Signal Recovery from Time and Frequency Samples

Abstract

We analyze signal recovery when samples are taken concomitantly from a signal and its Fourier transform. This two-sided sampling framework extends classical one-sided reconstruction and is particularly useful when measurements in either domain alone are insufficient because of sensing, storage, or bandwidth constraints. We formulate the resulting recovery problem in finite-dimensional spaces and reproducing kernel Hilbert spaces, and illustrate the infinite-dimensional setting in a Fourier-symmetric Sobolev space. Numerical experiments with sinc- and Hermite-based schemes indicate that, under a fixed sampling budget, two-sided sampling often yields better conditioned systems than one-sided approaches. A simplified spectrum-monitoring example further demonstrates improved reconstruction when limited time samples are supplemented with frequency-domain information.

Figures (6)

• Figure 1: Plot of $\log(\sigma_{\min}(A)/\sigma_{\max}(A))$ for the stacked $3\times 3$ system in $\mathcal{H}_2$, with $t_0=0$ fixed, as functions of $\omega_0$ and $\omega_1$. Numerically singular configurations (ratios below $1.85\times 10^{-5}$) occur along the diagonal $\omega_0=\omega_1$ (corresponding to repeated measurements) and the hyperbola $\omega_0\omega_1=-1$.
• Figure 2: Condition numbers for one-sided and two-sided sampling with fixed total budget $D=N$. Time samples are taken on $[1,2]$ and frequency samples on $[-1,0]$. In both cases, the samples are equispaced.
• Figure 3: Condition numbers when both time and frequency samples lie in $[-1,1]$. Equispaced two-sided sampling is singular for some $N$, whereas random sampling avoids these failures and remains better conditioned.
• Figure 4: Condition numbers are unchanged when part of the time-domain data is post-processed by the DFT and treated as frequency-domain data.
• Figure 5: Condition numbers in a finite-dimensional bandlimited space spanned by integer-shifted sinc functions. The time-sampling interval increases with the dimension of the function space, while the frequency-sampling interval is fixed at $[-3,3]$. Sampling points are chosen uniformly at random.

Theorems & Definitions (9)

• Definition 1: Uniqueness pair, e.g. kulikov2023fourier
• Example 1
• Proposition 1
• proof
• Proposition 2: Uniqueness pairs in $\mathcal{H}$ kulikov2023fourier
• Definition 2: Two-sided RKHS
• Theorem 1: Two-sided representer theorem
• proof
• Remark 1