山东大学2011-2012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 3)If the poles of a system function H (z)are all in the unit circle,then the system is 得分 Directions图 A.absolutely stable: B.maybe stable and causal; C.absolutely causal; D.maybe stable but not causal 好 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 4)A Type II FIR Linear-Phase System can be used as a( 人 record mark (20%). 3)Tables of properties of Discrete-time Fourier transform,z-transform and DFT are supplied to A.low-pass filter, B.low-pass or high-pass filter, C.high-pass filter; D.neither low-pass nor high-pass filter; you on the last page. 4)Unless otherwise indicated,answers must be derived or explained,not just simply written down. 5)For a N-point DFT(N=2")of x[n],the total number of computations is the same for the 得分 卷人 decimation-in-frequency and the decimation-in-time algorithms,the number of complex (10 pts,2 pts for each)Choose the best answer to fill in the blanks. multiplications is( 的 A.mN; B.mN/2: C.N2; D.m2; 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 ofN sequential values of the output, then the system 舞分 阕卷人 2.(10 pts)The following figure shows the impulse response h[n] for a FIR LTI system. A.must be LTI and causal;B.may not be LTI;C.must be LTI but noncausal;D.is causal. 2)If the poles of the system function H(z)ofa LTI system are as in the Figure,and the 101 system is stable,then h[n]is a -14 1)Is the system generalized linear-phase system?Tell the reason.If the system is Unit cirele z-plane generalized linear-phase system,then determine which type of FIR linear-phase system it is.(It is known that there are four types of FIR linear-phase systems.) (5 pts) 2)Determine the group delay associated with the system.(5pts) 器 A.two-sided sequence (may be h[n]=a"u[n]+b'u[n]+c"u[-n-1])B.two-sided sequence (may behn]=bu[n]+c"u[-n-1]):C.right-sided sequence (may be 暴 h[n]=b'u[n]).D.left-sided sequence (may be hn]=c"u[-n-1]). 第1负共4页
2011-2012 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 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. must be LTI and causal; B. may not be LTI; C. must be LTI but noncausal; D. is causal. 2) If the poles of the system function H (z) of a LTI system are as in the Figure, and the system is stable, then h[n] is a ( ). A. two-sided sequence (may be 1 n n n h n a u n b u n c u n = + + − − ); B. two-sided sequence (may be 1 n n h n b u n c u n = + − − ); C. right-sided sequence (may be n h n b u n = ); D. left-sided sequence (may be 1 n h n c u n = − − ); 3) If the poles of a system function H (z) are all in the unit circle, then the system is ( ). A. absolutely stable; B. maybe stable and causal; C. absolutely causal; D. maybe stable but not causal 4) A Type II FIR Linear-Phase System can be used as a ( ). A. low-pass filter; B. low-pass or high-pass filter; C. high-pass filter; D. neither low-pass nor high-pass filter; 5) For a N-point DFT ( 2 m N = ) of x[n], the total number of computations is the same for the decimation-in-frequency and the decimation-in-time algorithms, the number of complex multiplications is ( ). A. mN; B. mN/2; C. N2 ; D. m2 ; 2．(10 pts) The following figure shows the impulse response h[n] for a FIR LTI system. 1) Is the system generalized linear-phase system? Tell the reason.If the system is generalized linear-phase system, then determine which type of FIR linear-phase system it is. (It is known that there are four types of FIR linear-phase systems.) （5 pts） 2) Determine the group delay associated with the system. （5 pts） 得分 阅卷人 得分 阅卷人
