Spreading Factor Orthogonality

• Spreading factor orthogonality is a design principle ensuring that distinct user sequences produce ideally zero mutual interference under controlled conditions.
• The topic covers various constructions such as Hadamard matrices, Weyl/DFT-based sequences, and finite-field transforms applied in CDMA, OFDM, and LPWAN systems.
• Practical implementations face trade-offs due to nonidealities, prompting adaptive coding and interference cancellation to maximize spectral efficiency and capacity.

Spreading factor orthogonality is a foundational property in the design of multiuser and multichannel spread-spectrum systems, ensuring that transmissions encoded with distinct spreading signatures produce ideally zero mutual interference under synchronous or suitably controlled asynchronous conditions. The concept is realized in diverse physical and algorithmic forms—ranging from binary and non-binary code division multiple access (CDMA), finite-field spectral transforms, to modern orthogonal frequency division multiplexing (OFDM) systems with integrated sensing. This article details the formal mathematical definitions, construction principles, physical realizations, practical trade-offs, empirical limitations, and information-theoretic considerations underpinning spreading factor orthogonality.

1. Mathematical Definition of Spreading Factor Orthogonality

In canonical synchronous systems, orthogonality of spreading sequences entails the vanishing of (a)periodic cross-correlation functions for all sequence pairs assigned to distinct users or resource groups. Let $\mathcal{S} = \{s_i\}$ denote the set of user spreading sequences, each of length $N$ (the spreading factor). Orthogonality conditions are:

• Synchronous inner product (Hadamard-type):

$$\langle s_i, s_j \rangle = \sum_{n=0}^{N-1} s_i[n]\, s_j[n]=0,\quad i\neq j\;.$$

• Aperiodic cross-correlation (for delay spread/synchrony mismatches):

$$R_{ij}(\tau) = \sum_{n=0}^{N-1} s_i[n]\,s_j[n+\tau]$$

with strict orthogonality holding if $R_{ij}(\tau)=0$ for all $\tau$ and $i\neq j$ (Geng et al., 2010, Karrenbauer et al., 2018).

• Unitary matrix construction: Assign the columns of a unitary matrix $F\in\mathbb{C}^{N\times N}$, with $F F^H = I_N$, as user spreading sequences; this enforces perfect orthogonality (Said et al., 4 May 2025).

This definition generalizes to complex, finite-field, and multilevel codes (Oliveira et al., 2015, Oliveira et al., 2015), as well as to chirp-based or non-binary systems.

2. Classical and Finite Field Constructions

Orthogonal sequence sets are realized through several algebraic and analytic constructions:

• Sylvester–Hadamard matrices: Binary orthogonal codes of length $N=2^n$, recursively constructed and forming the basis for OVSF (orthogonal variable spreading factor) code trees (Karrenbauer et al., 2018).
• Weyl and DFT-based sequences: Roots of unity in $N$0, parameterized by index and phase, produce $N$1-user sets with zero cross-correlation under chip-synchronous conditions (Tsuda et al., 2016).
• Finite-field transforms: Finite Field Fourier Transforms (FFFT) and finite-field Hartley transforms generate orthogonal sequence sets over $N$2 of length $N$3 dividing $N$4, with explicit coset-leader reduction for bandwidth-efficient Galois-field Division Multiplex (GDM) (Oliveira et al., 2015, Oliveira et al., 2015).
• Plateaued and semi-bent Boolean families: Vectorial semi-bent function constructions admit $N$5 perfectly orthogonal binary codes of length $N$6, enabling maximal per-cell CDMA capacity under strict reuse and adjacency constraints (Zhang et al., 2016).
• DPSS (Slepian) eigenvectors: Carefully tailored “prolate” sequences minimize out-of-band leakage and are used as spreading bases atop OFDM for multiuser integrated sensing and communications (ISAC) systems (Said et al., 4 May 2025).

