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
Vector Basics - Drawing Vectors/ Vector Addition. In this video, I discuss the basic notion of a vector, and how to add vectors together graphically as well as what it means graphically to multiply a vector by a scalar.
This lecture introduces the idea of a path integral (scalar line integral). Dr Chris Tisdell defines the integral of a function over a curve in space and discusses the need and applications of the idea. Plenty of examples are supplied and special attention is given to the applications of path integrals to engineering and physics, such as calculating the centre of mass of thin springs.
Logistics, Convex Functions, Examples, Restriction Of A Convex Function To A Line, First-Order Condition, Examples (FOC And SOC), Epigraph And Sublevel Set, Jensen's Inequality, Operations That Preserve Convexity, Pointwise Maximum, Pointwise Maximum, Composition With Scalar Functions, Vector Composition
Showing that, unlike line integrals of scalar fields, line integrals over vector fields are path direction dependent.
A basic introduction on how to integrate over curves (line integrals). Several examples are discussed involving scalar functions and vector fields. Such ideas find important applications in engineering and physics.
Showing that if a vector field is the gradient of a scalar field, then its line integral is path independent.
Showing that the line integral of a scalar field is independent of path direction.
Vector Addition and Scalar Multiplication, Example 2. In this video I add two vectors in component form and also sketch the vectors to illustrate how to add vectors graphically (very useful stuff!).
Intuition of the gradient of a scalar field (temperature in a room) in 3 dimensions.
Vector Addition and Scalar Multiplication, Example 1. In this video, we look at vector addition and scalar multiplication algebraically using the component form of the vector. I do not graph the vectors in this video (but do in others).
Orthogonal Projections - Scalar and Vector Projections. In this video, we look at the idea of a scalar and vector projection of one vector onto another.