Please, I need some support (not solution) and input to know if the way is right to go on.
Consider the communication of binary messages in a transmission medium. Any message sent is selected from two possible symbols, $0$ or $1$. Each symbol occurs with equal probability. It is also known that any numerical value sent on that channel is subjected to distortion. If a value $x$ is transmitted, the $y$ value is received at its destination, described by $y = x + n$, where n represents a random variable that additive noise is independent of $x$. The noise has a normal distribution with parameters $σ^2 = 4$ and $\mu = 0$.
- Suppose the transmitter encodes the symbol $0$ with the value $x = -2$ and $1$ with the symbol value $x = 2$. At the destination, the received message is decoded according to the following rules:
not yet ....
- If $y ≥ 0$, one concludes that the symbol $1$ was sent.
- If $y <0$, conclude the symbol $0$ was sent.
Q: Determine the probability of error for this schema encoding / decoding.
So, I know that....
Bit error probability: $P(x=0|y=1) ~\&~ P(x=1|y=0)$. The probabilities of transmitting each signal $(0,1)$ are equal (i.e., $1/2$).
$y_t = x_t + n_t$ where $x_t$ is a free signal of noise.
Assuming that the noise has a Gaussian distribution.