Homework 8 (Due: April 6)
ECE 630: Statistical Communication Theory
Prof. B.-P. Paris
Due: April 6, 2017
- Madhow: Section 3.2 through 3.5.
- Binary Hypothesis Testing
The two hypotheses are of the form:
- Find the likelihood ratio.
- Compute the decision regions for various values of the
threshold in the likelihood ratio test (i.e., the MPE decision
- 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
where s is either s0 or s1.
- Show that the MPE decision rule has the form
- What is the optimum value of a?
- What is the optimum threshold γ?
- Compute the minimum probability of error.
- By how much would E need to be increased to achieve the
same probability of error if only R1 was observed?
- The following signals are used to transmit equally likely messages over
a channel corrupted by additive, white Gaussian noise of spectral
Compute the probability of error attained by the following
- Let g(t) = 1 -|t - 1|, 0 ≤ t ≤ 2.
- 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
The channel is corrupted by additive, white Gaussian noise of spectral
- Draw the block diagram of the optimum receiver.
- Compute the probability of error attained by this receiver.
- The threshold of the optimum receiver is given by γ =
0||2). What is the probability of error if instead
this threshold were chosen as γ = λ||s1||2 - (1 - λ)||s
- 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