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Optimal Signals and Detectors Based on Correlation and Energy

Yossi Marciano, Neri Merhav

TL;DR

This work generalizes likelihood-based detection to non-Gaussian noise by analyzing detectors based on correlation and energy under a Neyman–Pearson framework. It shows that for a fixed signal the optimal correlator is a low-complexity, at-most-three-point mapping, and that joint optimization drives the transmitted signal toward a balanced ternary form while tying the correlator to a three-point structure; allowing a linear combination with energy expands the parameter set to a five-parameter design, still solvable via Chernoff bounds and linear programming. The results provide explicit forms for the optimal test statistics and prove that the optimal joint signal-detector configurations remain highly structured and computationally feasible, with clear improvements in missed-detection exponents under FA constraints. The study also discusses extensions to broader noise models and potential applications to mismatched decoding in communications.

Abstract

In continuation of an earlier study, we explore a Neymann-Pearson hypothesis testing scenario where, under the null hypothesis ($\cal{H}_0$), the received signal is a white noise process $N_t$, which is not Gaussian in general, and under the alternative hypothesis ($\cal{H}_1$), the received signal comprises a deterministic transmitted signal $s_t$ corrupted by additive white noise, the sum of $N_t$ and another noise process originating from the transmitter, denoted as $Z_t$, which is not necessarily Gaussian either. Our approach focuses on detectors that are based on the correlation and energy of the received signal, which are motivated by implementation simplicity. We optimize the detector parameters to achieve the best trade-off between missed-detection and false-alarm error exponents. First, we optimize the detectors for a given signal, resulting in a non-linear relation between the signal and correlator weights to be optimized. Subsequently, we optimize the transmitted signal and the detector parameters jointly, revealing that the optimal signal is a balanced ternary signal and the correlator has at most three different coefficients, thus facilitating a computationally feasible solution.

Optimal Signals and Detectors Based on Correlation and Energy

TL;DR

This work generalizes likelihood-based detection to non-Gaussian noise by analyzing detectors based on correlation and energy under a Neyman–Pearson framework. It shows that for a fixed signal the optimal correlator is a low-complexity, at-most-three-point mapping, and that joint optimization drives the transmitted signal toward a balanced ternary form while tying the correlator to a three-point structure; allowing a linear combination with energy expands the parameter set to a five-parameter design, still solvable via Chernoff bounds and linear programming. The results provide explicit forms for the optimal test statistics and prove that the optimal joint signal-detector configurations remain highly structured and computationally feasible, with clear improvements in missed-detection exponents under FA constraints. The study also discusses extensions to broader noise models and potential applications to mismatched decoding in communications.

Abstract

In continuation of an earlier study, we explore a Neymann-Pearson hypothesis testing scenario where, under the null hypothesis (), the received signal is a white noise process , which is not Gaussian in general, and under the alternative hypothesis (), the received signal comprises a deterministic transmitted signal corrupted by additive white noise, the sum of and another noise process originating from the transmitter, denoted as , which is not necessarily Gaussian either. Our approach focuses on detectors that are based on the correlation and energy of the received signal, which are motivated by implementation simplicity. We optimize the detector parameters to achieve the best trade-off between missed-detection and false-alarm error exponents. First, we optimize the detectors for a given signal, resulting in a non-linear relation between the signal and correlator weights to be optimized. Subsequently, we optimize the transmitted signal and the detector parameters jointly, revealing that the optimal signal is a balanced ternary signal and the correlator has at most three different coefficients, thus facilitating a computationally feasible solution.
Paper Structure (10 sections, 3 theorems, 76 equations, 5 figures)

This paper contains 10 sections, 3 theorems, 76 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Problem Formulation and Preliminaries
  3. Detectors Based On Correlation
  4. Optimal Correlator for a Given Signal
  5. Joint Optimization of the Signal and the Correlator
  6. Detectors Based on Linear Combinations of Correlation and Energy
  7. Optimal Correlation and Energy Detector for a Given Signal
  8. Joint Optimization of the Signal and the Correlator
  9. Conclusion and Future Aspects
  10. Appendix

Key Result

Theorem 1

Let the assumptions of Section Section2, Section Section3 hold and let $N$ be a non-degenerate RV. Define the function $g$ as where $\dot{C}_{\hbox{\tiny{V}}}$ and $\dot{C}_{\hbox{\tiny{N}}}$ denote the derivatives of $C_{\hbox{\tiny{V}}}$ and $C_{\hbox{\tiny{N}}}$, respectively. Further assume that there exists $\varphi\ge 0$ (possibly, dependent on $\lambda$ and $\alpha$) that satisfies: where

Figures (5)

  • Figure 1: MD error exponents as functions of $E_{\hbox{\tiny{0}}}$ for the classical correlator and the optimal correlator with parameter values $a=4$, $z_0=7$ and $q=4$.
  • Figure 2: MD error exponent as a function of $E_{\hbox{\tiny{0}}}$ for optimal correlators. The lower (blue) curve is when the signal is 4-ASK signal and the upper (red) curve is for the optimal signal for detection with correlation. Parameter values are identical to Example 1.
  • Figure 3: MD error exponents as functions of $E_{\hbox{\tiny{0}}}$ for the optimal correlator and the optimal correlation-energy detector with parameter values $a=6$, $z_{\hbox{\tiny{0}}}=7$ and $B=5$.
  • Figure 4: The threshold $\theta$ as a function of $E_{\hbox{\tiny{0}}}$ for the classical optimal correlator for a given signal. (same parameter values as Figure \ref{['fig:Example5ROC']}).
  • Figure 5: MD Error exponents as a function of $E_{\hbox{\tiny{0}}}$ of the optimal correlator (with energy term) and the optimal signal-detector combination (with energy term).

Theorems & Definitions (6)

  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Lemma 1
  • proof