山东大学 2017-2018学年2学期数字信号处理（双语）课程试卷(A) 号 三 四 五大 七八九 总分 阅卷人 得分 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. 2) (5 pts)Shown in the folloing figure are eight different finite-duration sequences of 得分 卷人 (40 pts)Give brief answers for the following problems. which there is a minimum phase system and a maximum phase system).Each sequence is four points long.The magnitude of the Fourier transform is the same for all sequences. 家 Which of the sequences has all the zeros of its z-transform inside the unit circle?And 1)(5 pts)The folloing figure shows the pole-zero plots for four different LTI systems.Based which of the sequences has all the zeros of its z-transform outside the unit circle? on these plots,state whether or not each system is an all-pass system. 20.33 17.67 9.67 13.33 工m H1(z) Im H2(x) 331 65 15.33 -2067 (a) (b 04-3 21.33 Re 1 Re 13.33 m H3(z) Im H4(z) 0 器 Re Re 暴 第1项共5页
2017-2018 2 数字信号处理（双语） （A） 1 5 题号 一 二 三 四 五 六 七 八 九 十 总分 阅卷人 得分 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. 1．(40 pts) Give brief answers for the following problems. 1) (5 pts) The folloing figure shows the pole-zero plots for four different LTI systems. Based on these plots, state whether or not each system is an all-pass system. 2) (5 pts) Shown in the folloing figure are eight different finite-duration sequences ( of which there is a minimum phase system and a maximum phase system). Each sequence is four points long. The magnitude of the Fourier transform is the same for all sequences. Which of the sequences has all the zeros of its z-transform inside the unit circle? And which of the sequences has all the zeros of its z-transform outside the unit circle? 得分 阅卷人
山东大学2017-2018学年2学期数字信号处理（双语）课程试卷(A) 3)(5 pts)Determine the group delay for00,is the corresponding LTI system causal and stable?Justify your answer. 6)(10 pts)Draw the signal flow graph for an 8-point decimation-in-frequency FFT algorithm with normal ordering of the input time sequence x[n]and bit-reversed ordering of the output sequence X[k]. 器 第2项共5页
2017-2018 2 数字信号处理（双语） （A） 2 5 3) (5 pts) Determine the group delay for 0 < ω< π for the following sequence 4) (5 pts) If , is the corresponding LTI system causal and stable? Justify your answer. 5) (10 pts) The following figure is the impulse response of a FIR system, please determine the system function H(z) and draw the signal flow graph of direct form structure with the number of coefficient multipliers should be smallest. 6) (10 pts) Draw the signal flow graph for an 8-point decimation-in-frequency FFT algorithm with normal ordering of the input time sequence x[n] and bit-reversed ordering of the output sequence X[k]
山东大学2017-2018学年2学期数字信号处理（双语）课程试卷(A) 得分间港人 2.(15 pts)A causal LTI system has the system function Im 1+2z-1+z2 H(z)= （1+z-1)1- (a)Draw the signal flow graph of the system with 2ndorder transposed direct form II structure. (b)Find the impulse response ofthe system,h[n]. *15 (e)Find the output of this system,y[n,for the input=2. 家 得分侧人 4.(15 pts)Consider a causal continuous-time system with impulse response h(t)and system function ( 3+1 Assume the sampling time is T=Is. 1)Use the impulse invariance method to determine H(z)for a discrete-time system;Given the 器 rkion betwen the sep nd the pbee determine the step response s[n]of the system. 得分四卷人 3.(15 pts)A stable system with system function H(z)has the 2)If we use the bilinear transformation method to design H(z),determine H(z). pole-zero diagram shown in the following figure.It can be represented as the cascade of a stable minimum-phase system Hmin(z)and a stable all-pass system H(z).Determine the H(z),Hmin(z)and Hp(z),and draw the corresponding pole-zero plots for Hmin(z)and Hap(z).(Suppose H(z)=1 whenz=0.) 第3页共5页
