Analytic Phase Projection (APP)

• Analytic Phase Projection (APP) is a method that projects data onto analytic phase representations to improve reconstruction in signal processing and calibration in VLBI.
• In radio interferometry, APP establishes a universal small-baseline phase law that predicts key phase transitions, guiding optimal baseline design for imaging black hole photon rings.
• In time-series analysis, APP uses the Hilbert transform for hypersampling pseudo-periodic signals, enabling high-resolution reconstruction of fine-grained cycle features.

Analytic Phase Projection (APP) encompasses a distinct set of methodologies applied to both signal processing and radio interferometry domains, united by the central concept of projecting data onto analytic phase representations for improved reconstruction, calibration, or interpretation. In signal processing, APP refers to a framework for hypersampling pseudo-periodic signals using the analytic phase extracted from a reference via the Hilbert transform, enabling the reconstruction of fine-grained cycle features from undersampled measurements (Voss, 2018). In the context of radio interferometric imaging, APP defines a universal analytic law for the phase of the complex visibility function as a function of projected baseline in the small-baseline regime, with applications to the calibration of extreme-resolution Very Long Baseline Interferometry (VLBI), notably at space baselines exceeding several Earth diameters (Chernov, 16 Jan 2026).

1. Mathematical Foundations in Radio Interferometry

The core observable in radio interferometry is the complex visibility $V(\mathbf{b})$, given in its most general form as

$$V(\mathbf{b}) = \int_{\Omega} I(\mathbf{s})\, e^{-2\pi i\, \mathbf{b} \cdot \mathbf{s} / \lambda}\, d\Omega(\mathbf{s})$$

where $\mathbf{b}=(b_x, b_y, b_z)$ is the baseline vector, $\mathbf{s}=(l, m, n)$ with $n = \sqrt{1-l^2-m^2}$ is the sky direction, $I(\mathbf{s})$ is the sky brightness distribution, and $\lambda$ is the observing wavelength (Chernov, 16 Jan 2026). Expressing projected baselines as $(u, v, w) = \mathbf{b}/\lambda$, and invoking the small-angle approximation for ultra-high-resolution (sub-microarcsecond) VLBI, the visibility simplifies to the 2D Fourier transform:

$$V(u, v) = \iint I(l, m)\, e^{-2\pi i (u l + v m)}\, dl\, dm$$

The phase of the visibility, $\phi(u, v) = \arg[V(u, v)]$, encodes critical information for image reconstruction and astrophysical inference.

2. Universal Small-Baseline Phase Law

APP in radio interferometry establishes a universal analytic form for $$V(\mathbf{b}) = \int_{\Omega} I(\mathbf{s})\, e^{-2\pi i\, \mathbf{b} \cdot \mathbf{s} / \lambda}\, d\Omega(\mathbf{s})$$0 under small projected baselines, assuming the sky brightness is dominated by a thin ring of radius $$V(\mathbf{b}) = \int_{\Omega} I(\mathbf{s})\, e^{-2\pi i\, \mathbf{b} \cdot \mathbf{s} / \lambda}\, d\Omega(\mathbf{s})$$1 (a model well-suited to black hole photon rings):

Given an azimuthally modulated brightness,

$$V(\mathbf{b}) = \int_{\Omega} I(\mathbf{s})\, e^{-2\pi i\, \mathbf{b} \cdot \mathbf{s} / \lambda}\, d\Omega(\mathbf{s})$$2

and using the Bessel expansion for the visibilities, the phase for $$V(\mathbf{b}) = \int_{\Omega} I(\mathbf{s})\, e^{-2\pi i\, \mathbf{b} \cdot \mathbf{s} / \lambda}\, d\Omega(\mathbf{s})$$3 reduces to

$$V(\mathbf{b}) = \int_{\Omega} I(\mathbf{s})\, e^{-2\pi i\, \mathbf{b} \cdot \mathbf{s} / \lambda}\, d\Omega(\mathbf{s})$$4

Defining $$V(\mathbf{b}) = \int_{\Omega} I(\mathbf{s})\, e^{-2\pi i\, \mathbf{b} \cdot \mathbf{s} / \lambda}\, d\Omega(\mathbf{s})$$5, for $$V(\mathbf{b}) = \int_{\Omega} I(\mathbf{s})\, e^{-2\pi i\, \mathbf{b} \cdot \mathbf{s} / \lambda}\, d\Omega(\mathbf{s})$$6 one obtains the leading-order approximation

$$V(\mathbf{b}) = \int_{\Omega} I(\mathbf{s})\, e^{-2\pi i\, \mathbf{b} \cdot \mathbf{s} / \lambda}\, d\Omega(\mathbf{s})$$7

This "universal law" predicts phase zero-crossings and sign flips at Bessel function zeros, with locations insensitive to azimuthal detail except for the ring radius $$V(\mathbf{b}) = \int_{\Omega} I(\mathbf{s})\, e^{-2\pi i\, \mathbf{b} \cdot \mathbf{s} / \lambda}\, d\Omega(\mathbf{s})$$8 (Chernov, 16 Jan 2026).

