On the Impossibility of Lossless Waveform Rank Reduction for Certain Redundant Arrays

Piya Pal; Robin Rajam\"aki

On the Impossibility of Lossless Waveform Rank Reduction for Certain Redundant Arrays

T0 review · 0 major / 1 minor · reviewed 2026-05-24 · grok-4.3

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read Certain redundant arrays cannot achieve lossless waveform rank reduction due to their redundancy patterns.

desk verdict The paper shows that identical sum co-arrays do not guarantee the same identifiability at low waveform rank, because subspace properties of the redundancy pattern matter, and it gives a necessary condition based on that. read the letter →

arxiv 2305.06478 v2 pith:SSDAX46C submitted 2023-05-10 eess.SP

Robin Rajamäki , Piya Pal This is my paper

classification eess.SP

keywords arrayredundancywaveformrankparameteridentifiabilityMIMOactivesensingsumco-arraypattern

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper investigates the effects of array redundancy and waveform rank on identifiability in MIMO active sensing systems. It demonstrates that identifiability at reduced waveform rank depends critically on the subspace properties of the array redundancy pattern. Geometries with the same sum co-array can nevertheless display different identifiability at low waveform rank. A new necessary condition is presented that highlights how unfavorable redundancy patterns in some arrays restrict their performance.

What carries the argument

The subspace properties of the array redundancy pattern that control whether identifiability is preserved when waveform rank is reduced.

What would settle it

Finding that an array with an unfavorable redundancy pattern still achieves full parameter identifiability at reduced waveform rank would contradict the impossibility claim.

Watch

Extended reading notes

Core claim

Array geometries with identical sum co-arrays can exhibit markedly different identifiability properties at low waveform rank because parameter identifiability at reduced waveform rank depends on subspace properties of the array redundancy pattern; a novel necessary condition shows that the unfavorable redundancy patterns of certain redundant arrays fundamentally limit their performance.

Load-bearing premise

Parameter identifiability at reduced waveform rank is governed by the subspace properties of the array redundancy pattern.

Editorial extensions

If this is right

Identifiability at reduced WR depends on redundancy pattern subspaces rather than only the sum co-array.
Some arrays with identical sum co-arrays have different low-WR identifiability.
Maximizing identifiability at reduced WR requires satisfying a new necessary condition on the redundancy pattern.
The results motivate redundancy-aware design of arrays and waveforms for efficient sensing.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Array design for MIMO systems may need to prioritize specific redundancy patterns to enable low-rank operation.
Waveform design could potentially be adapted to mitigate unfavorable redundancy in existing arrays.
Simulations comparing identifiability for arrays like nested arrays versus others at reduced WR could test the condition further.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Desk Editor's Note

read the letter

The main thing to know is that array geometries with the same sum co-array can still differ sharply in identifiability once waveform rank drops, and the difference traces to the subspaces of their redundancy patterns. The authors derive a necessary condition that links those subspaces to whether identifiability survives the rank reduction. This is the concrete new piece: prior co-array work largely treats the sum co-array as the decisive object, but here the subspace view inside the redundancy pattern supplies an extra constraint that can rule out lossless reduction for some redundant arrays. That observation is stated clearly and follows from linear-algebraic properties that are standard in the area. The paper does a clean job of separating the sum co-array from the finer redundancy structure and showing why the latter can be the binding limit. The claim is grounded rather than circular; it does not appear to rest on fitted quantities or self-referential definitions. One soft spot is that the abstract gives no numerical example or explicit array pair that meets the condition versus one that violates it, so it is still unclear how often the necessary condition is tight or how much performance is lost in practice when it fails. The full derivation would need to be checked for edge cases where other factors, such as specific waveform choices, might override the subspace effect. This work is for people who design MIMO arrays and waveforms under hardware constraints on transmit rank. A reader already working on co-array identifiability or resource-efficient sensing will see a usable necessary condition to test against their own geometries. It is specific enough and grounded enough to merit a serious referee, even if the proofs need tightening and examples added. I would send it out for review.

