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# SFIxM: Flexible LoRa Modulations for Elastic Resource Allocation

Martin Heusse, G. Maury, Christelle Caillouet

► To cite this version:

Martin Heusse, G. Maury, Christelle Caillouet. SFIxM: Flexible LoRa Modulations for Elastic Resource Allocation. MSWiM 2025 - The 27th International Conference on Modeling, Analysis and Simulation of Wireless and Mobile Systems, Oct 2025, Barcelone, Spain. hal-05300178

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# SF/$xM$: Flexible LoRa Modulations for Elastic Resource Allocation

Martin Heusse

Univ. Grenoble Alpes,
CNRS, Grenoble INP, LIG
38000 Grenoble, France

Martin.Heusse@imag.fr

Ghislaine Maury

Univ. Grenoble Alpes,
CNRS, Grenoble INP, CROMA
38000 Grenoble, France

Ghislaine.Maury@grenoble-inp.fr

Christelle Caillouet

Université Côte d'Azur,
CNRS, Inria, I3S

06903 Sophia Antipolis, France

Christelle.Caillouet@univ-cotedazur.fr

**Abstract**—This paper presents SF/$xM$, a flexible evolution of the LoRa modulation scheme designed to facilitate the allocation of transmission parameters across nodes within a gateway's coverage area. SF/$xM$ increases the number of available quasi-orthogonal channels and, unlike standard LoRa, offers tunable robustness to noise and interference. Because SF/$xM$ introduces a new requirement for time-axis alignment between transmitter and receiver, we present a practical procedure to achieve this synchronisation, along with an analytical estimate of its failure probability, which we find negligible. Based on this foundation, we show that SF/$xM$ provides traffic capacity gains of 40 to 55%, comparable to the gains obtained with receiver diversity, a well-known approach to improving coverage and range, which also turns out to combine well with SF/$xM$.

**Index Terms**—LoRaWAN; Channel Capacity

## I. INTRODUCTION

LoRa (Long Range) is a proprietary modulation technique developed by Semtech that enables long-range communication with low power consumption, making it ideal for Internet of Things (IoT) applications. LoRa operates in unlicensed spectrum bands (such as 868 MHz in Europe and 915 MHz in North America) and uses a modulation method known as Chirp Spread Spectrum (CSS), which provides robustness against interference and allows for communication over several kilometers. Each CSS symbol is encoded as a unique chirp shifted in the frequency domain to convey information. Low data rates and long-range transmission capabilities make LoRa suitable for applications like smart agriculture, environmental monitoring, and industrial automation [1].

Building on the LoRa physical layer, LoRaWAN (Long Range Wide Area Network) is a communication protocol that defines the system architecture and network protocols between the devices, gateways, and servers. Standardized by the LoRa Alliance, LoRaWAN enables scalable, bi-directional communication and secure data transmission. In LoRaWAN networks, one of the most important transmission parameter is the spreading factor (SF), which controls the data rate and thus allows to adjust the signal robustness to the channel conditions between the sender and the gateway (GW). However, achieving greater resistance to attenuation and interference by increasing the spreading factor (SF) comes at the cost of longer transmission durations, which in turn reduces energy efficiency and exacerbates contention on the radio channel.

Current LoRa chips allow to use the spreading factors SF6 to SF12, with the data rate (roughly) doubling at each step SF($k$) to SF($k-1$), while the sensitivity increases by 2.5 dB to 3 dB [2], [3]. Transmissions using different spreading factors (SFs) are quasi-orthogonal. Croce et al. [4] experimentally showed that the immunity to inter-SF interference closely aligns with the commonly accepted SNR threshold for thermal noise for each SF [5], so that one modulation appears mostly as noise to the others, with a slight advantage when the SFs are far appart.

Thus, at the cell (or network) scale, SF allocation between the nodes needs to strike a balance between ensuring transmission reliability against thermal noise and maintaining an acceptable collision rate between transmissions using the same SF [6]. In general, the idea is to use a larger spreading factor for the more distant nodes, whereas power control only in the SF7 nodes reduces both intra and inter-SF interference throughout the entire cell [7]. Therefore, transmissions at different SFs are quasi-orthogonal and form virtual channels used by the nodes located in concentric annuli around the GW.

