#### Math Review

Basic Math

Inequalities, summation, calculus

Introduction to Probability

We first review the mathematical background on set theory and combinatorics which is required for classical probability. The formal definition of probability space is then defined. Various topics on basic probability are discussed including algebra and sigma algebra, Kolmogorov axioms, discrete and continuous random variables, independence, expectation and inequalities, transform methods (probability generating function, moment generating function, one-sided Laplace transform, characteristic function), conditional probability and expectation, convergence, law of large number, central limit theorem, function of random variables (transformations), random vectors.

#### Signals and Systems

My study on this topic started when I took classes offered by Prof. David Delchamps at Cornell as an undergrad. Later, I also had a chance to be a TA for his graduate-level class on theory of linear systems; most of the material in the second and third notes below are based on his lectures.

Signals and Systems

Fourier series, Fourier Transform, Sampling Theorem, Trig Identities.

Some 2D Fourier transform properties are also included.Continuous-time and Discrete-time State-space Linear Systems

Peano-Baker Series, Exponential of a matrix, Reachability, Controllability, Observability.Input-Output Linear Systems

Weighting pattern, Impulse response, Stability for time-invariant system.

#### CommunicationTheorys

- Analog Communication Systems: We review the basic properties of Fourier transform and then analyze analog communication systems.
- Communications, Detection, Estimation, and Signal Processing.

#### Communication Networks

Computer Networks (draft)

This article highlights several concepts usually taught in the introduction to computer network classes. It is a collection of personal notes from classes that I took and (later) worked as a teaching assistant at Cornell university. Most of the material is based on the book with the same title by Tanenbaum. It also contains material from the lectures given by Prof. Stephen B. Wicker, Prof. Toby Berger, and Prof. Zygmunt J. Haas.

#### Information Theory

Toby Berger, a renowned information theorist, is my advisor. However, my research focuses more on "communication" between neurons, and I don't have as much opportunities to use what I learned about information theory as I would love to. There are many useful identities, some of which are easy to prove but difficult to come up with in the first place. Therefore, I decided to collect them here. This article can also serve as a starting point for a second course on information theory.