Jointly optimal array geometries and waveforms in active sensing: New insights into array design via the Cram\'er-Rao bound

Geert Leus; Ids van der Werf; Robin Rajam\"aki

REVIEW 2 major objections 4 minor 31 references

Jointly optimal array geometries and waveforms in active sensing: New insights into array design via the Cram\'er-Rao bound

T0 review · 2 major / 4 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read Under aperture and sensor-number constraints, the receive array that minimizes the single-target angle Cramér–Rao bound is uniquely the two-ended clustered array, while the transmit array and waveform keep their freedom.

desk verdict Aperture-optimal receive array is clustered, but the uniqueness claim needs 'up to translation' and Lemma 1 needs a proof. read the letter →

arxiv 2501.00472 v2 pith:LV43QE7F submitted 2024-12-31 eess.SP

Ids van der Werf , Robin Rajamäki , Geert Leus This is my paper · ORCID

classification eess.SP MSC 94A12

keywords activesensingCramer-RaoboundMIMOradararraygeometrydesignsparsearrayswaveformsumco-arrayspatialvariance

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

Under the single-target active sensing model, this paper tries to pin down which transmit/receive array geometries and which transmit waveforms together minimize the Cramér–Rao lower bound for angle estimation. It claims that, given a fixed receive aperture and sensor count, the optimal receive array is unique: the clustered array $K_L^{N_r}=U_{N_r/2}\cup (L-U_{N_r/2})$, which packs sensors at the two edges of the aperture to maximize spatial variance $\chi_r$. The transmit array, by contrast, can be any geometry whose spatial variance stays below $\chi_r$, and the optimal waveform is coherent beamsteering toward the target. The paper then chooses a particular transmit array that keeps this optimality and makes the sum co-array contiguous and nonredundant, which matters for identifying multiple targets when independent waveforms are used. A sympathetic reader would care because this turns the CRB minimization into a concrete aperture-limited design rule: put receive sensors at the edges, and use the remaining transmit freedom to repair the co-array.

What carries the argument

The load-bearing object is the clustered array $K_L^N=U_{N/2}\cup(L-U_{N/2})$, together with the spatial variance $\chi(D)$ defined in (3). Lemma 1 asserts that, for even $N$, this two-ended cluster maximizes $\chi$ among all $N$-element subsets of the aperture $\{0,\dots,L\}$; the paper states that the proof follows directly by negation and omits it. Around that lemma the argument wraps two further mechanisms: the optimal-waveform reduction (from [12] and [13]) that collapses the CRB to $\sigma^2/(2|\gamma|^2N_tN_r\chi_r)$, and the sum co-array construction in Corollary 1 that rescales the transmit array by $N_r/2$ to tile $U_{N_tN_r}$ without redundancy.

What would settle it

Enumerate all subsets of $\{0,\dots,L\}$ of size $N$ for even $N$ and small $L$ (say $N=4$, $L=8$) and compute the spatial variance of each; if any subset beats $K_L^N$, Lemma 1 and hence Theorem 1 are false.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central discovery is Theorem 1: for even $N_r$, every solution of the joint optimization problem (9) has the form $S^\star=u a_t^H(\omega)/\sqrt{N_t}$, $D_r^\star=K_L^{N_r}$, and an arbitrary transmit array with $|D_t|=N_t$ and $\chi(D_t)<\chi(D_r^\star)$. The CRB reduces, after substituting the optimal waveform, to $\sigma^2/(2|\gamma|^2 N_t N_r \chi_r)$, so the receive geometry enters only through its spatial variance; maximizing that variance under the aperture constraint yields the two-ended clustered array. Corollary 1 then fixes $L=(N_t+1)N_r/2-1$ and chooses $D_t^\star=(N_r/2)U_{N_t}$, which satisfies the variance condition and produces the contiguous, nonredundant sum co-array $D_\Sigma=U_{N_tN_r}$. A parallel observation is that receive arrays with equal sums of squares of their centered positions give equal CRBs, so the CRB does not uniquely separate all geometries.

Load-bearing premise

The argument stands on the unproved combinatorial lemma that, for an even number of sensors in a fixed aperture, pushing the sensors to the two ends maximizes their spread; if any other placement had a larger spread, the claimed uniqueness of the optimal receive array would collapse.

Editorial extensions

If this is right

Any system that wants the lowest single-target angle CRB at a given aperture and receive sensor count must use the clustered receive array; no other receive geometry can match it.
The transmit array is not pinned down by the CRB, so designers can choose it to satisfy other criteria, such as co-array contiguity, without sacrificing the bound.
With the specific choice $D_t^\star=(N_r/2)U_{N_t}$ and $L=(N_t+1)N_r/2-1$, the sum co-array is the full contiguous ULA $U_{N_tN_r}$, allowing up to $N_tN_r/2$ targets to be identified when full-rank waveforms are transmitted.
Orthogonal transmit waveforms are not generally optimal for single-target CRB; coherent beamsteering is, as long as the receiver's spatial variance exceeds the transmitter's.
Distinct receive geometries with equal spatial variance (constructed from equal sums of squares) achieve identical CRBs but can differ substantially in threshold-region estimation behavior.

