IPUC: Iterative Projection onto the Unit Circle

• IPUC is an iterative method that normalizes vector entries to unit modulus, ensuring reliable state synchronization and CAZAC sequence formation.
• The algorithm alternates projections in time and frequency domains to enforce constant amplitude and zero autocorrelation constraints.
• Empirical studies show that IPUC yields almost-sure convergence in consensus protocols and near-optimal performance in radar signal design.

The Iterative Projection onto the Unit Circle (IPUC) is a mathematical and algorithmic procedure for projecting vector elements onto the complex or real unit circle, employed in discrete-time consensus protocols on the unit sphere and in the construction of constant-amplitude zero-autocorrelation (CAZAC) sequences for radar and communications applications. The central operation is the normalization of vector elements to have unit norm (magnitude one), performed either in state synchronization protocols or during alternating-projection algorithms to enforce structure in both time and frequency domains.

1. Mathematical Foundation and Core Operations

IPUC enforces that each entry $z$ of a vector, real or complex, satisfies $|z|=1$. The basic projection map $P_U$ for any $v\in\mathbb{C}^n$ (or $\mathbb{R}^2$) is component-wise normalization: $P_U(v)_k = v_k/|v_k|$. This projection is exploited in two main algorithmic settings:

• Consensus on $\mathbb{S}^1$: Given a network of $n$ agents with a state $x_i\in \mathbb{S}^1 \subset \mathbb{R}^2$, states are updated by forming a (conical) weighted sum of neighbors, followed by projection onto $\mathbb{S}^1$:

$|z|=1$0

where $|z|=1$1 encodes the adjacency and weights of a strongly connected directed graph (Thunberg et al., 16 Jan 2026).

• CAZAC sequence generation: For $|z|=1$2, IPUC alternates projections between time-domain and frequency-domain representations to meet simultaneous constant amplitude (CA) and zero circular autocorrelation (ZAC) constraints. At each iteration:
  1. Project $|z|=1$3 onto unit-modulus in time domain.
  2. Fourier transform to frequency domain, project to unit-modulus.
  3. Inverse transform back (Amis et al., 5 Sep 2025). This drives $|z|=1$4 toward CAZAC structure.

2. IPUC-Based Consensus on the Unit Circle

The discrete-time consensus protocol using IPUC considers a network graph $|z|=1$5 with each edge $|z|=1$6 weighted by $|z|=1$7. The update rule is given above. The weight matrix $|z|=1$8 is required to be strictly diagonally dominant with a $|z|=1$9 margin on $P_U$0 (i.e., $P_U$1 for all $P_U$2) and to respect the sparsity of the underlying graph.

Equilibrium Classification

All equilibria of the IPUC consensus protocol on $P_U$3 must take the form $P_U$4 for some $P_U$5 and $P_U$6. These split into:

• Consensus: $P_U$7 for all $P_U$8 ($P_U$9).
• Antipodal: Mixed signs, still all aligned or antialigned with $v\in\mathbb{C}^n$0 (i.e., a two-cluster structure).
• Other geometries are excluded: $v\in\mathbb{C}^n$1 (Thunberg et al., 16 Jan 2026).

Local Stability

• Consensus equilibria are maximizers of $v\in\mathbb{C}^n$2 and are non-unstable: no eigenvalues of the Jacobian exceed one in modulus.
• Antipodal equilibria are generically locally unstable: the Jacobian $v\in\mathbb{C}^n$3 has an eigenvalue strictly larger than one at such points.

Global Behavior

For almost every initial configuration (excluding a measure-zero set), convergence is to consensus. The set of initial conditions converging to unstable equilibria has Lebesgue measure zero, as established using the stable-manifold theorem applied to the smooth, nondegenerate map $v\in\mathbb{C}^n$4 induced by IPUC (Thunberg et al., 16 Jan 2026).

3. CAZAC Sequence Generation via IPUC

In CAZAC construction, the sequence $v\in\mathbb{C}^n$5 must satisfy both $v\in\mathbb{C}^n$6 (constant amplitude) and $v\in\mathbb{C}^n$7 for all $v\in\mathbb{C}^n$8 (zero circular autocorrelation). Since a sequence $v\in\mathbb{C}^n$9 is CAZAC iff its discrete Fourier transform $\mathbb{R}^2$0 is also CAZAC, the problem becomes one of finding a vector with unit modulus in both time and frequency domains.

