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

Václav Pavlíček, Ayush Bhandari

TL;DR

This work addresses sparse signal recovery in the Fractional Fourier Transform (FrFT) domain by proposing a time-domain recovery method that eliminates spectral leakage typical of transform-domain approaches. It provides a sparse-sampling theorem for arbitrary FrFT-bandlimited kernels and proves a practical recovery procedure that uses a polynomial whose roots yield spike locations, followed by linear inversion for amplitudes. Additionally, it derives Cramér–Rao Bounds (CRB) for the sparse-sampling problem, including a closed-form CRB in the single-spike case, offering performance guarantees under noise and a framework for PSNR-based error scaling. Hardware experiments validate the method’s robustness and demonstrate accurate recovery under realistic noisy conditions, highlighting the approach’s practical relevance for FrFT-domain sparse sensing and sub-Nyquist acquisition.

Abstract

Sampling theory in fractional Fourier Transform (FrFT) domain has been studied extensively in the last decades. This interest stems from the ability of the FrFT to generalize the traditional Fourier Transform, broadening the traditional concept of bandwidth and accommodating a wider range of functions that may not be bandlimited in the Fourier sense. Beyond bandlimited functions, sampling and recovery of sparse signals has also been studied in the FrFT domain. Existing methods for sparse recovery typically operate in the transform domain, capitalizing on the spectral features of spikes in the FrFT domain. Our paper contributes two new theoretical advancements in this area. First, we introduce 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 and is validated through a hardware experiment. Second, we present Cramér-Rao Bounds for the sparse sampling problem, addressing a gap in existing literature.

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

TL;DR

This work addresses sparse signal recovery in the Fractional Fourier Transform (FrFT) domain by proposing a time-domain recovery method that eliminates spectral leakage typical of transform-domain approaches. It provides a sparse-sampling theorem for arbitrary FrFT-bandlimited kernels and proves a practical recovery procedure that uses a polynomial whose roots yield spike locations, followed by linear inversion for amplitudes. Additionally, it derives Cramér–Rao Bounds (CRB) for the sparse-sampling problem, including a closed-form CRB in the single-spike case, offering performance guarantees under noise and a framework for PSNR-based error scaling. Hardware experiments validate the method’s robustness and demonstrate accurate recovery under realistic noisy conditions, highlighting the approach’s practical relevance for FrFT-domain sparse sensing and sub-Nyquist acquisition.

Abstract

Sampling theory in fractional Fourier Transform (FrFT) domain has been studied extensively in the last decades. This interest stems from the ability of the FrFT to generalize the traditional Fourier Transform, broadening the traditional concept of bandwidth and accommodating a wider range of functions that may not be bandlimited in the Fourier sense. Beyond bandlimited functions, sampling and recovery of sparse signals has also been studied in the FrFT domain. Existing methods for sparse recovery typically operate in the transform domain, capitalizing on the spectral features of spikes in the FrFT domain. Our paper contributes two new theoretical advancements in this area. First, we introduce 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 and is validated through a hardware experiment. Second, we present Cramér-Rao Bounds for the sparse sampling problem, addressing a gap in existing literature.
Paper Structure (5 sections, 1 theorem, 24 equations, 2 figures)

This paper contains 5 sections, 1 theorem, 24 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Mathematical Preliminaries
  3. Recovery Algorithm and Guarantees
  4. Cramér--Rao Bounds for Sparse Sampling
  5. Conclusion

Key Result

Theorem 2

Let $y\left( t \right) = \left( s_K *_\theta \phi_M \right)\left( t \right)$, as in eq:yt and $\phi_M,\theta$ be known. Suppose we are given $N$ samples of $y\left[n\right] = y\left( nT \right), T>0$, then, $N\geqslant 2KM$ guarantees recovery of the unknown signal $s_K\left( t \right)$ in eq:ksp.

Figures (2)

  • Figure 1: Visual illustration of "fractionalizing" the Fourier transform (FT) which introduces the FrFT for non-integer orders, $\theta\in\mathbb{R}$. While the traditional FT is a 4-periodic automorphism Condon:1937:J (applying FT 4 times returns the original function), this property holds only for integer orders, $n\in\mathbb{Z}$. The FrFT generalizes this concept to arbitrary real orders $\theta$. When $\theta=\pi/2$, the FrFT simplifies back to the FT.
  • Figure 2: Hardware-based experimental validation of sparse recovery method. Digital samples with $8$-bit resolution are marked by $\bullet$.

Theorems & Definitions (4)

  • Definition 1: Chirp Modulation
  • Definition 2: Fractional Convolution Operator Zayed:1998:J
  • Theorem 2: Sampling Criterion
  • proof