1. Consider the two-port communication system shown in Fig. 1. (LPF) If an input autocorrelation Rxx...

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1. Consider the two-port communication system shown in Fig. 1. (LPF) If an input autocorrelation Rxx (I) is given by Bxx (T) = BeITI, B'>0, 320, all I's 1.1. Determine the output autocorrelation using the Wiener- Khintchive theorem 1.2. Find the output noise, by XINPOT Y: OUTPUT Rxx (T) "Ryy (t) FIG. 1. 5y 2 =x 8 HINT: Ryy (t) = 3" { Syy(w)} = (syy (w) e+jww, Syy (w) = | H(w)|= 3x(w) ff(2)dz = lim [ {(w/dw + for fre)da = jam Res f (zi) R if(z)dz 2. A random process x(t) has an autocorrelation function. 100+25-151 Rxx (2) = 100 + 25e 3.1. Determine the mean value of the response of a system Maving a an impulse response h(t) = -wt, to 0 244 , 2.2. Find the variance of the response in (2.1) HINT: us the properties of a WSS autocorrelation E[x] = = MI 6x2 = m-m Q2. (25 pts) A random variable X has the probability density function, f(x)=41 and is defined for all real x. a) Find the value of the constant a. b) Find the distribution function which corresponds to the p.d.f. above. c) Find the probability that the stochastic variable, Y = X is between 1/4 and 1. c) Find the probability that the stochastic variable, Y = X2 is between 1/4 and 1. 1. Consider the two-port communication system shown in Fig. 1. (LPF) If an input autocorrelation Rxx (I) is given by Bxx (T) = BeITI, B'>0, 320, all I's 1.1. Determine the output autocorrelation using the Wiener- Khintchive theorem 1.2. Find the output noise, by XINPOT Y: OUTPUT Rxx (T) "Ryy (t) FIG. 1. 5y 2 =x 8 HINT: Ryy (t) = 3" { Syy(w)} = (syy (w) e+jww, Syy (w) = | H(w)|= 3x(w) ff(2)dz = lim [ {(w/dw + for fre)da = jam Res f (zi) R if(z)dz 2. A random process x(t) has an autocorrelation function. 100+25-151 Rxx (2) = 100 + 25e 3.1. Determine the mean value of the response of a system Maving a an impulse response h(t) = -wt, to 0 244 , 2.2. Find the variance of the response in (2.1) HINT: us the properties of a WSS autocorrelation E[x] = = MI 6x2 = m-m Q2. (25 pts) A random variable X has the probability density function, f(x)=41 and is defined for all real x. a) Find the value of the constant a. b) Find the distribution function which corresponds to the p.d.f. above. c) Find the probability that the stochastic variable, Y = X is between 1/4 and 1. c) Find the probability that the stochastic variable, Y = X2 is between 1/4 and 1.