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生物医学工程
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山东大学
题号
一
2014-2015
二
三
四
学年
五
2
六
学期
七
八
数字信号处理（双语） 课程试卷（A）
九
十
总分
阅卷人
得分
阅卷人
2．(45 pts) A causal LTI system is described by the difference equation
y[n] 
得分
1
3
3
y[n  1]  y[n  2]  x[n]  x[n  1]  x[n  2] .
4
8
2
注：本试卷考试允许使用计算器。
Directions:
1) The answers of this test should be in English.
2) The full mark of this test is 100. The final course mark is based on this test (80%) and class
record mark (20%).
3) Tables of properties of Discrete-time Fourier transform, z-transform and DFT are supplied to
you on the last page.
4) Unless otherwise indicated, answers must be derived or explained, not just simply written down.
得分
(a)(10 pts) Determine the system function H(z) and ROC, and determine the poles and zeros of
the system function H(z),
(b)(5 pts) Is the system a minimum-phase system (including explanation)? Is the system stable
(including explanation)?
(c)(5 pts) Draw the 2nd-order direct form II signal flow graph for the system function H(z).
(d)(7 pts) Draw the signal flow graph of 2nd–order parallel-form structure for H(z).
(e)(10pts) Determine the impulse response h[n].
(f)(8 pts) Determine expressions for a minimum-phase system H min  z  and an all-pass
system H ap  z  such that H  z   H min  z  Hap  z 
阅卷人
1．(10 pts, 2 pts for each) Choose the best answer to fill in the blanks.
1) For a system for which the input and output satisfy a linear constant-coefficient difference
equation, if the system is initially at rest, then the system (
)
A. is definitely not LTI system; B. may not be LTI;
C. must be LTI but noncausal;
D. must be LTI and causal;
2) For an LTI system of which the system function H(z) has three positive real poles: a, b, c,
and the poles are all in the unit circle, then the system is (
)
A. absolutely stable;
B. absolutely causal;
C. maybe stable and causal;
D. maybe stable but not causal.
3) If a rational system function H(z) has 3 poles, then the number of zeros of H(z) is
A. 3;
B. uncertain;
C. less than 3;
D. more than 3.
4) Consider an 8-point sequence x1[n] and a 4-point sequence x2 [n] , to guarantee the
N x [n] and linear convolution x [n]* x [ n] to be identical, the
circular convolution x1[n] ○
1
2
2
circular convolution must have a length N of at least (
A. 11;
B. 12;
C. 8;
D. 4;
) points.
5) The inverse minimum-phase system is (
).
A. stable and causal;
B. stable but not causal;
C. causal but not stable;
D. neither stable nor causal
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山东大学
2014-2015
学年
2
学期
数字信号处理（双语） 课程试卷（A）
得分
阅卷人
3．(15 pts) We wish to design an FIR lowpass filter satisfying the
specifications
0.95  H  e jw   1.05, w  0.25
H  e jw   0.1,
0.35  w  
by applying a window w[n] to the impulse response hd [n] for the ideal discrete-time lowpass
filter with cutoff ωc = 0.3π. (given log10 0.05  1.3 )
(a) （6 pts）Which of the windows listed in the table can be used to meet this specification?
And give explanation.
(b) （9 pts）For each window that you claim will satisfy this specification, give the minimum
length M + 1 required for the filter.
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山东大学
2014-2015
学年
2
学期
数字信号处理（双语） 课程试卷（A）
4．(20 pts) Let X (e j ) denote the Fourier transform of the
sequence x[n]  ( 1 )n u[n] . Let y[n] denote a finite-duration
2
sequence of length 10, i.e., y[n]=0, n<0, and y[n]=0, n≥10. The 10-point
DFT of y[n], denoted by Y[k], corresponds to 10 equally spaced samples of X (e j ) , i.e.
j
Y (k )  X (e j 2 k /10 ) . (a) Calculate X (e ) ; (b) Determine y[n] from Y (k).
得分
阅卷人
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山东大学
得分
2014-2015
阅卷人
学年
2
学期
数字信号处理（双语） 课程试卷（A）
5．(10 pts) The figure below shows the flow graph for an 8-point
decimation-in-time FFT algorithm. The heavy line shows a path from
sample x[7] to DFT sample X[2].
(a) What is the "gain" along the path that is emphasized in the Figure above?
(b) How many other paths in the flow graph begin at x[7] and end at X[2]? Is this true in general? That is,
how many paths are there between each input sample and each output sample?
(c) Now consider the DFT sample X[2]. By tracing paths in the flow graph of the Figure above, show that
each input sample contributes the proper amount to the output DFT sample X[2]; i.e., find the b[n]
(n=0, 1, .. ,7) in the equation:
7
X  2   x[n]b[n]
n 0
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山东大学
2014-2015
学年
2
学期
数字信号处理（双语） 课程试卷（A）
Properties of the DFT
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