Robust Minimum Redundancy Arrays

• RMRAs are sparse linear arrays that maximize aperture while ensuring a hole-free difference coarray and failure resilience.
• They achieve two-fold redundancy with doubly redundant core lags, maintaining performance even if any single non-end sensor fails.
• The design is cast as a combinatorial optimization problem, with scalable closed-form constructions available for arrays with N ≥ 8.

Searching arXiv for papers on Robust Minimum Redundancy Arrays and closely related sparse-array robustness work. Robust Minimum Redundancy Arrays (RMRAs) are sparse linear arrays that preserve the aperture-maximizing objective of classical minimum redundancy arrays while imposing explicit coarray redundancy and failure resilience. In the recent literature, an RMRA is treated as a specialized two-fold redundant sparse array: for a fixed number of sensors $N$, it maximizes aperture $L$, has a hole-free difference coarray (DCA), is doubly redundant over the core lag interval, and remains hole-free after failure of any single non-end sensor (Patwari et al., 30 Dec 2025, Kunchala et al., 14 Jul 2025).

1. Formal definition and coarray model

A linear sparse array is represented by sensor locations, normalized to $\lambda/2$, as

$$\mathbb{S} = \{s_0,s_1,\dots,s_{N-1}\}, \qquad s_i\in\mathbb{Z}, \qquad s_0=0,$$

with physical aperture

$L = s_{N-1}-s_0.$

Its difference set is

$$\mathbb{Z} = \{s_i-s_j \mid s_i,s_j\in\mathbb{S}\},$$

and the difference coarray is the set of unique lags

$\mathbb{D} = \mathrm{unique}(\mathbb{Z}).$

The multiplicity of lag $z$ is denoted $w(z)$, and the number of distinct lags,

$|\mathbb{D}|,$

is the number of degrees of freedom (DOFs). A DCA is hole-free on $L$0 if $L$1 with no missing integers in that interval (Patwari et al., 30 Dec 2025).

Classical minimum redundancy arrays (MRAs), going back to Moffet’s 1968 formulation, maximize the DCA span for a given $L$2, require a hole-free coarray over that span, and minimize unnecessary lag repetition. In the recent RMRA literature, MRAs are described as “maximally economic”: they deliver a large aperture and a hole-free DCA but possess little or no redundancy, which makes them fragile under sensor loss (Patwari et al., 30 Dec 2025, Kunchala et al., 14 Jul 2025).

Two-fold redundant sparse arrays (TFRSAs, or TFRAs in one of the papers) constitute the immediate superclass of RMRAs. In that terminology, a TFRSA is a sparse array whose healthy DCA is doubly redundant over its core span and whose DCA remains hole-free under any single non-end sensor failure. An RMRA is then the TFRSA that is optimal in aperture for fixed $L$3 (Patwari et al., 30 Dec 2025, Kunchala et al., 14 Jul 2025).

2. Robustness conditions, essential sensors, and fragility

The healthy-array condition is stated through the lag-weight function. If $L$4 denotes the multiplicity of lag $L$5, then a valid two-fold redundant design satisfies

$L$6

Because the weight function is even-symmetric, $L$7. These constraints imply a contiguous DCA over $L$8, with every lag in that interval generated by at least two sensor pairs, while the maximum lag $L$9 is produced exactly once (Patwari et al., 30 Dec 2025, Kunchala et al., 14 Jul 2025).

Robustness is enforced by removing any interior sensor $\lambda/2$0 and requiring the faulty-array multiplicities $\lambda/2$1 to satisfy

$\lambda/2$2

Equivalently, after any single non-end sensor failure, the DCA remains hole-free on $\lambda/2$3. This is the defining robustness property of the recent RMRA formulations (Patwari et al., 30 Dec 2025).

A closely related formalism uses the doubly redundant coarray

$\lambda/2$4

and the set of essential sensors $\lambda/2$5, defined as sensors whose removal changes either the span or the continuity of the DCA. One optimization statement requires

$\lambda/2$6

with only the two end sensors essential. Under that condition, fragility is

$\lambda/2$7

In the notation of the companion paper, the essential-sensor set is written as

$\lambda/2$8

which is equivalent to robustness under any single failure of a non-end sensor (Kunchala et al., 14 Jul 2025, Patwari et al., 30 Dec 2025).

The resulting distinction is strict. Conventional MRAs require only $\lambda/2$9 over the desired contiguous span and do not guarantee survival of that span after failure. RMRAs add both multiplicity and failure invariance constraints, so they are more constrained than classical MRAs and more specific than generic TFRSAs (Patwari et al., 30 Dec 2025).