While the gradation between the SFs allows to allocate more resources to distant nodes, there are situations where certain SFs become ineffective. For example, when all nodes are located far from the GW, where the sensitivity of lower SFs is insufficient, it renders those SFs —and the corresponding virtual channels— useless. Moreover, even with an homogeneous spatial distribution of nodes, the load across SFs can become highly imbalanced, with some underutilized SFs while others are heavily congested. Hence, the rigid $\approx 2.5$ dB sensitivity steps between LoRa SFs does not necessarily coincide with the spatial distribution of traffic demand and robustness requirements around the gateway, leading to suboptimal utilization of the available code-based multiplexing resources.

In this paper we propose SF/$xM$, an evolution of the LoRa modulation that offers greater flexibility for transmission parameters allocation, showing important traffic capacity benefits in LoRaWAN networks. In contrast with other works in the field, with SF/$xM$, we want to decrease the modulation spectral efficiency: the objective is to drastically increase resource allocation flexibility in LoRaWAN networks, by decoupling the definition of SF-based virtual channels from the associated sensitivity levels. The idea is to not only use individual chirps
<figure><img src="image_3.png" /><figcaption>Fig. 1. Example SF IxM symbol with M = 2, composed of 2 identical frequency ramps (the SF IxM base symbol precedes a different symbol represented with a dashed line)</figcaption></figure>

for data encoding but to use groups of M = 2^m chirps at the same SF I to form one SF IxM symbol. The sensitivity and data rate of the SF IxM modulation are the same as that of SFk with k = I + m, so that it can be used at a similar distance to the GW, while still forming a quasi-orthogonal virtual channel.

The paper is organized as follows: the next sections present the SF IxM modulation symbols and the additional synchronisation procedure that it entails (Sections II and III). In return, SF IxM increases the number of code-based virtual channels and allows to elastically distribute them across the cell, with significant benefits in terms of capacity (Section IV). In Section V, we discuss how SF IxM compares and complements other proposals, before concluding the paper.

## II. THE SF IxM SYMBOLS

Each SF IxM symbol is a series of M chirps and thus spans M × 2^I samples taken at the same frequency as the frequency channel width. Each symbol needs to be de-modulated by multiplying with M consecutive (complex conjugate) SF I “down” chirps, assuming the transmission uses “up” chirps (see Figure 1).

After de-chirping, the receiver takes 2^I × M samples per SF IxM symbol, and the Fast Fourier Transform (FFT) yields 2^I × M complex frequency domain samples. Thus, we expect one peak to emerge among 2^I × M samples and each SF IxM symbol allows to convey I + log₂(M) bits, so that for e.g. I = 7 and M = 2, we have a modulation which is quasi-orthogonal to SF8 and equivalent with respect to bitrate and robustness to noise.

So SF IxM symbols are mere groups of M identical SF I chirps, simply demodulated as one symbol. In other words, the reception procedure of SF IxM for M = 2, M = 4 etc. is very similar to the reception of SF I + 1, SF I + 2 chirps etc.

In a given cell, each value of I can be used only once, to ensure that transmissions with different parameters remain quasi-orthogonal.

## III. SYMBOL ALIGNMENT CHALLENGES

Whereas the demodulation of couples (or quartets etc.) of SF I chirps is not different from regular LoRa reception, initial receiver synchronisation to the incoming signal must change: the problem is to identify where the groups of chirps start, on top of identifying the chirp boundaries. Here, several approaches are possible: it would be possible to use longer chirps for the preamble (e.g. SF8 chirps for the preamble and

switch to SF IxM with (I, M) = (7, 2) for the payload), but then obviously it becomes impossible to assume that transmissions using different parameters form mostly independent virtual channels. Another option is to simply use an SF I LoRa preamble for any value of M. This approach does not break orthogonality, but then the preamble is less robust than data symbols and it could become the Achilles' heel of the reception process.

