hw9_fall10 - ECE 603 - Probability and Random Processes,...

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Unformatted text preview: ECE 603 - Probability and Random Processes, Fall 2010 Homework #9 Due 12/10/10 (in class) #$" !©§  ¥ ¡¨ ¦¥¤£¢ ¡ be a wide-sense stationary Gaussian random process with mean zero and autocorrelation be a white Gaussian noise process with power spectral density ¦¥¤£¢ ¡ ¥ &¤£¡ % . Let ¦¥¤£¢ ¡ 2 31 (a) Find , the power in (b) Is . strict-sense stationary? . ¥4 7 ¥ ¡ I¥ ¡ 6`YTSRQ¢XHG¢¤ ¡ 1 ¥¡U WVHB ¥ ¡ I¥ ¡ I¥ ¡ 658¢QPTSR¢QPHG¢9 F ¦¥¤£¡ % ¥ C5B¡ E &¤£¡ D ¥ ¦¥¤£¢ C5BA¨ @ ¡ ¥¡ ) that has input . Find (g) Find , the pdf of . . is a zero-mean Gaussian random process with autocorrelation function through an ideal lowpass filter with unity gain and bandwidth and zero otherwise) as shown below: Ideal LPF BW edcp bG edcG hf B f bg ¥¡ &¤£Q ¥ ¡ 0 (a) Suppose that I run is, = 1 for Hz &i¥ ¤£rq ¡ ¥¡ &¤£Q 2. Suppose that sinc . and output with power spectral denisty ¥ C5B¡ E ¥¡ &¤£Q i Find: ¥¡ &£¤Q ¥ ¥¡ 68`vaHGuq¡ 1 s t 31 s ¦¥¤£¡uq ¥t C5B¡ )@ s , the power in . ¦¥¤£uq ¡ , the power spectral density of . . through an ideal differentiator as shown below: ¥¡ &¤£rq i ¦xw w i (b) Suppose that I run edcG b (f) Let or . . ¥¡¨ a§ (e) Find a filter (give of F (d) Find the power spectral density ¥¡ C5BA¨ @ ¥8 7 ¥ ¡ 69!654¢¤ ¡ 1 (c) Find . )0( ' 1. Let Hz (that ¥¡ &¤£Q Find: ¥ ¥¡ 68`vaHGuq¡ 1 s t 31 s ¥¦¤£¡uq ¥t C5B¡ )@ s , the power in . ¦¥¤£uq ¡ , the power spectral density of . . G  ¦¥¤£¨ ¡ ¥¡ &¤£Q 3. Let be a wide-sense stationary Gaussian random process with mean and autocorrelation function , where is the Dirac delta function. The random process is run through a filter combination as shown below, where the first filter has frequency response: ¥ ¤£h¡ ¡ g 4 cf B f R § ¨  ¨ © G  ¥ 0  pC5B¡ 0 E   ¥ ¡¡ ¥ ¡ W¢G R ¨ §  C5B¡ ¦E ¥¥ . ¥ ¥ ¥¡ C5B¡ E i &¤£Q ¥¡ &¤£rq ¥ &¤£¡ F i C5B¡ 0 E i ¥  ¥ &¤£¡ F  ¦¥¤£¡ 0 F  ¦¥¤£uq  &¤£¡ 0 q  ¡¥ (b) Find the power ¥¡ &¤£Q and the second filter has frequency response (a) Find the power otherwise in . in . Be sure to simplify your answer as much as possible. 4. Per class, a main utility of wide-sense stationarity (WSS) is that it allows us to define the power spectral density for a given random process. However, there is a large class of random processes that are not wide-sense stationary for which we are still able to derive the power spectral density. ¥ &¤£¡ t ¥¡ &¤£uq A random process is defined to be cyclostationary if both its mean function and the autocorrelation function are periodic (in ) at some period ; that is, and for some . # &" is given by the Fourier trans- ¥ BB A @86¥ &S£©C©58¡ ¢97&¤£¡ 3 ¡ ¦¥¤£¢ is some constant. Use the definitions above to derive the power spectral density for ¥¡ &¤£Q be a zero-mean wide-sense stationary random process with autocorrelation function ¥4 ¡ 5§ ¦¥¤£¡ 3 ¥ ¡ t ' § 0 ¥¡ &£¤rq ¥t C5B¡ P@ ( 0( 1)# " £ ¨ I H¤2¦¥¦£¢P¤£¡ t § (  R ¡ t ' § ¥ ¥ &¤£¡rq #%" # I # I §   I ¨ )¦£¨ &" C¤£¡ t p¥ QP¦£!¤£¡ t § t ¥ %" P¤£¡ t  ¦¥¤£¡ t I   C¤¡ q &¤¡ q ¥ I £ ¥ £    P¦£!¤£¡ § ¥ I¨ ¥ ¢I B B where £ Let and let of a cyclostationary process # $" and the power spectral density form of . is given by: The “average autocorrelation” of a cyclostationary process . , ...
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This note was uploaded on 09/16/2011 for the course ECE ECE603 taught by Professor Dennisgoeckel during the Fall '10 term at UMass (Amherst).

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