Homework 3 (Due: February 16)

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ECE 630: Statistical Communication Theory
Prof. B.-P. Paris
Homework 3
Due: February 16, 2017

Reading
Madhow: Section 3.3.
Problems
  1. A stochastic process is defined by
    Xt = cos2 πF t

    where the frequency F is uniformly distributed over the interval [0,f0].

    1. Find the mean and correlation function of Xt.
    2. Show that this process is non-stationary.

    Now suppose we redefine the process Xt to be

    Xt = cos(2πF t + Θ )

    where F and Θ are statistically independent random variables. Θ is uniformly disributed over [-π,π) and F is distributed as before.

    1. Compute the mean and correlation function of Xt.
    2. Is Xt wide-sense stationary? Show your reasoning.
    3. Find the first order density pXt(x).
  2. The stationary random process Xt is passed through a linear filter with transfer function H(f),
            -j2πf-+--a
H (f) = j2πf  + 2a.

    The output process is labeled Y t. The mean of Y t is measured to be 1
2 and the covariance function of Y t is found to be

              2 -2a|τ|
KY (τ) = a e     .

    1. Compute the power spectral density of Y t.
    2. Find the second order description of Xt.
  3. In practice one often wants to measure the power spectral density of a stochastic process. For the purposes of this problem, assume the process Xt is wide-sense stationary, zero mean, and Gaussian. The following measurement system is proposed.

    PICT

    Here H1(f) is the transfer function of an ideal bandpass filter and H2(f) is an ideal lowpass,

             {
            1  for f0 - Δf-≤  |f | ≤ f0 + Δf
H1 (f) =    0  else      2               2

             {  -1--
H2 (f) =    2Δf  for |f | ≤ Δf
             0   else.

    Assume that Δf is small compared to the range of frequencies over which SX(f) varies, i.e., you may assume that SX(f) is constant over intervals of width Δf.

    1. Find the mean and correlation function of Y t2 in terms of the second order description of Xt.
    2. Compute the the power spectral density of the process Zt.
    3. Compute the expected value of Zt.
    4. By considering the variance of Zt, comment on the accuracy of this measurement of the power density of the process Xt.
  4. Let Wt (for t 0) be a Wiener process (Brownian motion) with variance σ2. Define the random process X t as the (runnning) integral over Wt, i.e., for t 0
          ∫
        t
Xt =     Wsds.
       0

    1. Find the mean of X(t).
    2. Compute the autocorrelation function of Xt.
    3. Is Wt wide-sense stationary?
    4. Compute the following probability for t 0
      Pr{|Xt| > σ ⋅ t}.