浙江科技學院考試試卷 專業班級 學號 姓名 裝訂線 浙江科技學院2012 -2013學年第一學期考試試卷A卷 考試科目 數字信號處理 考試方式 閉 完成時限 2小時 擬題人 審核人 批準人 2012 年 12 月 18 日信息與電子工程學院 2010年級 電子信息工程（互聯網方向）專業Solutions for Reference and Its Score CriteriaPart one Fundamentals (Total 8 Questions in This Part, Each Question Is Marked out of 8 )Question 1 (8 Marks)Discrete-time system differential equation is given by where x(n) and y(n) are the systems input and output respectively, determine the linearity of the system.Solution for ReferenceSuppose the system provides two outputs y1(n) and y2(n) given the inputs of x1(n) and x2(n) respectively, then we haveThus, it is nonlinear system（3 Marks+3 Marks +2 Marks）Question 2 (8 Marks)Determine the inverse z-transforms of the following z-transforms.(a), ROC （5 Marks）(b)（3 Marks）.Solution for Reference：(a) A partial-fraction expansion of X(z) is then of the form (2 Marks)The ROC of X(z) includes ，therefore is a causal sequence.The inverse z-transform of the above is given by (1 Marks )（2 Marks ）（b）The z-transform of a discrete sequence x(n), expressed as X(z), is defined as Thus the inverse z-transform of X(z) can be given by （1 Mark +2 Marks）Question 3 (8 Marks)Consider the connection of three LTI systems as follows:Let, and , then determine unit sample response of the whole system.Solution for Reference：The unit sample response of the whole system is of the form （3 Marks）(2 Marks)(3 Marks)Question 4 (8 Marks)Let the continuous-time signal is(a) Determine the Nyquist frequency of(4 Marks).(b) If is sampled at the double rate of the Nyquist frequency, what is the sampling period Ts of sine wave in the sampled sequence.? What is the sampled signal ? (4 Marks)Solution for Reference：(a) The frequency of signalis 325Hz (1 Mark )The frequency of signalis 360Hz (1 Mark )Thus, the highest frequency of is 360Hz (1 Mark )And the Nyquist frequency of is 360Hz (1 Mark )(b) The sampling frequency fs=360Hz*2=720 Hz, and the sampling period Ts=1/720 s (2 Marks)The sampled signal can be given by(2 Marks)Question 5 (8 Marks)Let x(n) be a length-3 sequence defined by (a) Determine the DTFT of x(n) (4 Marks).(b) Determine the N-point DFT of x(n) (4 Marks).Solution for Reference：(a) The DTFT of x(n) can be given as follows (2 Marks) (2 Marks) (b) The N-point DFT of x(n) is （2 Marks） (2 Marks ) Question 6 (8 Marks)Consider the cascade of the three causal first-order LTI discrete-time systems shown in the following structure, where, ,(a) Determine the transfer function of the overall system as a ratio of two polynomials in (2 Marks).(b) Determine the difference equation characterizing the overall system (3 Marks).(c) Develop the realization of the overall system with each section realized in direct form I (3 Marks).Solution for Reference：(a) The transfer function of the overall system is the following. (2 Marks)(b) The difference equation is given byCross-multiply the denominator to rewrite the H(z) asTo obtain the time-domain expression for the system, the inverse z-transform of the above equation has been done, and the result is (3 Marks)(c) The direct form II structure of the overall system is (3 Marks)Question 7 (8 Marks)The difference equation of a causal LTI discrete-time system is given by where the variables and represent systems input and output(a) Determine the transfer functionand its region of convergence (ROC) (3 Marks).(b) Determine the impulse response of the above system (3 Marks).(c) If the system is a digital filter actually, is it an IIR filter or FIR filter? (2Marks)Solution for Reference：（a）Take the z-transform of original equation in time-domain to get the Z-domain expression for the system And its transfer function and ROC are ，（1 Mark+2 Marks）（b）The impulse response of the system is （3 Marks，Only 1 mark can be given without writing）.(c) From the above, one can see that has a infinite length output, Thus the system is an IIR filter（1 mark +1 mark）.Question 8 (8 Marks)Let X(k), , be a 10-point DFT of with first 5 samples of X(K) given by ，. A N-point DFT property is well known, that is .（a）Determine the other 5 samples of X(K):（4 Marks）.（b）Determine by the formula of IDFT（4 marks）.Solution for reference（a），（1 Mark+1Mark+1 Mark+1Mark）（b）（2 Marks+2Marks）Part two Applications (Total 2 Questions in This Part, Each Question Is Marked out of 18 )Question 1（18 Marks）Design one low-pass IIR digital filter H(z) with a sampling frequency of 4KHz and digital filter requirements are specified by（1）The passband edge frequency ，the maximal passband attenuation .（2）The stopband edge frequency，the minuimum stopband attenuation.Please refer to the following procedures for the design:（a） Find the prewarp frequency andfor analog prototype filter design based on Butterworth method（6 Marks）；（b） Computer order N of Butterworth prototype filter and its cutoff frequency （6 Marks）.（c）Determine the transfer function for Butterworth filter (4 Marks）.（d）Transform the analog transfer function to digital transfer function by bilinear transform method（2 Marks）.Following data and formulas are provided for reference during your work. The denominator polynomial coefficients of Butterworth prototype filter versus to various order NsNa1a2a3a4a51121.414232.00002.000042.61313.41422.613153.23615.23615.23613.236163.86377.46419.14167.46413.8637，, where T is the sampling period.Solution for reference（a）Convert the analog frequency into digital frequencies （2 Marks+1 Marks）Transfer digital frequency to prewarp frequencies（2 Marks +1 Marks）（b）Calculate order N and cutoff frequency （2 Marks）An order of 6 is selected here, namely, N is 6（2 Marks） （2 Marks）(c) Refer to above Table, the transfer function for prototype filter is achieved as （2 Marks）The analog transfer function can be obtained by substituting in H(s). （2 Marks）（d）Substitute into achieve digital IIR transfer function H(z)（2 Marks）.Question 2（18 Marks）To design a low-pass FIR filter based on the sampling frequency 10 kHz and its following requirements:(1) Passband edge frequency, the maximal passband attenuation .(2) Stopband edge frequency , the minimum stopband attenuation .Following data and formulas are provided for reference