A biliary symmetric channel (BSC) has a transition probability of 1/8. If the binary transmit symbol X is such that P(X = 0) = 9/10, then the probability of error for an optimum receiver will bea)7/80b)63/80c)9/10d)1/10Correct answer is option 'D'. Can you explain this answer?

Question

Answer

Py = 1)/(x = 0P(x = 0)/Py = 1)/(x = 1P(x = 1)

P(x = 0) = 9/10

P(x = 0) + P(x = 1) = 1 SoP(x = 1) = 1/10

Transition Probability is nothing but Prabability af changing the value from 0 to 1 and 1 to 0

i.e.

P(0/1) = P(1/0) = 1/8

P_e= P(x = 0)Py = 1)/(x = 0 + P(x = 1)Py = 0)/(x = 1

= (9/10) × (1/8) + (1/10) × (1/8)

= 1/8

But this nat the P_e for optimum Receiver. So,P_e = 1 − P_c

where P_c is Probability of correct detection Use MAP detectionRule:

The received symbal m = m₁ if P(y/x₁)P(x₁)/P(y/x_j)P(x_j) ≥ 1

This is abtained from following:

P_c(m = m_i) = P(m_i/y)⋅P(y) = P(x_i/y)⋅P(y)

M is the estimated output signal Using Baye's thearem

P(x_i/y)⋅P(y) = P(y/x_i)⋅P(x_i)

when o/py = 0 then Py = 0)/(x = 0P(x = 0)/Py = 0)/(x = 1P(x = 1)

= (7/8⋅9/10)/(1/8⋅1/10) = 63 ≥ 1

So when y = 0,m = 0

Similarly when y = 1 then = (1/8⋅9/10)/(7/8⋅1/10) = 9/7 ≥ 1

So when y = 0 then alsa m^ = 0

So if we choose the rule such that the message signal is 0 for y =0 ( or ) 1 then the Prabability of error is nothing but P(x = 1) = (1/10)

Answer

Probability of Error for an Optimum Receiver in a Biliary Symmetric Channel

Given:
- Biliary Symmetric Channel (BSC) has a transition probability of 1/8.
- Binary transmit symbol X with P(X = 0) = 9/10.

To find:
Probability of error for an optimum receiver.

Solution:
1. Understanding Biliary Symmetric Channel (BSC):
- A Biliary Symmetric Channel is a type of binary channel where the probability of error is the same for both 0 and 1 symbols.
- In a BSC, the transition probability represents the probability that a transmitted bit will be received in error. In this case, the transition probability is 1/8.

2. Probability of Error for an Optimum Receiver:
- An optimum receiver makes a decision based on the received signal with the goal of minimizing the probability of error.
- The probability of error for an optimum receiver can be calculated using the probability of error for each symbol and the probability of each symbol.

3. Calculation of Probability of Error:
- Let's calculate the probability of error for the symbol 0 and the symbol 1 separately and then add them together.

Probability of error for symbol 0:
- The probability of error for symbol 0 is the probability that the received symbol is 1 given that the transmitted symbol was 0.
- This can be calculated using Bayes' theorem:
P(Error|X=0) = P(X=0|Error) * P(Error) / P(X=0)
P(Error|X=0) = P(X=0|Error) * P(Error) / (P(X=0|Error) * P(Error) + P(X=0|No Error) * P(No Error))

- In a BSC, the probability of error for both symbols is the same, so P(Error|X=0) = P(Error|X=1) = P(Error).
- P(X=0|Error) and P(X=0|No Error) can be calculated using the transition probability of the BSC:
P(X=0|Error) = 1 - P(X=0) = 1 - 9/10 = 1/10
P(X=0|No Error) = P(X=0) = 9/10

- Plugging in the values, we get:
P(Error) = (1/10 * P(Error)) / (1/10 * P(Error) + 9/10 * P(No Error))
Solving for P(Error), we get: P(Error) = 1/10

- Similarly, the probability of error for symbol 1 will also be P(Error) = 1/10.

4. Probability of Error for an Optimum Receiver:
- Since the probability of error for both symbols is the same, the total probability of error for an optimum receiver can be obtained by adding the probability of error for each symbol:
P(Total Error) = P(Error|X=0) + P(Error|X=1)
P(Total Error) = 1/10 + 1/10 = 2/10 = 1/5 = 0.2

5. Answer:
The correct answer is option D) 1/10.

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