We propose to use a synchronisation procedure that features equivalent robustness to noise as LoRa modulation with same data rate. For this, we propose that the preamble be composed of the same number of groups of M chirps as the number of chirps in the LoRa preamble (typically 8 base symbols and 2 sync words). Likewise, we use M times 2.25 conjugate chirps for frequency correction.

At this stage, there is an ambiguity on which chirp of the SF IxM symbol the receiver is aligned with: in Figure 2, at steps (A) or (B), receiver 1 and 2 are exactly in the same state, but if they used the same time offset for symbol alignment, and data reception started immediately, receiver 2 would suffer from inter-symbol interference at step (D), whereas receiver 1 is well aligned with the start of the SF IxM symbol at step (C). Thus, we need an additional reception process stage for finding the SF IxM symbol boundary after the frequency correction stage, inserted before the first data symbols in regular LoRa.

### A. SF IxM Symbol Alignment

In order to break the ambiguity, we propose to extend the preamble with an alignment stage in which, exceptionally, the chirps are not in groups of M identical SF I chirps. Consider Figure 3: for M = 2, the alignment stage would be composed of 3 (= 2M - 1) staggered chirps each shifted by BW/M = BW/2 relatively to the previous one (where BW is the channel bandwidth). Using the corresponding staggered chirps at the receiver for demodulation yields M identical spectral rays, at a position in the frequency domain defined by the initial time offset.

In effect, if the receiver is offset by k ∈ [0, M - 1] chirps from the SF IxM symbol boundary, the demodulated signal for chirp m ∈ [0, M - 1] of the alignment stage is the following:

$$ 
\begin{aligned}
A(t) &= e^{2\pi j t \left[ f_0 + \left(at + (k+m)\frac{BW}{M}\right) \mod BW \right]} \\
& \quad \times e^{-2\pi j t \left[ f_0 + \left(at+m\frac{BW}{M}\right) \mod BW \right]} \\
&= e^{2\pi j t \left[ k \frac{BW}{M} \mod BW \right]} \\
&= e^{2\pi j t \frac{kBW}{M}}
\end{aligned}
$$

where $f_0$ is the start of the occupied frequency channel and $a = \frac{BW^2}{2^I}$ for spreading factor SF I. Thus FFT yields a peak at frequency $\frac{kBW}{M}$, indicating the time offset.

Note that the receiver may either perform M chirp-long FFT computations for this stage (as on Figure 3), or a single SF IxM-long FFT, as the sought symbol is constant over M chirps.

There needs to be a sufficient number of chirps in the alignment stage of the preamble to accommodate a full SF IxM
<figure><img src="image_4.png" /><figcaption>Fig. 2. SF1xM synchronisation procedure. The dashed lines represent the demodulation SF1xM symbols (with $M = 2$) used by receiver 1 and 2 which are offset from each other, in time, by one SF1 chirp duration</figcaption></figure>

<figure><img src="image_5.png" /><figcaption>Fig. 3. Alignment stage for SF1xM modulation, $M = 2$</figcaption></figure>

symbol ($M$ chirps) complemented with $M - 1$ additional chirps for all non-null time offsets at the start of the alignment procedure. This additional physical layer header takes place just before the payload, as shown in Figure 5. Figure 4 shows the alignment stage for $M = 4$: the peak in the FFT at frequency $\frac{kBW}{M}$ informs the receiver that it needs to skip $M - 1 - k$ chirps before the start of the payload.

## B. Alignment Error Probability

Using the computation of symbol error probability derived by Ferré and Giremus [2] on a white Gaussian noise channel, it is relatively straightforward to extrapolate the alignment error probability. In effect, the decoding of the SF1xM symbols is similar to the decoding of a chirp of the same duration. Similarly, the $M$ staggered alignment chirps that the receiver is going to use for alignment computation form the input of an FFT on $M \times 2^I$ samples, the outcome of which is a single expected frequency peak. In presence of noise, an alignment error occurs if the highest peak is one of the other

<figure><img src="image_6.png" /><figcaption>Fig. 4. Alignment stage for SF1xM modulation, $M = 4$</figcaption></figure>

