Question

3. A transmitter sends the signals −1 and +1, with probabilities p and 1−p, respectively. The communication channel introduces additive noise that is normal with mean 0 and variance σ

2
. This noise is independent of the particular signal sent. The receiver at the other end of the channel receives a signal which is the sum of the transmitted signal and the channel noise. a. Let a be a constant between −1 and +1. The receiver interpretes its signal as −1 if it is less than a, and as +1 if it is greater than a. Find a formula (in terms of a,p and σ ) for the probability that the receiver makes an error. b. Determine the probability that the receiver makes an error if p=2/5,a=1/2 and σ
2
=1/4.

Answer 1

The probability that the receiver makes an error can be found by calculating the probability that the received signal is interpreted incorrectly. This is done by calculating the probabilities of a +1 signal being interpreted as -1 and a -1 signal being interpreted as +1 using the normal distribution. The probability of error is given by a formula that takes into account the probabilities of transmitting -1 and +1, as well as the value of a and the variance σ^2.

Step-by-step explanation:

The probability that the receiver makes an error can be found by calculating the probability that the received signal is interpreted incorrectly. Let's denote the probability of transmitting -1 as p and the probability of transmitting +1 as 1-p. The receiver interprets the signal as -1 if it is less than a, and as +1 if it is greater than a.

The probability of error can be calculated by finding the probabilities that a +1 signal is interpreted as -1 and that a -1 signal is interpreted as +1. These probabilities can be determined using the normal distribution. The probability of error is given by the equation:

Probability of error = p * P(X < a) + (1-p) * P(X > a)

Where X is a random variable that represents the sum of the transmitted signal and the channel noise, and P(X < a) and P(X > a) are calculated using the normal distribution with mean 0 and variance σ².