Spreading Factor (SF): Theory and Applications

• Spreading Factor (SF) is a key parameter defined as the ratio of chip rate to data symbol rate, impacting processing gain and throughput in communications.
• It governs system robustness and range by controlling symbol duration, SNR thresholds, and interference management in technologies like LoRa and CDMA.
• Dynamic selection methods using machine learning and rule-based approaches optimize SF assignment to enhance energy efficiency and link reliability under variable channel conditions.

The spreading factor (SF) is a fundamental parameter in spread spectrum and chirp spread spectrum (CSS) communication systems, controlling the degree of symbol spreading, processing gain, bitrate, and noise/interference resilience. SF appears centrally in physical-layer design (notably LoRa and CDMA), network-level adaptation, localization, and advanced coding techniques. It also arises in seismic imaging as the velocity-spreading factor in dynamic ray-tracing theory, quantifying wavefront curvature and anisotropy. Across these domains, the SF mediates the trade-offs between range, throughput, noise tolerance, coverage, and user population.

1. Mathematical Definition and Physical-Layer Role

In spread spectrum systems, the spreading factor is the ratio of the chip rate to the data symbol rate, equivalently the number of chips (or chirps) per symbol. In LoRa's CSS modulation, SF is an integer in the range 7–12; each symbol is encoded as $2^{\mathrm{SF}}$ chips. This directly determines the duration and spectral occupancy of symbols.

Formally, for bandwidth $\mathrm{BW}$ (Hz):

• Symbol rate: $R_s = \frac{\mathrm{BW}}{2^{\mathrm{SF}}}$ [Symbols/s]
• Symbol duration: $T_s = \frac{2^{\mathrm{SF}}}{\mathrm{BW}}$
• Bit rate: $$R_{\mathrm{b}} \approx \mathrm{SF} \cdot \mathrm{BW} / 2^{\mathrm{SF}} \cdot (\text{k}_\text{payload}/\text{k}_\text{total})$$ (with coding rate)
• Processing gain: $G_p = 2^{\mathrm{SF}}$ (linear), $G_p[\mathrm{dB}] \approx 3 \cdot \mathrm{SF}$

Increasing SF exponentially increases processing gain and symbol duration, directly lowering the minimum required SNR but also reducing throughput and increasing time-on-air ($\text{ToA}$), which scales as $l_{\mathrm{SF}}\propto2^{\mathrm{SF}}$ for fixed payload (Chen et al., 23 Nov 2025, Wijesuriya, 26 Jul 2025, Waret et al., 2018).

2. Network-Level Implications: Robustness, Interference, and SF Orthogonality

SF allocation manages the trade-off between communication robustness and network throughput:

• Robustness: Higher SFs enable reception at lower SNR due to greater processing gain (Chen et al., 23 Nov 2025). Typical minimum SNR thresholds fall from –7.5 dB (SF7) to –20 dB (SF12) (Wijesuriya, 26 Jul 2025, Waret et al., 2018).
• Range: Increases proportionally with SF, expanding coverage (Waret et al., 2018, Amichi et al., 2019).
• Throughput: Diminishes rapidly with increasing SF due to longer symbol durations and $\text{ToA}$ (Prakash et al., 12 Mar 2025, Bouazizi et al., 2020).

Orthogonality is not perfect in practice. Co-SF interference limits capacity; imperfect SF orthogonality introduces inter-SF interference, necessitating two capture conditions: co-SF ($q_{\mathrm{coSF}}$) and inter-SF ($q_{i\mathrm{SF}_m}$) thresholds. Empirical analysis reveals inter-SF collisions can halve throughput, particularly at high densities (Waret et al., 2018, Bouazizi et al., 2020). This requires joint optimization of SF and transmission power, typically through mixed-integer nonlinear programming decomposed into SF-assignment (e.g., many-to-one matching) and power allocation steps (Amichi et al., 2019).

3. Spreading Factor Adaptation and Selection Algorithms

Dynamic and static SF-selection strategies are central to efficient IoT/WAN deployments:

• Machine Learning for SF Prediction: ML classifiers (k-NN, Decision Tree, RF, MLR) can predict optimal SF from {RSSI, SNR} features. Two-feature models attain ≈64–66% accuracy, only marginally below full five-feature sets, greatly reducing measurement and energy costs (Prakash et al., 12 Mar 2025). Misclassifications mainly occur between adjacent SFs due to overlapping link metrics.
• Rule-Based and Weighted-Scoring Approaches: In constrained hardware scenarios (e.g., single-channel, mobile LoRa gateways), a two-phase approach first excludes infeasible SFs (by distance, margin, duty-cycle, datarate), then scores survivors on normalized $\text{ToA}$, energy, rate, and robustness (with application-specific weights). Field tests demonstrate 92% SF-selection matching to optimal across 672 scenarios (Wijesuriya, 26 Jul 2025).

