Homework 8 (Due: April 2)

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Reading

Madhow: Section 3.2 through 3.5.

Problems

1. Binary Hypothesis Testing
   The two hypotheses are of the form:
   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 = . There are two independent, Gaussian random variables, N₁ and N₂ each with zero mean and variance σ². The following inputs are observed:

   where s is either s₀ or s₁.

   1. Show that the MPE decision rule has the form

   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 :

   

   Compute the probability of error attained by the following receivers.

   1. 

   2. 

   3. 

   4. 

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

      

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,

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

   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 = (||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 = 1, E = 1, T = 1. You may approximate Q(x) by exp().