2017-2018 2 数字信号处理（双语） （A） 3 5 2．(15 pts) A causal LTI system has the system function (a)Draw the signal flow graph of the system with 2 nd -order transposed direct form II structure. (b)Find the impulse response of the system, h[n]. (c)Find the output of this system, y[n], for the input 3．(15 pts) A stable system with system function H(z) has the pole-zero diagram shown in the following figure. It can be represented as the cascade of a stable minimum-phase system Hmin(z) and a stable all-pass system Hap(z). Determine the H(z), Hmin(z) and Hap(z), and draw the corresponding pole-zero plots for Hmin(z) and Hap(z). (Suppose H(z)=1 when z=0.) 4．(15 pts) Consider a causal continuous-time system with impulse response hc(t) and system function 2 1 ( ) 2 2 a s H s s s + = + + . Assume the sampling time is T =1 s. 1) Use the impulse invariance method to determine H(z) for a discrete-time system; Given the relation between the step response s[n] and the impulse invariance h[n]: =− = n k s n h k , determine the step response s[n] of the system. 2) If we use the bilinear transformation method to design H(z), determine H(z). 得分 阅卷人 得分 阅卷人 得分 阅卷人
山东大学2017-2018学年2学期数字信号处理（双语）课程试卷（) n].N=2 1n1n1n1 210124561 签 家 得分圆卷人 5.(15 pts)Suppose]is a periodic sequence with period N. Then:]is also periodic with period 3N.Let]denote the DFS 器 coefficients of considered as a periodic sequence with period N. and let]denote the DFS coefficients of considered as a periodic sequence with period 3N. (a)Express]in terms of]. (b) By explicitly calculating]and],verify your result in part (a)when] is as given in the following figure. 第4项共5页
2017-2018 2 数字信号处理（双语） （A） 4 5 5．(15 pts) Suppose x n[ ] is a periodic sequence with period N. Then: x n[ ] is also periodic with period 3N. Let X k[ ] denote the DFS coefficients of x n[ ] considered as a periodic sequence with period N. and let 3 X k[ ] denote the DFS coefficients of x n[ ] considered as a periodic sequence with period 3N. (a) Express 3 X k[ ] in terms of X k[ ]. (b) By explicitly calculating X k[ ] and 3 X k[ ] , verify your result in part (a) when x n[ ] is as given in the following figure. 得分 阅卷人
山东大学2017-2018学年2学期数字信号处理（双语）课程试卷(A) TABLE 2.3 FOURIER TRANSFORM PAIRS TABLE 2.2 FOURIER TRANSFORM THEOREMS Sequence Fourier Transform Sequence Fourier Transform 1. 1 X(ej) 2.m-o】 e-jono 好 y Y(el) 1.ax(n]+byin] ax(el)+bY(el) 3.1(-o0n←0∞) ∑2红e+2的 2.x(n-ndl (n an integer) e-junX(ejo) 4.an(al1 游 3.-d-n- 1-x可 10) TABLE8.2 or oo (if m<0) Properties of the DFT 5.a"uln] 1-a可 la lal Finite-Length Sequence(Length N) N-point DFT (Length N) 6.-ad-n-1] la<lal 1.x(n] XI周 2.x[n].x2ln] X:[).X2[k] TABLE 3.2 SOME z-TRANSFORM PROPERTIES 说 3.ax[n]+bx2in] aX因+bXl附 Section Reference Sequence Transform ROC 4.X Nx[((-k))N] xin] X(2) Rr 5.x[((n-m))N] WmX[内 a X R 6.Wyx[n] X(-)》N X2(z) Rn N- 3.41 网+br aX(z)+bx2(z) Contains R:nR X内X2R 3.42 x[n -no] 刷X() R.except for the possible 器 ∑n(mxI(n-mjN m= addition or deletion of N-1 the origin oroo 8.x[n]x2In] x()x2((k-6)N] 3.43 网 X(z/zo) lzoRr 3.4.4 网 Rr.except for the possible 是 addition or deletion of the origin or oo 第5页共5页
2017-2018 2 数字信号处理（双语） （A） 5 5 Properties of the DFT