Expressing probability of bit error

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SUMMARY

The discussion focuses on calculating the probability of bit error (P_e) in a bipolar system using threshold detection. The key equation presented is P_e = Q(ρ), where ρ = A_p/σ_n. Participants express uncertainty about visualizing the bit error regions and how to express these probabilities in terms of integrals and conditional probabilities. The conversation highlights the relationship between bit error rate and probabilities of good bits and bytes, emphasizing the need for clarity in expressing these concepts mathematically.

PREREQUISITES
  • Understanding of probability theory, specifically conditional probabilities.
  • Familiarity with Q-function and its application in communication theory.
  • Knowledge of bipolar signaling systems and threshold detection methods.
  • Basic calculus skills for working with integrals in probability expressions.
NEXT STEPS
  • Study the Q-function and its properties in digital communication contexts.
  • Explore conditional probability and its applications in error analysis.
  • Learn about integral calculus as it applies to probability distributions.
  • Investigate the impact of noise on bit error rates in communication systems.
USEFUL FOR

Students in electrical engineering, communication system designers, and anyone involved in analyzing or optimizing digital communication protocols.

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Homework Statement


I need to the probability of bit error in terms of integrals and terms of conditional probabilities.

Its a bipolar system using threshold detection
https://www.physicsforums.com/attachment.php?attachmentid=58542&stc=1&d=1367888259


Homework Equations


probability of bit error → ##P_e=Q(\rho)##
##\rho = \frac{A_p}{\sigma_n}##


The Attempt at a Solution


Im not sure I understand what's going on in the figure for this question. I would think that the bit error would be the shaded regions correct? I might be able to figure something out for expressing it in terms of conditional probability but I don't think it would be 100% correct. On top of that I have no idea how to express them in terms of integrals.

Im kind of guessing here but would it be something like $$P_e=p(r|-a) \ + \ p(r|a)$$

Im thinking no...

Any help or hints are greatly appreciated!
 
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I've normally seen error rates based on bit error rate in this fashion:

bit error rate = p
probability of a good bit = 1 - p
probability of a good byte (8 good bits) = (1-p)^8
 

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