In a binary communication system the signals S1 t A 0 lessth

In a binary communication system the signals S_1 (t) {A, 0 lessthanorequalto t lessthanorequalto T 0, otherwise, S_2 (t) {0, 0 lessthanorequalto t lessthanorequalto T 0, otherwise, are transmitted with equal probabilities of 1/2 each. The noise in the channel is zero mean additive white Gaussian process with power spectral density N_0/2 Watts/Hz. Define Z = integral^infinity_-infinity Y (t)S_1 (t)dt. What is the optimal decision rule for this problem? Find the probability of error of this optimal receiver?

Solution

The Optimal Decision Rule: MAP
To begin with, let us list all the information that the receiver has.
1. The a priori probabilities of the two values the information bit can take. We will
normally consider these to be equal (to 0.5 each).
2. The received voltage y. While this is an analog value, i.e., any real number, in engineering
practice we quantize it at the same time the waveform is converted into a discrete
sequence of voltages. For instance if we are using a 16-bit ADC for the discretization,
then the received voltage y can take one of 216 possible values. We will start with this
discretization model first.
3. The encoding rule. In other words we need to know how the information bit is mapped
into voltages at the transmitter. For instance, this means that the mapping in illustrated
in Figure 2 should be known to the receiver. This could be considered part of the
protocol that both the transmitter and receiver subscribe to. In engineering practice,
all widespread communication devices subscribe to a universally known standard. For
example, Verizon cell phones subscribe to a standard known as CDMA.