<table><caption>Fig. 5. Physical layer headers for LoRa and SF1xM modulations, sizes in number of chirps</caption><tr><td>LoRa:</td><td>10</td><td>2</td><td>2.25</td></tr><tr><td>Preamble</td><td>Sync</td><td>Freq.</td><td>Payload</td></tr><tr><td>SF1xM:</td><td>10M</td><td>2M</td><td>2.25M</td><td>2M-1</td></tr><tr><td>Preamble</td><td>Sync</td><td>Freq.</td><td>Align</td><td>Payload</td></tr></table>

$M - 1$ frequency domain samples which correspond to the other alignment values. Alignment errors are less likely to occur than data symbol errors since only $M$ peaks may be valid; in contrast, for data symbols, there is a symbol error when the highest peak is one of the $M \times 2^I - 1$ frequency domain
<figure><img src="image_7.png" /><figcaption>Fig. 6. Bit error rate and Alignment error rate for SF1xM modulations equivalent to SF7-SF12 and M = 32. The dots highlight the BER for the generally accepted sensitivity of LoRa</figcaption></figure>

samples that are not the transmitted one.

We recall that the LoRa or SF1xM data symbol error probability, given the noise term w which adds to the expected peak, is:

$$S_{\text{Err}} = 1 - \left[ F_{\chi^2} \left( \frac{\sqrt{M 2^I} + w^2}{\sigma^2} \right) \right]^{M 2^I - 1},$$

where σ² is the noise variance, and Fχ² is the cumulative density function of the χ² distribution with 2 degrees of freedom [2].

Similarly, we derive the alignment error probability:

$$A_{\text{Err}} = 1 - \left[ F_{\chi^2} \left( \frac{\sqrt{M 2^I} + w^2}{\sigma^2} \right) \right]^{M-1}$$

We can estimate the probability of alignment error by Monte Carlo sampling, using a large number of generated values of w, which is Gaussian also with a variance of σ². As for LoRa, when a SF1xM symbol reception is wrong, not all bits are in error and so:

$$BER = \frac{M 2^I}{2(M 2^I - 1)} S_{\text{Err}}$$

Neglecting the LoRa error correcting code, successful frame reception is conditioned by preamble detection, proper alignment and by (1 - BER)² where L is the frame length in bits.

Figure 6 shows both the BER and alignment error probabilities against SNR for M = 32 and spreading factors equivalent to SF7 to SF12 for the LoRa modulation. The figure also shows (as circles) the BER values for the generally accepted sensitivity of LoRa on a white gaussian noise channel, between -7.5dB for SF7 to -20 for SF12, by increments of 2.5 dB, with the corresponding alignment error probabilities (stars). The theoretical curves are a bit more than 2.5 dB apart, so that BER decreases when the SF grows. In the worst case (for SF7 or SF2x32), we observe a BER of 3.3 × 10⁻⁴ and in this case an alignment error probability of 2.2 × 10⁻⁴ which will

be far from having any impact on frame reception success for all considered values.

For the largest M value considered below (SF2x128), the alignment error probability (black cross) remains well below the individual bit error probability.

## IV. CAPACITY BENEFITS AND DEPLOYMENT CONSIDERATIONS

For a group of nodes all located at the same (large) distance from the gateway, where e.g. only SF12 (or equivalent, like SF11x2) is useable, we expect major capacity improvements from using SF1xM, as it allows to build several virtual channels for the same transmission range.

We evaluate the performance by running a packet reception model which takes into account both attenuation g(d) (Okumura-Hata model), fast fading (propagation) and collisions, knowing that the probability to survive fading is not independent of overcoming collisions [7]. Without collision, the reception probability is H = e⁻$^{g_j}$ where threshold fast fading gain g_j is the value for which the reception power allows to reach the needed SNR q_j for SFj at distance d:

$$g_j = \frac{N q_j}{P_t g(d)},$$

where N is the in-band noise power, P_t the transmission power, and g(d), the average channel gain at distance d (which would e.g. follow the Hata model).

