Homework 8 (Due: April 6)

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ECE 630: Statistical Communication Theory
Prof. B.-P. Paris
Homework 8
Due: April 6, 2017

Reading
Madhow: Section 3.2 through 3.5.
Problems
  1. Binary Hypothesis Testing
    The two hypotheses are of the form:
    H  :  p (r) = 1-e-|r|
  0    R      2
H  :  p (r) = √-1--e- r22
  1    R        2π

    1. Find the likelihood ratio.
    2. Compute the decision regions for various values of the threshold in the likelihood ratio test (i.e., the MPE decision rule).
  2. A communications systems gives rise to the following decision problem. Equally likely signals take on the values s1 = -s0 = √--
 E. There are two independent, Gaussian random variables, N1 and N2 each with zero mean and variance σ2. The following inputs are observed:
    R  =   s + N
  1          1
R2 =   N1 + N2

    where s is either s0 or s1.

    1. Show that the MPE decision rule has the form
      R  + aR2  > γ  decide s
 1                     1
R1 + aR2  < γ  decide s0

    2. What is the optimum value of a?
    3. What is the optimum threshold γ?
    4. Compute the minimum probability of error.
    5. By how much would E need to be increased to achieve the same probability of error if only R1 was observed?
  3. The following signals are used to transmit equally likely messages over a channel corrupted by additive, white Gaussian noise of spectral height N0-
2:

    PICT

    Compute the probability of error attained by the following receivers.

    1. PICT

    2. PICT

    3. PICT

    4. PICT

    5. Let g(t) = 1 -|t - 1|, 0 t 2.

      PICT

  4. In this problem, we analyze the dependence of the probability of error on the threshold of the comparator in the optimum receiver. Assume one of two equally likely messages is transmitted using the following signals,
    s (t)  =   0     for 0 ≤ t ≤ T
 0
          ∘ --
s1(t)  =     E-  for 0 ≤ t ≤ T .
            T

    The channel is corrupted by additive, white Gaussian noise of spectral height N20-.

    1. Draw the block diagram of the optimum receiver.
    2. Compute the probability of error attained by this receiver.
    3. The threshold of the optimum receiver is given by γ = 1
2(||s1||2 -||s 0||2). What is the probability of error if instead this threshold were chosen as γ = λ||s1||2 - (1 - λ)||s 0||2?
    4. Plot the probability of error computed in part (c) for 0 λ 1. Use N0-
 2 = 1, E = 1, T = 1. You may approximate Q(x) by 1
2 exp(-x2
-2-).