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Random Processes Problem Set 6 - ECE 434 Spring 2003, Assignments of Electrical and Electronics Engineering

University of Illinois - Urbana-ChampaignElectrical and Electronics Engineering

Problem set 6 for the ece 434 - random processes course offered in spring 2003. The problems cover various topics such as cross spectral density, stationary markov processes, linear estimation, approximation of white noise, and simulating baseband random processes. Students are expected to solve these problems using the concepts learned in the course.

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ECE 434 RANDOM PROCESSES SPRING 2003
PROBLEM SET 6 Due Wednesday, April 23
Random Processes in Linear Systems and Spectral Analysis
Assigned Reading: Chapter 6 of the course notes.
Reminder: The second hour exam will be given Monday, April 14, 7-8:15 p.m. in Room 269 Everitt
Lab.
Problems to be handed in:
1. On the cross spectral density
Suppose Xand Yare jointly WSS and that SX(ω) = 0 for all ωin some open interval (a,b). Show
that SXY (ω) = 0 for all ωin the same interval (except possibly for a set of measure zero). Hint:
It is enough to show that the integral of SXY over any subinterval [c,d] of (a,b) is zero. To do so,
consider passing both Xand Ythrough a linear time-invariant system with narrowband transfer
function H[c,d]((ω) = I{c≤ω≤d}.
2. A stationary two-state Markov process
Let X= (Xk:k∈ZZ) be a stationary Markov process with state space S={1,−1}and one-step
transition probability matrix
P= 1−p p
p1−p!,
where 0 < p < 1. Find the mean, correlation function and power spectral density function of X.
(Hint: For nonnegative integers k:
Pk= 1
2
1
2
1
2
1
2!+ (1 −2p)k 1
2−1
2
−1
2
1
2!.
3. A linear estimation problem
Suppose Xand Yare possibly complex valued jointly WSS processes with known autocorrelation
functions, cross-correlation function, and associated spectral densities. Suppose Yis passed through
a linear time-invariant system with impulse response function hand transfer function H, and let
Zbe the output. The mean square error of estimating Xtby Ztis E[|Xt−Zt|2].
(a) Express the mean square error in terms of RX,RY,RXY and h.
(b) Express the mean square error in terms of SX,SY,SXY and H.
(c) Using your answer to part (b), find the choice of Hthat minimizes the mean square error. (Hint:
Try working out the problem first assuming the processes are real valued. For the complex case,
note that for σ2>0 and complex numbers zand zo,σ2|z|2−2Re(zzo) is equal to |σz −zo
σ|2−|zo|2
σ2,
which is minimized with respect to zby z=zo
σ2.)
1
pf3
pf4

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Download Random Processes Problem Set 6 - ECE 434 Spring 2003 and more Assignments Electrical and Electronics Engineering in PDF only on Docsity!

ECE 434 RANDOM PROCESSES SPRING 2003

PROBLEM SET 6 Due Wednesday, April 23

Random Processes in Linear Systems and Spectral Analysis

Assigned Reading: Chapter 6 of the course notes.

Reminder: The second hour exam will be given Monday, April 14, 7-8:15 p.m. in Room 269 Everitt Lab. Problems to be handed in:

  1. On the cross spectral density Suppose X and Y are jointly WSS and that SX (ω) = 0 for all ω in some open interval (a,b). Show that SXY (ω) = 0 for all ω in the same interval (except possibly for a set of measure zero). Hint: It is enough to show that the integral of SXY over any subinterval [c,d] of (a,b) is zero. To do so, consider passing both X and Y through a linear time-invariant system with narrowband transfer function H[c,d]((ω) = I{c≤ω≤d}.
  2. A stationary two-state Markov process Let X = (Xk : k ∈ ZZ) be a stationary Markov process with state space S = { 1 , − 1 } and one-step transition probability matrix

P =

( 1 − p p p 1 − p

) ,

where 0 < p < 1. Find the mean, correlation function and power spectral density function of X. (Hint: For nonnegative integers k:

P k^ =

( 1 2

1 1 2 2

1 2

)

  • (1 − 2 p)k

( 1 2 −^

1 2 − (^1212)

) .

  1. A linear estimation problem Suppose X and Y are possibly complex valued jointly WSS processes with known autocorrelation functions, cross-correlation function, and associated spectral densities. Suppose Y is passed through a linear time-invariant system with impulse response function h and transfer function H, and let Z be the output. The mean square error of estimating Xt by Zt is E[|Xt − Zt|^2 ]. (a) Express the mean square error in terms of RX , RY , RXY and h. (b) Express the mean square error in terms of SX , SY , SXY and H. (c) Using your answer to part (b), find the choice of H that minimizes the mean square error. (Hint: Try working out the problem first assuming the processes are real valued. For the complex case, note that for σ^2 > 0 and complex numbers z and zo, σ^2 |z|^2 − 2 Re(zzo) is equal to |σz − z σo |^2 − |zo|

2 σ^2 , which is minimized with respect to z by z = z σo 2 .)

