Reference no: EM133025712

Question 1 (CO4 - Understand propagation of random signals in LTI systems.)

Consider a stochastic process, X(t) = A sin(2πt + φ), where φ is a uniform random variable such that φ~U(0, 2π). Determine the following for the given stochastic process, X(t)

(i) the mean, μx(t), (ii) the ACF (Auto-Correlation Function), Rxx(t1, t2),

(iii) Whether the process X(t) is WSS (Wide Sense Stationary) or not and why?

Consider a zero-mean stochastic process, X(t) with the ACF Rxx(t1, t2) = 1-|t1-t2|, for |t1-t2|≤1, and Rxx(t1, t2) = 0 for |t1-t2|>1.

Suppose a random variable Z is obtained by integration of the given process X(t), over the time interval of (-1, 1), then determine the following for the random variable, Z

(i) the mean of Z, (ii) the variance of Z 4N

Question  2 (CO4 - Understand propagation of random signals in LTI systems.)

Consider a zero-mean, white and Gaussian Stationary stochastic process, X(t) 

(b) with the average instantaneous power of 2 units. Suppose the given process X(t) is the input of a memory-less system and the corresponding output of the memory-less system is the process Y(t), the determine the cross- correlation function of X(t) and Y(t), Rxy(t1, t2) for the following memory-less systems.

(i) Y(t) = 2X(t) + 5, (ii) Y(t) = X(t) X(t), (iii) Y(t) = X(t) X(t) X(t) / 4,

Consider a zero-mean, white and WSS (Wide Sense Stationary) stochastic process, X(t) with the average instantaneous power of 2 units. Suppose the given process X(t) is the input of an LTI (Linear Time Invariant) system and the corresponding output of the LTI system is the process Y(t), and given that the impulse response of the LTI system is h(t) = exp(-t)u(t), then determine the following for the output process Y(t)

(i) the mean, μy(t), (ii) the ACF (Auto-Correlation Function), Ryy(t1, t2),
(iii) Whether Y(t) is a white process or not and why?
(c) Consider a zero-mean, and WSS (Wide Sense Stationary) discrete-time stochastic process, X[n] with the ACF Rxx[n] having the r[0]= 1, r[1] = 2/3, r[2]
= 1/6, and r[k] = 0 for 3≤k. Suppose the process Y[n] = X[n] + X[n-1], then determine the following for the output process Y[n]
(i) the mean, μy[n], (ii) the ACF (Auto-Correlation Function), Ryy[n1, n2],
(iii) Whether X[n] and Y[n] are white processes or not and why?

OR

(c) Consider a zero-mean WSS (Wide Sense Stationary) discrete-time stochastic process, W[n] with the ACF (Auto-Correlation Function) Rww[n]=δ[n]. Suppose W[n] is filtered through a second-order AR (Auto-regression) system to obtain an output process X[n], such that
X[n] = -0.25 X[n-2] + W[n], then determine the following
(i) the expression for ρ[n], the normalized ACF of X[n]
(ii) the values of ρ[n] for |n|≤6

Attachment:- Propagation of random signals.rar