610 ANSWERS
3. 3[xn] = L::°=O XnZ-n = 0 +Oz-I+ Oz-Z + ... + Ozm-l + L::"=m. lz- n,
3[ J- m"oo ( -l)n - m 1 z - m z . •-m
Xn = Z L.m=O Z = Z 1 -.-' - >=I - z-1 ·
- 3 -1( ';=tb• 1-a.tl ) - J = 3-1( a=rab z=a:-z a -b lz- 1 z ) -- aab -1 3-1( z=a z ) - a - 1 b 3-1( ;=]• ) )
3-1 (..£!.... I ) _ ....!!!!... n __ b_l = (a•+n_l)b
z - 11 - az-I - a- 1 a a-1 a - 1 '7a. 3[xn] = L::::'=o Xnz- " valid for iz l > R1, and 3[Yn] = L::::'=o YnZ-n valid for
iz l > R2. Hence, 3[cxn + dyn] = L::::°= 0 (cxn + dyn)z-n = c L::°=o Xnz-n +
dL:~=oYnZ-n = cX(z) + dY(z) is valid for lzl > R = ma.x{R1>R2}.
7 c. 3[Xn+1J = L::::'=o Xn+1Z- n = z L:::'=o Xn+1Z-n-l = z L::°=i x,.z-n = z(xo +
L::::'=i Xnz- n - xo) = z(L:::°=oxnz- n - Xo) = z(X(z) -xo).
9a. Using a table of z-transforms we get x[n] = 3-^1 [ 5 ;: 2 ] = 3-^1 [, 2 i ] = (~) ".
9c. Using a table of z-tra.nsforms we get
[ J 3 -1[ so.• J 3 - 1[ 2» J 3 -1^1 2z(^2) ]
X n = 25z2- 9 = ~ = (•-!)(•+~) =^3 - 11 z-! z + •+i z I =
3 - '1.:i 1 + 3 - '1.~ 11 = G)" + (5
3
)".
Using residues we get
R Ix( ) n- 1 3] l' ( 3) 2z(^2) n - 1 l ' 2z2 n - 1
es z z , 5 = im.-~ z- 5 (•-~H>+~)z = 1m·-~ (•+!)z -
5 3 (3) 5 +n- l - (3)n 5 ,an d
R es Ix( z z ) n - 1 , - 5 3] = l' tmz--~ ( z + 3) 5 (•-n2z' (>+i)z n - 1 _ -
im·--~ (•-_1L i)z n - 1 - - 3 s ( -s 3)n- 1 _ - (~s. )"
Therefore, x [n ] = Res[X(z)z"-^1 , ~] + Res[X(z)zn-^1 , - ~] = G)" + (5^3 )".
Use t he recursive formula yin + l] = ay[n] + b to find the solution with
initial condition y[OJ = y 0. The first few terms look like y[l] = yoa + b,
y[2] = a(yoa + b) + b = y 0 a^2 + (1 + a)b, yl3J = a(yoa^2 + (1 + a)b) + b =
yoa^3 + (1 + a + a^2 )b.
Assume that vln - 1] has the form y[n - l ] = y 0 a"-^1 + (1 +a+ a^2 + ... +
an-^3 +an-^2 )b, then the next step is yin] = ay[n-1] + b = a(yoan-I + (1 +a+
a^2 + ... + a"-^3 +an-^2 )b) +b, yin]= yoan+ (1 + a + a^2 + ... + an-Z +an-l )b =
Yoa n + ("nL-i=O - 1 a i)b = yoa n + an-lb a.:1.
Therefore, we have established the formula by mathemat ical induction.Note: If we observe that x\n -i) = b then the equation y[n] = yoan +
(L:::-~ ai)b can be writ ten as yin] = Yoan - x[O]a" + L:::-~ xln - i]ai +
x[n -n ]an. Now use c 1 = Yo - x [OJ and combine terms to get yin] =
(vo-x\O])a"+ L:::':o x ln - i]ai, which is the convolution form of the solution.