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Chapter 2: Modelling in the frequency domain

Ch2p1

Bit strings will be used to identify parts of this tutorial on the computer output. Bit strings arerepresented by the text enclosed in apostrophes, such as ab. Comments begin with % and

are ignored by MATLAB. Numbers are entered without any other characters. Arithmetic canbe performed using the proper arithmetic operator. Numbers can be assigned using a left-hand argument and an equals sign. Finally, we can find the magnitude andangle of a

complex number, Q usingabs(Q)and angle(Q), respectively.

Ch2p2

Polynomials in s can be represented as row vectors containing the coefficients. Thus3 2

1 7 3 23P s s s can be represented by the vector shown below with elements separated by a

space or comma. Bit strings can be used to identify each section of this tutorial.

Ch2p3

Running the previous statements causes MATLAB to display the results. Ending thecommand with a semicolon suppresses the display. Typing an expression without a left-handassignment and without a semicolon causes the expression to be evaluated and the result

displayed. EnterP2in the MATLAB Command Window after execution.

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Ch2p4

An F(s) in factored form can be represented in polynomial form. Thus 3 2 5 6P s s s

can be transformed into a polynomial using poly(V), where Vis a row vector containing the

roots of the polynomial and poly(V)forms the coefficients of the polynomial.

Ch2p5

We can find roots of polynomials using the roots(V)command. The roots are returned as

a column vector. For example, find the roots of 4 3 25 7 9 3 2 0s s s s

Ch2p6

Polynomials can be multiplied together using the conv(a,b)command (standing forconvolve). Thus, 3 2 4 3 25 ( 7 10 9)( 3 6 2 1)P s s s s s s s is generated as follows:

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Ch2p7

The partial-fraction expansion for ( ) ( ) / ( )F s b s a s can be found using the

[K, p, k ]= residue (b, a) command (K = residue; p = roots of denominator; k =

direct quotient, which is found by dividing polynomials prior to performing a partial fraction

expansion). We expand 2 2

( ) 7 9 12 / 7 10 100F s s s s s s s

as an example. Using the resultsfrom MATLAB yields

( ) 0.2554 0.3382 / 5.0 8.6603 0.2554 0.3382 / 5.0 8.6603

0.5280 / 7 0.0171 /

F s i s i i s i

s s

Ch2p8

Solve for y(t).