Teacherbot 27 Aug, 14:49:13

To derive the average bit error probability for this system, we need to calculate the probability of error for each possible transmitted symbol and then take the average.

Let’s consider the case when the transmitted symbol is +sqrt(Eb). The received signal can be written as:

rn = 0.8sn + 0.5sn-1 + zn = 0.8sqrt(Eb) + 0.5sqrt(Eb) + zn = 1.3*sqrt(Eb) + zn

The decision threshold for the correlator receiver is 0. If the received signal is greater than 0, the receiver decides that the transmitted symbol is +sqrt(Eb), otherwise it decides that the transmitted symbol is -sqrt(Eb).

The probability of error for the case when the transmitted symbol is +sqrt(Eb) can be calculated as:

P(error|+sqrt(Eb)) = P(rn < 0 | +sqrt(Eb)) = P(1.3sqrt(Eb) + zn < 0 | +sqrt(Eb)) = P(zn < -1.3sqrt(Eb))

Since zn is a zero-mean complex Gaussian random variable with variance 0, its real and imaginary parts are independent and normally distributed with zero mean and variance 0. Therefore, P(zn < -1.3*sqrt(Eb)) can be calculated as:

P(zn < -1.3sqrt(Eb)) = P(Re(zn) < -1.3sqrt(Eb)) * P(Im(zn) < -1.3sqrt(Eb)) = P(Re(zn) < -1.3sqrt(Eb))^2 = (1/2)[1 - erf(1.3sqrt(Eb)/sqrt(2))]^2

Similarly, the probability of error for the case when the transmitted symbol is -sqrt(Eb) can be calculated as:

P(error|-sqrt(Eb)) = P(rn > 0 | -sqrt(Eb)) = P(1.3sqrt(Eb) + zn > 0 | -sqrt(Eb)) = P(zn > -1.3sqrt(Eb))

P(zn > -1.3sqrt(Eb)) = P(Re(zn) > -1.3sqrt(Eb)) * P(Im(zn) > -1.3sqrt(Eb)) = P(Re(zn) > -1.3sqrt(Eb))^2 = (1/2)[1 - erf(1.3sqrt(Eb)/sqrt(2))]^2

The average bit error probability can be calculated as the average of the two error probabilities:

P(error) = (1/2)[P(error|+sqrt(Eb)) + P(error|-sqrt(Eb))] = (1/2)[(1/2)[1 - erf(1.3sqrt(Eb)/sqrt(2))]^2 + (1/2)[1 - erf(1.3sqrt(Eb)/sqrt(2))]^2] = [1 - erf(1.3*sqrt(Eb)/sqrt(2))]^2

Therefore, the average bit error probability for this system as a function of Eb/N0 is:

P(error) = [1 - erf(1.3*sqrt(Eb/N0)/sqrt(2))]^2

You 27 Aug, 15:25:34

Make this simpler to understand

Teacherbot 27 Aug, 15:25:34

To simplify the expression for the average bit error probability, let’s break it down step by step.

1. The received signal at the demodulator is given by: rn = 0.8sn + 0.5sn-1 + zn

2. The decision threshold for the correlator receiver is 0. If the received signal is greater than 0, the receiver decides that the transmitted symbol is +sqrt(Eb), otherwise it decides that the transmitted symbol is -sqrt(Eb).

3. Let’s consider the case when the transmitted symbol is +sqrt(Eb). The received signal can be written as: rn = 1.3*sqrt(Eb) + zn

4. The probability of error for this case can be calculated as the probability that the received signal is less than 0: P(error|+sqrt(Eb)) = P(rn < 0 | +sqrt(Eb))

5. Since zn is a zero-mean complex Gaussian random variable with variance 0, its real and imaginary parts are independent and normally distributed with zero mean and variance 0. Therefore, P(zn < -1.3sqrt(Eb)) can be calculated as: P(zn < -1.3sqrt(Eb)) = (1/2)[1 - erf(1.3sqrt(Eb)/sqrt(2))]^2

6. Similarly, the probability of error for the case when the transmitted symbol is -sqrt(Eb) can be calculated as: P(error|-sqrt(Eb)) = (1/2)[1 - erf(1.3sqrt(Eb)/sqrt(2))]^2

7. The average bit error probability can be calculated as the average of the two error probabilities: P(error) = (1/2)[P(error|+sqrt(Eb)) + P(error|-sqrt(Eb))] = [1 - erf(1.3sqrt(Eb/N0)/sqrt(2))]^2

So, the average bit error probability for this system as a function of Eb/N0 is: P(error) = [1 - erf(1.3*sqrt(Eb/N0)/sqrt(2))]^2

This expression gives us the average probability of error for the system, which represents the likelihood of incorrectly decoding a bit.