5.3 Maximum Likelihood Decoding

Maximum likelihood decoding picks the most probable codeword based on what's received. It's like choosing the best guess when you're not sure what was sent. This method minimizes decoding errors, making it super useful in noisy channels.

The Hamming distance between codewords is key here. It measures how different two codewords are. The bigger the minimum distance in a code, the more errors it can fix. This is crucial for keeping messages clear in messy transmissions.

Maximum Likelihood Decoding

Decoding Principles

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• Maximum likelihood decoding selects the codeword that maximizes the probability of receiving the observed word given the codeword was sent
• Nearest neighbor decoding chooses the codeword closest to the received word in terms of Hamming distance
  • Hamming distance measures the number of positions in which two codewords differ
  • For example, the Hamming distance between

    1011

    and

    1001

    is 1 because they differ in only one position
• Minimum distance decoding selects the codeword with the smallest Hamming distance from the received word
  • If multiple codewords have the same minimum distance, the decoder may choose arbitrarily or report a decoding failure

Optimality and Performance

• Maximum likelihood decoding is optimal in the sense that it minimizes the probability of decoding error
  • It selects the most likely codeword based on the observed received word
  • Assuming all codewords are equally likely to be sent, maximum likelihood decoding is equivalent to minimum distance decoding
• The performance of maximum likelihood decoding depends on the code's minimum distance and the channel's error characteristics
  • A code with a larger minimum distance can correct more errors
  • For example, a code with a minimum distance of 3 can correct any single-bit error, while a code with a minimum distance of 5 can correct any double-bit error

Error Analysis

Hamming Distance and Error Correction

• Hamming distance is a fundamental concept in error analysis and correction
  • It quantifies the dissimilarity between two codewords
  • The minimum Hamming distance of a code determines its error-correcting capability
    • A code with a minimum distance of $d$ can correct up to $\lfloor\frac{d-1}{2}\rfloor$ errors
    • For instance, a code with a minimum distance of 7 can correct up to 3 errors
• The error-correcting capability of a code is crucial in noisy communication channels
  • Errors can occur during transmission due to various factors (noise, interference, or hardware faults)
  • By using codes with sufficient minimum distance, the receiver can recover the original message even in the presence of errors

Decoding Regions and Error Probability

• Decoding regions are subsets of the vector space associated with each codeword
  • A received word is decoded to the codeword whose decoding region it falls into
  • The shape and size of the decoding regions depend on the code's structure and the decoding algorithm
• The probability of decoding error is related to the decoding regions and the channel's error characteristics
  • A larger decoding region for a codeword reduces the chances of decoding errors
  • The error probability can be calculated using the noise distribution and the geometry of the decoding regions
  • For example, in a binary symmetric channel with bit error probability $p$, the error probability for a code with minimum distance $d$ is approximately $\sum_{i=t+1}^{n} \binom{n}{i} p^i (1-p)^{n-i}$, where $t = \lfloor\frac{d-1}{2}\rfloor$ and $n$ is the codeword length