# Probability in Communication

## Probability in Communication Engineering

The field of communication engineering deals with wired and wireless transmission of information. Examples of systems are land-line phones, cellular networks, satellite networks  and many more. The field of communication is very wide and deals both with the information theory aspect, how information is formatted and modulated. However it also deals with models of the channel, what happens between the transmitter and receiver.

## Communication models and noise

In the most simple case of a communication system with one transmitter and one receiver, we could have the following $y(t) = x(t) + w(t)$

Where $y(t)$ is the received signal, $x(t)$ is the transmitted one, and $w(t)$  is noise in our system. Noise always exists in all electronic systems, and it is specifically important in communication since it will determine what performance we will have. In the usual case, the noise is modeled as being Gaussian with zero mean, and some variance $N_0$.

So basically, this model describes that we transmit our signal, and some Gaussian noise is added on the way to our receiver. The question arises, how will the noise impact our system?

Imagine that we want to transmit one unit of information, one bit. We decide that if we send ‘a’ it represents a 1, and if we send ‘-a’ it represents a 0. Hence if we do this repeated times we can send a sequence of 1’s and 0’s, a binary sequence. This is the basis of all digital communication, such as cellular and satellite networks.

If we assume that we have no noise, a noiseless system, then we can always determine what we sent with 100% accuracy, since in that case $y(t) = x(t) + w(t) = x(t)$

Since $w(t) = 0$ for the noiseless case. In this ideal communication system, we have no errors, and we always know exactly what we sent. However, if we now include some noise, we realize that sometimes it can be difficult to decide what we actually sent.

Lets assume that the signal is either {a} or {-a}, let a = 5 in this case. Hence the values that x(t) can take, the constellation, is equal to x(t)= {-5, 5}. Let the noise have mean zero and a variance of 1.  Assume that we for example receive 2.3. In this case it is easy to realize that our best guess would be x(t) = 5! Since it is the closest value. However the more interesting question is, how large is the probability that we guess incorrectly on what was actually transmitted! We can actually not know exactly was transmitted, since the receiver will only see what was transmitted, plus the random noise! Since the noise is random, and unknown, this might be tricky.

Let’s try to calculate how large the probability is, that we “guess” wrong on what was sent at the transmitter. If we assume that we sent an ‘a’, the probability that it is mistaken for a ‘-a’ is

P(‘incorrect guess’) = P( y(t) < 0, given x(t) = 5 ) = P(x(t) + w(t) < 0, given x(t) = 5)

Since we are given x(t) = 5, we can replace x(t) by 5 in the expression and obtain

P(‘incorrect guess’) = P(5 + w(t) < 0) = P(w(t) < -5)

Hence, the probability of an errors is the probability that the noise will be less than -5. Since we know that the noise is Gaussian with mean 0 and variance $N_0$, we get

P(w(t) < -5)

Where we in this case need to calculate the probability that a normal Gaussian random variable with mean 0 and variance $N_0$ is less than -5.

## Summary – Noise in Communication

In this chapter we discussed the basics of how probability is used within communication engineering. We saw that the noise, being modeled as Gaussian, will impact the system performance. How many errors we have, the bit errors rate, can be exactly calculated with our statistical tools. As we delve deeper into areas such as wireless communication, we will extensively use statistics and probability.