The table below classifies key constructions by sequence type, field, and orthogonality properties:

Construction	Field	Maximum Orthogonal Set Size
Hadamard/Sylvester	$N$7	$N$8
Weyl/DFT exponentials	$N$9	$$\langle s_i, s_j \rangle = \sum_{n=0}^{N-1} s_i[n]\, s_j[n]=0,\quad i\neq j\;.$$0
FFFT/FFHT (Galois transform)	$$\langle s_i, s_j \rangle = \sum_{n=0}^{N-1} s_i[n]\, s_j[n]=0,\quad i\neq j\;.$$1	$$\langle s_i, s_j \rangle = \sum_{n=0}^{N-1} s_i[n]\, s_j[n]=0,\quad i\neq j\;.$$2 ($$\langle s_i, s_j \rangle = \sum_{n=0}^{N-1} s_i[n]\, s_j[n]=0,\quad i\neq j\;.$$3)
Semi-bent vectorial Boolean	$$\langle s_i, s_j \rangle = \sum_{n=0}^{N-1} s_i[n]\, s_j[n]=0,\quad i\neq j\;.$$4	$$\langle s_i, s_j \rangle = \sum_{n=0}^{N-1} s_i[n]\, s_j[n]=0,\quad i\neq j\;.$$5 (with spatial reuse)
DPSS/Prolate sequences	$$\langle s_i, s_j \rangle = \sum_{n=0}^{N-1} s_i[n]\, s_j[n]=0,\quad i\neq j\;.$$6	$$\langle s_i, s_j \rangle = \sum_{n=0}^{N-1} s_i[n]\, s_j[n]=0,\quad i\neq j\;.$$7 (practical, spectral)

3. Physical Realizations and Analytical Effects

The realization of spreading factor orthogonality has immediate consequences for the mitigation of multiuser interference across application domains:

• CDMA: Orthogonal sequences (Hadamard, Weyl, FFFT, semi-bent) enable simultaneous communication by up to $$\langle s_i, s_j \rangle = \sum_{n=0}^{N-1} s_i[n]\, s_j[n]=0,\quad i\neq j\;.$$8 users per cell with zero cross-talk in synchronous transmission (Tsuda et al., 2016, Oliveira et al., 2015, Zhang et al., 2016).
• OFDMA and ISAC: In multi-band scenarios, inter-band (IB) cross-correlation can be nonzero under aperiodic operations such as radar-range estimation, requiring a spreading layer (e.g., via a semi-unitary matrix) to null leakage and minimize integrated sidelobe level (ISL) (Said et al., 4 May 2025). Simulations confirm reductions of ISL by $$\langle s_i, s_j \rangle = \sum_{n=0}^{N-1} s_i[n]\, s_j[n]=0,\quad i\neq j\;.$$9 dB at the cost of proportional spectral efficiency losses as the number $$R_{ij}(\tau) = \sum_{n=0}^{N-1} s_i[n]\,s_j[n+\tau]$$0 of employed spreading vectors per band is reduced.
• LoRa/LPWAN: In CSS-based systems, “spreading factor orthogonality” is realized among chirps of different SF, in principle supporting coexistence of multiple users with high density. However, real-world imperfections (chirp misalignment, Doppler, filter truncation) yield only quasi-orthogonality, quantified by nonzero “capture thresholds” or cross-correlation coefficients; this translates to observable inter-SF interference (Waret et al., 2018, Mahmood et al., 2018, Amichi et al., 2019).
• Grant-Free Random Access: In compressive sensing-based GF-RA, code diversity is engineered by using multiple independently chosen spreading sequences per user (MSRA). This reduces averaged Babel mutual coherence, transforming the multiuser detection problem into a well-conditioned MMV estimation and supporting an $$R_{ij}(\tau) = \sum_{n=0}^{N-1} s_i[n]\,s_j[n+\tau]$$1 increase in supported active users at low misdetection rates (Abebe et al., 2021).

4. Limitations of Practical Orthogonality: Imperfect SF Isolation

Physical-layer impairments, symbol asynchrony, and implementation constraints inevitably degrade perfect orthogonality:

• LoRa networks: Empirically measured cross-correlation between different SFs result in nonzero inter-SF “capture thresholds”—i.e., the SIR required to successfully demodulate a target packet in the presence of an interfering SF. Typical measured thresholds range from $$R_{ij}(\tau) = \sum_{n=0}^{N-1} s_i[n]\,s_j[n+\tau]$$2 dB for SF7 up to $$R_{ij}(\tau) = \sum_{n=0}^{N-1} s_i[n]\,s_j[n+\tau]$$3 dB for SF12 (Amichi et al., 2019). Inclusion of inter-SF interference in network models shows throughput and coverage losses of $$R_{ij}(\tau) = \sum_{n=0}^{N-1} s_i[n]\,s_j[n+\tau]$$4--$$R_{ij}(\tau) = \sum_{n=0}^{N-1} s_i[n]\,s_j[n+\tau]$$5 over ideal models, and a critical reduction in maximum deployable device density per cell (Mahmood et al., 2018, Bouazizi et al., 2020).
• Multipath and low-SF UWB: For short spreading factors, residual cross-correlation $$R_{ij}(\tau) = \sum_{n=0}^{N-1} s_i[n]\,s_j[n+\tau]$$6 leads to increased inter-path, inter-chip, and inter-symbol interference in multipath-rich channels. Iterative interference cancellation and appropriate parameter selection restore effective orthogonality but require careful system-level optimization (Geng et al., 2010).
• Partly-overloaded CDMA: Structures such as column-permuted Hadamard matrices enable the coexistence of globally orthogonal code subsets and overloaded (quasi-orthogonal) subsets; the latter permit controlled interference to flexibly support best-effort users while ensuring zero collision for machine-type users (Karrenbauer et al., 2018).

