After reviewing Fourier analysis, Sampling theorem and Discrete time Fourier transform as the common languages in signal processing, the instructor first introduces first several classical solutions, e.g., Generalized inverses and Best Linear Unbiased Estimator, for the linear inverse problems based on the orthogonal projection theorem and the singular value decomposition. Next, the instructor presents a unified view for many adaptive filtering algorithms and/or Online learning algorithms based on the convex projection theorem and fixed point theorems in Hilbert space. The instructor also introduces powerful ideas in fixed point approximations that are common principles in convex optimization algorithms and their applications to inverse problems. Finally, the instructor introduces several advanced topics, e.g., Hierarchical convex optimization, Subspace tracking and Phase unwrapping, etc.
Along with rapid progress in the computational technology, many powerful algorithms in modern signal processing have been established in the last two decades.
By the end of this course, students will be able to:
1) understand such algorithms in unified ways.
2) understand mathematical ideas behind such algorithms.
3) understand how such algorithms can be applied to real world problems.
Fourier analysis, Sampling theorem, Discrete time Fourier transform, Linear inverse problems, Generalized inverse, Best Linear Unbiased Estimator, Adaptive filtering, Convex optimization, Fixed point theorems
✔ Specialist skills | Intercultural skills | Communication skills | ✔ Critical thinking skills | ✔ Practical and/or problem-solving skills |
After reviewing Fourier analysis, Sampling theorem and Discrete time Fourier transform as the common languages in signal processing, the instructor explains systematically powerful signal processing algorithms and their applications to modern inverse problems and adaptive learning problems.
Course schedule | Required learning | |
---|---|---|
Class 1 | Introduction to Signal Processing -From classic and modern | What is Signal processing ? |
Class 2 | Review of Fourier analysis | Explain about the basic ideas in Fourier analysis |
Class 3 | Sampling Theorem, DFT and FFT | How are Fourier series expansion and Fourier transform used to derive Sampling theorem ? |
Class 4 | Hilbert space - Mathematical stage for modern approach | What is Hilbert space ? |
Class 5 | Projection theorem, Generalized inverses, Best linear unbiased estimator | Explain about |
Class 6 | Singular Value Decomposition (SVD) and low rank estimator | Explain about |
Class 7 | Fixed point theorems and convexity | Explain about Fixed point theorems and convexity |
Class 8 | Adaptive learning based on projection theorems 1: Algorithms | Explain about Adaptive learning based on projection theorems. |
Class 9 | Adaptive learning based on projection theorems 2: Applications to classification problems. | Explain about applications to classification problems. |
Class 10 | Fixed point algorithms for nonexpansive operators. | Explain about Fixed point algorithms for nonexpansive operators. |
Class 11 | Convex optimization and its image recovery applications | Explain about convex optimization and its image recovery applications |
Class 12 | Inverse problems and hierarchical optimization | Explain about Inverse problems and hierarchical optimization |
Class 13 | Subspace tracking 1: Algorithms of Subspace tracking | Explain about Algorithms of Subspace tracking |
Class 14 | Subspace tracking ２： Applications to Directions of arrivals estimation | Explain about applications to Directions of arrivals estimation |
Class 15 | Phase unwrapping: Algorithms and applications | Explain about phase unwrapping: and applications |
None
Handouts will be distributed at the beginning of class if necessary.
Learning achievement is evaluated by the quality of the written reports.
Firm understanding is required on linear algebra and multivariate calculus.
isao[at]sp.ce.titech.ac.jp
Contact by e-mail in advance to schedule an appointment.