3. Comparison with MHD Simulations and Validity Domain

Direct numerical comparison with 3D general relativistic magnetohydrodynamics (GRMHD) and ray-tracing/radiative transfer models demonstrates that the analytic APP phase closely matches simulated visibilities for projected baselines $$V(\mathbf{b}) = \int_{\Omega} I(\mathbf{s})\, e^{-2\pi i\, \mathbf{b} \cdot \mathbf{s} / \lambda}\, d\Omega(\mathbf{s})$$9 Earth diameters, typically agreeing within a few degrees and replicating zero crossings and $\mathbf{b}=(b_x, b_y, b_z)$0 flips at Bessel function nodes. Beyond this regime, higher-order azimuthal structure, non-ringlike emission, and finite ring thickness introduce corrections. Applicability is thus limited to $\mathbf{b}=(b_x, b_y, b_z)$1–2, corresponding to $\mathbf{b}=(b_x, b_y, b_z)$2 ED for $\mathbf{b}=(b_x, b_y, b_z)$3 μas (Chernov, 16 Jan 2026). For larger baselines, full numerical or higher-order analytic approaches are required.

4. Applications to Calibration and Array Design in Space-VLBI

APP provides a closed-form phase template for calibrating and interpreting closure phases in VLBI arrays with sparse and extended $\mathbf{b}=(b_x, b_y, b_z)$4-coverage, including baselines up to six Earth diameters. Analytical loci of visibility phase sign changes specify optimal baseline spacings for sampling critical structure in ringlike sources, directly guiding baseline selection in space-VLBI design.

The table below summarizes key design implications:

Application	APP Role	Characteristic Formula/Limit
Phase Calibration	Predict closure phases	$\mathbf{b}=(b_x, b_y, b_z)$5 as above
Baseline Selection	Maximize sensitivity	$\mathbf{b}=(b_x, b_y, b_z)$6 at first zero: $\mathbf{b}=(b_x, b_y, b_z)$7
Validity Limit	Universal law holds	$\mathbf{b}=(b_x, b_y, b_z)$8 for $\mathbf{b}=(b_x, b_y, b_z)$9as

Practical utility is greatest for designing and calibrating arrays targeting the direct imaging of supermassive black hole photon rings at mm/sub-mm wavelengths (Chernov, 16 Jan 2026).

5. Analytic Phase Projection for Hypersampling in Time-Series

A second, independently developed form of APP is the technique for hypersampling pseudo-periodic signals, primarily in biomedical signal processing (Voss, 2018). Here, the objective is to reconstruct high-resolution cycle waveforms $\mathbf{s}=(l, m, n)$0 from undersampled time series $\mathbf{s}=(l, m, n)$1, by phase-aligning each target sample with the instantaneous analytic phase of a high-rate reference $\mathbf{s}=(l, m, n)$2, obtained via Hilbert transform:

• Compute analytic signal: $\mathbf{s}=(l, m, n)$3
• Instantaneous phase: $\mathbf{s}=(l, m, n)$4
• Map each $\mathbf{s}=(l, m, n)$5 to phase $\mathbf{s}=(l, m, n)$6, bin, and average in phase domain:

$\mathbf{s}=(l, m, n)$7

with $\mathbf{s}=(l, m, n)$8.

This enables effective temporal upsampling of physiological events (e.g., intracranial pulse waves) by orders of magnitude, provided the reference is monocomponent and well-resolved in phase (Voss, 2018). The method outperforms retrospective gating by leveraging continuous, non-template-based phase mapping and adaptation to cycle-to-cycle variability.

6. Limitations and Extensions

APP in radio interferometry is limited by its derivation under small projected baselines and ring-dominated morphologies. At larger $\mathbf{s}=(l, m, n)$9 or for sources with strong non-ringlike structure, deviations necessitate direct numerical simulation or extension of the analytic framework to higher-order harmonics. In signal hypersampling, APP requires a monocomponent reference; non-monotonic or multi-component reference signals can compromise phase assignment. Ongoing research seeks to generalize APP to multicomponent settings using variants of empirical mode decomposition or synchrosqueezing techniques (Voss, 2018).

7. Broader Significance

APP provides a set of analytically grounded, computationally efficient tools for phase-centric analysis in both imaging and temporal reconstruction. In VLBI, APP underpins precision calibration and informs baseline design in ultra-high-resolution observations of black holes (Chernov, 16 Jan 2026). In biomedical signal analysis, APP enables the discovery of fine-grained physiological dynamics from undersampled or multimodal measurements, facilitating new forms of hybrid imaging (Voss, 2018). The theoretical concordance of APP forms across distinct domains highlights the broad utility of analytic phase as a representational bridge between measured data and underlying source structure.