Representations and Transform Domains (Wavelets, DCT, Time–Frequency)
- 3. Mathematical Preliminaries: Linear Algebra, Probability, and Performance Metrics
- 4. Key Concepts: Incoherence, Sparsity Models, and the Restricted Isometry Property
- 5. Convex Approaches: L1 Minimization and Basis Pursuit
- 6. Greedy and Combinatorial Algorithms: OMP, CoSaMP, Subspace Pursuit
- 7. Iterative and Proximal Methods: ISTA, FISTA, and ADMM
- 8. Bayesian and Statistical Methods: Sparse Priors and Probabilistic Inference
- 9. Adaptive Sparse Filtering and Online Algorithms
- 10. Applications — Audio and Speech Processing
- 11. Applications — Radar and Remote Sensing
- 12. Applications — Communications Systems and Channel Estimation
- 13. Practical Issues: Noise, Quantization, Sampling Hardware and Simulations
- 14. Case Studies, Toolboxes, and Implementation Notes
- 15. Outlook: Emerging Directions and Open Problems
Languages, Platforms & Tools
How It Compares
Compared with Foucart & Rauhut's mathematically rigorous A Mathematical Introduction to Compressive Sensing (2013), this title is more application-driven and DSP-oriented; compared with Eldar & Kutyniok's edited Compressed Sensing: Theory and Applications (2012), it places greater emphasis on practical algorithms and engineering use cases.
