356 CHAPTER 9 • Z-TRANSFORMS AND APPLI CATIONS
(c) Time forward 1 tap: 3[xn+1) = z(X(z) - xo).
(d) Differentiation: 3[nx.,J = -zX '(z).- Find x{n] = 3-^1 [X(z)] using two methods: (i) partial fractions and Table 9.1, and
(ii) using residues.
( ) X( ) - z2 - •2
a z - •'--t•+3 - (• 1)(. a) -- (1-• - 1)(1^1 -a.-•)·(b} X(z} - » - •
(^2) - i
- z2-4.z+ 4 - (z-~)2 - (1-~z-l )^2 '
( } X( ) •(^2) .. I • I •
c z = •"+I = ( •-i)(z+i) = 2 • - i + 2 •+'.
- Find x(n] = 3-^1 (X(z}] using two methods: (i} partial fractions and Table 9.1, and
(ii) using residues.
( ) X( )
_ Sz _ z _ I
a z - 5z=2 - .-_a.. - 1 _1,-1 ·(b) X(z) = 2s,>25;;,+12 = ·•-f:+M
1(t-i• 1)(1-~• 1 )"
(c) X(z) _ wz(^2) 2z2 2 _ 2
- 2 s.•- 9 - ~ - i-?!i• 2 - (1-~.-1)(1+i• t}"
4z2 z2 z2 1
(d} X(z} = 4z2 + 1 = z2 + ~ = (z - M (z + ~i) - 1 + ~Z-^2
110. Use direct substitution and trigonometric identities to show the following:(a.) y{n] = x[n] + x[n - 2] will filter out the sequences x [n ] = cos(~n) and
x [n] = sin( In), and(b} y(n] = x[n] - vl2x[n - 1] + x [n-2] will filter out the sequences x[n] =
cos( in} and x[n] =sin( in).
- Solve the difference equation y [n + l] = ay[n J+b with t he initial condition y[O] =yo.
Use recursion (and mathematical induction) to find the solution. That is, compute
y[l] = yoa + b, y[2] = yoa^2 + (1 + a)b, y(3] = yoa^3 + (1 + a + a^2 )b, t hen find the
general term. - Solve the exponential growth model y[n + 1] = (1 + r)y[nJ using the parameter
r = fa and initial condition y(O] = 100.
(a} Use the z-tra.nsfor m and Tables 9.1-9.3 to find t he solution.
(b) Use residues to find t he solution.13. In the exponential growth model y[n+ 1] = (1 + r)y[n] use the parameter r = -~
and initial condition y[O] = 1000.