3. RMRA synthesis as a combinatorial optimization problem

The recent RMRA literature formulates synthesis as a discrete aperture-maximization problem over integer sensor placements. One representative statement is

$$\mathbb{S} = \{s_0,s_1,\dots,s_{N-1}\}, \qquad s_i\in\mathbb{Z}, \qquad s_0=0,$$0

subject to fixed sensor count, hole-free DCA, doubly redundant core lags, exactly two essential sensors, and $$\mathbb{S} = \{s_0,s_1,\dots,s_{N-1}\}, \qquad s_i\in\mathbb{Z}, \qquad s_0=0,$$1. In the alternate notation,

$$\mathbb{S} = \{s_0,s_1,\dots,s_{N-1}\}, \qquad s_i\in\mathbb{Z}, \qquad s_0=0,$$2

subject to $$\mathbb{S} = \{s_0,s_1,\dots,s_{N-1}\}, \qquad s_i\in\mathbb{Z}, \qquad s_0=0,$$3, $$\mathbb{S} = \{s_0,s_1,\dots,s_{N-1}\}, \qquad s_i\in\mathbb{Z}, \qquad s_0=0,$$4, and $$\mathbb{S} = \{s_0,s_1,\dots,s_{N-1}\}, \qquad s_i\in\mathbb{Z}, \qquad s_0=0,$$5 (Kunchala et al., 14 Jul 2025, Patwari et al., 30 Dec 2025).

For fixed $$\mathbb{S} = \{s_0,s_1,\dots,s_{N-1}\}, \qquad s_i\in\mathbb{Z}, \qquad s_0=0,$$6 and $$\mathbb{S} = \{s_0,s_1,\dots,s_{N-1}\}, \qquad s_i\in\mathbb{Z}, \qquad s_0=0,$$7, the search space consists of choosing $$\mathbb{S} = \{s_0,s_1,\dots,s_{N-1}\}, \qquad s_i\in\mathbb{Z}, \qquad s_0=0,$$8 internal sensor positions from $$\mathbb{S} = \{s_0,s_1,\dots,s_{N-1}\}, \qquad s_i\in\mathbb{Z}, \qquad s_0=0,$$9, with

$L = s_{N-1}-s_0.$0

Search is staged over increasing aperture $L = s_{N-1}-s_0.$1, typically starting from $L = s_{N-1}-s_0.$2. At each stage, every candidate is checked for DCA continuity, healthy-state lag weights, and robustness under all single non-end sensor failures. If no valid candidate exists for a given $L = s_{N-1}-s_0.$3, the previous successful aperture is declared optimal for that $L = s_{N-1}-s_0.$4 (Patwari et al., 30 Dec 2025, Kunchala et al., 14 Jul 2025).

The computational burden is explicitly described as NP-hard. One paper relates RMRA design to minimum-redundancy rulers and Golomb-ruler-type placement problems, while adding the further global constraints of lag multiplicity and failure robustness. The combinatorial growth is illustrated by the $L = s_{N-1}-s_0.$5, $L = s_{N-1}-s_0.$6 stage, which already yields

$L = s_{N-1}-s_0.$7

candidate arrays (Kunchala et al., 14 Jul 2025).

Two search accelerations are reported. The first is structural pruning: fixing the first three and last two sensors,

$L = s_{N-1}-s_0.$8

because those positions were observed to be important for preserving the multiplicities of the corner lags $L = s_{N-1}-s_0.$9, $$\mathbb{Z} = \{s_i-s_j \mid s_i,s_j\in\mathbb{S}\},$$0, and $$\mathbb{Z} = \{s_i-s_j \mid s_i,s_j\in\mathbb{S}\},$$1. This reduces the combinations from

$$\mathbb{Z} = \{s_i-s_j \mid s_i,s_j\in\mathbb{S}\},$$2

The second is the Leap-on-Success exhaustive search (LoSES), which terminates the search at a given aperture as soon as the first valid RMRA is found, then “leaps” to the next aperture $$\mathbb{Z} = \{s_i-s_j \mid s_i,s_j\in\mathbb{S}\},$$3. Optimality is still certified at the first aperture for which no valid configuration exists (Patwari et al., 30 Dec 2025, Kunchala et al., 14 Jul 2025).

Implementation details are also reported. The exhaustive-search study used MATLAB, with computationally heavy components implemented as MEX in C/C++, while the LoSES study likewise used MATLAB validation under all single-element failure scenarios (Patwari et al., 30 Dec 2025, Kunchala et al., 14 Jul 2025).

