Optimal Signal Detection Using Correlation and Energy Metrics

Core Concepts

The optimal detector for distinguishing between a null hypothesis (no signal) and an alternative hypothesis (signal corrupted by noise) is derived by jointly optimizing the transmitted signal and the detector parameters. The detector is based on a linear combination of the correlation and energy of the received signal.

Abstract

The paper explores a Neyman-Pearson hypothesis testing scenario where the received signal under the null hypothesis is a white noise process and under the alternative hypothesis is a deterministic transmitted signal corrupted by additive white noise. The authors focus on detectors based on the correlation and energy of the received signal, motivated by implementation simplicity. The key insights are: For a given signal, the optimal correlator weights are derived as a non-linear function of the transmitted signal. When jointly optimizing the transmitted signal and the detector parameters, the optimal signal is shown to be a balanced ternary signal and the correlator has at most three different coefficients, enabling a computationally feasible solution. The authors extend the analysis to consider detectors that are a linear combination of correlation and energy, deriving the optimal detector parameters for a given signal and the jointly optimal signal and detector. The analysis provides a comprehensive framework for designing optimal signal detectors under practical constraints, balancing detection performance and implementation complexity.

Stats

The received signal under the null hypothesis is a white noise process Nt. The received signal under the alternative hypothesis is Xt = st + Zt, where st is a deterministic waveform and Zt is an i.i.d. zero-mean noise process. The test statistic is T = Σ_t w_t Y_t + γ Σ_t Y_t^2, where w_t are the correlator weights and γ is the energy coefficient.

Quotes

"The need to place detectors in small sensors and mobile platforms, with severe constraints on power, weight, budget and size, motivates one to examine some classes of simpler detectors and find the optimal ones within these classes." "When joint signal-detector optimization was carried out, {w_t} and {s_t} did not sustain a linear relation in general. Interestingly, the favorable property that the set {w_t, s_t} has a simple structure was preserved, with just three parameters to be optimized, which makes a numeric solution feasible."

Key Insights Distilled From

Optimal Signals and Detectors Based on Correlation and Energy

by Yossi Marcia... at arxiv.org 05-07-2024

https://arxiv.org/pdf/2405.02931.pdf

Deeper Inquiries

How can the proposed detection framework be extended to handle non-stationary or time-varying signal and noise models

To extend the proposed detection framework to handle non-stationary or time-varying signal and noise models, we can introduce adaptive algorithms that update the detector parameters in real-time based on the changing characteristics of the signals and noise. This can involve techniques such as recursive estimation, Kalman filtering, or adaptive filtering. By continuously adjusting the detector parameters to adapt to the evolving signal and noise conditions, the detection system can effectively handle non-stationary scenarios. Additionally, incorporating machine learning algorithms, such as neural networks or deep learning models, can also enhance the adaptability of the detector to time-varying environments by learning patterns and trends in the signal and noise data.

What are the potential tradeoffs between detection performance and implementation complexity for the jointly optimized signal and detector

The jointly optimized signal and detector framework presents a tradeoff between detection performance and implementation complexity. By optimizing the signal and detector parameters together, the system can achieve improved detection accuracy by tailoring the signal characteristics to enhance the detection process. However, this joint optimization may increase the computational complexity and memory requirements of the system, as it involves solving optimization problems for both the signal and detector simultaneously. The tradeoff lies in balancing the performance gains from the optimized signal and detector with the additional computational resources and complexity needed for implementation.

Can the insights from this work be applied to other signal processing applications beyond detection, such as estimation or classification tasks

The insights from this work can be applied to various signal processing applications beyond detection, such as estimation or classification tasks. For estimation tasks, the optimized signal and detector parameters can be leveraged to improve the accuracy and efficiency of parameter estimation algorithms. By jointly optimizing the signal and detector for estimation purposes, the system can enhance the estimation performance by leveraging the correlation and energy characteristics of the signals. Similarly, in classification tasks, the optimized signal and detector framework can be utilized to improve the classification accuracy by tailoring the signal features to enhance the discrimination between different classes. This approach can lead to more robust and accurate classification models in various applications, such as image recognition, speech processing, and pattern recognition.