hw8_fall10 - ECE 603 - Probability and Random Processes,...

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Unformatted text preview: ECE 603 - Probability and Random Processes, Fall 2010 Homework #8 Due: 12/06/10, in class ¥I£#F 30HG&¤£¡ 5 E ¥ ¦¤£¡ 5 D ¥ 75 C¤B¡ A@9864 ¥)#)   3210¡ ( '&$"¡ © ¥%# !   and  £ © §¥¡ ¨¦¤£¢ (a) Find the probability density function (b) Find the mean function , and and are independent. . . (c) Find the autocorrelation function . wide-sense stationary (WSS)? 2. In a simple quadrature amplitude modulation (QAM) communication system, ¥¡ ¦¤£Q© ¥¡ ¦¤£P (d) Is © , where  1. Let is defined as: d S d S ¥ b a %¡ e  ¥ b a %¡ X V U S § ¥ ¡ H£ 14 1`CAf¦X d cH£ 14 1`Y'WTR¦¤£Q© where is the “carrier” frequency, and and are independent random variables with mean zero and variance . Assume all of the moments of and , except the first moment (mean), are nonzero. ¥¡ ¦¤£Q© . b g4 h (b) Find the autocorrelation function of wide-sense stationary? strict-sense stationary? (Hint: To show that it is not, try showing that is a discrete random variable.) ¥ †„‚ c¤B¡ ‡€ …ƒ4 . £ ) for all . (b) Find the mean function and autocorrelation function for is a Gaussian random process. ¥¡ ¦¤£Q© ¥7 C¤B¡ A@98’‚ 4 ˆ ‘H¤£¡ © x £ § ˆ¥ (a) Find the probability density function and variance . I observe the random process ‰ t v # pD ‚ E t pv ‚ D be a Gaussian random variable with mean . ¥¡ ¦¤£Q© (c) Determine whether for all . (c) Find the autocorrelation function 4. Let (Note that t pv © v (b) Find the mean function )§ (Ct pv © (a) Find the probability density function for all . If “tails” appears, v £ 3. A fair coin is tossed. If “heads” appears, . depends €¥) x¡ § 3y0wCt pv © (d) Is on .). . t¥ ¡ r p uH¤£sQ© qi ¥H¤£¡ © ¥ H¤£¡ © (c) Is ¥¡ ¦¤£Q© I (a) Find the mean function of . be a Gaussian random process with zero-mean and autocorrelation function ¥¡ H¤£P . . I run this sample through a square-law ¥ 3) I ¤©¡ ¥)¡ 30P FI ‘ƒ"P FI § I ©  ¥ !¡ FI ¤¢ ¥£ %  IF © w § (b) Suppose that I form the linear combination density function (pdf) of , and calculate . . Find ¥ C¤B¡ ‚ 4 ¥   '"¡  ! §£ ; call it . ¥ !¡ ƒ"P § F © © ¦ ¨§ ¦ (a) Suppose I take a sample at time device to yield . Find § ¥ ¡¡ 5 E 5. Let , the probability  I© ...
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