4. Reported configurations and catalog expansion

The two 2025 RMRA studies agree on the newly found optimal configurations for $$\mathbb{Z} = \{s_i-s_j \mid s_i,s_j\in\mathbb{S}\},$$4 through $$\mathbb{Z} = \{s_i-s_j \mid s_i,s_j\in\mathbb{S}\},$$5, thereby extending the Liu–Vaidyanathan catalog beyond $$\mathbb{Z} = \{s_i-s_j \mid s_i,s_j\in\mathbb{S}\},$$6 (Patwari et al., 30 Dec 2025, Kunchala et al., 14 Jul 2025).

$$\mathbb{Z} = \{s_i-s_j \mid s_i,s_j\in\mathbb{S}\},$$7	Optimal $$\mathbb{Z} = \{s_i-s_j \mid s_i,s_j\in\mathbb{S}\},$$8	Configuration
11	22	$$\mathbb{Z} = \{s_i-s_j \mid s_i,s_j\in\mathbb{S}\},$$9
12	26	$\mathbb{D} = \mathrm{unique}(\mathbb{Z}).$0
13	32	$\mathbb{D} = \mathrm{unique}(\mathbb{Z}).$1
14	36	$\mathbb{D} = \mathrm{unique}(\mathbb{Z}).$2

These arrays satisfy the healthy and faulty weight constraints, have only two essential sensors, and attain the maximum aperture found by exhaustive search for the corresponding $\mathbb{D} = \mathrm{unique}(\mathbb{Z}).$3 (Patwari et al., 30 Dec 2025, Kunchala et al., 14 Jul 2025).

Beyond that shared core, the two papers diverge in scope and classification. One reports new optimal RMRAs for $\mathbb{D} = \mathrm{unique}(\mathbb{Z}).$4 to $\mathbb{D} = \mathrm{unique}(\mathbb{Z}).$5, with the $\mathbb{D} = \mathrm{unique}(\mathbb{Z}).$6 optimal configuration

$\mathbb{D} = \mathrm{unique}(\mathbb{Z}).$7

and aperture $\mathbb{D} = \mathrm{unique}(\mathbb{Z}).$8, then reports near-optimal configurations for $\mathbb{D} = \mathrm{unique}(\mathbb{Z}).$9 to $z$0 (Kunchala et al., 14 Jul 2025). The other reports optimal RMRAs for $z$1 to $z$2 and near-/sub-optimal arrays for $z$3 to $z$4, obtained under partially constrained search because MATLAB limits prevented fully exhaustive enumeration at larger sizes (Patwari et al., 30 Dec 2025).

The near-/sub-optimal catalogs are extensive in both studies. One paper gives representative near-optimal arrays for $z$5 through $z$6, including a $z$7-sensor configuration with aperture $z$8, and also presents pattern-based extrapolations for $z$9 and $w(z)$0 with larger apertures than those found in the direct search (Kunchala et al., 14 Jul 2025). The other paper lists validated robust arrays for every $w(z)$1 from $w(z)$2 through $w(z)$3, but explicitly labels them near-/sub-optimal because the search space was restricted and global optimality was not certified (Patwari et al., 30 Dec 2025).

Both studies emphasize that these catalogs are valuable beyond the single best array per $w(z)$4. They provide a substantial repository of valid robust configurations for later pattern mining, benchmarking, and practical deployment when exhaustive optimization is infeasible (Patwari et al., 30 Dec 2025).

5. Closed-form sub-optimal RMRA and TFRSA family

A principal contribution of the December 2025 paper is a closed-form family derived by pattern mining over the exhaustively discovered arrays. Defining

$w(z)$5

the proposed family for $w(z)$6 is

$w(z)$7

The first $w(z)$8 sensors form a dense ULA-like segment, while the last six sensors constitute a sparse tail whose positions scale linearly with $w(z)$9. The same family is also written in inter-element-spacing form as

$|\mathbb{D}|,$0

This construction is presented as a TFRSA and therefore as a sub-optimal RMRA family: it is robust and scalable, but not aperture-optimal relative to the best exhaustively found RMRAs (Patwari et al., 30 Dec 2025).

Its aperture is

$|\mathbb{D}|,$1

and the cited validity condition for genuine sparsity is

$|\mathbb{D}|,$2

so the construction is taken as valid for

$|\mathbb{D}|,$3

The same source states that the array is hole-free over $|\mathbb{D}|,$4, with

$|\mathbb{D}|,$5

Those formulas provide closed-form sensor positions, aperture, and DCA size for arbitrary $|\mathbb{D}|,$6 (Patwari et al., 30 Dec 2025).