In presence of concurrent traffic, collisions will cause packet losses and we model the overall probability of reception P_C with a single GW or with two collocated GWs: P_C+D [7].

### A. Low Node Density

We first consider sensors placed in the plane around one GW following a Poisson point process (PPP). All senders generate frames with an average intensity of 1 packet every 12.3 min, that would saturate the 1% duty cycle at SF12 with 3 frequency channels in the band, for 59B packets.

We do SF allocation iteratively with the objective to maintain at least 40% PDR (packet delivery rate) so that an inter-frame error correcting code of coding rate 1/3 would result in reliable data delivery. With basic SF7 to SF12 LoRa modulations and a single receiver, the results appear in Figure 7(a). In this figure, the sharp contrast between the operating point for low and high spreading factors is apparent: for SF7 and 8, the load (thin line) is very light and the switch to the next SF is mostly constrained by the modulation sensitivity. In contrast, for SF11, the channel load is significantly higher. This elevated usage leads to an increased probability of collisions among transmissions. As a result, nodes operating at SF11 are more likely to experience interference, which in turn necessitates the transition to a higher spreading factor, such as SF12, in order to maintain reliable communication.

With SF1xM modulations, we can use more quasi-orthogonal modulations and we have flexibility to decide on their robustness. In this way, the load may be more evenly distributed. Figure 7(b) illustrates how, with SF1xM, the load
<figure><img src="image_8.png" /><figcaption>Fig. 7. Target PDR 40%, density 10 nodes/km², no receiver diversity — How to interpret this graph: in Figure 7(a), SF7 is used in the disk of radius 3.94 km around the GW which covers 488 nodes; in the SF11 annulus, the load is 0.46 Erlang and at the outer edge (7.14 km), the loss rate due to only fading is 57% and overall packet loss is 40% by construction. SF1xM increases the capacity by 55%.</figcaption></figure>

<figure><img src="image_9.png" /><figcaption>Fig. 8. Target PDR 40%, density 10 nodes/km², with receiver diversity</figcaption></figure>

is well balanced throughout the cell. In this case, the limiting factor becomes the SF12 (or SF1xM equivalent) sensitivity, which we choose not to alter in the context of this paper.

The modulations in this case are: SF2x128, SF3x128, SF4x128, SF5x128, SF6x64, SF7x32, SF8x16, SF9x8, SF10x4, SF11x2, SF12. The least robust modulation used here is equivalent to SF9 (= 2 + log₂(128)).

## B. Is SF1xM the Best Usage of Additional Demodulators?

In essence, SF1xM requires a number of additional demodulators and preamble detection engines at the GW, one for each additional SF value used. It is legitimate to question if SF1xM is the best possible usage of this additional hardware in terms of capacity benefits.

In particular, receiver diversity, which boils down to adding a second GW with a second antenna not far from the original one, is another way of making use of additional demodulators. The benefits of receiver diversity are well established in presence of fast fading, as a second antenna reshuffles multipath fading and improves the PDR. In an ALOHA context with capture, the benefits are even greater as for two colliding frames, the loser may not be the same on the different receivers.

The benefits of diversity are apparent in Figure 8(a), compared to Figure 7(a): with receiver diversity, the range and thus

the number of covered nodes significantly improves (in this Figure, the probability H of reception regardless of collision comprises the effect of diversity). Nevertheless the benefits are lower than that of SF1xM, which also does not require a second antenna.

Furthermore, SF1xM is not mutually exclusive with receiver diversity: in Figure 8(b), the combined use of additional modulations and receiver diversity more roughly doubles the number of covered nodes – at the cost of a multiplication by 4 of the number of demodulators, in all fairness.

Here, the flexibility brought by the addition of SF1xM modulations is apparent by considering e.g. the load throughout the cell. The channel load is far less contrasted with SF1xM modulations, whereas for LoRa, the load increases sharply when moving away from the GW.

