Sparse Signal Recovery Guarantees and Cramér-Rao Bounds in the Fractional Fourier Domain

Centrala begrepp

A novel time-domain sparse recovery method that avoids the limitations of transform domain approaches, along with Cramér-Rao Bounds for the sparse parameter estimation problem in the fractional Fourier domain.

Sammanfattning

This paper introduces a new time-domain method for recovery of sparse signals from low-pass filtered measurements in the Fractional Fourier Transform (FrFT) domain. The key advantages of this method are:

It does not require any Discrete Fractional Fourier Transform (DFrFT) operations, thereby eliminating the issues of spectral leakage that plague transform domain approaches.
The time-domain recovery approach is backed by a sampling theorem, providing theoretical guarantees on the required number of samples for exact recovery.

Additionally, the paper derives Cramér-Rao Bounds (CRB) for the sparse parameter estimation problem in the FrFT domain, which was previously missing in the literature. This serves as a performance guarantee for the recovery problem in the presence of noise.
The key steps are:

The authors present a novel time-domain sparse recovery method that avoids the typical bottlenecks of transform domain methods, such as spectral leakage. This method is backed by a sparse sampling theorem applicable to arbitrary FrFT-bandlimited kernels.
The authors derive Cramér-Rao Bounds for the sparse sampling problem, addressing a gap in existing literature.
The authors validate the empirical robustness of their algorithm through a hardware experiment, demonstrating accurate recovery of the sparse signal despite the presence of noise and quantization effects.

Statistik

The sparse signal parameters are:
c0 = 0.748
c1 = 0.891
t0 = 0.500 s
t1 = 0.830 s
The recovered sparse signal parameters are:
c̃0 = 0.733
c̃1 = 0.843
t̃0 = 0.501 s
t̃1 = 0.834 s

Citat

None

Viktiga insikter från

Sparse Sampling in Fractional Fourier Domain: Recovery Guarantees and Cramér-Rao Bounds

by Václ... på arxiv.org 04-30-2024

https://arxiv.org/pdf/2404.18850.pdf

Djupare frågor

How can the proposed time-domain sparse recovery method be extended to handle non-bandlimited sampling kernels?

The proposed time-domain sparse recovery method can be extended to handle non-bandlimited sampling kernels by incorporating a more flexible interpolation scheme that can adapt to the characteristics of the specific kernel. One approach could involve developing a generalized interpolation formula that can accommodate a wider range of kernel shapes and bandwidths. By incorporating the properties of the non-bandlimited sampling kernels into the interpolation process, the method can effectively reconstruct sparse signals sampled using these kernels.
Additionally, the method can be enhanced by incorporating adaptive techniques that adjust the interpolation parameters based on the specific characteristics of the sampling kernel. This adaptability can improve the accuracy and robustness of the sparse signal recovery process when dealing with non-bandlimited kernels. By optimizing the interpolation process to suit the properties of the sampling kernel, the method can achieve more accurate and reliable sparse signal recovery results.

What are the potential applications of the derived Cramér-Rao Bounds beyond the sparse sampling problem in the fractional Fourier domain?

The derived Cramér-Rao Bounds (CRB) can have various applications beyond the sparse sampling problem in the fractional Fourier domain. Some potential applications include:

Performance Evaluation: The CRB can serve as a benchmark for evaluating the performance of other signal processing algorithms in the fractional Fourier domain. By comparing the estimation accuracy of different algorithms to the CRB, researchers can assess the efficiency and effectiveness of their methods.

Optimal Sensor Placement: In sensor network applications, the CRB can be used to determine the optimal sensor placement for signal estimation tasks. By analyzing the CRB values at different sensor locations, researchers can identify the configurations that provide the most accurate estimation results.

Signal Processing System Design: The CRB can guide the design of signal processing systems by setting performance targets based on the theoretical limits defined by the bounds. System designers can use the CRB as a reference to optimize the design parameters and improve the overall performance of the system.

Noise Analysis: The CRB can be utilized for noise analysis in signal processing applications. By comparing the noise levels in the system to the CRB values, researchers can assess the impact of noise on signal estimation and make informed decisions about noise mitigation strategies.

How can the insights from this work on sparse recovery in the fractional Fourier domain be leveraged to develop new sampling and reconstruction techniques for other generalized transform domains?

The insights gained from the work on sparse recovery in the fractional Fourier domain can be leveraged to develop new sampling and reconstruction techniques for other generalized transform domains by applying similar principles and methodologies to different transform spaces. Here are some ways in which these insights can be utilized:

Adaptation of Interpolation Methods: The interpolation methods developed for sparse recovery in the fractional Fourier domain can be adapted and extended to suit the characteristics of other transform domains. By understanding the underlying principles of sparse signal recovery, researchers can tailor interpolation techniques to specific transform spaces.

Exploration of Sampling Criteria: The sampling criteria established for sparse recovery in the fractional Fourier domain can be explored in the context of other transform domains. By studying the requirements for accurate signal recovery in different transform spaces, researchers can develop new sampling strategies that optimize signal reconstruction.

CRB Analysis: The concept of Cramér-Rao Bounds can be applied to other generalized transform domains to establish performance benchmarks and evaluate the estimation accuracy of signal processing algorithms. By deriving CRB for different transform spaces, researchers can assess the theoretical limits of signal estimation and guide the development of new reconstruction techniques.

By leveraging the insights and methodologies from sparse recovery in the fractional Fourier domain, researchers can advance the field of signal processing and develop innovative sampling and reconstruction techniques for a wide range of transform domains.