CAZAC Sequences: Design & Applications

• CAZAC sequences are complex-valued sequences defined by constant amplitude and zero periodic autocorrelation, ensuring robust synchronization and interference suppression.
• Classical families like Zadoff-Chu and Björck, along with permutation polynomial interleaving, expand the design space for optimal waveform performance in communications and radar.
• Numerical and algorithmic synthesis methods enable novel CAZAC sequences that address PAPR, Doppler resilience, and orthogonality challenges for emerging applications.

A Constant Amplitude Zero Auto-Correlation (CAZAC) sequence is a complex-valued sequence of length $N$ that satisfies the unimodular condition $|x[k]| = 1$ for all $k$ and perfect periodic autocorrelation so that $$\theta(d) = \sum_{k=0}^{N-1} x[k] x^*[k+d \mod N] = N \delta[d]$$ for $d = 0,\ldots,N-1$, and $\theta(d) = 0$ otherwise. CAZAC sequences exhibit zero autocorrelation sidelobes and are fundamental to waveform design for wireless communications, radar, synchronization, channel sounding, and coded acquisition. The archetypal constructions rely on quadratic phase laws (e.g., Zadoff-Chu), but algebraic generalizations, nonlinear numerical synthesis, and interleaved polynomial constructions have vastly expanded the available sequence classes.

1. Mathematical Definition and Properties of CAZAC Sequences

A length-$N$ CAZAC sequence $x = (x[0], \ldots, x[N-1])$ is defined by:

• Constant amplitude: $|x[k]| = 1$ for all $k$.
• Zero periodic autocorrelation: $$\theta(d) = \sum_{k=0}^{N-1} x[k] x^*[k + d \pmod N] = 0$$ for $d \neq 0$, and $\theta(0) = N$.

These conditions ensure a constant envelope (unit modulus) and aperiodic perfect autocorrelation—key for high-fidelity transmission and ideal synchronization (Berggren et al., 2023). The DFT of a CAZAC is also unimodular, manifesting mutual unbiasedness and facilitating tight frame constructions (Magsino, 2016).

2. Classical Families: Zadoff-Chu and Björck Constructions

The canonical Zadoff-Chu (ZC) sequence of length $N$ and root $u$ ($\gcd(u,N)=1$) is given by:

$$x[k] = e^{-j\frac{2\pi u}{N}\frac{(k + q + N \bmod 2)k}{2}}, \quad k=0,\dots,N-1,$$

with $q$ as integer phase offset. ZC sequences achieve perfect cyclic autocorrelation and are widely deployed for reference and random access in 4G/5G systems (Berggren et al., 2023).

Björck sequences, constructed for prime $p$, utilize the Legendre symbol $\chi(k)$ to define $u_p(k) = \exp(i \theta \chi(k))$ ($\theta$ selected per residue class of $p$), and satisfy $|u_p(k)| = 1$, yielding autocorrelation zeros at every nonzero shift (Benedetto et al., 2011). Björck's construction leverages deep exponential sum bounds (Weil) to ensure minimal ambiguity function sidelobes, which scale as $O(p^{-1/2})$—optimal up to constants.

3. Algebraic and Permutation Polynomial-Based CAZAC Constructions

A substantial expansion of CAZAC sequence families has been achieved via permutation polynomial (PP) interleaving. Let a degree-$v$ PP $\pi[k] = f_v k^v + \cdots + f_0\ (\bmod N)$ permute the index set $\{0, ..., N-1\}$. Interleaving a ZC by $\pi$ gives $y[k] = x[\pi(k)]$, where $x[n]$ is ZC. For quadratic permutation polynomials (QPP), $v=2$, irreducible $\pi$ preserves CAZAC properties; the autocorrelation vanishes everywhere off the origin via cubic-phase exponential sum arguments (Berggren et al., 2023).

The extension to high-degree PPs allows even greater sequence diversity: for $N=p^n N_1$, the map $\pi(x) = x^p + a x + b$ (with specified constraints on $a$ and $b$ per factorization) permutes $\mathbb{Z}_N$ and yields fundamentally inequivalent CAZACs when used for ZC index interleaving. Both $\pi$ and its compositional inverse produce sequences outside the QPP/ZC equivalence classes, as proven by explicit contradiction in Vandermonde systems modulo $N$ (Yuan et al., 17 Jan 2026).