Alternating Projections Algorithm

IPUC is implemented as a sequence of alternating projections:

1. Start with a random unit-modulus vector $\mathbb{R}^2$1 in the frequency domain.
2. At each iteration:
   • Project $\mathbb{R}^2$2 to enforce $\mathbb{R}^2$3.
   • Transform to frequency domain ($\mathbb{R}^2$4), project $\mathbb{R}^2$5 to unit modulus.
   • Inverse transform to obtain new $\mathbb{R}^2$6.
   • Stop if the discrepancy $\mathbb{R}^2$7 (as defined below) drops below tolerance (Amis et al., 5 Sep 2025).

Discrepancy Metrics

• $\mathbb{R}^2$8
• $\mathbb{R}^2$9
• $P_U(v)_k = v_k/|v_k|$0

Empirically, convergence is typically achieved to $P_U(v)_k = v_k/|v_k|$1 within a few hundred to thousands of iterations for moderate $P_U(v)_k = v_k/|v_k|$2.

Practical Algorithm Overview

Step	Operation	Domain
Time-domain projection	$P_U(v)_k = v_k/|v_k|$3	$P_U(v)_k = v_k/|v_k|$4
FFT	$P_U(v)_k = v_k/|v_k|$5	$P_U(v)_k = v_k/|v_k|$6
Frequency-domain projection	$P_U(v)_k = v_k/|v_k|$7	$P_U(v)_k = v_k/|v_k|$8
IFFT	$P_U(v)_k = v_k/|v_k|$9	$\mathbb{S}^1$0

(Amis et al., 5 Sep 2025)

4. Equilibrium and Classification in IPUC-Generated CAZAC Sequences

IPUC-generated CAZAC sequences of length $\mathbb{S}^1$1 were empirically found to yield exactly four equivalence classes under standard CAZAC-invariant operations (rotation, shift, decimation, modulation, conjugation, DFT). Each family is described by specific structural or parametric form:

• Palindromic (decimation $\mathbb{S}^1$2 invariant)
• Decimation $\mathbb{S}^1$3 and $\mathbb{S}^1$4 invariants
• Special “generic” forms with intricately parameterized phase relations

In explicit terms, representatives of each class (e.g., for the “Popovic-type” class $\mathbb{S}^1$5, $\mathbb{S}^1$6) are given, with closed-form algebraic or explicit trigonometric parameterizations for all but one class, where only a numerical solution is provided for representative phases (Amis et al., 5 Sep 2025).

5. Simulated Annealing Integration for Autocorrelation Optimization

In radar applications, the goal is often not just a CAZAC sequence but one with exceptionally low non-circular autocorrelation sidelobes. IPUC is integrated as an inner loop within a Simulated Annealing (SA) process:

• Outer SA loop: Perturbs phase seeds, seeks to minimize non-circular autocorrelation cost $\mathbb{S}^1$7 over the set of near-CAZACs deliverable by IPUC.
• Inner IPUC loop: Enforces projection to near-CAZAC at each SA proposal.

Empirical results show this hybrid yields sequences with $\mathbb{S}^1$8-scaled main-to-side lobe ratios close to theoretical maxima, e.g., with $\mathbb{S}^1$9, $n$0 dB compared to the bound $n$1 dB (Amis et al., 5 Sep 2025).

6. Measure-Zero and Generic Convergence Results

Fundamental results establish that, for strictly diagonally-dominant weights (with the specified margin), the set of matrices $n$2 admitting non-consensus equilibria for IPUC on $n$3 is of Lebesgue measure zero in parameter space. This ensures that both in consensus and CAZAC contexts, almost all initializations and almost all admissible weightings converge to the “generic” or “trivial” fixed points: all-equal consensus in networks or pure CAZACs in signal construction (Thunberg et al., 16 Jan 2026, Amis et al., 5 Sep 2025).

7. Summary of Theoretical Properties and Practical Performance

IPUC, as formalized in the referenced works, delivers:

• Rigorous equilibrium characterization and almost-sure global convergence for consensus on $n$4 under strict diagonal dominance with network-theoretic constraints (Thunberg et al., 16 Jan 2026).
• Broad applicability in generating CAZAC and near-CAZAC sequences of arbitrary length, with efficient alternating projections leveraging unit circle projections in both domains, and classification of output sequences into known equivalence classes (Amis et al., 5 Sep 2025).
• Compatibility with stochastic global optimization for advanced autocorrelation profile shaping, extending its utility in modern radar sequence design.

No rigorous proof of global convergence is available for CAZAC construction outside network consensus, but empirical results indicate rapid and reliable reduction of discrepancy metrics to practical levels. The projection operation $n$5 remains fundamental to all variants, ensuring that the iterative process robustly maintains unit-modulus constraints throughout the dynamical evolution in state- or signal-space.