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arxiv: 1907.06227 · v1 · pith:JAFQRRDOnew · submitted 2019-07-14 · 📡 eess.SP

Designing Unimodular Sequences with Optimized Auto/cross-correlation Properties via Consensus-ADMM/PDMM Approaches

Yongchao Wang , Jiangtao Wang This is my paper

Pith reviewed 2026-05-24 21:35 UTC · model grok-4.3

classification 📡 eess.SP
keywords unimodular sequencesauto-correlationcross-correlationADMMPDMMconsensus optimizationradar applicationswireless communication
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The pith

Reformulating unimodular sequence design as a consensus nonconvex problem enables ADMM and PDMM algorithms that reach stationary points with improved correlation properties or lower cost.

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

The paper seeks to create unimodular sequences with strong auto- and cross-correlation properties for radar and wireless uses. It casts the task as minimizing a quartic polynomial under constant modulus constraints. Introducing auxiliary phase variables converts this into an equivalent consensus nonconvex optimization problem. Efficient solutions come from two new algorithms based on ADMM and PDMM, which are shown to reach stationary points. Simulations indicate these approaches improve on prior methods in cost or correlation quality.

Core claim

The designing problem is reformulated as a consensus nonconvex optimization problem via auxiliary phase variables. Consensus-ADMM and consensus-PDMM algorithms are proposed to solve it efficiently, with proofs that they converge to stationary points of the original problem. The approaches also include analysis of local optimality and complexity, and simulations confirm better performance than state-of-the-art.

What carries the argument

Consensus nonconvex optimization problem formed by auxiliary phase variables, solved via alternating direction method of multipliers (ADMM) and parallel direction method of multipliers (PDMM).

If this is right

  • The consensus-ADMM algorithm converges to a stationary point of the original nonconvex problem.
  • The consensus-PDMM algorithm's output is a stationary point of the original nonconvex problem if convergent.
  • Local optimality of the nonconvex optimization model can be analyzed.
  • Computational complexity of the consensus-ADMM/PDMM approaches can be characterized.
  • The proposed approaches can outperform state-of-the-art methods in either computational cost or correlation properties.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The reformulation approach could extend to other constant-modulus design tasks in signal processing.
  • Adopting the resulting sequences might improve range resolution or interference rejection in deployed radar systems.
  • The parallel structure of PDMM suggests scalability for designing very long sequences on distributed hardware.
  • Testing the methods under additive noise or hardware constraints would reveal practical robustness.

Load-bearing premise

The original designing problem can be equivalently reformulated as a consensus nonconvex optimization problem by introducing auxiliary phase variables.

What would settle it

A simulation on standard sequence design benchmarks where the ADMM or PDMM outputs fail to match or exceed the correlation properties or computational efficiency of existing methods would challenge the performance claims.

Figures

Figures reproduced from arXiv: 1907.06227 by Jiangtao Wang, Yongchao Wang.

Figure 1
Figure 1. Figure 1: Comparisons of convergence performance with [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Comparisons of convergence performance with [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Correlation levels with N = 256, M = 3, T = [0, 39]. -300 -200 -100 0 100 200 300 Time lag -300 -250 -200 -150 -100 -50 0 Correlation level (dB) WeCAN MM-WeCorr consensus-ADMM consensus-ADMM-SBCD-50% consensus-ADMM-AGD consensus-ADMM-AGD,SBCD-50% -300 -200 -100 0 100 200 300 Time lag -300 -250 -200 -150 -100 -50 0 Correlation level (dB) WeCAN MM-WeCorr consensus-PDMM consensus-PDMM-SBCD-50% consensus-PDMM-… view at source ↗
Figure 4
Figure 4. Figure 4: Correlation levels with N = 256, M = 3, T = [90, 128]. -2500 -2000 -1500 -1000 -500 0 500 1000 1500 2000 2500 Time lag -55 -50 -45 -40 -35 -30 -25 -20 -15 -10 Correlation level (dB) MM-WeCorr consensus-ADMM consensus-ADMM-SBCD-50% consensus-ADMM-AGD consensus-ADMM-AGD,SBCD-50% -50 0 50 -46 -44 -42 -40 -2500 -2000 -1500 -1000 -500 0 500 1000 1500 2000 2500 Time lag -55 -50 -45 -40 -35 -30 -25 -20 -15 -10 Co… view at source ↗
Figure 5
Figure 5. Figure 5: Correlation levels with N = 2048, M = 256, T = [0, 39] [PITH_FULL_IMAGE:figures/full_fig_p007_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Correlation levels with N = 2048, M = 256, T = [90, 128]. TABLE III THE MINIMUM AND AVERAGE VALUES OF THE CORRELATION LEVEL IN dB FOR INTERVAL [0, 39] ACHIEVED BY DIFFERENT ALGORITHMS WeCAN MM-WeCorr consensus-ADMM consensus-PDMM N M average minimum average minimum average minimum average minimum 256 3 -41.4 -43.0 -247.8 -251.7 -279.4 -285.8 -291.7 -295.3 4 -34.3 -35.1 -44.2 -44.7 -44.1 -44.5 -44.3 -44.7 5… view at source ↗
read the original abstract