## C. High Node Density

For 100 nodes per km² (so that the sensors are on a 100 m grid), SF1xM allows to distribute resources differently than in the low density case, where the focus in on the farthest nodes. Whereas SF1xM increases the number of covered nodes by 43% compared to LoRa, here, receiver diversity –in Figure 10(a)– gives slightly better results than SF1xM. Nevertheless, again, combining SF1xM and diversity yields
<figure><img src="image_10.png" /><figcaption>Fig. 9. Target PDR 40%, density 100 nodes/km², no diversity</figcaption></figure>

<figure><img src="image_11.png" /><figcaption>Fig. 10. Target PDR 40%, density 100 nodes/km², with receiver diversity</figcaption></figure>

approximately a doubling in the number of served nodes compared to single receiver LoRa (in Figure 9(a)).

The modulation used with SF1xM in Figures 9(b) and 10(b) are: SF2x32, SF3x32, SF4x32, SF5x16, SF6x8, SF8x4, SF8x4, SF9x2, SF10, SF11, SF12.

## D. 80% PDR

In many cases, LoRaWAN traffic does not use inter-frame error correcting codes, meaning reliable data delivery must rely on conservative SF allocation and frame replication. With 3 transmissions of each data frame, a radio frame delivery rate of 80% results in less than 1% data loss.

For a density of 100 nodes per km², SF1xM improves greatly the cell coverage, in particular in conjunction with receiver diversity (Figures 11 and 12). Without receiver diversity, SF1xM increases the number of covered nodes by 34%; with diversity, the improvement is 52%.

At relatively low node densities, achieving the target packet delivery ratio (PDR) relies heavily on receiver diversity and modulation robustness. In this context, the communication range is limited not only by the sensitivity of the most robust modulation, but also by the availability of multiple modulation options to carry traffic from distant nodes. This is where SF1xM offers a clear advantage over standard LoRa modulation, as illustrated in Figure 13 (40% more covered nodes).

## E. SF1xM Deployment Problematics

This section discusses the costs and constraints associated with deploying SF1xM in a LoRaWAN network.

1) Hardware Requirements: Implementing SF1xM necessitates to enhance the gateway hardware: first, if we use other frequency ramp slopes than the basic values of $I \in [7..12]$, the preamble detection engine needs to be able to detect the corresponding new preambles. Moreover, on top of carrying out the proposed operations during the alignment phase (Section III-A), the receiver may have to deal with a larger number of simultaneous parallel packet receptions. More packet demodulators are thus needed if increasing capacity is the objective. The purpose of section IV-B is to assess in which cases SF1xM may be beneficial, knowing that e.g. adding a second gateway is typically a more straightforward way of deploying more demodulators.

2) Power consumption overhead: The alignment phase of SF1xM increases the packet time on air, although the additional overhead remains smaller than e.g. the sync word duration (see Figure 5). So, on the one hand, LoRa modulations tend to offer higher energy efficiency. And on the other hand, SF1xM provides additional flexibility, as its broader set of modulation options allows fine-tuning of transmission robustness and helps avoid the use of unsuitable spreading factors.
<figure><img src="image_12.png" /><figcaption>Fig. 11. PDR 80%, density 100 nodes/km², LoRa modulations</figcaption></figure>

<figure><img src="image_13.png" /></figure>

<figure><img src="image_14.png" /><figcaption>Fig. 12. PDR 80%, density 100 nodes/km², with SF1xM modulations</figcaption></figure>

<figure><img src="image_15.png" /></figure>

<figure><img src="image_16.png" /><figcaption>Fig. 13. PDR 80%, density 10 nodes/km², with receiver diversity</figcaption></figure>

<figure><img src="image_17.png" /></figure>

3) SF1xM and LoRa Cohabitation: In an environment where LoRa and SF1xM-enabled senders would coexist, the receiver has to quickly determine what modulation a given transmission uses. The sync word may serve this purpose, just like it currently allows to differentiate between the transmission on private links and to public networks. Using a new sync word value makes SF1xM backward compatible: a legacy LoRa-only gateway would simply ignore SF1xM frames, even if they use an *I* value that it demodulates.

