Conjugate phase retrieval in shift-invariant spaces generated by a Gaussian

Conjugate phase retrieval considers the recovery of a function, up to a unimodular constant and conjugation, from its phaseless measurements. In this paper, we explore the conjugate phase retrieval in a shift-invariant space generated by a Gaussian funciton. First, we show that the modulus function in the Gaussian shift-invariant space can be determined from the phaseless Hermite samples taken on a discrete sampling set. We then show that a function in the shift-invariant space generated by a Gaussian can be uniquely determined, up to a unimodular constant and conjugation, from its phaseless Hermite samples on a discrete set. For the functions with finite coefficient sequences, we provide an explicit reconstruction procedure.

💡 Research Summary

The paper addresses the problem of recovering complex‑valued functions from magnitude‑only measurements in the shift‑invariant space generated by a Gaussian kernel, a setting where classical phase retrieval fails because the space is invariant under complex conjugation. The authors introduce the notion of “conjugate phase retrieval,” which seeks to determine a function up to a global unimodular factor and possible conjugation, using only the absolute values of the function and its first derivative (Hermite samples).

The main functional setting is
(V_{\infty}^{\beta ,\lambda}=\Big{\sum_{k\in\mathbb Z}c_k,e^{-\lambda (x-\beta k)^2};:; {c_k}\in\ell^\infty(\mathbb Z)\Big}),
with step size (\beta>0) and decay parameter (\lambda>0). The authors assume a mild growth condition ({k,c_k}\in\ell^\infty) (equation 2.3) to guarantee that the derivative also belongs to a similar shift‑invariant space.

Key technical contributions

1. Phaseless Hermite sampling (Theorem 2.1).
   By exploiting the fact that (|f|^2) lies in the shift‑invariant space (V_{\infty}^{2\beta,2\lambda}) (Lemma 2.1) and that any set (\Gamma\subset\mathbb R) with lower Beurling density (D^{-}(\Gamma)>2\beta-1) is a uniqueness set for (V_{\infty}^{\beta ,\lambda}) (Lemma 2.2), the authors prove that the discrete samples ({|f(\gamma)|,|f’(\gamma)|}_{\gamma\in\Gamma}) uniquely determine the full modulus functions (|f|) and (|f’|) on the whole real line. The proof introduces an auxiliary function (\omega(x)=\sum_k k c_k e^{-\lambda (x-k)^2}) and derives an explicit relation (equation 2.7) linking (|f’|^2) to (|f|^2) and (|\omega|^2). Since the coefficients of (|\omega|^2) can be expressed through the same convolutional structure as those of (|f|^2), they are recoverable from the same sampling set.

2. Conjugate phase retrieval (Theorem 3.1).
   With the full modulus functions known, the problem reduces to identifying the coefficient sequence ({c_k}) up to a global phase and conjugation. The authors map the coefficient sequence to an entire function (\widetilde C(z)=\sum_{k}\widetilde c_k e^{2\pi i k z}) where (\widetilde c_k=c_k e^{-\lambda k^2}). Lemma 3.2 guarantees that (\widetilde C) extends to an entire function of order 2, allowing the use of complex‑analytic tools. Lemma 3.3 (proved in Appendix B) shows that if two entire functions of finite order satisfy (ff^\natural = gg^\natural) and (f’f’^\natural = g’g’^\natural) (where (\natural) denotes the involution (f^\natural(z)=f(-\overline z))), then they differ only by a unimodular constant or by conjugation. Applying this to (\widetilde C) and the analogous function for a second candidate (g) yields the desired uniqueness: any (g) sharing the same Hermite magnitude samples must be either (\alpha f) or (\alpha \overline f) with (|\alpha|=1).

3. Finite‑support reconstruction algorithm.
   For signals whose coefficient sequence has finite support, the authors give an explicit reconstruction scheme. By truncating the Fourier series to a finite sum, the unknown coefficients become solutions of a finite system of quadratic equations derived from the sampled magnitudes. In the canonical case (\beta=1,\lambda=1), they present a step‑by‑step algorithm, discuss numerical stability, and provide simulation results confirming exact recovery under the prescribed density condition.

Implications and novelty

• The paper establishes that, contrary to earlier results showing impossibility of classical phase retrieval for complex‑valued Gaussian shift‑invariant spaces, conjugate phase retrieval is feasible with a modest sampling density (just above (2\beta-1)).
• The use of Hermite samples (both function and derivative magnitudes) is crucial; it supplies enough algebraic structure to recover both (|f|) and (|f’|), which together encode the phase information up to conjugation.
• The analytic bridge from discrete coefficient sequences to entire functions of finite order enables a powerful uniqueness argument that does not rely on oversampling or additional measurements.
• The finite‑support algorithm makes the theory applicable to practical signal processing tasks where signals are naturally truncated or windowed.

Overall, the work advances the theory of phaseless signal reconstruction for complex‑valued functions, providing both rigorous mathematical foundations and constructive algorithms for a class of signals that are highly relevant in time‑frequency analysis, optics, and communications.