山东大学 2015-2016学年2学期数字信号处理（双语）课程试卷(A) 号 三 四 五 七八 九 总分 阅卷人 得分 侧卷人 2.(45 pts)The impulse response h[n]of an LTI system is 得分 M=12.5×n+36.5×0.2"n-48×0.4". Directions图 (1)(10 pts)Determine the system function H(z)of the LTI system,determine the poles and 1)The answers of this test should be in English. zeros of H(z). 好 (2)(8 pts)Determine the ROC of H(z).Is the system stable?Why?. 2)The full mark of this test is 100.The final course mark is based on this test (80%)and class (3)(7 pts)Determine the difference equation relating input x[n]and output y[n]. record mark (20%). (4)(5 pts)Draw the 2nd-order direct form II signal flow graph for the system function H(Z). 3)Tables of properties of Discrete-time Fourier transform,z-transform and DFT are supplied to (5)(5 pts)Draw the signal flow graph of 2nd-order parallel-form structure for H(z). you on the last page. (6K10 pts)Determine expressions for a minimum-phase system Hlm()and an all-pass system 4)Unless otherwise indicated,answers must be derived or explained,not just simply written down. Hap(z)such that H()=H(2H() 得分 阿卷人 (10 pts,2 pts for each)Choose the best answer to fill in the blanks. 的 1)For a causal LTI system for which the input and output satisfy a linear constant-coefficient 游 difference equation,the output of the system can be uniquely specified if the input is given. ( A.The statement is true;B.The statement is false 2)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 家 A.is definitely not LTI system;B.may not be LTI; C.must be LTI but noncausal:D.must be LTI and causal: 3) Consider an L-point sequence x[n and a P-point sequence x,[n,for the circular convolution x [n]x,In]and linear convolutionxn]x,[n]to be identical,the circular convolution must have a length N of at least )points. A.L+P+1; B.L+P.1: C.L+P: D.L; 4)The all-pass system is 器 A.absolutely stable: B.maybe not stable; C.absolutely causal; D.neither stable nor causal 5)A Type IV FIR Linear-Phase System can be used as a( A.low-pass filter; B.Band-stop filter, 暴 C.high-pass filter, 第1负共4页
2015-2016 2 数字信号处理（双语） （A） 1 4 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．(10 pts, 2 pts for each) Choose the best answer to fill in the blanks. 1) For a causal LTI system for which the input and output satisfy a linear constant-coefficient difference equation, the output of the system can be uniquely specified if the input is given. ( ) A. The statement is true; B. The statement is false 2) 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; 3) Consider an L-point sequence 1 x n[ ] and a P-point sequence 2 x n[ ] , for the circular convolution 1 x n[ ] ○N 2 x n[ ] and linear convolution 1 2 x n x n [ ]* [ ] to be identical, the circular convolution must have a length N of at least ( ) points. A. L+P+1; B. L+P-1; C. L+P; D. L; 4) The all-pass system is ( ). A. absolutely stable; B. maybe not stable; C. absolutely causal; D. neither stable nor causal 5) A Type IV FIR Linear-Phase System can be used as a ( ). A. low-pass filter; B. Band-stop filter; C. high-pass filter; 2．(45 pts) The impulse response h[n] of an LTI system is [ ] 12.5 [ ]+36.5 0.2 [ ] 48 0.4 [ ] n n h n n u n u n = − . (1)（10 pts）Determine the system function H(z) of the LTI system, determine the poles and zeros of H(z) . (2)（8 pts）Determine the ROC of H(z). Is the system stable? Why?. (3)（7 pts）Determine the difference equation relating input x[n] and output y[n]. (4)（5 pts）Draw the 2nd-order direct form II signal flow graph for the system function H(z). (5)（5 pts）Draw the signal flow graph of 2nd–order parallel-form structure for H(z). (6（) 10 pts）Determine expressions for a minimum-phase system Hmin(z) and an all-pass system Hap(z) such that ( ) min ( ) ( ) H z H z H z = ap . 得分 阅卷人 得分 阅卷人
山东大学2015-2016学年2学期数字信号处理（双语）课程试卷(A) 海分侧卷人 4.(15 pts)The system functionofa discrete-time filter is 4 2 H()=1e41-e Assume that this discrete-time filter was designed by the impulse invariance method with Ta=2;i.e.,h[n]=2/c(2n),where real hc(t)is the unit impulse response of the continuous-time filter that coul have been the basis for the design.Find the system function Hc(s)of ho(t).Is 都 your answer unique?Ifnot,find another system function Hc(s). 分侧卷人 3.(5 pts)A sequence x[n]is as the following figure: x[n] ◆一 0 Determine 5-point DFT X[k]ofthe sequence x[n]. 器 第2项共4页
2015-2016 2 数字信号处理（双语） （A） 2 4 3．(5 pts) A sequence x[n] is as the following figure : Determine 5-point DFT X[k] of the sequence x[n]. 4．(15 pts) The system function of a discrete-time filter is ( ) 0.6 1 0.4 1 4 2 1 1 H z e z e z − − − − = − − − . Assume that this discrete-time filter was designed by the impulse invariance method with Td=2; i.e., h[n] =2hC(2n), where real hC(t) is the unit impulse response of the continuous-time filter that could have been the basis for the design. Find the system function Hc(s) of hC(t). Is your answer unique? If not, find another system function Hc(s). 得分 阅卷人 得分 阅卷人 x n[ ]
山东大学2015-2016学年2学期数字信号处理（双语）课程试卷(A) 得分间港人 5.(15 pts)Give proof of the DFT periodic convolution property:Le. 得分 港人 6.(10 pts)The figure below shows the flow graph for an 8-point Letand be two finite-duration sequences,both of length N, decimation-in-time FFTalgorithm.Let x[n]be the sequence whose DFT is X[k].In the flow graph,A[].B[-].C[.],and D[]represent separate with N-point DFs]and]respectively.If we form the product] arrays that are indexed consecutively in the same order as the indicated nodes.Determine and sketch the sequence C[r],0,1,...7,if the output Fourier transform is X[k]=1,k=0,1,..,7. then prove that the finite-duration sequencecorresponds tois A周 D间 间-2网[a-m,]0sn≤N-l. A。 PD川 回w装 "PD 4。爬 同w 7D网 4 Cw 7 家 。 CI]W D 6肉W S倒w房 D7 器 第3项共4页
2015-2016 2 数字信号处理（双语） （A） 3 4 5．(15 pts) Give proof of the DFT periodic convolution property: i.e. Let x n 1 and x n 2 be two finite-duration sequences, both of length N, with N-point DFTs X k 1 and X k 2 respectively. If we form the product X k X k X k 3 1 2 = , then prove that the finite-duration sequence x n 3 corresponds to X k 3 is (( )) 1 3 1 2 0 , 0 1 N N m x n x m x n m n N − = = − − . 6．(10 pts) The figure below shows the flow graph for an 8-point decimation-in-time FFT algorithm. Let x[n] be the sequence whose DFT is X[k]. In the flow graph, A[·], B[·], C[.], and D[·] represent separate arrays that are indexed consecutively in the same order as the indicated nodes. Determine and sketch the sequence C[r], r=0, 1, ... ,7, if the output Fourier transform is X[k]=1, k=0, 1, .. ,7. 得分 阅卷人 得分 阅卷人
山东大学2015-2016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4页共4页
2015-2016 2 数字信号处理（双语） （A） 4 4 Properties of the DFT