學號
九十五學年度 第二學期 數位訊號處理
班級
期末考
姓名
1. Determine the inverse z-transform of the following;
a. H a 
1  z 1
1  0.8 z 1
b. H b 
1  0.8 z 1
1  0.9 z 1
2 z 2
1  0.9 z 1
H d  1  z 1  2 z 2  3z 3
c. H c 
d.
e. H e 
1  z 1
1
1
1  z 1  z  2
6
6
2. A linear time-invariant filter is described by the difference equation
y[n] = x[n]-3x[n-1]+3x[n-2]-x[n-3]
(a) if x[n]=δ[n], what is h[n]?
(b) The frequency response   , use (1-a)3 = 1-3a+3a2-a3
(c) What is the output if the input is x[n] = 10+4cos(0.6πn + π/3)?
(d) Use the superposition to find the output if x[n] = 10 + 4cos(0.6πn + π/3) +5δ[n]
3. Given an IIR filter defined by the difference equation
y[n]= -0.5y[n-1]+x[n]+x[n-1]
a. Determine the system function H(z).
b. What are the poles and zeros?
c. When the input to the system is x[n]=u[n], determine the output y[n]. Assume that y[n] is zero for n<0.
d. When the input to the system is x[n]=δ[n]-δ[n-1]+ δ[n-2], determine the output y[n]. Assume that y[n] is
zero for n<0.
4.The frequency response of a linear time-invariant filter is given by the formula
H ( w)  (1  e jw )(1  e j 2 / 3e jw )(1  e j 2 / 3e jw )
a. Write the difference equation that gives the relation between the input x[n] and the output y[n].
b. What is the output if the is x[n]=δ[n]?
c. If the input is of the form x[n]= Ae j e jwn , for what values of    w   will y[n]=0 for all n?