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Unformatted text preview: ECE 421 - Sum 2010 Notes Set 10: Vectors and Matrices 1 INTRODUCTION Vectors and matrices are very important math tools in modern digital signal processing systems and digital communication systems. In this course we will find that vectors and matrices are very useful for understanding a common method of implementing two-way, simultaneous transmission. This particular application we examine is called Echo Cancel- lation. The geometrical concept of orthogonal vectors will also be important for under- standing Linear Prediction Coding (LPC), which we will also study later. Methods such as LPC are commonly used in speech and audio digital compression systems. The definition of a vector in this course is somewhat different from what you may have learned in physics or electromagnetics. In those courses, vectors are frequently defined as a structure in a three-dimensional space, where the three dimensions correspond to x , y , and z in Cartesian coordinates. In this course, and in much of digital communications and digital signal processing, a vector is a special case of a matrix. We will often refer to this structure as a signal vector, to distinguish it from a three-dimensional vector, like force or the electric field. Since matrices can have high dimensions (like a 20 50 matrix) a signal vector can like- wise exist in a high-dimension vector space. We will develop a way of looking at a sampled time-domain signal as a geometrical vector. This vector has length, an angle with respect to the origin, and other geometrical properties. When viewed in this manner, the powerful techniques of matrix analysis become available for designing digital signal processing and digital communication systems. We will see an introduction to this approach in this set of Notes. ECE 421 - Sum 2010 Notes Set 10: Vectors and Matrices 2 SIMULTANEOUS EQUATIONS: MATH One of the first matrix applications we see is in solving a set of simultaneous equations. Suppose we have two equations in two unknowns, v 1 and v 2 . For example: 3 v 1 2 v 2 = 7 v 1 4 v 2 = 5 1 There are many ways of solving (1) for v 1 and v 2 . One way is to rewrite (1) in terms of a matrix M and vectors v and w : M v = w 2 where in equation (2) we have defined M = 3 2 1 4 ; v = v 1 v 2 ; w = 7 5 3 a ; 3 b ; 3 c We can solve (3) analytically by premultiplying both sides of equation (3) by the inverse matrix M 1 . By definition of the inverse matrix we know that M 1 M = I , where I is the identity matrix. Premultiplying both sides of equation (3) by the inverse matrix M 1 then gives v = M 1 w 4 The inverse of a 2 2 matrix has a useful formula for its inverse. If M = a b c d 5 a then the inverse is given by M 1 = 1 ad bc d b c a 5 b You can easily verify (5b) by multiplying M in (5a) by M 1 in (5b) and showing that the result is the identity matrix. ECE 421 - Sum 2010...
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