The binary communications system under consideration uses the following signals
to transmit equally likely messages over an additive, white Gaussian noise
channel (spectral height )

where is a zero-mean, white Gaussian noise process with autocorrelation function .

Assume first that the following receiver front-end is being used, where denotes an arbitrary signal of unit energy, i.e., .

- Compute the conditional distribution of the random variable for each of the two signals and .
- Assume now that the conditional distributions for the random variable
are given by

Find the likelihood-ratio test that minimizes the probability of error. Simplify as much as possible. - Find the probability of error for your test.
- Consider now the following receiver front-end

The functions and are orthonormal. Find the joint distribution of random variables and .

- For the receiver front-end above, devise the minimum probability of error test for determining which of the two signals and was transmitted. You do not have to compute the probability of error for your test, but you must provide a convincing argument if the probability of error will be larger or smaller than with the first receiver front-end.
- What is the optimum receiver for this problem?

2003-01-28