A Generalized Nyquist-Shannon Sampling Theorem Using the Koopman Operator

TL;DR

This work extends classical sampling theory by proposing a generalized Nyquist-Shannon sampling theorem based on the Koopman operator for signals in generator-bounded spaces. By linking CT signal evolution to the Koopman generator $L$ and its semigroup $U^\\tau$, it derives an aliasing bound tied to the imaginary part of the Koopman spectrum and provides a reconstruction framework that reduces to classical forms for band-limited signals. The authors present infinite- and finite-dimensional signal spaces, including inverse Laplace-type signals and polynomial-exponential signals, with corresponding sampling bounds and reconstruction formulas. A Koopman-based reconstruction (KR) method with convergence guarantees is introduced, demonstrated numerically to be robust to low sampling frequencies and noise, and capable of handling non-band-limited signals beyond Nyquist limitations.

Abstract

In the field of signal processing, the sampling theorem plays a fundamental role for signal reconstruction as it bridges the gap between analog and digital signals. Following the celebrated Nyquist-Shannon sampling theorem, generalizing the sampling theorem to non-band-limited signals remains a major challenge. In this work, a generalized sampling theorem, which builds upon the Koopman operator, is proposed for signals in a generator-bounded space. It naturally extends the Nyquist-Shannon sampling theorem in that: 1) for band-limited signals, the lower bounds of the sampling frequency and the reconstruction formulas given by these two theorems are exactly the same; 2) the Koopman operator-based sampling theorem can also provide a finite bound of the sampling frequency and a reconstruction formula for certain types of non-band-limited signals, which cannot be addressed by Nyquist-Shannon sampling theorem. These non-band-limited signals include, but are not limited to, the inverse Laplace transform with limit imaginary interval of integration, and linear combinations of complex exponential functions. Furthermore, the Koopman operator-based reconstruction method is supported by theoretical results on its convergence. This method is illustrated numerically through several examples, demonstrating its robustness against low sampling frequencies.

Figures (14)

• Figure 1: Impulse-train sampling oppenheim1997signals. The sampling of the signal $g(t)$ leads to $g_p(t)$, which is represented by the multiplication of $g(t)$ and sampling function $p(t) = \sum_{k=-\infty}^\infty \delta(t-kT_s)$, where $T_s$ is the sampling period.
• Figure 2: The framework to investigate the sampling theorem of signals by the Koopman operator. The analysis consists of two steps: (1) Considering the sampling problem in the generator-bounded space $\mathcal{F}_e$ of signal $g(t)$, (2) obtaining the generator $L|_{\mathcal{F}_e}$ uniquely from DT Koopman operator $U^{T_s}|_{\mathcal{F}_e}$, (3) reconstructing the signal $g(t)$ by the generator $L|_{\mathcal{F}_e}$.
• Figure 3: Reconstruction of the band-limited signal from samples of sampling period $T_s = 0.2$s (a), $T_s = 0.4$s (b), and $T_s=0.6$s (c) .
• Figure 4: Reconstruction of the signal with exponential growth from samples of sampling period $T_s = 0.2$s (a), $T_s = 0.4$s (b), and $T_s=0.6$s (c)
• Figure 5: Reconstruction of the signal with polynomial growth from samples of sampling period $T_s = 0.2$s (a), $T_s = 0.4$s (b), and $T_s=0.6$s (c)

Theorems & Definitions (47)

• Definition 1: Koopman eigenvalues and eigenfunctions
• Definition 2: Generator-bounded space $\mathcal{F}_e$
• Remark 1
• Lemma 1: Principal logarithmkrabbe1956logarithm
• Theorem 1: Koopman operator-based sampling theorem
• proof
• Remark 2: The reason for generalization
• Theorem 2: Reconstruction requirement
• proof
• Remark 3: The comparison with shift-invariant (SI) space