1549312215-Complex_Analysis_for_Mathematics_and_Engineering_5th_edition__Mathews

9.1 • THE Z-TRANSFORM 349

(iii) Use partial fractions to expand Y (z ) in a sum of terms and look up

the inverse z-transform(s), using Table 9.1, to get

y[n] = 3-^1 [Y(z)].

Residue method

Perform steps (i) and (ii) of the above z..transform method. Then find the solu-

tion using the formula in step (iii),

k

y[n] = 3-^1 [Y(z)] = ERes[Y(z)zn-^1 ,z;J,

i = l

where zi, z2, .. ., Z k are the poles of f(z) = Y(z)z" -^1.

Convolution method

(i) Solve the homogeneous equation Yh[n + 1] - a Yh[n] = 0 and get

Yh[n] = c1a".

(ii) Use the transfer function H(z) = i-~.- 1 and construct the unit-

sample response h[n] = 3-^1 [H(z)) =an.

(iii) Construct the particular solution yp[n] = 3-^1 [X(z)H(z)J in convo-

lution form yp[n] = L:~o x [n - i]h[i) = L:~o x[n -i ]ai.

(iv) The general solution to the nonhomogeneous difference equation is

n

y[n] = Yh[n) + Yp[n] = c1a" + E x [n -i]ai.

i=O

(v) The constant Ct =Yo -x[OJ will produce the proper initial condition

y[O] =Yo· Therefore,

n

y[n] = (Yo -x[O])an + L x [n -i]ai.

i= O

Remark 9.6
The particular solution yp{n] obtained by using convolution has the initial con-

dition yp[O] = L:?= 0 x[O-i]h[i] = x[O)h[O] = x[O]. •

• EXAMPLE 9.9 Solve the difference equation y[n+ 1J-2y[nj = 3" with initial
  condition y[OJ = 2.

(a) Use the z..transform and Tables 9.1 and 9.2 to find the solution.

(b) Use residues to find the solution.