Fundamentals of Circuits and Filters pptx

[...]... Processing Technical University of Munich Munich, Germany Stanisław Nowak Institute of Electronics University of Mining and Metallurgy Krakow, Poland Alison Payne Institute of Biomedical Engineering Imperial College of Science, Technology and Medicine London, England Tomasz W Postupolski Institute of Electronic Materials Technology Warsaw, Poland P K Rajan Department of Electrical and Computer Engineering... Institute of Biomedical Engineering Imperial College of Science, Technology and Medicine London, England P P Vaidyanathan Department of Electrical Engineering California Institute of Technology Pasadena, California Jiri Vlach Department of Electrical and Computer Engineering University of Waterloo Waterloo, Ontario, Canada Bogdan M Wilamowski Alabama Nano=Micro Science and Technology Center Department of. .. intention of this section is to provide a unified view of key concepts in the theory of linear circuits and filters, to emphasize interrelated concepts, to provide a mathematical reference to the handbook itself, and to illustrate methodologies through the use of many and varied examples This chapter begins with a basic examination of vector spaces over fields In relating vector spaces, the key ideas of linear... It is common to interpret I as an identity matrix of appropriate size, without explicitly denoting the number of its rows and columns The transpose MT 2 Fn3m of a matrix M 2 Fm3n is found by interchanging the rows and columns The first column of M becomes the first row of MT, the second column of M becomes the second row of MT, and so on The notations MT and M0 are also used If M ¼ MT the matrix is called... of rows equal to the number of rows in M and its number of columns equal to the number of columns of N Thus, for M 2 Fm3n and N 2 Fn3p, MN ¼ P 2 Fm3p Elements in the resulting matrix P may be determined by pij ¼ n X mik nkj (1:21) k¼l Matrix multiplication involves one row and one column at a time To compute the pij term in P, choose the ith row of M and the jth column of N Multiply each element in... 2 examines algebras in greater detail Examples  (F ¼ R): M ¼ 4 2   1 3 3 , N¼ 2 4 1 5 6  Fundamentals of Circuits and Filters 1-8  MN ¼ P ¼ 10 4 24 10 38 16    1 , and evaluate Equation To find p11, take the first row of M, [4 3], and the first column of N, 2 1.21: 4(1) þ 3(2) ¼ 10 Continue for all i and j NM does not exist because that product is not comformable Any matrix M 2 Fm3n multiplied... 0!V !W ~ L W !U!0 (1:12) (1:13) ~ Sequence (Equation 1.12) is exact if and only if L is monic, whereas Equation 1.13 is exact if and only if L is epic Further, let L: V ! W be a linear mapping between finite-dimensional vector spaces The rank of L, r(L), is the dimension of the image of L In such a case Fundamentals of Circuits and Filters 1-6 r(L) þ dim ( ker L) ¼ dim V (1:14) Linear operators commonly... removal of row i and column j Thus, the minors of m11, m12, m21, and m22 are m22, m21, m12, and m11, respectively To calculate the determinant of this M, (1) choose any row i (or column j), (2) multiply each element mik (or mkj) in that row (or column) by its minor and by (À1)iþk (or (À1)kþj), and (3) add these results Note that the product of the minor with the sign (À1)iþk (or (À1)kþj) is called the cofactor... element s gives t as a solution ( f(s) ¼ t), and each s 2 S as a first element in U appears exactly once 1-1 Fundamentals of Circuits and Filters 1-2 A binary operation is a function acting on a Cartesian product set S 3 T When T ¼ S, one speaks of a binary operation on S 1.2 Vector Spaces over Fields A field F is a nonempty set F and two binary operations, sum (þ) and product, such that the following properties... frequently encounter linear methodologies, and bits and pieces of mathematics that aid in problem solution are stored away Unfortunately, in memorizing the process of finding the inverse of a matrix or of solving a system of equations, the essence of the problem or associated knowledge may be lost For example, most engineers are fairly comfortable with the concept of a vector space, but have difficulty in . alt="" Fundamentals of Circuits and Filters The Circuits and Filters Handbook Third Edition Fundamentals of Circuits and Filters Feedback, Nonlinear, and. by Wai-Kai Chen University of Illinois Chicago, U. S. A. The Circuits and Filters Handbook Third Edition Fundamentals of Circuits and Filters CRC Press Taylor