山东大学2011-2012学年2学期数字信号处理（双语）课程试卷(A) 得分侧普人 3.(15 pts)Given two finite sequences: gm={-3,2,4n=0,1,2,Mm={2,4,0,5n=0,1,2,3, 架 (a)Compute yu[n]=gin*h[n]:(5 pts) (b)Compute ydn]=gin④hln：(5pts) (c)Can yL[n]be computed by using DFT?IfyL[n]can be computed,then write down the steps.(5 pts) 家 得分圆卷人 5.(15 pts)The system function of a stable LTI system is H(e)= 2/5 25 1-2:-）(1+0.5） (a)Plot the poles and zeros of the system function in the z-plane.(5 pts (b)Determine the ROC and the impulse response h[n].(10 pts) 器 得分卷人 4.(15 pts)Consider a causal continuous-time system with impulse response he(t)and system function ( 3+2 Assume the sampling time is 7=0.Is,use impulse invariance to determine H(z)for a discrete-time system such that hin]=he(nT). 第2项共4页
2011-2012 2 数字信号处理（双语） （A） 2 4 3．(15 pts) Given two finite sequences: g[n] = { -3, 2, 4; n = 0, 1, 2 }, h[n] = { 2, -4, 0, 1; n = 0, 1, 2, 3 }, (a) Compute yL[n] = g[n]*h[n] ； （5 pts） (b) Compute yC[n] = g[n] ④h[n] ；（5 pts） (c) Can yL[n] be computed by using DFT? If yL[n] can be computed, then write down the steps.（5 pts） 4．(15 pts) Consider a causal continuous-time system with impulse response hc(t) and system function 2 3 2 ( ) 2 3 1 a s H s s s + = + + . Assume the sampling time is T = 0.1 s, use impulse invariance to determine H(z) for a discrete-time system such that h[n] = hc(nT). 5．(15 pts) The system function of a stable LTI system is ( ) ( ) ( ) 1 1 2 5 2 5 1 2 1+0.5 H z z z − − = − − . (a) Plot the poles and zeros of the system function in the z-plane. （5 pts） (b) Determine the ROC and the impulse response h[n]. （10 pts） 得分 阅卷人 得分 阅卷人 得分 阅卷人
山东大学2011-2012学年2学期数字信号处理（双语）课程试卷(A) 得分港人 6.(15 pts)The system function of a FIR filter is He)=0-X0+6:-0- (a)Determine the difference equation relating input x[n]and output y[n].(5 pts) (b)Draw the signal flow graph of direct form structure of the FIR filter.(5 pts) 签 (c)Draw the signal flow graph of cascade structure of the FIR filter with Ist-order sections.(5 pts) 家 海分圆卷人 8.(10 pts)Draw the flow graph for an 8-point decimation-in-time FFTalgorithm with bit-reversed ordering of the input time sequence x[n] and normal ordering of the output sequence X[k]. 得分圆卷人 7. (10 pts)A discrete-time causal LTI system has the system function 器 1+3红） H(e)= 140.5= Determine expressions for a minimum-phase system (and an all-pass system H()such that H()=H()H(2) 第3项共4页
2011-2012 2 数字信号处理（双语） （A） 3 4 6．(15 pts) The system function of a FIR filter is 1 1 1 1 ( ) (1 )(1 6 )(1 ) 2 H z z z z − − − = − + − (a) Determine the difference equation relating input x[n] and output y[n]. （5 pts） (b) Draw the signal flow graph of direct form structure of the FIR filter. （5 pts） (c) Draw the signal flow graph of cascade structure of the FIR filter with 1st-order sections. （5 pts） 7．(10 pts) A discrete-time causal LTI system has the system function ( ) ( ) 1 1 1 3 . 1 0.5 z H z z − − + = + Determine expressions for a minimum-phase system H z min ( ) and an all-pass system ( ) H z ap such that ( ) min ( ) ( ) H z H z H z = ap . 8．(10 pts) Draw the flow graph for an 8-point decimation-in-time FFT algorithm with bit-reversed ordering of the input time sequence x[n] and normal ordering of the output sequence X[k]. 得分 阅卷人 得分 阅卷人 得分 阅卷人
山东大学2011-2012学年2学期数字信号处理（双语）课程试卷(A) TABLE 2.2 FOURIER TRANSFORM THEOREMS TABLE 2.3 FOURIER TRANSFORM PAIRS Sequence Fourier Transform Sequence Fourier Transform xn X(e) 1 y Y(ei) 1. 2.m-o】 e-jono 好 1.ax[n]+byn] ax(el)+bY(ei) 3.1(-o0n←0∞) 2.x[n-na](na an integer) e-junX(eiw) +2a k一x 3.ejoonx[n] X(ef-wo)) 4.an(al1 游 3.-4-n- 1-x可 10) or oo (if m<0) Finite-Length Sequence(Length N) N-point DFT(Length N) 5.a"uln] 1-a可 la lal 1.x(n] XI内 6.-ad-n-1] la<lal 2.xiln].x2ln] X:[).X2[] TABLE 3.2 SOME z-TRANSFORM PROPERTIES 的 3.ax[n]+bx2in] aX内+bX周 4.X[n] Section Nx[((-k))N] Reference Sequence Transform ROC 5.x[((n-m))N] w装mX xin] X(2) R: 6. Wnxinl X(k-)》N a X R N- ∑mala-m al X2(z) Rn 7. X肉X2肉 3.41 网+br aX(z)+bx2(z) Contains R:nR 3.42 x[n -no] :X( R.except for the possible 器 addition or deletion of 8.xIn]x2In] X1()X2(k-)N the origin or oo 3.43 网 X(z/zo) lzoRr 3.4.4 网 Rr.except for the possible addition or deletion of the origin or oo 第4页共4页
2011-2012 2 数字信号处理（双语） （A） 4 4 Properties of the DFT