Question

A binary signal detector is being built. When the signal A is sent, a sequence of i.i.d. samples Xj​∼N(−1,1) is received. When the signal B is sent, a sequence of Xj​∼N(+1,1) is received. Given a very large number n of observations (X1​,⋯,Xn​) propose a detector for deciding between A and B. Consider two separate design cases:

(a) Misdetecting A for B or B for A are equally bad.

Answer 1

The design of a binary signal detector for differentiating between two normally distributed signals can use signal detection theory by setting a threshold for the average value of observations, typically at 0, to minimize decision errors when the cost of misdetecting signals is the same.

Step-by-step explanation:

Designing a binary signal detector for distinguishing between two normally distributed signals, one for signal A (Xj​​≈N(−1,1)) and another for signal B (Xj​​≈N(+1,1)). Given a large number of observations, the detector can be built using the principle of signal detection theory. This can be approached by using the average value of the observations as the decision statistic.

In the case where misdetecting A for B or B for A are equally bad, a threshold can be set for the average value of received observations such that:

• If the average is closer to -1, then signal A is detected.
• If the average is closer to +1, then signal B is detected.

Since the variance is the same for both signals, a threshold of 0 would be a logical choice, as it would minimize the probability of decision errors over a large sample size.