---
title: Minimum-Redundancy Arrays (MRAs)
url: https://www.emergentmind.com/topics/minimum-redundancy-arrays-mras
type: topic
---

# Minimum-Redundancy Arrays (MRAs)

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 [2601.15126][2507.10706][2512.24155][2010.08977][2208.05263].

## 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, $R=O(1)$ [2601.15126][2010.08977]. 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 $[-L, L]$ [2208.05263].

**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 [2601.15126]. 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 $i$ [2507.10706][2512.24155].

## 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 $R=1$ [2601.15126].

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 $=65$ [2512.24155][2507.10706]. 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 [2512.24155]. 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 [2512.24155].

## 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 $\mathbb{G}+\mathbb{G} \supset [0:\ell-1]$ [2601.15126][2010.08977].

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 [2010.08977]. 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 [2208.05263].

## 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 [2601.15126][2010.08977].

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 [2208.05263].

## 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 [2512.24155][2507.10706]. 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 $N$ [2507.10706]. For larger $N$, closed-form but sub-optimal constructions balance robustness, scalability, and performance [2512.24155]. Mutual coupling is mitigated by array designs that minimize the weight of small-lag redundancies [2208.05263].

## 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 $N$ [2512.24155][2507.10706]. 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.

Source: https://www.emergentmind.com/topics/minimum-redundancy-arrays-mras