Minimum-Redundancy Arrays (MRAs)

• Minimum-Redundancy Arrays (MRAs) are sparse sensor arrays defined by their ability to form the largest contiguous co-array, yielding up to O(N^2) virtual sensors.
• They employ combinatorial design strategies to optimize sensor placement, significantly outperforming Uniform Linear Arrays in degrees of freedom and mutual coupling reduction.
• Robust MRAs (RMRAs) enhance reliability by ensuring a hole-free co-array under any single-sensor failure, critical for applications in radar, sonar, and medical ultrasound.

Minimum-Redundancy Arrays (MRAs) are sparse sensor array configurations designed to maximize the number of contiguous virtual sensors—equivalently, degrees of freedom (DOF)—achievable via their (sum or difference) co-array, for a given number of physical elements. The classical MRA design problem is intrinsically combinatorial and non-convex, with significant implications for array signal processing, particularly in direction-of-arrival (DOA) estimation and active sensing modalities such as radar, sonar, and medical ultrasound (Rajamäki et al., 21 Jan 2026, Kunchala et al., 14 Jul 2025, Patwari et al., 30 Dec 2025, Rajamäki et al., 2020, Zhang et al., 2022).

1. Foundational Definitions and Redundancy Metrics

MRAs are defined by their ability to produce the largest possible contiguous co-array (either sum or difference) for a fixed number of sensors. For an array of $N$ sensors with normalized integer positions $\mathbb{D} = \{d_1,\dots,d_N\}$, the sum co-array is

$$\mathbb{D}_\Sigma = \mathbb{D} + \mathbb{D} = \{d_i + d_j : d_i, d_j \in \mathbb{D}\}.$$

When $\mathbb{D}_\Sigma = [0:h-1]$ is contiguous, the redundancy metric is

$R(\mathbb{D}, \mathbb{D}) = \frac{N(N+1)}{2h}.$

Redundancy $R$ quantifies wasted sensor-pair contributions: lower $R$ means more virtual sensors per physical sensor. For a Uniform Linear Array (ULA), $R\to\infty$ as $N\to\infty$, whereas for MRAs, $R=O(1)$ (Rajamäki et al., 21 Jan 2026, Rajamäki et al., 2020). In the difference co-array setting, the redundancy ratio is similarly given by $\frac{\tfrac12 N(N-1)}{L}$, where $L$ is the aperture and the co-array covers $[-L, L]$ (Zhang et al., 2022).

Table: Redundancy Comparison

Array Type	Asymptotic Redundancy $R_\infty$	DOF Scaling
ULA	$\infty$	$O(N)$
MRAs	$1.2$–$1.9$	$O(N^2)$
Symmetric Nested Array (CNA)	$2$	$O(N^2)$
Kløve Array (KA)	$\approx 1.917$	$O(N^2)$

2. Formulation of the MRA and RMRA Design Problems

The restricted MRA problem for fully overlapping transmit–receive (Tx/Rx) positions of cardinality $N$ seeks

$$\underset{\mathbb{D}\subset\mathbb{N}, h\in\mathbb{N}}{\text{maximize}} \quad h \quad \text{subject to} \quad \mathbb{D} + \mathbb{D} = [0:h-1],\; |\mathbb{D}|=N.$$

For general active sensing with distinct Tx/Rx sets $\mathbb{D}_{\mathrm{tx}}, \mathbb{D}_{\mathrm{rx}}$, the problem seeks maximal contiguous sum co-array with prescribed cardinalities and overlap (Rajamäki et al., 21 Jan 2026). These formulations are combinatorial (search grows super-exponentially), rendering global optimum designs computationally intractable beyond $N\gtrsim 50$.

Robust MRAs (RMRAs) extend this by requiring that the difference co-array remain hole-free under any single-sensor failure (excluding endpoints), i.e., a two-fold redundancy constraint: \begin{align*} \text{Healthy:} \quad & w(i) \geq 2,\quad i\in[0, L-1];\ w(L) = 1, \ \text{Failure:} \quad & \forall n\notin{0,L},\, w_{S\setminus{n}}(i) \geq 1,\ i\in[0,L], \end{align*} where $w(i)$ is the weight function at lag $i$ (Kunchala et al., 14 Jul 2025, Patwari et al., 30 Dec 2025).

3. Solutions, Closed-Form Constructions, and Catalogued MRAs

For small $N$, exhaustive or branch-and-bound search yields optimal MRAs. Kohonen et al. catalogued fully overlapping MRAs up to $N=48$. For example, for $N=11$: $\mathbb{D} = \{0,1,2,3,7,11,15,17,20,21,22\}.$ Non-overlapping/nested MRAs have closed-form constructions for particular configurations: $$\mathbb{D}_{\mathrm{tx}} = \{0,N_{\mathrm{rx}},2N_{\mathrm{rx}},\dots,(N_{\mathrm{tx}}-1)N_{\mathrm{rx}}\},\; \mathbb{D}_{\mathrm{rx}} = \{0,1,2,\dots,N_{\mathrm{rx}}-1\},$$ which achieves $R=1$ (Rajamäki et al., 21 Jan 2026).

