This course is the second of a two-term sequence. The focus is on coding techniques for approaching the Shannon limit of additive white Gaussian noise (AWGN) channels, their performance analysis, and design principles. After a review of Principles of Digital Communication I and the Shannon limit for AWGN channels, the course begins by discussing small signal constellations, performance analysis and coding gain, and hard-decision and soft-d...more
This course serves as an introduction to the theory and practice behind many of today's communications systems. 6.450 forms the first of a two-course sequence on digital communication. The second class, 6.451, is offered in the spring. Topics covered include: digital communications at the block diagram level, data compression, Lempel-Ziv algorithm, scalar and vector quantization, sampling and aliasing, the Nyquist criterion, PAM and QAM m...more
A special event to mark the publication of Professor Barrow's new book, 'The Book of Universes'. This is a book about universes, a story that revolves around a single unusual and unappreciated fact: that Einstein’s famous theory of relativity describe universes – entire universes. Not many solutions of Einstein’s tantalizing universe equations have ever been found, but those that have are all very remarkable. Some of them describe universe...more
Model Predictive Control, Linear Time-Invariant Convex Optimal Control, Greedy Control, 'Solution' Via Dynamic Programming, Linear Quadratic Regulator, Finite Horizon Approximation, Cost Versus Horizon, Trajectories, Model Predictive Control (MPC), MPC Performance Versus Horizon, MPC Trajectories, Variations On MPC, Explicit MPC, MPC Problem Structure, Fast MPC, Supply Chain Management, Constraints And Objective, MPC And Optimal Trajectori...more
Stochastic Model Predictive Control, Causal State-Feedback Control, Stochastic Finite Horizon Control, 'Solution' Via Dynamic Programming, Independent Process Noise, Linear Quadratic Stochastic Control, Certainty Equivalent Model Predictive Control, Stochastic MPC: Sample Trajectory, Cost Histogram, Simple Lower Bound For Quadratic Stochastic Control, Branch And Bound Methods, Methods For Nonconvex Optimization Problems, Branch And Bound A...more
Linear Systems: Basic Definitions, Direct Proportionality As Example, Special Cases Of Linear Systems, Eigenvectors And Eigenvalues, The Spectral Theorem And Finding A Basis Of Eigenvectors, Matrix Multiplication = Only Example Of Finite Dimensional Linear Systems, Integration Against A Kernel Generalizing Matrix Multiplication, Example: The Fourier Transform
Bias/variance Tradeoff, Empirical Risk Minimization (ERM), The Union Bound, Hoeffding Inequality, Uniform Convergence - The Case of Finite H, Sample Complexity Bound, Error Bound, Uniform Convergence Theorem & Corollary
State-action Rewards, Finite Horizon MDPs, The Concept of Dynamical Systems, Examples of Dynamical Models, Linear Quadratic Regulation (LQR), Linearizing a Non-Linear Model, Computing Rewards, Riccati Equation
Theory of Generalization - How an infinite model can learn from a finite sample. The most important theoretical result in machine learning.
A look ahead, final review, other statistics courses, regression example, sampling from a finite population example.