## V. RELATED WORK

Given that LoRa is fundamentally based on the Chirp Spread Spectrum (CSS) technique, it is somewhat ironic that many

research efforts to improve its focus primarily on increasing spectral efficiency. In principle, this can be achieved simply by lowering the spreading factor (SF). Therefore, the true value of alternative approaches lies elsewhere—specifically in addressing the core challenges of coverage and capacity, while preserving receiver simplicity. If increased receiver complexity is acceptable, switching from CSS to frequency hopping can yield substantial improvements, as demonstrated by technologies such as LR-FHSS [8] and mioty [9].

PSK-LoRa for instance, consists in using the phase information in the received signal to code more information, whereas in LoRa, only the position of the dominant peak after the FFT
counts. Using the phase imposes to find and track it, though, a problem that the DPSK-MM-CSS scheme tackles [10].

Another way of increasing the number of bits carried by each symbol is to send several chirps with different frequency shifts in parallel, so that the receiver reads the combination of peaks after FFT to make a decision [11], [12]. Dual Orthogonal LoRa Modulation [13] is simpler, as the proposition is to interleave the samples of two different chirps at the even and odd positions to double the data rate at the cost of the expected loss of robustness (3 dB). In the context of building new modulation schemes to help where there is need to offload a given area around the gateway, these proposals would allow to build e.g. an enhanced SF13 which would be quasi-orthogonal to SF11 and/or SF12 and have similar sensitivity. One question these papers leave unanswered is what kind of physical preamble should be used to not make it become problematic and what would be the corresponding overhead. Nevertheless, these approaches are complementary to SF1xM by reinforcing available modulation “from above” instead of by lowering the spectral efficiency of smaller SFs.

More disruptive proposals explore the use of chirps with nonlinear frequency sweeps, as in BIC-LoRa [14], while others leverage combinations of up-chirps and down-chirps to encode additional bits [15], [16], [17]. It is worth noting that LoRaWAN networks already exploit chirp direction to distinguish between uplink and downlink transmissions.

## VI. CONCLUSION

Some of the key features of LoRa, especially in a LoRaWAN context, are flexibility as well as low-cost devices (25% of e.g. NBIoT [18]) and gateways. SF1xM builds on this foundation by further enhancing flexibility with minimal added complexity, enabling more efficient resource allocation to the cell configuration. When the nodes are far from the gateway, SF1xM enables to dedicate resources to the edge of the cell. When the density is higher, then the load can be made more even throughout the cell. While SF1xM requires an additional alignment step after the LoRa preamble, its overhead is minimal and outweighed by the resulting performance gains. In future work, we plan to explore if alignment could be carried out in the sync word part of the preamble.

This study assumes SF12 data rate as the lowest usable data rate. Under this condition, SF1xM demonstrates strong benefits. However, relaxing this assumption could allow SF1xM to extend the cell edge even further in sparse deployments, unlocking additional coverage potential.

This paper shows the promises of SF1xM, but there are many outstanding questions: what would be the benefits of considering to use e.g. SF3 with BW=250kHz instead of SF2 and BW=125 kHz, as these two modulation have the same chirp slope and they are not orthogonal? Finally, although it is easy to find a working vector of M values given the traffic intensity, it is clear that parameter allocation in SF1xM has more degrees of freedom than LoRa, and we would need a method to find the optimal one. Furthermore, in real networks, LoRaWAN Network Servers do not have precise knowledge

of node channel conditions or of node distribution. So rate adaptation is necessarily ad hoc, using ADR (Adaptive Data Rate) mechanisms. With SF1xM, the ADR would need to deal with more degrees of freedom, which leaves room for much future investigations.

## ACKNOWLEDGMENTS

This work has been partially supported by the French Ministry of Research projects PERSYVAL-Lab under contract ANR-11-LABX-0025-01, and PEPR DONUTS 5G.

## REFERENCES

[1] L. Vangelista, "Frequency shift chirp modulation: The lora modulation," *IEEE Signal Processing Letters*, vol. 24, no. 12, pp. 1818-1821, 2017.

