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

22.3 Rewriting difference equations 687
Example22.4 Write the following equations so that the highest argument of the dependent variable is
n+2.
(a) 3x[n +4] −2nx[n +2] = (n −1) 2 x[3] = 6 x[4] = −7
(b) z[n−2]+z[n−1]+z[n]=1+n z[0]=1 z[1]=0
Solution (a) Thehighestargument,n+4,mustbereplacedbyn+2,thatisnisreplacedbyn−2
throughout the equation.
3x[n +2] −2(n −2)x[n] = (n −3) 2 x[3] = 6 x[4] = −7
(b) The highest argument,n, isincreased ton +2,thatisnisreplaced byn+2.
z[n]+z[n+1]+z[n+2]=n+3 z[0]=1 z[1]=0
Example22.5 Write the following equations so that the highest argument of the dependent variable
isk:
s[k +2] −2s[k +1] +s[k]
(a) a[k +2] =
15
whereaisthe dependent variable.
(b) a[k +3] = l[k+3]+l[k+2]+l[k+1]+l[k]+l[k−1]
5
whereaisthe dependent variable.
Solution (a) The highest argument,k+2, must be replaced byk, that iskis replaced byk−2
throughout the equation:
a[k] =
s[k] −2s[k −1] +s[k −2]
15
(b) The highest argument,k+3, must be replaced byk, that iskis replaced byk−3
throughout the equation:
a[k] = l[k]+l[k−1]+l[k−2]+l[k−3]+l[k−4]
5
EXERCISES22.3
1 Writeeachequation sothat the highestargument of
the dependent variableisasspecified:
(a) p[k] −3p[k +1] = p[k −2],highestargumentof
the dependent variableisto bek+2.
(b) R[n−1]−R[n−2]−R[n−3]=n,R[0]=1,
R[1] = −2, highestargumentofthe dependent
variableisto ben.
(c) q[t] +tq[t −1] = 3q[t +1],q[1] = 0,
q[2] = −2,highestargumentofthe dependent
variableis to bet −1.
(d) T[m]+(m−1)T[m−2]=m 2 ,
T[0] =T[1] = 1,highestargumentofthe
dependent variableis to bem +2.
(e) y[k +1] −y[k +3] = (s[k +2] −s[k +4])/2,
whereyisthe dependentvariable andthe highest
argumentofthe dependent variableisto bek.