The paper gives explicit examples. For the smallest valid case, $|\mathbb{D}|,$7 and $|\mathbb{D}|,$8,

$|\mathbb{D}|,$9

and the healthy and single-failure conditions are checked directly. For $L$00, the family becomes

$L$01

obtained without any search (Patwari et al., 30 Dec 2025).

Validation is algorithmic rather than analytic. The reported pipeline evaluates the closed-form family for every $L$02 up to $L$03, checks the healthy-state two-fold redundancy, then tests every single non-end sensor failure. The reported Mega Count is $L$04, so no violation was found in that range; on that basis the closed-form family is declared valid for all $L$05 tested (Patwari et al., 30 Dec 2025).

6. Verification, hidden dependencies, applications, and related directions

A persistent theme in the robustness literature is that nominal lag multiplicity does not by itself guarantee failure resilience. One of the RMRA papers explicitly shows that some DDB-based 2FRAs possess hidden essential sensors. For the $L$06 array

$L$07

the healthy weight function appears doubly redundant, but removing the sensor at $L$08 creates a hole at lag $L$09. The array then has a third essential sensor and fragility

$L$10

rather than the RMRA target $L$11 (Kunchala et al., 14 Jul 2025).

The same phenomenon extends to higher-order redundancy. A 2025 framework for testing three-fold redundant sparse linear arrays against all two-sensor failures introduces essential sensor pairs and hidden essential sensor pairs (HESPs), then applies exhaustive failure analysis to existing TRSLA families. The study reports that almost $L$12 of the tested 3FRA configurations suffer from hidden vulnerabilities, and that all tested TRAs contain two or more HESPs; in both cases, nominal multi-fold redundancy does not guarantee actual robustness under the designated failure budget (Patwari et al., 9 Sep 2025). For RMRAs, that result has a direct methodological implication: robustness must be verified by explicit failure enumeration, not inferred solely from healthy-state multiplicity counts.

The application space is broad. The RMRA papers explicitly place robust sparse arrays in direction-of-arrival estimation for radar, sonar, and wireless systems, in automotive MIMO radar, and in MIMO-SAR and high-resolution mapping, where coarray continuity under hardware faults is operationally significant (Patwari et al., 30 Dec 2025, Kunchala et al., 14 Jul 2025). The broader multifold-redundancy literature adds radio astronomy, aperture synthesis radiometers, medical imaging arrays, and integrated sensing and communications (ISAC) as domains where failure-tolerant sparse geometries are valuable (Patwari et al., 9 Sep 2025).

Adjacent sparse-array research broadens the context even when the RMRA label is absent. Active-sensing work formulates minimum-redundancy design in terms of the sum co-array, introduces symmetric low-redundancy families such as the Concatenated Nested Array and Kløve Array, and studies redundancy pattern matrices and image addition for Tx–Rx beam synthesis (Rajamäki et al., 21 Jan 2026, Rajamäki et al., 2020). Planar active-sensing work on the Concentric Rectangular Array treats minimum-redundancy structure together with a robustness objective tied to mutual coupling, measured through the number of unit-spaced pairs rather than sensor-failure tolerance (1803.02219). This suggests that RMRA methodology can plausibly be extended beyond the 1-D difference-coarray setting by reformulating robustness in the appropriate virtual-array domain.

Estimator robustness forms a further, complementary layer. The maximum-likelihood method MESA is reported to be usable for arbitrary sparse linear arrays, including minimum redundancy arrays, nested arrays, and coprime arrays, and is proved robust to source correlations though derived under uncorrelated-source assumptions (Yang et al., 2022). This suggests that RMRA research naturally decomposes into a geometry problem—maintaining hole-free virtual aperture under hardware faults—and a processing problem—maintaining inference quality under model mismatch and source correlation.

Taken together, the recent literature establishes RMRAs as a sharply defined subclass of robust sparse arrays: they are not merely low-redundancy arrays, and not merely arrays with repeated lags, but aperture-optimal TFRSAs whose coarray continuity survives the designated failure model. The main open pressure points are equally clear in the cited works: combinatorial complexity, the need for stronger search and optimization machinery, the elimination of hidden dependencies, and the development of scalable closed-form constructions that retain certified robustness (Patwari et al., 30 Dec 2025, Kunchala et al., 14 Jul 2025).