5. Trade-offs, Spectral Efficiency, and Capacity Scaling

The spreading factor $$R_{ij}(\tau) = \sum_{n=0}^{N-1} s_i[n]\,s_j[n+\tau]$$7 fundamentally determines the system’s user capacity, processing gain, and spectral utilization, but optimal trade-offs must be made to balance these metrics:

• GDM and cyclotomic compression: Only $$R_{ij}(\tau) = \sum_{n=0}^{N-1} s_i[n]\,s_j[n+\tau]$$8 cyclotomic coset leaders need be transmitted, yielding a bandwidth compactness factor $$R_{ij}(\tau) = \sum_{n=0}^{N-1} s_i[n]\,s_j[n+\tau]$$9 and a proportional boost in spectral efficiency without sacrificing in-cell orthogonality (Oliveira et al., 2015, Oliveira et al., 2015).
• OFDM-ISAC: Reducing the fraction $R_{ij}(\tau)=0$0 of in-band Slepian vectors employed per group reduces inter-band interference and ISL but at the price of linearly reduced data rate (symbols per channel use) (Said et al., 4 May 2025).
• Adaptive assignment in LoRa: Proper spreading factor and power allocation, aware of imperfect orthogonality, substantially ameliorates fairness and aggregate throughput, maintaining performance for higher active device numbers per cell than legacy random or range-based allocation (Amichi et al., 2019).

6. Information-Theoretic and Analytical Measures of Sequence Spreading

Orthogonality ensures that the ensemble of spreading sequences is tight and evenly distributed in the underlying function space. Information-theoretic spreading measures such as the Rényi and Shannon lengths, as well as the Fisher information length, provide complementary quantifications of sequence “spread”:

• Fisher length captures local oscillatory (gradient) content; Rényi and Shannon lengths probe global concentration and delocalization. For density functions on the orthogonality domain (e.g., wavefunctions or spreading signatures), the inequalities $R_{ij}(\tau)=0$1 hold for $R_{ij}(\tau)=0$2, reflecting the hierarchy of uncertainty measures and implications for the robustness of orthogonal sequence sets against noise and interference (Dehesa et al., 2013).
• Cramér–Rao and Shannon bounds formalize the fundamental limits: $R_{ij}(\tau)=0$3 and $R_{ij}(\tau)=0$4 for continuous domains.

7. Applications and System Design Implications

Spreading factor orthogonality underpins the architecture of contemporary wireless systems:

• Dense CDMA systems: Key to scalable multi-cell, multiuser overlays, with vectorial semi-bent constructions (for $R_{ij}(\tau)=0$5) enabling maximal per-cell capacity with provably zero-interference assignments even in regular hexagonal cellular tessellations with minimum frequency reuse distance $R_{ij}(\tau)=0$6 (Zhang et al., 2016).
• OFDMA/ISAC: Essential for enabling joint communications and ranging with minimized sidelobe artifacts and robust multiuser interference suppression. The use of nearly-orthogonal spreading (e.g., DPSS bases) allows ISAC to approach the integrated sidelobe and mutual interference floor dictated by the physics of time-frequency concentration (Said et al., 4 May 2025).
• LPWAN/IoT and LoRa: Orthogonality, or its breakdown, critically impacts the practical capacity and reliability of massive IoT deployments, mandating precise models that incorporate empirical SF cross-interference in both physical and network layer design and planning (Waret et al., 2018, Mahmood et al., 2018, Amichi et al., 2019, Bouazizi et al., 2020).

In conclusion, while the theoretical underpinnings of spreading factor orthogonality are algebraically exact in idealized mathematical constructions, deployment-scale systems must continually negotiate a trade space defined by physical nonidealities, desired user capacity, interference tolerance, and spectral efficiency. Ongoing research develops adaptive, robust, and spectrally efficient spreading protocols and codes to approach the performance limits imposed by practical orthogonality impairments.