没有合适的资源？快使用搜索试试~ 我知道了~

文库首页开发技术其它信号处理中的矩阵论（英文）

信号处理中的矩阵论（英文）

信号处理

4星 · 超过85%的资源需积分: 32 60 下载量 172 浏览量 2008-12-17 22:15:00 上传评论 11 收藏 8.2MB PDF 举报

温馨提示

试读

220页

Matrix Computations for Signal Processing 英文版的，对于想深入学习信号处理各种算法的人了解一些矩阵论方面的东西还是很必要的。

资源推荐

资源详情

资源评论

EE731 Lecture Notes: Matrix Computations

for Signal Pro cessing

James P. Reilly



Department of Electrical and Computer Engineering

McMaster University

February 2, 2000

0 Preface

This collection of ten chapters of notes will give the reader an intro duc-

tion to the fundamental principles of linear algebra for application in many

disciplines of mo dern engineering and science, including signal pro cessing,

control theory, pro cess control, applied statistics, robotics, etc. We assume

the reader has an equivalentbackground to a freshman course in linear alge-

bra, some introduction to probability and statistics, and a basic knowledge

of the Fourier transform.

The rst chapter, some fundamental ideas required for the remaining p ortion

of the course are established. First, we lo ok at some fundamental ideas of

linear algebra such as linear indep endence, subspaces, rank, nullspace, range,

etc., and how these concepts are interrelated. The idea of auto correlation,

and the covariance matrix of a signal, are then discussed and interpreted.

In chapter 2, the most basic matrix decomposition, the so{called eigendecom-

p osition, is presented. The fo cus of the presentation is to give an intuitive

insight into what this decomp osition accomplishes. We illustrate how the

eigendecomp osition can be applied through the Karhunen-Lo eve transform.

In this way, the reader is made familiar with the imp ortant prop erties of this

decomp osition. The Karhunen-Loeve transform is then generalized to the

broader idea of transform co ding.

In chapter 3, we develop the

singular value decomposition

(SVD), which is

closely related to the eigendecomp osition of a matrix. We develop the rela-

tionships between these two decomp ositions and explore various prop erties

of the SVD.

Chapter 4, etc.

1 Fundamental Concepts

The purpose

of this lecture is to review imp ortant fundamental concepts in

linear algebra, as a foundation for the rest of the course. We rst discuss the

fundamental building blo cks, such as an overview of matrix multiplication

from a \big block" p ersp ective, linear independence, subspaces and related

ideas, rank, etc., up on which the rigour of linear algebra rests. We then

discuss vector norms, and various interpretations of the matrix multiplication

op eration. We close the chapter with a discussion on determinants.

1.1 Notation

Throughout this course, we shall indicate that a matrix

is of dimension



, and whose elements are taken

from the set of real numbers, by the

notation



. This means that the matrix

b elongs to the Cartesian

pro duct of the real numbers, taken



times, one for each element of

In a similar way, the notation

2 C

mn

means the matrix is of dimension



, and the elements are taken from the set of complex numb ers. By the

matrix dimension \



", we mean

consists of

rows and

columns.

Similarly, the notation

2 <

(

) implies a vector of dimension

whose

elements are taken from the set of real (complex) numbers. By \dimension

of a vector", we mean its length, i.e., that it consists of

elements.

Also, we shall indicate that a scalar

is from the set of real (complex)

numb ers by the notation

2 <

(

). Thus, an upp er case bold character

denotes a

matrix

, a lower case bold character denotes a vector, and a

lower

case non-b old character denotes a scalar.

By convention, a vector by default is taken to be a

column

vector. Further,

for a matrix

, we imply that its

th column is

. We also imply that its

th row is

, even though this notation may be ambiguous, since it may

also be taken to mean the transp ose of the

th column. The context of the

discussion will help to resolve the

ambiguit

1.2 \Bigger-Blo ck" Interpretations of Matrix Multi-

plication

Let us dene the matrix pro duct



(1)

The three interpretations of this op eration now follow:

1.2.1 Inner-Pro duct Representation

and

are column vectors of the same length, then the scalar quantity

is referred to as the

inner product

and

. If we dene

2 <

the

th rowof

and

as the

th column of

, then the element

is dened as the inner pro duct

. This is the conventional small-blo ck

representation of matrix multiplication.

1.2.2 Column Representation

This is the next biggest{blo ck view of matrix multiplication. Here we lo ok

at the op eration column{wise. The

th column

may be expressed

as a linear combination of columns

with co ecients which are the

elements of the

th column of

. Thus,

 j

:::n:

(2)

This op eration is identical to the inner{pro duct representation ab ove, except

we form the pro duct one column at a time. For example, if we ev

aluate

only the

th element of the

th column

, we see that (2) degenerates into

. This is the inner pro duct of the

th row and

th column of

and

resp ectively, which is the required expression for the (

p j

)th element

1.2.3 Outer{Pro duct Representation

This is the largest{blo ck representation. Let us dene a column vector

and a rowvector

. Then the

outer pr

oduct

and

is an



matrix of rank one and is dened as

Nowlet

and

b e the

th column and rowof

and

resp ectively. Then

the pro duct

may also b e expressed as

(3)

By lo oking at this op eration one column at a time, we see this form of

matrix multiplication performs exactly the same op erations as the column

representation ab ove.