Reading between the lines

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

The variance-maximizing lemma is stated only for even $N_r$; an analogous one-sided cluster likely covers odd $N_r$, which would make the uniqueness statement in Theorem 1 hold for all sensor counts.
Because the optimal waveform (5) needs the true target angle, a practical realization would first estimate the angle and then reconfigure; the paper notes this but does not analyze the two-stage or iterative version.
The equal-sum-of-squares degeneracy offers a controlled experimental handle: pick two arrays with identical CRB but different beampatterns and probe where maximum-likelihood estimation breaks down, isolating geometry effects beyond the bound.
For automotive radar, the result suggests placing receive elements near the array ends rather than in a uniform line; whether this survives near-field, multipath, and mounting constraints is an open engineering question.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Punchline: the core result is right—under an aperture constraint, the CRB-optimal receive array is the clustered array—but the uniqueness claim is stated too strongly because spatial variance is translation-invariant.

What is actually new: the paper shows that array geometries with equal sums of squares achieve equal CRBs under the optimal waveform, which connects array design to number theory. The bigger contribution is Corollary 1: a Tx array that, together with the optimal clustered Rx array, gives a contiguous nonredundant sum co-array while still minimizing the single-target CRB. That specific construction is new and practically useful. The derivation from the known optimal waveform (Forsythe-Bliss, Li et al.) to the CRB depending only on receive spatial variance is clean and transparent. The appendix proof of the co-array property checks out.

Soft spots, in proportion: the load-bearing Lemma 1 is stated without proof ('follows directly via negation'). For a uniqueness claim, that is a real gap; a citation or a short proof is needed. More importantly, the stress-test is correct: problem (9) constrains only aperture, and chi is invariant under translation, so every translate of K_L^Nr is also a solution. Theorem 1's statement that the solutions are precisely D_r* = K_L^Nr is literally false; the proof silently normalizes the array window to [0,L]. This is fixable by saying 'unique up to translation,' but it is a genuine formal error. Also, Lemma 1 writes 'arg min' where the text says maximizing; that sign error should be corrected. Feasibility is only characterized by a sufficient condition, but the authors acknowledge that, so it is a minor limitation.

None of this overturns the main message: put receive sensors at the two ends of the aperture. The paper is a solid theoretical contribution to MIMO radar and ISAC array design. I'd send it to a serious referee; with the translation caveat and a proof of Lemma 1, it should be publishable. The numerical study is illustrative rather than exhaustive, which is appropriate here.

Referee Report

2 major / 4 minor

Summary. The paper studies joint transmit/receive array geometry and transmit waveform design for a single-target active sensing system, with the goal of minimizing the Cramér-Rao lower bound (CRB) of the target angle. Using prior optimal waveform results, the authors show that for a class of optimal waveforms corresponding to transmit beamforming, the CRB depends only on the receive array's spatial variance. Under a constraint on receive aperture and an even number of receive sensors, they claim that the unique CRB-optimal receive geometry is a clustered array with sensors placed at the two ends of the aperture, while the transmit array can be chosen freely subject to a spatial-variance inequality. They then propose a specific sparse transmit array that yields a contiguous and nonredundant sum co-array, and illustrate the results with numerical MLE comparisons.

Significance. If the central claims hold, the paper provides a clean characterization of CRB-optimal receive array geometry in active sensing and introduces a sparse array configuration with attractive identifiability properties. The derivation builds on externally established CRB formulas (Stoica-Nehorai, Forsythe-Bliss, Li et al.) and the variance computations in Eqs. (8) and (12) are straightforward and correct. The equal-sums-of-squares perspective linking equal CRBs to Diophantine solutions is a nice insight. However, the uniqueness statement in Theorem 1 is too strong as written because it ignores translation invariance, and the key Lemma 1 is asserted without proof or citation. These are load-bearing issues, but they are fixable by reformulating the theorem and providing a proof of Lemma 1, so the paper's geometric insight is likely salvageable.

major comments (2)

