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Homework 4 (Due: February 26)

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ECE 630: Statistical Communication Theory
Prof. B.-P. Paris
Homework 4
Due: February 26, 2019

Reading
Madhow: Section 3.3.
Problems
  1. The stationary random process Xt is passed through a linear filter with transfer function H(f),
    H (f) = -j2πf-+--a. 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.
  2. 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-- for |f | ≤ Δf H2 (f) = 2Δf 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. The following may be helpful — this is known as Isserlin’s Theorem: If X and Y are jointly Gaussian, then E[X2Y 2] = E[X2]E[Y 2] + 2E2[XY ]
    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.
  3. 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 X = W ds. t 0 s

    1. Find the mean of Xt.
    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}.

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