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Introductory Circuit Analysis (13th Edition)

Introductory Circuit Analysis (13th Edition)

13th Edition

ISBN:9780133923605

Author:Robert L. Boylestad

Publisher:Robert L. Boylestad

Chapter1: Introduction

Section: Chapter Questions

Problem 1P: Visit your local library (at school or home) and describe the extent to which it provides literature...

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Consider the system
* = Ax + Bu
y = Cx
1
B=
where A=
(a) Find a matrix F such that the eigenvalues of A+BF are both at zero.
(b) Find a matrix L such that the eigenvalues of A-LC are both at zero.

Expert Solution

Step 1

Given:

$x = A x + B u$

$y = C x$

where,

$A = [\begin{matrix} 0 & 1 \\ - 1 & - 2 \end{matrix}]$ , $B = {[\begin{matrix} 0 \\ 1 \end{matrix}]}_{2 \times 1}$ , $C = [\begin{matrix} 1 & 1 \end{matrix}]$

Step 2

a) For $A + B F$ eigen values to be zero 'F' matrix should be of $1 \times 2$

$F = [\begin{matrix} a & b \end{matrix}]$

$B F = [\begin{matrix} 0 \\ 1 \end{matrix}] [\begin{matrix} a & b \end{matrix}]$

$B F = [\begin{matrix} 0 & 0 \\ a & b \end{matrix}]$

$A + B F = [\begin{matrix} 0 & 1 \\ - 1 & - 2 \end{matrix}] + [\begin{matrix} 0 & 0 \\ a & b \end{matrix}]$

$A + B F = [\begin{matrix} 0 & 1 \\ a - 1 & b - 2 \end{matrix}]$

For eigen values of $A + B F$ to be zero,

$b - 2 = 0$

$b = 2 a n d a \in R$

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we are going to construct a (7, 3) linear block code by extending a (6, 3) linear blockcode. The generator matrix of the (6, 3) code is, G = 1 0 0 1 1 00 1 0 1 0 10 0 1 0 1 1 The (7, 3) code is extended from the (6, 3) code by appending one extra bit to into the codeword suchthat the Hamming weight of the resulting codeword is even. For example, for an input information sequence of [111], the (6, 3) code produces an codeword of [111000] which is then extended by appendingan extra bit “1” at the end to form the final codeword of [1110001]. The reason for appending a bit “1”(instead of bit “0”) is to make the Hamming weight of the resulting codeword in this example be aneven number. Another way to describe the (7, 3) code would be that it consists of a (6, 3) code with generator matrixG followed by another single parity check code with even parity a) What is the code rate of this (7, 3) code? b) What is the generator matrix of this (7, 3) code? c) What is the minimum Hamming distance of…
b)
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