………………………………………………密………………………………封………………………………线……………………………………………… 山东大学 2013-2014 学年 2 学期 数字信号处理（双语） 课程试卷（A） 控制 学院 生物医学工程 专业 级 学号 姓名 题号 得分 一 二 三 四 五 六 七 八 九 十 总分 阅卷人 得分 阅卷人 2．(40 pts) The system function of a causal LTI system is H?z??6?4z?1?1??z?1?5? 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 (70%) and class record mark (30%). 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 阅卷人 ?z?1?2??z?3??1. (a)（5 pts）Determine the poles and zeros of the system, is it a minimum-phase system? (b)（5 pts）Determine the difference equation relating input x[n] and output y[n]. (c)（5 pts）Draw the 2nd-order direct form II signal flow graph for the system function. (d)（5 pts）Draw the signal flow graph of parallel-form structure for the system function. (e)（5 pts）Determine the ROC. Is the system stable? Why? (f)（7 pts）Determine the impulse response h[n]. (g) (8 pts) Determine expressions for a minimum-phase system Hmin?z? and an all-pass system Hap?z? such that H?z??Hmin?z?Hap?z? 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 auxiliary information is in the form of N sequential values of the output, then the system ( ). A is LTI but noncausal system;; B. is LTI and causal system;; C. may not be LTI; D. is causal. 2) If all the three real poles (a, b, c) of a system function H (z) satisfy the condition: 00; D. right-sided sequence, and h[n]=0, for n<0; 3) The minimum-phase system is ( ). A. stable and causal; B. stable but not causal; C. causal but not stable; D. neither stable nor causal 4) Consider an L-point sequence x1[n] and a P-point sequencex2[n], for the circular Nx[n]and linear convolution x[n]*x[n] to be identical, the circular convolutionx1[n]○122 convolution must have a length N of at least ( ) points. A. L+P+1; B. L+P-1; C. L+P; D. L; 5) A Type III FIR Linear-Phase System can be used as a ( ). A. low-pass filter; B. Band-stop filter; C. high-pass filter; D. Band-pass filter; 第 1 页 共 1 页 ………………………………………………密………………………………封………………………………线……………………………………………… 山东大学 2013-2014 学年 2 学期 数字信号处理（双语） 课程试卷（A） 控制 学院 生物医学工程 专业 级 学号 姓名 得分 阅卷人 得分 阅卷人 3．(15 pts) A discrete-time lowpass filter is to be designed by applying the impulse invariance method to a continuous-time Butterworth filter 1having magnitude-squared functionHc?j??2?. 2N1??j?j?c?The specifications for the discrete-time system are：0.89125?H?ej???1,0???0.2? H?ej???0.17783,0.3?????. Assume that aliasing will not be a problem, and suppose the integer order N and ?c of the continuous-time Butterworth filter have been determined with Td=1, i.e. N=6 ?c=0.7032; and we have determined the six poles (sk,k?1,2,3,4,5,6) for system function Hc?s? in 1the magnitude-squared function Hc?s?Hc??s?? with Td=1. 2N1??sj?c?4．(15 pts) Give proof of the DFS periodic convolution property: i.e. ?2?n?be two periodic sequences, each with period N and ?1?n?andxLet x??k? respectively. If we form the product ??k?and Xwith DFS coefficients denoted by X21If Td≠1, it also can be determined that N=6, ?cTd=0.7032, and the six poles are skTd,k?1,2,3,4,5,6. Prove that the discrete-time filter system function H (z) which results from impulse invariance design with Td≠1 is the same as the result for Td=1. ??k??X??k?X??k?，then prove that the periodic sequence x?3?n?with Fourier series X312??k?is x?3?n???x?1?m?x?2?n?m?. coefficients X3m?0N?1 第 2 页 共 2 页 ………………………………………………密………………………………封………………………………线……………………………………………… 山东大学 2013-2014 学年 2 学期 数字信号处理（双语） 课程试卷（A） 控制 学院 生物医学工程 专业 级 学号 姓名 得分 阅卷人 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 5．(10pts) Two 8-point sequences x1[n] and x2[n] shown in the following figures have 8-point DFTs X1[k] and X2[k], respectively. Determine the relationship between X1[k] and X2[k]. 第 3 页 共 3 页 ………………………………………………密………………………………封………………………………线……………………………………………… 山东大学 2013-2014 学年 2 学期 数字信号处理（双语） 课程试卷（A） 控制 学院 生物医学工程 专业 级 学号 姓名 Properties of the DFT 第 4 页 共 4 页