1549312215-Complex_Analysis_for_Mathematics_and_Engineering_5th_edition__Mathews

354 CHAPTER 9 • Z.TRANSFORMS ANO APPLICATIONS

The particular solution is calculated with t he formula Yp [n) = 3 -^1 (X ( z )H ( z)]

as follows:

I I

_1 [ bz 1 ]

Yp n =^3 (z - 1 )(1- az-1)

3 _

(^1) [b b a
2
b ]
= - (-l+a)(-l+z) + (-l+ a)(-a+z)
b al+nb
= M[n) - - u [n - l ] + --u(n - 1)

a - 1 a-1

which can be simplified to obtain

In convolution form Yv(n) = x[n) * h[n) = E:= 0 x[n - i]h(i), and we

have

n (al+n_l)b

Yp(n] =~=)a•=.

i=O a - 1

The particular solution Yp(n) obtained by using convolution has the

initial condition yp[O) E~=O x[O - i]h[i] = x(O]h(OJ = x[O] = b. The

total solution to the nonhomogeneous difference equat ion is

n

y[n] = y1;[n] + yp[n] = c1an + l:)ai

i=O

Now we compute Yo= y[OJ = C1a^0 + <:-::_Vb = C1 +band solve for the

constant c 1 = Yo - b, which will produce the proper initial condit ion.

Therefore,

(al+n_l)b

y[n] =(Yo -b)an + ,

a - 1

which can be manipulated to yield y(n] = yo an + a,.~/ b.

An illustration of the dosage model using the parameters a = ~, b = 1

and initial condition y 0 = 0 is shown in Figure 9.1.