[2] G. Ferré and A. Giremus, "LoRa physical layer principle and performance analysis," in *2018 25th IEEE International Conference on Electronics, Circuits and Systems (ICECS)*, 2018, pp. 65-68.

[3] T. Elshabrawy and J. Robert, "Closed-Form Approximation of LoRa Modulation BER Performance," *IEEE Communications Letters*, vol. 22, no. 9, pp. 1778-1781, Sep. 2018.

[4] D. Croce, M. Gucciardo, S. Mangione, G. Santaromita, and I. Tinnirello, "LoRa Technology Demystified: From Link Behavior to Cell-Level Performance," *IEEE Transactions on Wireless Communications*, vol. 19, no. 2, pp. 822-834, 2020.

[5] "SX1272 — Long Range, Low Power RF Transceiver 860-1000MHz — Semtech," https://www semtech fr/products/wireless-rf/lora-connect/ sx1272#documentation, Jan. 2019.

[6] C. Caillouet, M. Heusse, and F. Rousseau, "Optimal SF Allocation in LoRaWAN Considering Physical Capture and Imperfect Orthogonality," in *IEEE Global Communications Conference (GLOBECOM)*, Dec. 2019.

[7] M. Heusse, C. Caillouet, and A. Duda, "Performance of unslotted ALOHA with capture and multiple collisions in LoRaWAN," *IEEE Internet of Things Journal*, vol. 10, no. 20, pp. 17824-17838, 2023.

[8] Semtech, "Application Note: LR-FHSS System Performance," [Online]. Available: https://www.mouser.com/pdfDocs/AN1200-64_ LR-FHSS_system_performance_V1_2.pdf

[9] J. Robert and T. Lauterbach, "MIOTY Comparative Study Report," Technische Universität Ilmenau, Tech. Rep., 2023. [Online]. Available: https://mioty-alliance.com/mioty-vs-lora-study-report/

[10] X. Yu, X. Cai, W. Xu, H. Sun, and L. Wang, "Differential phase shift keying-aided multi-mode chirp spread spectrum modulation," *IEEE Wireless Communications Letters*, pp. 1-1, 2023.

[11] M. Hanif and H. H. Nguyen, "Methods for improving flexibility and data rate of chirp spread spectrum systems in lorawan," US Patent 0358475, Sep. 2020.

[12] Q. Yu, H. Wang, Z. Lu, and S. An, "Group-based CSS modulation: A novel enhancement to LoRa physical layer," *IEEE Wireless Communications Letters*, vol. 11, no. 3, pp. 660-664, 2022.

[13] L. Vangelista and A. Cattapan, "A new LoRa-compatible modulation improving the LoRaWAN network level performance," in *2019 IEEE Latin-American Conference on Communications (LATINCOM)*, 2019, pp. 1-6.

[14] G. Lee, E. Park, M. Park, J. Paek, and S. Bahk, "BIC-LoRa: Bits in chirp shapes to boost throughput in LoRa," in *2024 23rd ACM/IEEE International Conference on Information Processing in Sensor Networks (IPSN)*, 2024, pp. 237-248.

[15] A. Mondal, M. Hanif, and H. H. Nguyen, "SSK-ICS LoRa: A LoRa-based modulation scheme with constant envelope and enhanced data rate," *IEEE Communications Letters*, vol. 26, no. 5, pp. 1185-1189, 2022.

[16] T. Elshabrawy and J. Robert, "Interleaved chirp spreading LoRa-based modulation," *IEEE Internet of Things Journal*, vol. 6, no. 2, pp. 3855-3863, 2019.

[17] A. W. Azim, A. Bazzi, R. Shubair, and M. Chafii, "Dual-mode chirp spread spectrum modulation," *IEEE Wireless Communications Letters*, vol. 11, no. 9, pp. 1995-1999, 2022.

[18] M. Ballerini, T. Polonelli, D. Brunelli, M. Magno, and L. Benini, "NB-IoT versus LoRaWAN: An experimental evaluation for industrial applications," *IEEE Transactions on Industrial Informatics*, vol. 16, no. 12, pp. 7802-7811, 2020.