COSC 3213: Computer Networks
COSC 6221: Statistical Signal Processing Theory
Assignment # 5: Random Sequences: Response to LTI Systems, Stationarity, and Power Spectral Density
Due Date: November 21, 2003
In the last week, we introduced the concept of random sequences and defined several statistical parameters including correlation, covariance and power spectral density (PSD) that are frequently used to represent random sequences. We reviewed linear shift invariant (LSI) systems and studied how the mean, covariance function, and PSD of a random sequence applied at the input of a LSI system is modified at the output of the system. We also covered important categories of random sequences including wide sense stationary (WSS) sequences, stationary sequences, and ergodic sequences. Finally, we covered Markov random sequences and Markov chains that are used in several signal processing application such as speech modeling and image processing.
Please review sections 6.1 to 6.5 of the Woods text before attempting the following questions.
1.(Random Vectors) Let be a random vector whose components satisfy the following expression
where ’s are jointly independent and Bernoulli distributed variables taking on values of 0 and 1 with equal probability.The initial condition for the above equation is By substituting’s in a random vector and in a random vector , solve the following questions.
(a)Write the above relationship for some constant matrix and determine .
(b)Find the mean vector
(c)Find the covariance matrix for random vector
(d)Find the covariance matrix for random vector .
2.(Conditional pdf) Let and be independent identically distributed (IID) random variables with the exponential probability density function
(a)Determine the probability density function for the ratio
(b)Let A be the event X < 1/Y. Determine the conditional pdf of X given that A occurs and that Y = y; that is determine
(c)Using the definitions of (b), what is the minimum mean square estimator of X given that the event A occurs and that Y = y?
3.(Response of a LTI system to a random sequence) Consider a random sequence X[n]as the input to a linear filter with impulse response
Given the mean and correlation function of the input X[n], express the following quantities in terms of these parameters:
(a)Mean function of the output Y[n].
(b)Autocorrelation function of the output Y[n].
(c)Autocovariance function of the output Y[n].
(d)Assuming that the input X[n] is a Gaussian random sequence, determine the joint pdf at arbitrary instants
4.(Linear CCD Modeling) A charge coupled device (CCD) array with ``leaky” cells is modeled by a random sequence
where is the k’th component of , the infinite dimensional outcome of the experiment. The random variables are jointly independent and Gaussian distributed with mean and variance . The width-3 pulse function h[n] is defined as follows:
(a)Find the mean function .
(b)Compute the first order pdf
(c)Find the second order pdf
5.(LTI Systems) Consider the difference equation model for a LTI system
where is a constant such that || < 1.
(a)Compute the output y[n] for the input x[n] = nU[n], || < 1, assuming causality applies, i.e., y[n] = 0 for n < 0.
(b)Compute the output y[n] for the input x[n] = -nU[-n], assuming anticausality applies, i.e., y[n] = 0 for n > 0.
6.(Asymptotic Stationarity) Let W[n] be an independent random sequence with mean W = 0 and variance 2. Define a new random variable X[n] as follows:
(a)Find the mean value of X[n] for n > 0.
(b)Find the covariance of X[n].
(c)For what values of does tend to as become large with G being a finite-valued function. This situation is called asymptotic stationarity.
7.(Wide Sense Stationary) Let the random variables A and Bbe independent and identically distributed (iid) with zero mean 0, variance 2, and third order moment E[A3] = E[B3] = 3 ≠ 0. Consider the random sequence
where o is a fixed frequency in radian/s.
(a)Show that X[n] is wide-sense stationary (WSS).
(b)Prove that X[n] is not stationary by presenting a counter example.
8.(Power Spectral Density) Consider the following system
where the input X[n] and the noise V[n] are WSS and mutually uncorrelated with zero mean and power spectral density (PSD) given by Sxx() and Svv() respectively.
(a)Find the PSD Syy() of the output Y[n] in terms of the transfer function H() of the system and the PSD’sSxx() and Svv().
(b)Find the cross-power spectral density Sxy() between the input X[n] and output Y[n].
9.(Power Spectral Density) Consider the DT system with input random sequence X[n] and output Y[n] given as
Assume that the autocorrelation RXX[n] = 2[n] for the input sequence X[n].
(a)Calculate the PSD of the output random sequence Y[n].
(b)Calculate the output correlation function RYY[n] for the output sequence Y[n].
10.(Markov Chains) Let X[n] be a Markov chain on n ≥ 0 taking values 1 and 2 with one-step transition probabilities,
given in the matrix form as
We describe the state probabilities at time n by the vector
(a)Show that
(b)Draw a two state transition diagram and label the branches with the one-step transition probabilities .
(c)Given that X[0] = 1, find the probability the probability that the first transition to state 2 occurs at time n.