[Section 3.2, Theorem 1 (Eq. (10))] The statement that the solutions to (9) are precisely D_r* = K_L^{Nr} is not correct as written. The constraint in (9) is max D_r - min D_r <= L, not fixing the array to lie in {0,...,L}, and the CRB in Eq. (6) depends on the spatial variance chi(D_r), which is invariant under translation of D_r. Hence, for any integer a, the array D_r(a) = {a,...,a+Nr/2-1} union {a+L-Nr/2+1,...,a+L} has the same aperture and the same spatial variance as K_L^{Nr}, so it also achieves the same CRB and satisfies all constraints. The proof silently normalizes the aperture window to [0,L] without stating this w.l.o.g. step. The theorem should be reformulated as 'unique up to translation', or the optimization problem should be stated over translation equivalence classes.
[Section 3.2, Lemma 1 (Eq. (7))] Lemma 1 is the sole basis for the optimality and claimed uniqueness of the clustered receive array in Theorem 1, but the proof is omitted ('follows directly via negation') and no citation is provided. This is a nontrivial combinatorial extremal claim and cannot be accepted without proof. In addition, Eq. (7) writes K_L^N as the arg min of chi(D), while the surrounding text and Theorem 1 require arg max; this sign error must be corrected. The authors should supply a complete proof (e.g., an exchange argument or an explicit derivation) or replace the lemma with a referenced result.

minor comments (4)

[Section 2.2] The sentence 'can be show to reduce' should read 'can be shown to reduce'.
[Figure 1 caption] The word 'nonredudant' should be 'nonredundant'.
[Section 3.1] The example '12+82 = 42+72' would be clearer if typeset as 1^2 + 8^2 = 4^2 + 7^2; the current plain-text representation is ambiguous.
[Section 3.2, Lemma 1 setup] The notation 'Let L = U_{L+1}' is confusing because the letter L is used both for the aperture constraint and for the set. Consider denoting the set by U_{L+1} directly.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the CRB and optimal-waveform results come from external references, and no fitted input is relabeled as a prediction.

full rationale

The derivation is self-contained with respect to circularity. The single-target CRB expression (6) is obtained by substituting the externally established optimal waveform result (5) from Forsythe and Bliss [12] and Li et al. [13] into the CRB formula (2) from Stoica and Nehorai [22]; no parameter is fitted to data and no quantity is defined in terms of the claimed result. The optimization in (9) then reduces to maximizing the spatial variance chi(Dr), and Lemma 1 supplies that maximum; although the proof is omitted, this is a missing proof rather than a circular reduction. The contiguous nonredundant sum co-array in Corollary 1 is an algebraic construction from the chosen Dt* and Dr*, not a fitted reproduction of a dataset. Self-citations appear only peripherally: [16] is an introductory pointer to earlier waveform design, and [23] (with coauthor Rajamaki) supplies an identifiability sufficient condition that is not needed to derive the CRB-optimal geometry. The formal concern that Theorem 1 states uniqueness of Dr* up to translation rather than exact uniqueness, together with the sign typo in Lemma 1 ('arg min' versus the required 'arg max'), is a rigor issue rather than a circularity issue: the optimality conclusion follows from variance maximization independent of any fitted or self-imported premise.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The paper's central claim rests on a standard CRB model, prior optimal-waveform results, and an unproved variance-maximization lemma. There are no fitted free parameters and no invented physical entities. The Diophantine equal-variance examples are illustrative, not fitted.

assumptions (6)

domain assumption Single-target narrowband monostatic MIMO model (Eq. 1) with white Gaussian noise and unknown reflection coefficient.
The CRB analysis starts from this model in Section 2.1.
domain assumption Optimal waveform takes the Tx beamforming form (5) when chi_r > chi_t, as established in [12, 13].
Theorem 1 and problem (9) restrict attention to this waveform family and this variance condition.
ad hoc to paper Lemma 1: for even N, clustered array K_L^N maximizes spatial variance among N-element subsets of {0, ..., L}.
The proof is omitted ('follows directly via negation') and no prior reference is given; this lemma underpins the uniqueness of the optimal receive array.
domain assumption Sensor positions are restricted to integer multiples of half wavelength, and the aperture constraint applies only to the receive array.
The integer grid is required for the Diophantine and sum co-array arguments; the transmit aperture is unconstrained in (9).
standard math CRB expression (2) and its reduction to (6) from [12, 22] are correct.
These formulas are standard CRB results used without rederivation in this paper.
domain assumption Nr is even.
Lemma 1 and Corollary 1 require even Nr; the odd case is not treated.