Unimodular sequences with good auto/cross-correlation properties are favorable in wireless communication and radar applications. In this paper, we focus on designing these kinds of sequences. The main content is as follows: first, we formulate the designing problem as a quartic polynomial minimization problem with constant modulus constraints; second, by introducing auxiliary phase variables, the polynomial minimization problem is equivalent to a consensus nonconvex optimization problem; third, to achieve its good approximate solution efficiently, we propose two efficient algorithms based on alternating direction method of multipliers (ADMM) and parallel direction method of multipliers (PDMM); fourth, we prove that the consensus-ADMM algorithm can converge to some stationary point of the original nonconvex problem and consensus-PDMM's output is some stationary point of the original nonconvex problem if it is convergent. Moreover, we also analyze the nonconvex optimization model's local optimality and computational complexity of the proposed consensus-ADMM/PDMM approaches. Simulation results demonstrate that the proposed ADMM/PDMM approaches outperform state-of-the-art ones in either computational cost or correlation properties of the designed unimodular sequences.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 0 minor

Summary. The paper formulates unimodular sequence design as a quartic polynomial minimization subject to constant-modulus constraints, reformulates it as an equivalent consensus nonconvex problem by introducing auxiliary phase variables, develops consensus-ADMM and consensus-PDMM algorithms to find approximate solutions, proves that consensus-ADMM converges to a stationary point of the original problem (and consensus-PDMM does so if convergent), analyzes local optimality and complexity, and reports simulations showing outperformance versus prior art in computational cost or correlation metrics.

Significance. If the claimed equivalence holds without relaxation of the feasible set and the stationary-point guarantees transfer to the original quartic objective, the work would supply practical, theoretically grounded algorithms for a core problem in radar and communications waveform design, potentially improving upon existing methods in either speed or achieved correlation sidelobes.

major comments (2)
  1. [Abstract] Abstract (second and third steps): The assertion that introducing auxiliary phase variables renders the original quartic minimization 'equivalent' to a consensus nonconvex program is stated without an explicit bijection, constraint-set identity proof, or demonstration that every stationary point of the lifted problem maps back to a feasible point of the constant-modulus quartic. Because the convergence claims and simulation comparisons rest on the algorithms solving the original problem, this mapping must be shown to be one-to-one on the feasible set.
  2. [Convergence analysis] Convergence analysis section: The proof that consensus-ADMM reaches a stationary point of the 'original nonconvex problem' must explicitly verify that the auxiliary-phase lifting does not enlarge the feasible set or alter the objective value at recovered points; otherwise the reported stationary-point guarantee does not apply to the quartic formulation used for benchmarking.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and for highlighting the need for explicit verification of the equivalence between the original quartic formulation and the consensus lifting. We agree that these details should be stated more rigorously and will revise the manuscript accordingly.

read point-by-point responses

  1. Referee: [Abstract] Abstract (second and third steps): The assertion that introducing auxiliary phase variables renders the original quartic minimization 'equivalent' to a consensus nonconvex program is stated without an explicit bijection, constraint-set identity proof, or demonstration that every stationary point of the lifted problem maps back to a feasible point of the constant-modulus quartic. Because the convergence claims and simulation comparisons rest on the algorithms solving the original problem, this mapping must be shown to be one-to-one on the feasible set.

    Authors: We acknowledge that the abstract and introductory description of the reformulation would benefit from an explicit statement of the bijection. In the revised manuscript we will insert a dedicated proposition (new Proposition 1 in Section II) that (i) constructs the auxiliary phase vector θ from any feasible unimodular x such that the consensus constraints hold with equality, (ii) shows the converse mapping recovers a feasible constant-modulus x from any feasible (x,θ) pair, and (iii) proves that the lifted objective equals the original quartic polynomial at every such pair. The same proposition will also establish that every stationary point of the consensus problem projects to a stationary point of the original quartic problem, thereby justifying the convergence claims. revision: yes

  2. Referee: [Convergence analysis] Convergence analysis section: The proof that consensus-ADMM reaches a stationary point of the 'original nonconvex problem' must explicitly verify that the auxiliary-phase lifting does not enlarge the feasible set or alter the objective value at recovered points; otherwise the reported stationary-point guarantee does not apply to the quartic formulation used for benchmarking.

    Authors: We agree that the convergence theorem should contain an explicit verification step. In the revised Section IV we will augment the proof of Theorem 1 with two additional lemmas: Lemma 1 will show that the consensus constraints together with the auxiliary-phase definition enforce that the feasible set of the lifted problem is in one-to-one correspondence with the constant-modulus set of the quartic problem (hence no enlargement occurs), and Lemma 2 will demonstrate that the objective value of any recovered point equals the original quartic value. These lemmas will be invoked directly in the stationarity argument so that the guarantee applies unambiguously to the quartic formulation. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper's chain proceeds from a standard quartic constant-modulus formulation, through an auxiliary-variable reformulation asserted as equivalent, to consensus-ADMM/PDMM algorithms whose convergence to stationary points is proved directly, followed by simulation comparisons against external benchmarks. No step reduces by construction to its own inputs: the equivalence is not shown to be tautological, the algorithms are not fitted parameters renamed as predictions, and no load-bearing uniqueness or ansatz is imported via self-citation. The reported outperformance rests on numerical experiments rather than internal redefinition, satisfying the default expectation of non-circularity.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work relies on standard nonconvex optimization theory for ADMM convergence; no free parameters, new entities, or ad-hoc axioms are introduced in the abstract.

axioms (1)
  • standard math ADMM and PDMM methods converge to stationary points of nonconvex problems under suitable conditions.
    Invoked to support the convergence claims in the abstract.

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Reference graph

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