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 X  [ X 1 X 2  X 5 ] be a random vector whose components satisfy the following expression
X i  X i 1  Bi
for (1  i  5)
where Bi ’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 X 1  B1 . By substituting Bi ’s in a random vector B and
X i in a random vector X , solve the following questions.
(a) Write the above relationship X  AB for some constant matrix A and determine A .
(b) Find the mean vector μ X  E{X}.
(c) Find the covariance matrix K B for random vector B.
(d) Find the covariance matrix K B for random vector X .
2.
(Conditional pdf) Let X and Y be independent identically distributed (IID) random variables with the exponential
probability density function
f X ()  f Y ()  e U ()
(a)
Determine the probability density function f R (r ) for the ratio
0R
X
1
X Y
(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
f X ( x | A, Y  y).
(c)
3.
Using the definitions of (b), what is the minimum mean square estimator of X given that the event A occurs and
that Y = y?
(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
1 / 2 n  0

h[n]  1 / 2 n  1
 0
else

Given the mean  X [n] and correlation function R XX [n1 , n2 ] of the input X[n], express the following quantities in
terms of these parameters:
(a) Mean function  Y [n] of the output Y[n].
(b) Autocorrelation function RYY [n1 , n2 ] of the output Y[n].
(c) Autocovariance function K YY [n1 , n 2 ] of the output Y[n].
(d) Assuming that the input X[n] is a Gaussian random sequence, determine the joint pdf f Y ( y1 , y 2 ; n1 , n 2 ) at
arbitrary instants n1  n2 .
4.
(Linear CCD Modeling) A charge coupled device (CCD) array with ``leaky” cells is modeled by a random sequence

X [n,  ]   A( k )h[n  k ]
k  
where  k is the k’th component of  , the infinite dimensional outcome of the experiment. The random variables
A( k ) are jointly independent and Gaussian distributed with mean  and variance . The width-3 pulse function h[n]
is defined as follows:
1 / 2 n  0

h[n]  1 / 4 n  1
 0
else,

(a)
Find the mean function  X [n] .
(b) Compute the first order pdf f X ( x; n).
(c) Find the second order pdf f X ( x1 , x2 ; n1 , n2 ).
5.
(LTI Systems) Consider the difference equation model for a LTI system
y[n]  y[n  1]  x[n],   n  
where  is a constant such that || < 1.
(a) Compute the output y[n] for the input x[n] =  n U[n], || < 1, assuming causality applies, i.e., y[n] = 0 for n < 0.
(b) Compute the output y[n] for the input x[n] = -n U[-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:
X [0]  0
X [n]  X [n  1]  W [n] for n  1
(a) Find the mean value of X[n] for n > 0.
(b) Find the covariance K XX [n1 , n 2 ] of X[n].
(c) For what values of  does K XX [n1 , n 2 ] tend to G[n1  n 2 ] as n1 , n2 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 B be 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
X [n]  A cos(o n)  B sin(o n)
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
X[n]
+
h[n]
Y[n]
V[n]
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’s Sxx()
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
Y [ n] 
1 2
 X [n  k ]
5 k  2
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,
Pij  P{X [n]  j | X [n  1]  i}, 1  i, j  2
given in the matrix form as
0.9 0.1
P  {Pij }  

 0 . 2 0. 8 
We describe the state probabilities at time n by the vector
p[n]  [ P{ X [n]  1}, P{ X [n]  2}].
(a) Show that p[n]  p[0]P n .
(b) Draw a two state transition diagram and label the branches with the one-step transition probabilities Pij .
(c) Given that X[0] = 1, find the probability the probability that the first transition to state 2 occurs at time n.