1.2.4 Matrix Pre{ and Post{Multiplication

Let us now lo ok at some fundamental ideas distinguishing matrix pre{ and

p ost{multiplication. In this respect, consider a matrix

pre{multiplied

to give

. (All matrices are assumed to have conformable di-

mensions). Then we can interpret this multiplication as

op erating on the

columns

to givethecolumns of the pro duct. This follows b ecause each

column

of the pro duct is a transformed version of the corresponding col-

umn of

 i.e.,

. Likewise, let's consider

post{multiplied

by a

matrix

to give

. Then, we interpret this multiplication as

op-

erating on the

rows

, b ecause eachrow

of the pro duct is a transformed

version of the corresp onding row of

 i.e.,

, where we dene

as the

th row of

Example:



Consider an orthonormal matrix

of appropriate dimension. Weknow

that multiplication by an orthonormal matrix results in a rotation op-

eration. The operation

rotates each column of

. The op eration

rotates each row.

There is another way to interpret pre{ and p ost{multiplication. Again con-

sider the matrix

pre{multiplied by

to give

. Then according

to (2), the

th column

is a

linear combination

of the

columns

whose coecients are the

th column of

. Likewise, for

, we can

saythatthe

th row

is a

linear c

ombination

of the

rows

,whose

co ecients are the

th row of

Either of these interpretations is equally valid. Being comfortable with the

representations of this section is a big step in mastering the eld of linear

algebra.

剩余219页未读，继续阅读

评论收藏

内容反馈

资源评论

资源反馈

评论星级较低，若资源使用遇到问题可联系上传者，3个工作日内问题未解决可申请退款~

zjaixy

2013-05-26

理论要求比较高，而且是英文版的，还要求英语比较好
linuxluo2011

2011-10-05

纯数学的东西，要有高等数学，矩阵论的基础
crazy31415926

2016-03-31

一本很专业的参考书，并且对于提升英文阅读能力很有帮助
caicailoving

2013-09-28

不错的资源。就是比较难懂

石膏灰

粉丝: 87
资源: 18

上传资源快速赚钱

我的内容管理展开

我的资源快来上传第一个资源

我的收益

登录查看自己的收益

我的积分登录查看自己的积分

我的C币登录后查看C币余额

我的收藏

我的下载

下载帮助

前往需求广场，查看用户热搜

信号处理中的矩阵论（英文）

矩阵论（英文课件全）.zip

矩阵论（英文版）matrix

矩阵在信息处理中的应用

Matrix Computations for Signal Processing (X.L.Ma)

矩阵理论在计算机视觉专业方面的应用

Matrix_Computations_for_Signal_Processing

矩阵论在多矩阵论在多变量控制系统中可控性及客观性方面的应用

机器学习中的矩阵论

矩阵论 课后习题答案_2019矩阵论第二版课后题

矩阵论考试试题及答案

矩阵论 《矩阵论》戴华-科学出版社-课后习题答案

矩阵论讲义(西工大版).rar_MSU_矩阵论_矩阵论应用_矩阵论西工大_矩阵论讲义

矩阵论课后习题答案

西北工业大学矩阵论试题

矩阵论 程云鹏 第三版 答案

矩阵论习题解答矩阵论习题解答矩阵论习题解答

南京邮电大学 矩阵论 PPT

信号处理中的非负矩阵分解

华中科技大学矩阵论课件 经典的矩阵论教程

矩阵论 戴华 课后答案 研究生数学

矩阵论期末试题和复习资料_中科院矩阵论期末,中国科学院矩阵论

矩阵论复习知识PDF.pdf

矩阵论札记

Qt 5实现串口调试助手 （源工程文件、0积分下载）

【SystemVerilog】路科验证V2学习笔记（全600页）.pdf

AutoSAR标准协议4.2.2

光伏-储能并网系统仿真.rar

NPPJSONViewer.zip

GD32替换STM32注意事项.pdf

最新资源

矩阵论课后习题答案_2019矩阵论第二版课后题

矩阵论《矩阵论》戴华-科学出版社-课后习题答案

矩阵论程云鹏第三版答案

南京邮电大学矩阵论 PPT

华中科技大学矩阵论课件经典的矩阵论教程

矩阵论戴华课后答案研究生数学

Qt 5实现串口调试助手（源工程文件、0积分下载）