Glossary

9.2 Error exponents and reliability functions

2 min read•july 25, 2024

Error exponents and reliability functions are crucial tools in channel coding. They measure how quickly decoding errors decrease as grows, helping us understand the trade-off between and error probability.

These concepts provide bounds on for different channel models. By analyzing error exponents, we can determine optimal coding strategies and assess performance limits for various communication systems.

Error Exponents and Reliability Functions in Channel Coding

Error exponents and reliability functions

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  • Error exponents measure decrease rate of decoding error probability as block length increases, denoted E(R)E(R) for coding rate RR
  • Reliability functions represent inverse of error exponents, maximum achievable rate for given error probability, denoted E−1(R)E^{-1}(R)

Relationship to decoding error probability

  • Decoding error probability decreases exponentially with block length for rates below : Pe≈2−nE(R)P_e \approx 2^{-nE(R)} (n = block length)
  • Error exponents and reliability functions provide bounds on decoding error probability, determining rate-error probability trade-off
  • positive for rates below capacity, approaches zero as rate nears capacity

Calculation for simple channel models

  • (BSC): E(R)=D(p∣∣q)E(R) = D(p || q), D = , p = , q maximizes D(p || q)
  • (AWGN) channel: E(R)=(C−R)24CE(R) = \frac{(C-R)^2}{4C}, C = channel capacity
  • : E(R)=1−R/CE(R) = 1 - R/C for 0≤R≤C0 \leq R \leq C

Behavior vs coding rate and parameters

  • Error exponent decreases as coding rate increases, reaches zero at channel capacity
  • Better channel conditions (higher ) yield larger error exponents, worse conditions smaller exponents
  • Critical rate divides error exponent curve into regions with different slopes
  • provides upper bound, achievable performance for random codes
  • offers tighter bound for low rates, improving upon random coding exponent

Key Terms to Review (14)

Additive white gaussian noise: Additive white Gaussian noise (AWGN) refers to a type of noise that is characterized by its statistical properties, where the noise is added to a signal and has a constant spectral density. This noise is 'white' because it contains equal power across all frequencies, and 'Gaussian' because its amplitude follows a Gaussian distribution. AWGN is crucial in analyzing communication systems and understanding error rates and reliability functions in the context of transmitting information.
Binary symmetric channel: A binary symmetric channel is a communication model that transmits binary data (0s and 1s) with a certain probability of introducing errors during the transmission process. In this model, each bit sent can be flipped with a fixed probability, which represents the noise in the channel. This concept is fundamental to understanding how information is affected by noise and lays the groundwork for concepts like channel capacity and error rates in reliable communication.
Block Length: Block length refers to the number of bits or symbols in a single unit of data that is transmitted over a communication channel. It plays a crucial role in determining the efficiency and reliability of coding schemes, especially in the context of transmitting information over noisy channels, where longer block lengths can lead to better error correction and capacity utilization.
Channel Capacity: Channel capacity is the maximum rate at which information can be reliably transmitted over a communication channel without errors, given the channel's characteristics and noise levels. Understanding channel capacity is essential for optimizing data transmission, developing efficient coding schemes, and ensuring reliable communication in various technologies.
Coding rate: Coding rate is the ratio of the number of information bits to the total number of bits transmitted, reflecting the efficiency of a coding scheme in transmitting data over a communication channel. A higher coding rate indicates a more efficient use of bandwidth, while a lower coding rate typically corresponds to greater redundancy used for error correction. Understanding coding rate is crucial in the context of designing systems that balance data integrity and transmission efficiency.
Crossover Probability: Crossover probability is the likelihood that a transmitted signal will be incorrectly decoded due to noise or other disturbances in a communication channel. This concept is crucial for evaluating the performance and reliability of communication systems, particularly in relation to channel capacity and error rates, as it directly influences how much information can be reliably transmitted.
Decoding error probability: Decoding error probability refers to the likelihood that a decoding process will incorrectly interpret a transmitted message, resulting in an erroneous output. This concept is crucial in assessing the reliability of communication systems, as it helps determine how well a coding scheme can recover the original information from a received signal, especially in the presence of noise. Understanding this probability connects to error exponents and reliability functions, which quantify the performance and limits of coding strategies in mitigating errors.
Erasure Channel: An erasure channel is a communication channel where some of the transmitted symbols may be lost or erased during transmission, but the receiver is aware of which symbols were lost. This characteristic allows the receiver to treat the lost symbols as 'erased' rather than incorrectly received, significantly impacting error correction strategies and the overall reliability of the communication system.
Error Exponent: The error exponent quantifies the rate at which the probability of error decreases as the length of the code increases, giving insight into the reliability of a communication system. It is crucial in understanding how efficiently a coding scheme can transmit information over a noisy channel while maintaining acceptable error rates. By analyzing the error exponent, one can derive key performance metrics that help in designing robust coding strategies and optimizing system performance.
Expurgated Exponent: The expurgated exponent is a measure in information theory that quantifies the exponential decay of the error probability in the context of communication over noisy channels. It provides insights into how reliably a message can be transmitted as the block length increases, taking into account the worst-case scenarios for decoding errors. This concept connects closely with error exponents and reliability functions, helping to assess the performance of coding schemes under different noise conditions.
Kullback-Leibler Divergence: Kullback-Leibler divergence is a measure of how one probability distribution diverges from a second, expected probability distribution. This concept is fundamental in understanding how information is processed and represented, particularly in the context of comparing distributions, quantifying information loss, and establishing a framework for data analysis and coding theory.
Random coding exponent: The random coding exponent is a measure that quantifies the exponential decay of the probability of error in the context of coding for communication over noisy channels. It provides insights into how efficiently a coding scheme can approach the channel's capacity and how well it can perform with respect to reliability as the block length increases. This concept is closely linked to the performance of random codes, which are used to achieve the bounds set by information theory.
Reliability function: The reliability function is a measure used to quantify the probability that a communication system will perform without failure over a specified time period, given a certain level of noise and interference. This concept connects deeply with error exponents, as it helps to describe how likely it is for the system to maintain its integrity in the presence of errors. In essence, the reliability function provides insight into the performance limits of communication channels under various conditions.
SNR: SNR, or Signal-to-Noise Ratio, is a measure used to quantify the level of a desired signal to the level of background noise. A higher SNR indicates a clearer and more reliable signal, which is crucial when assessing the performance of communication systems and the reliability of data transmission.
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