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This study addresses recovering Boolean signals from noisy data. We derived a formula for signal recovery, revealing the number of measurements needed based on signal properties in low signal-to-noise ratio conditions.

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Area of Science:

  • Signal processing
  • Information theory
  • Statistical inference

Background:

  • The Boolean multireference alignment problem involves reconstructing a Boolean signal from multiple observations that are shifted and corrupted by noise.
  • Understanding the fundamental limits of signal recovery is crucial for designing efficient algorithms.

Purpose of the Study:

  • To derive an expression for the error exponent of the maximum A posteriori (MAP) decoder in Boolean multireference alignment.
  • To characterize the number of measurements required for accurate signal recovery in the low signal-to-noise ratio (SNR) regime.

Main Methods:

  • Derivation of the error exponent for the MAP decoder.
  • Analysis of the relationship between measurement count and signal properties.

Main Results:

  • An explicit expression for the error exponent of the MAP decoder was obtained.
  • The number of measurements for signal recovery in the low SNR regime is characterized by higher-order autocorrelations of the signal.
  • This characterization is detailed for signals with prime and even dimensions.

Conclusions:

  • The study provides a theoretical framework for understanding signal recovery in Boolean multireference alignment.
  • Higher-order signal autocorrelations are key determinants of measurement requirements for accurate recovery, particularly in low SNR environments.