数字信号处理(英文版)复习重点题型和答案.pdf

Digital Signal Processing Review Questions Apr 15, 2012 pp. 73, Q1.1 Given the sequences shown, write them in terms of unit impulses -0.5 1 0.5 1 -0.8 x n n Ans: 0.5 1 0.5 120.8 3 x nnn nnn = + + pp.74 Q1.2 Compute the following integrals a. (1) t etdt d. 32 (1) ( )tttt dt + + e. 2 cos (20.1 ) (1)ttdt + Ans a: 1 (1) t etdte = Ans. d. 32 (1) ( )1tttt dt + += Ans. e. 22 2 cos (20.1 ) (1)cos (20.1 ) cos (0.1 ) ttdt += = pp75 Q1.7 Given the discrete time sinusoid 3cos(0.10.2 )x nn=, can you determine another sinusoid with digital frequency 02 with the same samples Ans: yes. 0 3cos(0.10.2 ) 3cos(0.2 ) x nn n = = ,where 0 0.1= 0 20.12kk=+=+ , where k is an integer. Select 1k = , and 0.121.9= 3cos( 1.90.2 ) 3cos(1.90.2 ) a x nn n = =+ pp.75 Q1.8 Write the following sinusoids in terms of complex exponentials a. ( )3cos(10015 ) o x tt=+ b. ( )2cos(100.1 )x tt= Ans. a. 1515 18012 o = 100100 1212 100100 1212 3 ( ) 2 3 2 jtjt jj jtjt x tee eeee + =+ =+ b. (omit) pp76 Q1.13 Given the discrete time sinusoid 2cos(0.20.1 )x nn=+ and the sampling frequency 2 s FkHz= a. Determine the corresponding continuous time sinusoid in the frequency range 0 0/ 2FF b. Determine two other continuous time sinusoids with the same sample values Ans. a. 2cos(0.20.1 )x nn=+ Digital frequency: 00 0.22 s FT= Therefore, the continuous time frequency: 0 0 0.2 20.2 22 s FkHz T = Ans. b. 2cos(0.1 ) x x nn=+, where 0.22 x k=+ and k is an integer Select 1k = , 1 0.222.2=+= 2 0.221.81.8= The corresponding continuous time frequencies are: 1 1 2.2 22.2 22 s FkHz T = 2 2 1.8 21.8 22 s FkHz T = pp.76 Q1.16 In each of the following systems let ( )x t or x n be the input and ( )y t or y n be the output. Determine whether each system is (1) linear, (2) time invariant, (3) causal, (4) BIBO stable. g. 12y nx nx nx n=+ i. 2 12y nnx nx nx n=+ Ans. g. linear, time invariant, causal, BIBO stable. i. non-linear, time variant, causal, BIBO unstable Hints: to show y n is time variant, substitute x n with x nL into the right-hand side of the equation. Can we gety nL? Pp77 Q1.17 A linear time invariant system has impulse response 0.5 n h nu n=. Determine the output sequence y n for each of the following input signals: b. 2x nu n= Ans: 0.5 2 k n k k y nh nx n h nk x k u nk u k = = = Since 0u n =for 0n , 0u nk for kn, and 20u k for 2k Therefore, when 2n = = ( )( ) 0.5 j j j z e e XX z e = = pp82. Q1.42 You know that ( )sin ( )FT rect tc F=. Determine the following: i. (2 ) n FTrect tn = Ans: 000000 ()() () kk FTx tkTFXkFFkF = , or 00 00 ( )( ) TF FT rep x tF comb XF= ,where 00 ( )( )XFFT x t= 0 0 (2 )( ) T n FTrect tnFT rep x t = = ,where 0 2T = and 0( ) ( )x trect t= Therefore, 00 000 ( )( ) TF FT rep x tF comb XF= , where 00 1/FT= and 00 ( )( )( )sin ( )XFFT x tFT rect tc F= 00 000 1 ()sin () () 1 sin ( ) () 222 kk k kk FTx tkTcF TTT kk cF = = Chapter 2. 2.1 Consider a sinusoidal signal ( )3cos(10000.1 )x tt=+that is sampled at a frequency 2 s F =kHz (a). determine an expression for the sampled sequence () s x nx nT=, and determine its discrete time Fourier transform ( ) XDTFT x n= (b). determine ( ) ( )X FFT x t= (c). Re-compute ( )X from ( )X F, and verify that you obtain the same expression as in (a) Ans: a) 0 3cos(10000.1 )3cos(0.1 ) s x nnTn=+=+ where 0 1000 s T= Using the formula given in page 53, we can get 00 ( )3()3() jj Xee =+ Ans. b). Using the formula given in page 65, we can get the Fourier transform of ( )3cos(10000.1 )x tt=+ is: 000 ( )( ) cos(2)()() 2 jj X FFT x t A FT AF teFFeFF = +=+ where 0.1= and 0 500F =. Let 2/ ( )() s ssF F XFX FkF = = ? , it is easily shown that 2/ ( )( )( ) s sF F XXF X F = = for ()/ 2,/ 2 Q2.3 For each ( ) ( )X FFT x t= shown, determine ( ) XDTFT x n=, where d () s x nx nT=, is the sampled sequence. The sampling frequency s Fis given for each case (b) ( )(500)(500)X FFF=+, 1200 s F = Hz Ans (b): Since ()2max( ) s FX F, 2/ ( )( ) s sF F XF X F = = 2 ( (1000/)(1000/) ss FF =+ (d) ( )2 1000 F X Frect = , 1000 s F = Hz Ans: (omitted) Q 2.5 We want to digitize and store a signal on a CD, and then reconstruct it at a later time. Let the signal ( )x t be ( )2cos(500)3sin(1000)cos(1500)x tttt=+ and let the sampling frequency be 2000 s F = Hz. (a) Determine the continuous time signal ( )y t after the reconstruction (b) Notice that ( )y t is not exactly equal ( )x t. How could you reconstruct the signal ( )x t exactly from its samples x n ? Ans: (a) Hint: Step 1: find DTFT of x n using Fourier transform of ( )x t Step 2: Let impulse response of ZOH be( )g t. Find the Fourier transform of ( )g t, i.e., ( ) ( )G FFT g t= Step 3: Assume the LPF is ideal with the frequency response ( )(/) s H Frect F F= Step 4: Find the Fourier transform of ( )y t using the results from the steps above. Specifically 2/ ( )( )( ) s F F Y FXG F = =i Step 5: Take the inverse Fourier transform of ( )Y Fto get ( )y t Q.2.7: Suppose in DAC you want to use a linear interpolation between samples as shown in accompanying figure. This reconstructor can be called a first order hold., because the equation of a line is a polynomial of degree 1. (a) Show that ( ) () s n y tx n g tnT= , with ( )g t a triangular pulse as shown in the figure. (b) Determine an expression for ( ) ( )Y FFT y t= in terms of ( ) ( )YDTFT y n= and ( ) ( )G FFT g t= Ans: similar to the ans in Q2.5 Q2.9 In the following system, let the signal x n be affected by some random error e n, as shown. The error is white, zero mean, with variance 2 1.0 e =. Determine the variance of the error n after the filter for each of the following filter ( )H z (a) ( )H z, an ideal l