Phaseless STLCT Sampling Insights

• The paper establishes that phaseless STLCT sampling guarantees unique signal recovery up to a global phase using nonuniform rectangular square-root lattices.
• Its analytic framework converts magnitude-only measurements into entire functions, enabling explicit reconstruction and stability analysis in Gaussian shift-invariant spaces.
• Stability and robustness are achieved under specific sampling geometries and anchor-point conditions, addressing phase retrieval challenges in both L2 and band-limited signal classes.

Phaseless STLCT sampling is the problem of recovering a signal from magnitude-only samples of its short-time linear canonical transform (STLCT). For a window $\varphi$, an LCT parameter matrix $A=(a,b,c,d)$ with $ad-bc=1$ and $b\neq 0$, and a sampling set $\Lambda$, the measurements are $\{|V_\varphi^{(A)}f(\lambda)|:\lambda\in\Lambda\}$. Because multiplication by a unimodular constant does not change these magnitudes, recovery can at best be unique up to a global phase factor. In the current literature, the subject has two principal strands: uniqueness theory for phaseless STLCT sampling on nonuniform sets such as rectangular square-root lattices, and explicit reconstruction, stability, and noisy finite-sample recovery for special signal classes, notably Gaussian shift-invariant spaces (Dong et al., 26 Aug 2025, Cheng et al., 8 May 2026).

1. Formal setting and transform model

For $f\in L^2(\mathbb{R}^{d'})$, the STLCT with window $g$ is defined by

$$V_{g}^{(A)}f(x,\mu)=L_A\!\bigl(f\,\overline{T_x g}\bigr)(\mu),$$

where $T_x g(t)=g(t-x)$ is translation, $A=(a,b,c,d)$0 is the linear canonical transform, and the generalized modulation operator is

$A=(a,b,c,d)$1

When $A=(a,b,c,d)$2, the STLCT reduces to the ordinary short-time Fourier transform (STFT) (Dong et al., 26 Aug 2025).

The phaseless sampling problem asks whether

$A=(a,b,c,d)$3

forces

$A=(a,b,c,d)$4

This is the canonical uniqueness formulation for STLCT phase retrieval (Dong et al., 26 Aug 2025).

The recent STLCT literature distinguishes sharply between unrestricted $A=(a,b,c,d)$5-signals and structured subspaces. In full $A=(a,b,c,d)$6, sampling geometry is decisive; in structured Gaussian shift-invariant spaces, semi-discrete magnitude measurements can already determine every signal uniquely up to a unimodular constant and admit explicit inversion formulas (Dong et al., 26 Aug 2025, Cheng et al., 8 May 2026).

2. Sampling geometries: square-root lattices, uniform lattices, and restricted signal classes

The central positive uniqueness theorem for general $A=(a,b,c,d)$7-signals uses rectangular square-root lattices. These are sampling sets of the form

$A=(a,b,c,d)$8

with diagonal generating matrix

$A=(a,b,c,d)$9

If $ad-bc=1$0 and the spacings satisfy

$ad-bc=1$1

then

$ad-bc=1$2

holds if and only if $ad-bc=1$3. Thus phaseless STLCT samples on suitable rectangular square-root lattices uniquely determine every square-integrable signal up to global phase (Dong et al., 26 Aug 2025).

The negative result is equally sharp. For general $ad-bc=1$4, uniform lattices and parallel-line sampling geometries do not, in general, support STLCT phase retrieval. With a Gaussian window, the paper constructs

$ad-bc=1$5

for which $ad-bc=1$6 but

$ad-bc=1$7

on a uniform-line-type sampling geometry. This establishes genuine nonuniqueness in the full $ad-bc=1$8-setting (Dong et al., 26 Aug 2025).

The failure of uniform lattices is not absolute. For band-limited spaces, phase retrieval can be restored on a uniform lattice. Specifically, for $ad-bc=1$9, $b\neq 0$0, and

$b\neq 0$1

the following are equivalent for $b\neq 0$2: $b\neq 0$3 and

$b\neq 0$4

Phaseless STLCT sampling therefore exhibits a three-way division: square-root lattices give uniqueness on all of $b\neq 0$5, uniform lattices fail on all of $b\neq 0$6, and uniform lattices become admissible again on restricted band-limited classes (Dong et al., 26 Aug 2025).

