082-Engineering-Mathematics-Anthony-Croft-Robert-Davison-Martin-Hargreaves-James-Flint-Edisi-5-2017

686 Chapter 22 Difference equations and the z transform
(iv) 7(n −1)x[n −1] − (n +1)x[n −2] =0
(vi) and (vii)are already instandard form.
Hence wefind the following:
(i) Inhomogeneous
(ii) Homogeneous
(iv) Homogeneous
(vi) Inhomogeneous
(vii) Inhomogeneous
EXERCISES22.2
1 Foreach ofthe followingequations (a)--(e): (c)y[n −2] +y[n −1] +y[n] =n 2
(i) Statethe order ofthe equation. (ii)State whether
each equation islinear ornon-linear. (iii)Foreach
linear equation, state whether itis homogeneous or
inhomogeneous.
(a)n(3n +x[n]) =x[n −1]
2z[k −4]
(b)
z[k −3] =z[k−2]
(d) √ n +x[n] =x[n −2] +e[n], wherexis the
dependent variable.
(e) (2w[n−1]+1) 2 = w[n−2]+s[n−1]−s[n−2],
where w isthe dependent variable.
Solutions
1 (a) First order; linear;inhomogeneous
(b) Second order; non-linear
(c) Second order; linear;inhomogeneous
(d) Second order; non-linear
(e) Firstorder; non-linear
22.3 REWRITINGDIFFERENCEEQUATIONS
Sometimes an equation or expression can be written in different ways. At first sight, it
may appear there are two independent equations when in fact there is only one. Thus
weneedtobeabletorewriteequationssothatcomparisonscanbemade.Whengeneral
solutions of equations are to be found, usually the equation is first written in a standard
form. Soonceagain thereisaneedtorewrite equations.
Example22.3 Rewrite the equation sothat the highestargumentofthe dependentvariable isn +1.
x[n+3]−x[n+2]=2n
x[2]=7
Solution Thehighestargumentinthegivenequationisn +3;thismustbereducedby2ton +1.
To do thisnisreplaced byn −2.Theequation becomes
x[n +1] −x[n] =2(n −2) x[2] =7
Note,however,thattheinitialcondition,x[2] = 7,isnotchanged.Thisissimplystating
thatxhas a valueof7when the independentvariablehas a value of2.