Question

Analog Signal Processing & Filter Design

Please answser problem 1 below for parts a, b, c, and d. Please show all work written out, and matlab code for checking the problems with matlab.

1. Find the roots of the following polynomials and write them in factored form:

            a. s² + 4s + 4 (Use the quadratic equation and check with Matlab)

            b. s² + 2s + 10 (Use the quadratic equation and check with Matlab)

            c. s³ + 3s² + 3s + 1 (check with Matlab)

            d. s⁵ + 5s⁴ + 5s³ + 5s² + 5s + 5 (check with Matlab)

Answer 1

Answer:Roots of quadrartic equation

(a.) s2 + 4s + 4

( b.) s2 + 2s + 10

( c.) s3 + 3s2 + 3s + 1

(d ) s5 + 5s4 + 5s3 + 5s2 + 5s + 5

Factored form

For irreducible expressions

>> syms x
>> factor(x^2 + 2*x++10, x, 'FactorMode', 'complex')

ans =

[ x + 1.0 - 3.0i, x + 1.0 + 3.0i]

>> syms x
>> factor(x^5 + 5*x^4+5*x^3+5*x^2+5*x+5, x, 'FactorMode', 'complex')

ans =

[ x + 4.0048532978402229828370560613657, x + 0.85551861251863940993560935374244 + 0.67897350403368406304597156425586i, x - 0.35794526143875090135413738442528 - 0.95836154763395495528090502329203i, x - 0.35794526143875090135413738442528 + 0.95836154763395495528090502329203i, x + 0.85551861251863940993560935374244 - 0.67897350403368406304597156425586i]