What Is Nonuniform Sampling?

Nonuniform sampling is a signal acquisition strategy in which samples are taken at irregular time intervals rather than at the fixed periodic rate required by classical Nyquist-Shannon sampling theory. It arises in contexts where physical constraints or deliberate design choices make regular spacing impractical: sensor dropout, asynchronous event-driven acquisition, missing data in legacy recordings, and intentional randomized sampling schemes all produce nonuniform sample sets. The central challenge is reconstructing the original signal from these irregularly spaced observations without introducing distortion or aliasing.

The theoretical basis for nonuniform sampling was established alongside the Nyquist-Shannon framework in the mid-twentieth century, with results showing that a bandlimited signal can be exactly recovered from a nonuniform sample sequence provided the average sampling density meets the Nyquist criterion. In practice, however, reconstruction from nonuniform data requires substantially more computation than the simple inverse discrete Fourier transform applicable to uniform data, and the sensitivity to noise and the selection of reconstruction algorithm depend strongly on how the irregular spacings are distributed.

Sampling Theory and Reconstruction

The foundational result governing nonuniform sampling is the Beurling-Landau density theorem, which states that a set of sampling times is sufficient for exact reconstruction of a bandlimited signal if and only if the lower Beurling density of the set exceeds the signal bandwidth. For finite-length data in practice, reconstruction is typically posed as a matrix inversion problem or a least-squares minimization. Iterative algorithms such as the Papoulis-Gerchberg method and frame-theoretic reconstruction approaches compute the missing spectral content by alternating projections between the time-domain sample constraints and the known bandwidth limits. The Rice University Compressive Sensing resources provide a detailed account of how these reconstruction ideas connect to the wider field of sparse signal recovery.

Compressive Sensing and Randomized Sampling

Compressive sensing, developed in the early 2000s by Donoho, Candès, and Romberg, reframed nonuniform sampling as a means to acquire sparse signals at rates well below the Nyquist limit. The key insight is that a signal with a sparse representation in some transform domain (such as wavelet or Fourier) can be recovered from far fewer measurements than its bandwidth would suggest, provided those measurements are incoherent with the sparsifying transform. Nonuniform random sampling naturally satisfies the incoherence condition; the resulting measurement process is equivalent to taking random projections of the signal, and recovery is accomplished by convex optimization such as basis pursuit or greedy algorithms such as orthogonal matching pursuit. A foundational exposition appears in an introduction to compressive sampling published in IEEE Signal Processing Magazine and a systematic review of compressive sensing implementations surveys the hardware and algorithm variants that have emerged from this framework.

Spectral Estimation from Nonuniform Data

Estimating the frequency content of a nonuniformly sampled signal requires specialized spectral methods. The Lomb-Scargle periodogram, developed independently in astronomy and signal processing, computes a least-squares fit of sinusoids to irregularly spaced data and produces spectral estimates that approach the quality of the standard discrete Fourier transform for approximately uniform data. Nonuniform fast Fourier transform (NUFFT) algorithms generalize the FFT to arbitrary sample locations using interpolation and gridding steps, enabling efficient spectral computation for large datasets. In cognitive radio and dynamic spectrum access applications, nonuniform sampling of the wideband radio spectrum reduces analog-to-digital converter requirements by concentrating samples in spectrally active subbands.

Applications

Nonuniform sampling has applications in a range of fields, including:

• Magnetic resonance imaging (MRI) acquisition with variable k-space trajectories
• Wideband radar and cognitive radio systems operating at sub-Nyquist rates
• Seismic data acquisition with irregular sensor spacing
• Neuroscience recordings from spike-timing-dependent neural data
• Astronomical light-curve analysis from telescopes with observation gaps
• Power-grid monitoring with asynchronous phasor measurement units