英文版数字信号处理去年考试题.doc

1.Consider an LTI system with input xn and output yn that satisfies the difference equation a. Find the system function H(z). How many ROCs are associated with H(z)? For each case ,determine what type of the corresponding impulse response hn?b. If this system is causal, then is it stable ? Justify your answer .And whether exists or not ? If it exists , determine .2.Consider the finite-length sequence xn=1,0,2,1,0n3 , with an 4-point DFT given by xk .a. If the 4-point DFT yk of length-4 sequence yn is given by y(k)=, determine yn.b.If the 4-point DFT wk of length-4 sequence wn is denoted by w(k)=xk,determine wn.c. If N-point DFTs are used in the two-step procedure , how should we choose N so that wn=xn*xn for 0nN-1 ? Determine wn in the case also .Note: using the DFT properties without computing xk.3.Given xn=0,1,2,3,4,5,6 be a length-7 sequence defined for 0n6 , with X denoting its DTFTa) Evaluate the following function without computing the transform itself : X() ; ;b) Define Yk =X(e), 0n4, with yn denoting its 5-point IDFT.Determine yn without computing Yk and its IDFT .Can you recover Xn from yn.4. An IIR filter is described by the following system function :H(z)=Determine and draw the following structures:(a) Direct from II(b) Cascade form.5.Verify the identity equation 6.A continuous-time signal X=cos(2x300t)+ (2x500t)+ (2x1200t) is sampled at a 2kHz rate , and the sampled sequence is passed through an ideal lowpass filter with a cutoff frequency of 900Hz , generating a continuous-time signal y .a. Determine the discrete-time signal xn generated by periodically sampling X at F=2khzb.What are the frequency componts present in the reconstructed signal y.c.If y is equal to the original continuous signal ,determine the sampling frequency in this case d.If we want to pass the frequency components at 300Hz ,what type filter we should choose ? If the transition-width is assumed to as 100Hz, the minimum stopand attenuation , which window functions can we choose from Table1 ? Determine the length of the filter for the window you selectd.Table1Window NameTransition Width Min. stopband attenuationRectangular20.9dBHanning43.9dBHamming54.5dBBlackman75.3dB一、(20 points) 1. The input-output pair of a stable LTI system is shown in Fig.1(a).(a) Determine the response to the input x1n in Fig.1(b).(b) Determine the impulse response and frequency response of the system.(c) Sketch the magnitude-frequency response of the system.(a)(b)Fig.1 二、(20 points)Let hn be a Type-4 real-coefficient linear-phase FIR filter . (1)If this filter has the following zeros:, please determine the locations of the remaining zeros.(2)Please determine the FIR transfer function and realize it in cascade form and direct form I.三、(20 points) The difference equation of a LTI discrete-time system is : where xn and yn are the input and output sequences respectively.(a) Please give the transform function H(z) as well as its poles and zeros;(b) If the system is casual and stable, please give the ROC of the z-transform;(c) Give the impulse response hn of this casual stable system.四、(20 points) Design a DIGITAL low pass filter to meet the following requirements:Select a window to design a linear-phase FIR filter, using the lowest order filter to meet the specifications.五、(20 points) Fig.2(a) shows a 6-point discrete time sequence xn. Assume that xn = 0 outside the interval shown. The value of x4 is not known and is represented as b. Let X(ej) be the DTFT of xn, and X1k be samples of X(ej) with sampling interval /2, i.e., The 4-point sequence x1n is the 4-point Inverse DFT of X1k, and is shown in Fig.2(b). (a) (b)Fig.2(a) Please determine the value of b.(b) Let X2k be samples of X(ej) with sampling interval /3, i.e.,Please determine and sketch the finite-length sequence yn whose 6-point DFT is Yk = W64kX2k.Determine and sketch the finite-length seque4