Referee Report

0 major / 1 minor

Summary. The paper examines the effects of array redundancy and waveform rank (WR) on parameter identifiability in MIMO active sensing. It claims that identifiability at reduced WR depends critically on subspace properties of the array redundancy pattern rather than solely on the sum co-array. Geometries sharing identical sum co-arrays are shown to exhibit different identifiability at low WR. A novel necessary condition is derived for maximizing identifiability under WR reduction, indicating that unfavorable redundancy patterns in certain redundant arrays impose fundamental performance limits.

Significance. If the central claims hold, the work offers useful insights for resource-efficient MIMO sensing by demonstrating that redundancy patterns affect identifiability beyond standard co-array properties. The emphasis on subspace properties and the necessary condition could guide redundancy-aware array and waveform design. The observation that identical co-arrays yield differing performance is a potentially valuable distinction if supported by the derivations.

minor comments (1)

The title emphasizes 'impossibility of lossless waveform rank reduction,' but the abstract uses the milder phrasing 'fundamentally limits their performance'; clarifying this distinction in the introduction would improve consistency.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for reviewing our manuscript and for their summary recognizing the potential insights into redundancy patterns and identifiability beyond standard co-array properties. No specific major comments were provided in the report.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; derivation grounded in standard subspace analysis

full rationale

The paper's central claim derives a necessary condition for identifiability at reduced waveform rank from subspace properties of the array redundancy pattern, as stated in the abstract. This follows directly from linear-algebraic properties of sum co-arrays without reduction to fitted parameters, self-definitional loops, or load-bearing self-citations. Array geometries with identical co-arrays yielding different performance is presented as an observation from those properties, not a renaming or ansatz smuggled via prior work. No equations or steps in the provided text equate a prediction to its input by construction. The result is self-contained against external benchmarks of array signal processing.

Assumptions & free parameters 0 free parameters · 1 assumptions · 0 invented entities

The claim depends on standard linear-algebra definitions of sum co-arrays and subspace properties; no free parameters, ad-hoc axioms, or invented entities are mentioned in the abstract.

assumptions (1)

domain assumption Parameter identifiability is determined by the dimension or rank of certain subspaces formed by the array manifold and waveform matrix.
Invoked implicitly when linking redundancy pattern to identifiability at reduced WR.

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Cite this review

Pith. "Pith review of On the Impossibility of Lossless Waveform Rank Reduction for Certain Redundant Arrays." pith.science (2026). https://pith.science/paper/SSDAX46C

@misc{pith2026230506478,
  author       = {Pith},
  title        = {Pith review of: On the Impossibility of Lossless Waveform Rank Reduction for Certain Redundant Arrays},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/SSDAX46C}},
  note         = {Machine review of arXiv:2305.06478}
}

read the original abstract

Efficient use of spatio-temporal resources, including sensor arrays and transmit waveforms, is a key challenge in modern MIMO active sensing systems. This paper studies the impact of array redundancy and waveform rank (WR) on active sensing performance. Specifically, we show that parameter identifiability at reduced WR critically depends on subspace properties of the so-called array redundancy pattern. We show that array geometries with identical sum co-arrays can exhibit markedly different identifiability properties at low WR. We derive a novel necessary condition for maximizing identifiability at reduced WR, which reveals that the unfavorable redundancy patterns of certain redundant arrays fundamentally limits their performance. The results yield new insights into resource-efficient sensing systems, motivating redundancy-aware array and waveform design.

Figures

Figures reproduced from arXiv: 2305.06478 by the authors.

**Figure 2.** ULA and NA—two prototypical Tx-Rx array configuratio [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗

**Figure 4.** Minimum waveform rank yielding maximal identifiabil [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗

Figures from the paper (1 more)

**Figure 5.** Figure 5: Tx beampatterns in Examples 3 and 4 (ULA, [PITH_FULL_IMAGE:figures/full_fig_p010_5.png]

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