Metric	Impact of Higher SF	Reference
Symbol Duration, $\text{T}_s$	Doubles per SF increment	(Chen et al., 23 Nov 2025)
Processing Gain, $G_p$	Doubles with each SF increment	(Chen et al., 23 Nov 2025)
Min. Decodable SNR	Improves by ≈3 dB per SF	(Chen et al., 23 Nov 2025)
Data Rate, $R_{\mathrm{b}}$	Halves with each SF increment	(Waret et al., 2018)
Time-on-Air, ToA	Roughly doubles per SF	(Prakash et al., 12 Mar 2025)

4. Limits of SF and Advanced Coding Techniques

SF is capped at 12 in LoRa PHY to maintain channel coherence and mitigate hardware-induced phase/frequency drift. SF$>$12 leads to excessively long symbols (e.g., $T_s \gtrsim 32$ ms at 125 kHz), exceeding typical coherence times and leading to loss of processing gain due to time-varying impairments (Chen et al., 23 Nov 2025). SFusion techniques circumvent this by software-emulating larger SF (quasi-SF$(k+m)$): grouping and repeating $2^m$ standard SF$k$ symbols, compensating hardware offsets, aggregating energy, and applying opportunistic Hamming-code block decoding. SFusion achieves up to 15 dB gain in packet reception rate over SF12 in ultra-low SNR scenarios, demonstrating SF extension via combined PHY and coding-level mechanisms (Chen et al., 23 Nov 2025).

5. Dynamical Effects: Time-Variability and SF Robustness

Conventional wisdom posits that higher SF always increases robustness. However, under time-varying (e.g., Rayleigh-fading) channels with short coherence time relative to frame length, higher SFs become counterproductive. Simulations show that, for payloads above a critical length (e.g., $B \gtrsim 14$ bytes at $\tau_c\approx20$ ms), the frame-error rate of SF12 can exceed that of lower SFs by up to an order of magnitude. This effect emerges due to inter-symbol de-correlation, which erodes demodulation gain. Optimal SF allocation must consider the relationship $T_{\mathrm{frame}}\ll \tau_c$ and avoid automatic maximization of SF, instead selecting the smallest SF achieving the target FER under prevailing channel dynamics and payload size (Bapathu et al., 2020).

6. SF in Multi-User and Multi-Service Contexts: CDMA/VSF, QoS, and Fairness

In CDMA, SF equals the length $N$ of the spreading sequence. Variable SF (VSF) architectures assign strictly orthogonal codes (upper tree) to critical users and quasi-orthogonal, overloaded codes (lower tree) to best-effort users. SF/VSF enables flexible support for varying bitrates, multi-QoS, and diverse user types (e.g., reliability/latency-sensitive vs. bursty sensors). The system organizes codes in an OVSF-tree, balancing throughput, BER, and latency according to SF allocation and overload region (Karrenbauer et al., 2018).

Multi-device optimization in PHY and MAC layers—including SF-assignment, power control, and matching—improves minimum user throughput and fairness by adapting SFs to link conditions and network density, subject to imperfect orthogonality and coexistence constraints (Amichi et al., 2019, Waret et al., 2018, Bouazizi et al., 2020).

7. Velocity-Spreading Factor in Seismic Imaging

In paraxial ray theory, the velocity-spreading factor $\mathbf{Q}_F$ generalizes the notion of spreading factor to wavefront propagation in anisotropic media:

$$[\mathbf{Q}_F]^2 = \mathbf{Q}_1^{T}\,[\mathbf{C}_0/v_0^2]^{-1}\,\mathbf{Q}_1$$

where $\mathbf{Q}_1$ is the plane-wave paraxial propagator, $\mathbf{C}_0$ the wave-propagation metric tensor, and $v_0$ the local phase velocity. The Dix velocity is then formulated

$$\mathbf{V}_{\mathrm{Dix}}^2 = v_0^2\, [\mathbf{Q}_F]^{-2}$$

Deviations of $Q_F$ from unity quantify anisotropy and spreading. This parameter is essential for accurate time-to-depth conversion, model building, and correcting velocity estimations in time-migration imaging, especially in general anisotropic media (Coimbra et al., 2023).

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The spreading factor is thus a multi-domain construct, with rigorous mathematical foundations and critical engineering implications in physical-layer modulation, network optimization, multi-service wireless MAC, deep-learning-based localization, robust coding, and geophysical wave-propagation. Its practical impact lies in mediating fundamental trade-offs among rate, noise-robustness, capacity, and energy efficiency; its optimal selection and exploitation require an integrated perspective on PHY, MAC, environment dynamics, and, in seismic imaging, anisotropy and wavefront curvature.