This geometry-driven dichotomy parallels the earlier STFT theory, where ordinary lattices were shown to be insufficient for phase retrieval on $b\neq 0$7, whereas square-root lattices restore uniqueness for a large analytic window class including Gaussians and Hermite functions (Grohs et al., 2022).

3. Analytic mechanism of uniqueness

The square-root-lattice theorem is based on an analytic-function framework. The relevant window class is

$b\neq 0$8

For windows in this class, the STLCT spectrogram inherits an entire extension with controlled growth (Dong et al., 26 Aug 2025).

A key structural identity rewrites the STLCT in terms of another windowed transform: $b\neq 0$9 where

$\Lambda$0

This identity is the canonical reduction from STLCT phase retrieval to an analytic uniqueness problem for transformed functions (Dong et al., 26 Aug 2025).

The decisive analytic step is that, for $\Lambda$1,

$\Lambda$2

with parameter vector

$\Lambda$3

Hence sampled magnitudes determine an entire function of $\Lambda$4 complex variables. The square-root lattice is then shown to be a uniqueness set for that entire-function class; once the full spectrograms agree, a continuous uniqueness theorem yields $\Lambda$5 (Dong et al., 26 Aug 2025).

This mechanism extends the analytic continuation strategy developed for STFT square-root lattices. In that setting, the proof uses the fact that $\Lambda$6 belongs to a suitable entire-function growth class and that square-root-distributed sampling sets are uniqueness sets, established via Jensen’s formula and zero counting (Grohs et al., 2022). A complementary STFT line replaces entire-function arguments by completeness of discrete translates on compact supports: if the families $\Lambda$7, with $\Lambda$8, are complete and the frequency sampling set is a uniqueness set for $\Lambda$9, then phaseless lattice STFT samples determine every $\{|V_\varphi^{(A)}f(\lambda)|:\lambda\in\Lambda\}$0 up to global phase (Grohs et al., 2022). This suggests that phaseless STLCT sampling belongs to a broader phase-retrieval pattern in which analyticity or translate completeness substitutes for direct phase information.

4. Explicit reconstruction and stability in Gaussian shift-invariant spaces

The first STLCT phase-retrieval theory that includes explicit inversion, stability, and a robust reconstruction algorithm is formulated in the Gaussian shift-invariant space

$\{|V_\varphi^{(A)}f(\lambda)|:\lambda\in\Lambda\}$1

The measurements are phaseless STLCT samples on the semi-discrete set

$\{|V_\varphi^{(A)}f(\lambda)|:\lambda\in\Lambda\}$2

taken with the chirp-modulated Gaussian window

$\{|V_\varphi^{(A)}f(\lambda)|:\lambda\in\Lambda\}$3

For this model, every signal in $\{|V_\varphi^{(A)}f(\lambda)|:\lambda\in\Lambda\}$4 is uniquely determined, up to a global unimodular constant, by its phaseless STLCT measurements on $\{|V_\varphi^{(A)}f(\lambda)|:\lambda\in\Lambda\}$5, and the paper derives an explicit reconstruction formula (Cheng et al., 8 May 2026).

The reconstruction theory is built around the auxiliary functions

$\{|V_\varphi^{(A)}f(\lambda)|:\lambda\in\Lambda\}$6

for which

$\{|V_\varphi^{(A)}f(\lambda)|:\lambda\in\Lambda\}$7

The coefficients of the biorthogonal expansion of $\{|V_\varphi^{(A)}f(\lambda)|:\lambda\in\Lambda\}$8 are encoded in the phaseless STLCT data through a Fourier identity involving $\{|V_\varphi^{(A)}f(\lambda)|:\lambda\in\Lambda\}$9. This converts magnitude-only measurements into a linear reconstruction mechanism inside a Gaussian shift-invariant space (Cheng et al., 8 May 2026).

The stability theory is conditional and local. It assumes an anchor-point condition: there exist points

$f\in L^2(\mathbb{R}^{d'})$0

and $f\in L^2(\mathbb{R}^{d'})$1 such that

$f\in L^2(\mathbb{R}^{d'})$2

Let

$f\in L^2(\mathbb{R}^{d'})$3

The principal theorem shows that the stability constant is governed by the maximal spacing between adjacent anchor points rather than by the radius of the whole interval. This prevents exponential deterioration with respect to the interval size (Cheng et al., 8 May 2026).

