[ ] sin [ ] - Oregon State University
Transcription
[ ] sin [ ] - Oregon State University
4.4 The continuous-time signal xc t sin 20 t cos 40 t is sampled with a sampling period T to obtain n 2 n the discrete-time signal x n sin cos 5 5 (a) Determine a choice for T consistent with this information. (b) Is your choice for T in part (a) unique? If so, explain why. If not, specify another choice of T consistent with the information given. 4.11 The following continuous-time input signals xc t and corresponding discrete-time output signals x n are those of an ideal C/D as shown in figure 4.1: Specify a choice for the sampling period T that is consistent with each pair of xc t and x n . In addition, indicate whether your choice of T is unique. If not, specify a second possible choice of T consistent with the information given. n (a) xc t sin 10 t , x n sin 4 sin 10 t sin n 2 , x n (b) xc t 10 t n 2 4.21 Consider a continuous-time signal xc t with Fourier transform X c j shown in Figure P4.21-1. (a) A continuous-time signal xr t is obtained through the process shown in Figure P4.21-2: Frist, xc t is multiplied by an impulse train of period T1 to produce the waveform xs t , i.e., xs t x n t nT . Next, x t is passed through a low pass filter with frequency response n 1 s H r j . H r j is shown in Figure P4.21-3: Determine the range of values for T1 for which xr t xc t . (b) Consider the system in Figure P4.21-4: The system in this case is the same as the one in part (a), except that the sampling period is now T2 . The system H s j is some continuous-time ideal LTI filter. We want xo t to be equal to xc t for all t , i.e., xo t xc t for some choice of H s j . Find all values of T2 for which xo t xc t is possible. For the largest T2 you determined that would still allow recovery of xc t , choose H s j so that xo t xc t . Sketch H s j . 4.24 Consider the system shown in Figure P4.24-1: The anti-aliasing filter is a continuous-time filter with the frequency response L j shown in Figure P4.24-2: The frequency response of the LTI discrete-time system between the converters is given by: Hd e j e j 3 , (a) What is the effective continuous-time frequency response of the overall system, H j ? (b) Choose the most accurate statement: d d T 1 (i) yc t xc 3t (ii) yc t xc t (iii) yc t xc t 3T (iv) yc t xc t dt dt 3 3 (c) Express yd n in terms of yc t . (d) Determine the impulse response h n of the discrete-time LTI system. 4.25 Two bandlimited signals, x1 t and x2 t , are multiplied, producing the product signal w t x1 t x2 t . This signal is sampled by a periodic impulse train yielding the signal wp t w t t nT n w nT t nT . Assume that x t is bandlimited to , and x t is n 1 1 2 bandlimited to 2 ; that is X 1 j 0, 1 X 2 j 0, 2 . Determine the maximum sampling interval T such that w t is recoverable from wp t through the use of an ideal lowpass filter. 4.32 Consider the discrete-time system shown in Figure P4.32-1: where (i) L and M are positive integers. x n L n kL ,k Z xe n (ii) , otherwise 0 (iii) y n ye nM (iv) M j H e 0 4 4 (a) Assume that L 2 and M 4 , and that X e j , the DTFT of x n , is real and is as shown in Figure P4.32-2: Make an appropriately labeled sketch of X e e j , Ye e j , and Y e j , the DTFTs of xe n , ye n , and y n , respectively. Be sure to clearly label salient amplitudes and frequencies. (b) Now assume L 2 and M 8 . Determine y n in this case. Hint: see which diagrams in your answer to part (a) change. 4.33 For the system shown in Figure P4.33: z1 n z2 n z3 n , find an expression for y n in terms of x n . Simplify the expression as much as possible. 4.38 Consider the two system of Figure P4.38: (a) For M 2 , L 3 , and any arbitrary x n , will y A n yB n ? If your answer is yes, justify your answer. If your answer is no, clearly explain or give a counterexample. (b) How much M and L be related to guarantee y A n yB n for arbitrary x n ?