Table: CAZAC Construction Types, Sequence Formula, Orthogonality

Construction Type	Sequence Definition	Orthogonality/Expansion
Zadoff-Chu (ZC)	$x[k]=e^{-j2\pi u(k+q+(N\,\bmod\,2))k/2N}$	$\phi(N)$, root-variant
Björck	$u_p(k)=e^{i\theta\chi(k)}$	$p$ (prime size), shifts
PP-Interleaved ZC (QPP or higher)	$y[k]=x[\pi(k)]$, $\pi$ permutes indices	Enlarged, inequivalent

While the classical ZC root-index method offers up to $\phi(N)$ distinct sequences, permutation polynomial interleaving multiplicitly expands the available CAZAC set. The orthogonal set size achievable by QPP interleaving generally satisfies $I<N$, with full orthonormal bases rarely obtained (Berggren et al., 2023, Yuan et al., 17 Jan 2026).

4. Numerical and Algorithmic CAZAC Synthesis

Numerical optimization can explore CAZACs outside algebraically prescribed families. Nonlinear sum-of-squares minimization encodes constant amplitude ($a_j^2+b_j^2=1$ for $j$) and autocorrelation zero constraints as polynomial equations. Running trust-region reflective solvers (Python, SciPy) with random restarts has led to full enumeration (e.g., 532 inequivalent length-7 CAZAC sequences, $3040$ for length-10), covering ZC, Björck, and hybrid types as well as entirely new solutions (Magsino et al., 2022).

Iterative projection onto the unit circle (IPUC) further enables length-agnostic CAZAC construction: for arbitrary initial phase vectors, alternated frequency/time domain projections are applied until the discrepancy (modulus deviation + autocorrelation error) is below a chosen threshold, often converging in $10^3$-$10^4$ iterations. Classification for $n=8$ yields four distinct solution classes: one Popović-like and three new families (Amis et al., 5 Sep 2025).

5. CAZACs in Spectrally Sparse and Block-Repetitive Frameworks

Modulatable CAZAC (MCAZAC) sequences are constructed in frequency via the componentwise product of a long CAZAC and a periodically extended orthogonal "short" sequence. By embedding in block-repetitive DFT support, these yield time-domain CAZACs with 0 dB PAPR and designer-controlled zero correlation zones (ZCZs)—key for interference-free synchronization in systems with spectral constraints (Popovic et al., 2020).

Zak-domain frameworks produce optimal multiple ZCZ sets, enforcing CAZAC and perfect cross-correlation properties through sparse index and phase matrices, with design bounds meeting Tang-Fan-Matsufuji and Sarwate equalities (Peng et al., 9 Feb 2025). Zak-OTFS systematics inherit constant-amplitude, mutual-unbiasedness, and delay-Doppler isolation from these design principles (Mehrotra et al., 30 Mar 2025).

6. Applications: Wireless, Sensing, Synchronization, Coding

CAZAC sequences underpin 4G/5G random access preambles, synchronization signals, and massive-MIMO sounding due to their constant envelope (minimizing PA distortion), autocorrelation zeros (ideal timing), and orthogonality (multiplexing) (Berggren et al., 2023). Björck sequences demonstrate optimal ambiguity behavior under Doppler-rich conditions, supporting low-Earth orbit PNT, radar, and non-terrestrial localization (Dureppagari et al., 31 May 2025).

Pulse trains constructed from CAZACs with tailored Doppler resilience (root-index, phase parameters) minimize false alarm rates under mobility, as required in mmWave/THz sensing (Zhang et al., 2023). Repetitive CAZAC sequences, common in PRACH/PUCCH for NR-U, are susceptible to high cubic metric (CM). Phase-rotation, cyclic-shift, and blockwise modulation schemes restore low CM without compromising autocorrelation or detection performance (Zhao et al., 2019).

7. Limitations, Orthogonality Bounds, and Open Directions

• The CAZAC property—with perfect periodic autocorrelation and constant amplitude—is not preserved for all permutation polynomials beyond degree 2; QPPs, and PPs with QPP inverses, are provably safe classes.
• Orthogonal CAZAC set size via QPP interleaving remains below $N$ for tested $N \le 128$; full orthonormality is rare except in classical cases.
• Some interleaved CAZACs coincide due to central symmetry in ZC, especially when $N$ decomposes with repeated prime factors; uniqueness depends on factorization (Berggren et al., 2023).
• Numerical and geometric approaches (IPUC, Zak domain) suggest a nearly continuous design space for long CAZACs—contrasting to root/character algebraic constraints—but equivalence class classification and complexity limitations remain subjects of research (Amis et al., 5 Sep 2025).
• Doppler-induced ambiguity and cyclic-shift misidentification in high mobility must be mitigated by sequence subset selection, coarse Doppler compensation, or assignment protocols (Dureppagari et al., 31 May 2025, Zhang et al., 2023).

CAZAC sequences remain a foundational tool for integrated communications and sensing, with ongoing research extending their algebraic, algorithmic, and application scope.