For RMRAs, optimal solutions for $N=6$ to $N=14$ have been obtained via exhaustive search. For $N=13$: $\{0,1,2,4,5,9,14,19,24,25,30,31,32\}$ with $A=32$ and DOFs $=65$ (Patwari et al., 30 Dec 2025, Kunchala et al., 14 Jul 2025). Near-optimal and sub-optimal RMRAs for $N$ up to $20$, and scalable closed-form expressions for sub-optimal RMRAs with $N\ge8$, are now available (Patwari et al., 30 Dec 2025). A typical closed-form for the sub-optimal RMRA uses $p=N-6$: $$\mathbb{S}(N) = \{0,1,\dots,p-1,2p,2p+1,3p+1,3p+2,4p+1,4p+2\}$$ yielding aperture $A = 4N - 22$ and $D(N) = 8N - 43$ DOFs (Patwari et al., 30 Dec 2025).

4. Scalable Symmetric and Low-Redundancy Array Designs

Given the computational challenge of true MRAs for large $N$, symmetric array frameworks—such as the Concatenated Nested Array (CNA) and the Kløve Array (KA)—have been proposed. The general symmetric array with generator $\mathbb{G}$ is

$$\mathbb{S} = \mathbb{G} \cup \left(\max \mathbb{G} - \mathbb{G} + \ell\right),$$

with the co-array $\mathbb{S} + \mathbb{S}$ fully contiguous if $\mathbb{G}-\mathbb{G} \supset [0:\max \mathbb{G}]$ and $\mathbb{G}+\mathbb{G} \supset [0:\ell-1]$ (Rajamäki et al., 21 Jan 2026, Rajamäki et al., 2020).

For the CNA (symmetric nested array): $$\mathbb{G} = \{0,1,\dots,N_1-1\} \cup \{N_1 + k(N_1+1):k=0,\dots,N_2-1\},$$ with $N=N_1+N_2$ and $R_\infty=2$. The KA attains $R_\infty\approx1.917$ and generalized scalability (Rajamäki et al., 2020). These designs achieve $O(N^2)$ virtual sensors, albeit with slightly higher redundancy than the strict MRAs.

Further, low-redundancy arrays with closed-form expressions (e.g., Type-(4r) arrays) can maintain redundancy $R<1.5$ for $N\geq18$ and deliver hole-free co-arrays, outperforming super-nested and classical maximum-inter-element-spacing-constraint (MISC) arrays in both redundancy and mutual coupling (Zhang et al., 2022).

5. Degrees of Freedom, Aperture Scaling, and Performance

An MRA of size $N$ with contiguous co-array $[0:h-1]$ achieves $N_\Sigma=h\propto N^2$ virtual elements, resulting in $\mathcal{O}(N^2)$ degrees of freedom. The maximum physical aperture is $\max\mathbb{D}\approx h/2 \propto N^2$, far exceeding that of a ULA ($\propto N$). MRAs can, in principle, resolve up to $\tfrac12 N(N+1)$ scatterers, compared to at most $N$ for a ULA (Rajamäki et al., 21 Jan 2026, Rajamäki et al., 2020).

Closed-form, order-optimal arrays (e.g., CNA, KA, Type-(4r) arrays) provide scalable solutions for large $N$, while retaining most of the DOF/sparsity tradeoff of the exact (but intractable) MRA. In simulation, new low-redundancy arrays exhibit superior DOA estimation accuracy, lower root mean square error (RMSE), and more robust performance under increasing mutual coupling, compared to other state-of-the-art sparse array designs (Zhang et al., 2022).

6. Robustness and Practical Implementation Considerations

While classical MRAs maximize DOFs for a given $N$, they are fragile: failure of a single sensor introduces holes, breaking the contiguous co-array property and reducing resolvable DOFs (Patwari et al., 30 Dec 2025, Kunchala et al., 14 Jul 2025). RMRAs, constructed with two-fold redundancy, retain a hole-free co-array under any single-sensor failure, important for practical deployment in radar, MIMO, sonar, and wireless communications.

Algorithms such as Leap-on-Success Exhaustive Search (LoSES) reduce the computational burden of RMRA design, especially for small to moderate $N$ (Kunchala et al., 14 Jul 2025). For larger $N$, closed-form but sub-optimal constructions balance robustness, scalability, and performance (Patwari et al., 30 Dec 2025). Mutual coupling is mitigated by array designs that minimize the weight of small-lag redundancies (Zhang et al., 2022).

7. Current Limitations and Future Directions

Despite advancements, exact MRA and RMRA designs for large $N$ remain open due to exponential search complexity. Current RMRA catalogues are optimal only up to $N=14$, with near-optimal and closed-form sub-optimal solutions for larger $N$ (Patwari et al., 30 Dec 2025, Kunchala et al., 14 Jul 2025). Future work targets:

• Closed-form robust arrays with $O(N^2)$ DOFs and quantifiable redundancy,
• Extensions to multi-sensor-failure tolerance or higher-fold redundancy,
• Improved mutual-coupling resistance for large-scale arrays,
• Algorithmic seeding and constraint programming for efficient exploration of the high-$N$ design space.

A plausible implication is that order-optimal, symmetric, and closed-form array families will underpin next-generation sparse array design, enabling robust, scalable, and high-resolution sensing in diverse active and passive contexts.