[next] [prev] [prev-tail] [tail] [up]

Homework 9 (Due: April 9)

[PDF]

ECE 630: Statistical Communication Theory
Prof. B.-P. Paris
Homework 9
Due: April 9, 2019

Reading
Madhow: Section 3.4 through 3.6.
Problems
  1. Hypothesis Testing with Laplacian Noise
    A random variable N is said to be Laplacian distributed if its probability density function is given by
    1-- |x| pN (x) = 2e for - ∞ < x < ∞.

    Consider the following decision problem involving the observed random variable Z:

    H0: Z = 3 + N H1: Z = - 1 + N,

    where the two hypotheses are equally likely.

    1. Provide expressions for the probability density function for Z for each of the two hypotheses.
    2. Show that the maximum likelihood decision rule can be simplified to
      H0 Z ≷ γ. H1

      Determine the value of the optimum threshold γ.

    3. Compute the probability of error for this decision rule.

      For the remainder of the problem, consider a two-dimensional random vector N⃗ = (

      N1
      N2
      ) with independent Laplacian distributed components, i.e.,
      p (⃗x) = p (x )p (x ) = 1e-(|x1|+ |x2|) for - ∞ < x < ∞. N⃗ N1 1 N2 2 4 i

      with ⃗x = (

      x1
      x2
      ).

      Consider the following decision problem involving the observed random vector Z⃗ = (

      Z1
      Z2
      ):
      ( ) ⃗ 2 ⃗ H0: Z = 1 + N ( ) H : ⃗Z = - 2 + N⃗, 1 - 2

      where again the two hypotheses are equally likely.

    4. Provide expressions for the probability density function for ⃗Z for each of the two hypotheses.
    5. Show that the maximum likelihood decision rule can be simplified to
      H0 |Z1 + 2| - |Z1 - 2| ≷ |Z2 - 1| - |Z2 + 2|. H1

    6. The absolute values in the decision rule induce three distinct intervals for Z1 (Z1 < -2, -2 ≤ Z1 ≤ 2, and Z1 > 2) and three intervals for Z2 (Z2 < -2, -2 ≤ Z2 ≤ 1, and Z2 > 1).

      Consider all nine regions formed by combinations of these intervals (e.g., the region with Z1 < -2 and Z2 < -2) and simplify the decision rule for each of these combinations.

    7. Draw a two-dimensional signal-space diagram with axes Z1 and Z2. Mark the locations of E[⃗Z|Hi] for the two hypotheses. Then, draw the decision boundary formed by the optimal decision rule using the results from part (f).
  2. Binary Signal Sets
    The following signal set is employed to transmit equally likely signals over an additive white Gaussian noise channel with spectral height N02-.
    ( ( ) ) s0(t) = A 1 - 2t2 for - T-≤ t ≤ T- T 2 2 ( ( ) ) s1(t) = - A 1 - 2t-2 for - T-≤ t ≤ T- T 2 2

    1. Sketch and accurately label the block diagram of a receiver that minimizes the probability of error.
    2. Compute the energy of each of the two signals.
    3. Compute the probability of error for your receiver from part (a).
    4. Consider now the following receiver:

      SVG-Viewer needed.

      Find the conditional distribution of the random variable R at the output of the integrator for each of the two signals s0(t) and s1(t).

    5. Compute the probability of error achieved by the suboptimum receiver.
    6. Compare the probability of error for the suboptimum receiver to that of the optimum receiver. Express your answer in the form: “to achieve the same probability of error as the optimum receiver, the suboptimum system requires a times more energy.” Determine the factor a.
    7. Assume now that the received signal is corrupted by an interfering signal x(t) = A2, for -T2- ≤ t ≤T2- so that the received signal under the i-th hypotesis (i = 0, 1) is given by
      H : R = s (t) + A-+ N for - T- ≤ t ≤ T- i t i 2 t 2 2

      Compute the probability of error by the optimum receiver in the presence of the interfering signal.

    8. Explain your result in part (g).

[next] [prev] [prev-tail] [front] [up]