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Pith. "Pith review of Jointly optimal array geometries and waveforms in active sensing: New insights into array design via the Cram\'er-Rao bound." pith.science (2026). https://pith.science/paper/LV43QE7F

@misc{pith2026250100472,
  author       = {Pith},
  title        = {Pith review of: Jointly optimal array geometries and waveforms in active sensing: New insights into array design via the Cram\'er-Rao bound},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/LV43QE7F}},
  note         = {Machine review of arXiv:2501.00472}
}

read the original abstract

This paper investigates jointly optimal array geometry and waveform designs for active sensing. Specifically, we focus on minimizing the Cram\'er-Rao lower bound (CRB) of the angle of a single target in white Gaussian noise. We first find that several array-waveform pairs can yield the same CRB by virtue of sequences with equal sums of squares, i.e., solutions to certain Diophantine equations. Furthermore, we show that under physical aperture and sensor number constraints, the CRB-minimizing receive array geometry is unique, whereas the transmit array can be chosen flexibly. We leverage this freedom to design a novel sparse array geometry that not only minimizes the single-target CRB given an optimal waveform, but also has a nonredundant and contiguous sum co-array, a desirable property when launching independent waveforms, with relevance also to the multi-target case.

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Reference graph

Works this paper leans on

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INTRODUCTION Multisensor active sensing systems have recently experienced an increased research interest due to emerging application s such as automotive radar, and integrated sensing and com- munications [1, 2]. An important goal of such systems is high spatial resolution, including unambiguous and accura te direction-of-arrival (DoA) estimation, whereas...

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JOINTL Y OPTIMAL ARRA Y -W A VEFORM PAIRS 3.1. Equal CRB via Rx arrays with equal sums of squares Substituting the optimal waveform in (5) into (2) can be shown to simplify the single-target CRB into [12] CRB(ω ) = σ 2 2|γ|2 1 NtNr χ −1 r . (6) Hence, the CRB (given an optimal waveform) is independent of the target angle ω and only depends on the Tx array...

work page
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We focus on the four array geometries depicted in Fig

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Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.

[1] [1]

INTRODUCTION Multisensor active sensing systems have recently experienced an increased research interest due to emerging application s such as automotive radar, and integrated sensing and com- munications [1, 2]. An important goal of such systems is high spatial resolution, including unambiguous and accura te direction-of-arrival (DoA) estimation, whereas...

work page

[2] [2]

spatial variances

BACKGROUND 2.1. Measurement model We consider a monostatic active sensing multiple-input multi- ple output (MIMO) system consisting of Nt transmit (Tx) sen- sors collocated with Nr receive (Rx) sensors. Thus, the angle of incidence on the Rx array equals the angle of departure of the Tx array. Assuming a single target located in the far ﬁeld of linear Tx ...

work page

[3] [3]

JOINTL Y OPTIMAL ARRA Y -W A VEFORM PAIRS 3.1. Equal CRB via Rx arrays with equal sums of squares Substituting the optimal waveform in (5) into (2) can be shown to simplify the single-target CRB into [12] CRB(ω ) = σ 2 2|γ|2 1 NtNr χ −1 r . (6) Hence, the CRB (given an optimal waveform) is independent of the target angle ω and only depends on the Tx array...

work page

[4] [4]

We focus on the four array geometries depicted in Fig

NUMERICAL EXAMPLES Next, we illustrate the results of Section 3 numerically. We focus on the four array geometries depicted in Fig. 1, where (a) shows the optimal array deﬁned by (10) and (11). We consider the maximum-likelihood estimator (MLE) of ω for a ﬁxed waveform matrix S. Given (1), the MLE can be shown to reduce to the following joint Tx-Rx beamfo...

work page

[5] [5]

Since S is a function of the true angle, an ini- tial estimate of ω would be needed in practice; see [13] for a discussion and examples

We as- sume the optimal waveform in (5) is used, with u = 1 √ T 1T and T = Nt. Since S is a function of the true angle, an ini- tial estimate of ω would be needed in practice; see [13] for a discussion and examples. The ground truth target angle and reﬂectivity are set to ω = 0 and γ = 1 , respectively, whereas entries of noise vector n are drawn from an ...

work page

[6] [6]

Sparsity-enforcing sensor selection for DOA estimation,

V . Roy, S. P . Chepuri, and G. Leus, “Sparsity-enforcing sensor selection for DOA estimation,” in Proc. of the IEEE Int. W orkshop Computat. Advances Multi-Sensor Adaptive Process. (CAMSAP), 2013, pp. 340–343

work page 2013

[7] [7]

CONCLUSION This paper investigated waveform-array geometry pairs min - imizing the single-target CRB in active sensing. Focusing o n a family of optimal waveforms [12, 13] corresponding to Tx beamforming in the target direction, we showed that the opti - mal linear Rx array places sensors at the edges of its apertur e to maximize its spatial variance—the ...

work page

[8] [8]

MIMO radar for advanced driver-assistance systems and autonomous driving: Advantages and challenges,

S. Sun, A. P . Petropulu, and H. V . Poor, “MIMO radar for advanced driver-assistance systems and autonomous driving: Advantages and challenges,” IEEE Signal Pro- cess. Mag., vol. 37, no. 4, pp. 98–117, 2020

work page 2020