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Tham khảo tài liệu 'proakis j. (2002) communication systems engineering - solutions manual (299s) episode 6', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 2 Consider the sample sequence of An s 1 1 1 1 1 1 1 1 1 1 . Then the corresponding sequence of Bn s is 0 2 0 2 2 0 0 0 0 . The following figure depicts the corresponding sample function X t . If p t n T 2 then P f 2 T2sinc2 Tf and the power spectral density is Sx f Tsinc2 Tf 2 2cos 2nfT In the next figure we plot the power spectral density for T 1. 3 If Bn An aAn-1 then 1 a2 k l 0 Rb k l a k l 1 0 otherwise The power spectral density in this case is given by Sx f P T 1 a2 2acos 2nfT Problem 4.48 In general the mean of a function of two random variables g X Y can be found as E g X Y E E g X Y X where the outer expectation is with respect to the random variable X . 1 my t E X t e E E X t e e where E X t e e Jx t e fx t e xự0 dx Ị X t 0 fx t x dx mx t 0 where we have used the independence of X t and e. Thus 1 fT my t E mx t 0 dp mx t 0 d0 my T J0 98 where the last equality follows from the periodicity of mX t Ớ . Similarly for the autocorrelation function Ry t T t E E X t T e X t e I e E Rx t T 0 t Ớ 1 fT Rx t T 0 t ỡ dỡ T Jo 1 rT _ y Rx t T t dt T Jo where we have used the change of variables t t 0 and the periodicity of RX t T t 2 Sy f 4lim 1 1 E E lim M e tX0 T tF0 T E E lim Xt f ej2nĩeI2 1 e E E lim IXT f 2 y TX0 T T o T E Sx f Sx f 3 Since SY f F T fT RX t T t dt and SY f Sx f we conclude that Sx f F RX t T t dt Problem 4.49 Using Parseval s relation we obtain Zo f 2 Sx f df -o r F-1 f 2 F-1 Sx f dT J o Zo 1 o - rx t dT 1 d2 - à -1 2 RX T 0 4n2 dT2 Rx t 0 Also Zo Sx f df Rx 0 -o Combining the two relations we obtain 1 d2 . 00 f 2Sx f df . . _si Wrms 0 Sxfdf - WM0 T Rx T 1 0 Problem 4.50 Rxy ti t2 E X ti Y t2 E Y t2 X ti Ryx t2 ti If we let T t1 12 then using the previous result and the fact that X t Y t are jointly stationary so that Rxy t1 t2 depends only on T we obtain RxY t1 t2 RxY ti t2 RyX t2 ti Ryx T 99 Taking the Fourier transform of both sides of the previous relation we obtain SXY f F AxY T F YX T i Ryx T e j2nfTdT J r RyxT e j2nfT dT J SY X f Problem .