The same paper develops a finite noisy-sample algorithm on

$f\in L^2(\mathbb{R}^{d'})$4

with noisy data

$f\in L^2(\mathbb{R}^{d'})$5

Its stages are anchor detection, anchor selection, local reconstruction with phase transitions, and global phase propagation. Under explicit lower bounds on $f\in L^2(\mathbb{R}^{d'})$6, $f\in L^2(\mathbb{R}^{d'})$7, and $f\in L^2(\mathbb{R}^{d'})$8, and a corresponding noise constraint, the output $f\in L^2(\mathbb{R}^{d'})$9 satisfies

$g$0

In the Fourier case, these results recover the corresponding Gabor phase retrieval results of Grohs and Liehr and improve the stability constants (Cheng et al., 8 May 2026).

5. Relation to broader phaseless sampling theory

Phaseless STLCT sampling is part of a broader sampling-theoretic landscape in which identifiability is controlled by structural constraints. In real-valued shift-invariant spaces

$g$1

the central uniqueness class is the class of nonseparable signals, and graph connectivity characterizes whether a signal is determined, up to a sign, by its magnitude data. Under local complement property, one can construct a discrete set with finite sampling density such that nonseparable signals can be reconstructed stably from phaseless samples; the MAPSET algorithm then performs minimization, adjusting phases, piecewise sewing, and thresholding (Cheng et al., 2017).

A parallel exact theory exists in real spline spaces. For the spline space

$g$2

the paper on local and global phaseless sampling gives necessary and sufficient counting conditions for a sequence of distinct points to be a local or global phaseless sampling sequence for nonseparable functions. It also shows that almost phaseless sampling requires fewer points than exact phaseless sampling (Sun, 2017).

Randomization provides another route. For nonseparable causal signals in complex-generated shift-invariant spaces, if the generalized Haar condition holds, then with probability $g$3, random phaseless samples of density $g$4 are sufficient in the complex case, while density $g$5 is sufficient for real-valued nonseparable causal signals in real-generated spaces (Li et al., 2019). This suggests that the nonuniformity appearing in square-root-lattice STLCT sampling is one instance of a wider phenomenon: irregular sampling can overcome phase-retrieval obstructions that remain present for rigid deterministic lattices.

These adjacent theories are not STLCT theorems, but they illuminate recurring technical motifs in phaseless sampling: nonseparability, local complementarity, entire-function uniqueness, translate completeness, graph connectivity, and phase synchronization from local data (Cheng et al., 2017, Sun, 2017, Li et al., 2019).

6. Scope, misconceptions, and current boundaries

A common misconception is that a standard sampling theorem for a short-time canonical transform automatically yields a phaseless sampling theorem. That is not the case. The convolution-based short time offset linear canonical transform (STOLCT) has a continuity theory, orthogonality relations, inversion formulas, a range theorem, a convolution theorem, Poisson summation, a Paley–Wiener criterion, and a Shannon-type sampling theorem. However, it does not study phaseless sampling explicitly: there is no theorem reconstructing a signal from $g$6 or $g$7 alone, and no explicit phase-retrieval algorithm (Mahato et al., 12 Jun 2026).

A second misconception is that uniqueness settles the practical inverse problem. The 2025 STLCT paper establishes uniqueness and counterexamples, but it does not provide stability estimates or a reconstruction algorithm for arbitrary $g$8-signals (Dong et al., 26 Aug 2025). Those issues are addressed only later, and then only in the Gaussian shift-invariant setting, where explicit inversion, interval stability under anchor points, and a finite noisy-sample algorithm become available (Cheng et al., 8 May 2026).

The present state of the subject is therefore differentiated rather than uniform. For general $g$9-signals with analytic windows, phaseless STLCT sampling is now understood at the level of uniqueness and sampling geometry: rectangular square-root lattices succeed, uniform lattices fail in general, and band-limited subclasses restore uniform-lattice recovery (Dong et al., 26 Aug 2025). For Gaussian shift-invariant signals, the theory extends further to explicit formulas, quantitative stability, and robust reconstruction from finitely many noisy measurements (Cheng et al., 8 May 2026). By contrast, adjacent canonical transforms such as STOLCT presently supply transform-domain sampling infrastructure but not magnitude-only inversion (Mahato et al., 12 Jun 2026).

Within that boundary, phaseless STLCT sampling has emerged as a distinct branch of phase retrieval: it is no longer merely an STLCT analogue of Gabor phase retrieval, but a sampling theory with its own nonuniform geometries, analytic uniqueness classes, semi-discrete reconstruction formulas, and stability mechanisms.