  1. An approximation of white noise White noise in continuous time can be approximated by a piecewise constant process as follows. Let T be a small positive constant and let (Bk : k ∈ ZZ) be a discrete-time white noise process with RB (k) = σ^2 I{k=0}. Define (Nt : t ∈ IR) by

Nt = AT

∑^ ∞ −∞

pT (t − nT )Bn

where

pT (s) =

{ 1 0 ≤ s ≤ T 0 else

and AT is a constant depending on T. (a) Sketch a typical sample path of N and express E[|

∫ (^1) 0 Nsds|

(^2) ] in terms of AT , T and σ (^2). For simplicity assume that T = (^) K^1 for some large integer K. (b) What choice of AT makes the expectation found in part (a) equal to σ^2? This choice makes N a good approximation to a continuous-time white noise process with autocorrelation function σ^2 δ(τ ). (c) What happens to the expectation found in part (a) as T → 0 if AT = 1 for all T?

  1. Simulating a baseband random process Suppose a real-valued Gaussian baseband process X = (Xt : t ∈ IR) with mean zero and power spectral density

SX (2πf ) =

{ 1 if |f | ≤ 0. 5 0 else

is to be simulated over the time interval [− 500 , 500] through use of the sampling theorem with sampling time T = 1. (a) What is the joint distribution of the samples, Xn : n ∈ Z? (b) Of course a computer cannot generate infinitely many random variables in a finite amount of time. Therefore, consider approximating X by X(N^ )^ defined by

X( t N^ )=

∑^ N

n=−N

Xnsinc(t − n)

Find a condition on N to guarantee that E[(Xt − X( t N^ ))^2 ] ≤ 0 .01 for t ∈ [− 500 , 500]. (Hint: Use |sinc(τ )| ≤ (^) π^1 |τ | and bound series by an integral. Your choice of N should not depend on t because the same N should work for all t in the interval [− 500 , 500] ).

  1. A narrowband Gaussian process Let X be a real-valued stationary Gaussian process with mean zero and RX (τ ) = cos(2π(400τ ))(sinc(6τ ))^2. (a) Find and carefully sketch the power spectral density of X. (b) Sketch a sample path of X. (c) The process X can be represented by Xt = Re(Zte^2 πj^400 t), where Zt = Ut + jVt for jointly stationary narrowband real-valued random processes U and V. Find the spectral densities SU , SV , and SU V. (d) Find P [|Z 33 | > 5]. Note that |Zt| is the real envelope process of X.

Extra Credit Problems 6 (Not Required) Prof. B. Hajek Same due date as Problem Set 6 Spring 2003

  1. Cyclostationary random processes A random process X = (Xt : t ∈ IR) is said to be cyclostationary with period T , if whenever s is an integer multiple of T , X has the same finite dimensional distributions as (Xt+s : t ∈ IR). This property is weaker than stationarity, because stationarity requires equality of finite dimensional distributions for all real values of s. (a) What properties of the mean function μX and autocorrelation function RX does any second order cyclostationary process possess? A process with these properties is called a wide sense cyclostationary process. (b) Suppose X is cyclostationary and that U is a random variable independent of X that is uniformly distributed on the interval [0, T ]. Let Y = (Yt : t ∈ IR) be the random process defined by Yt = Xt+U. Argue that Y is stationary, and express the mean and autocorrelation function of Y in terms of the mean function and autocorrelation function of X. Although X is not necessarily WSS, it is reasonable to define the power spectral density of X to equal the power spectral density of Y. (c) Suppose B is a stationary discrete-time random process and that g is a deterministic function. Let X be defined by

Xt =

∑^ ∞ −∞

g(t − nT )Bn.

Show that X is a cyclostationary random process. Find the mean function and autocorrelation function of X in terms g, T , and the mean and autocorrelation function of B. If your answer is complicated, identify special cases which make the answer nice. (d) Suppose Y is defined as in part (b) for the specific X defined in part (c). Express the mean μY , autocorrelation function RY , and power spectral density SY in terms of g, T , μB , and SB.

  1. Zero crossing rate of a stationary Gaussian process Consider a zero-mean stationary Gaussian random process X with SX (2πf ) = |f | − 50 for 50 ≤ |f | ≤ 60, and SX (2πf ) = 0 otherwise. Assume the process has continuous sample paths (it can be shown that such a version exists.) A zero crossing from above is said to occur at time t if X(t) = 0 and X(s) > 0 for all s in an interval of the form [t − , t) for some  > 0. Determine the mean rate of zero crossings from above for X. If you can find an analytical solution, great. Alternatively, you can estimate the rate (aim for three significant digits) by Monte Carlo simulation of the random process.