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 Ans: Formula ( ), 1( ), n n z x na u nX zza za z x na unX zza za = = = ( )( ) 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 lowpass filter with bandwidth / 4 (b) ( ) 0.5 z H z z = (c) () 1 123 4 y ns ns ns ns n=