Question

In a binary communication system two equiprobable messages s1 = (1, 1) and s2 = (−1, −1) are used. The received signal is r = s + n, where n = (n¹, n²). It is assumed that n1 and n2 are independent and each is distributed according to:

f (n) = 1/2 x eⁿ

Determine and plot the decision regions D1 and D2 in this communication scheme.

Answer 1

Decision regions D1 and D2 in a binary communication system with equiprobable messages (1, 1) and (-1, -1) are determined by the points closer to s1 and s2, respectively, with the decision boundary at the line x = 0.

Step-by-step explanation:

In a binary communication system using messages s1 = (1, 1) and s2 = (-1, -1), the received signal r is the sum of the sent message s and the noise vector n = (n1, n2), where n1 and n2 are independent and identically distributed with a probability density function f(n) = 1/2 x e⁻|n|.

To determine and plot the decision regions D1 and D2, we consider the distance of the received signal r to both s1 and s2. The decision regions are defined as the collection of points r that are closer to s1 than to s2 for D1, and vice versa for D2. The boundary between D1 and D2 would be the set of points equidistant to s1 and s2, which, in this case, is the line x = 0, as both s1 and s2 have equal probability.

The decision region D1 will consist of all points to the right of this boundary line while D2 will consist of all points to the left. No distribution curves are necessary to plot these decision regions since we are given a binary system with equiprobable messages and the noise distribution's specifics do not affect the location of the decision boundary in this particular case, but only the error probability.