Chengwen Zhang，Liankai Wang，Libin Jiao，Shipeng Wang，Jun Shi，Jia Yue

1 School of Electronics and Information Engineering，Harbin Institute of Technology，Harbin 150080，China

2 Science and Technology on Communication Networks Laboratory，Shijiazhuang 050081，China

3 Key Laboratory of Police Wireless Digital Communication，Ministry of Public Security，Harbin 150001，China

*The corresponding author，email: junshi@hit.edu.cn

Abstract: In recent years，LoRa has been extensively researched in the satellite Internet of Things (IoT).However，the multiple access technology of LoRa is still one of the bottlenecks of satellite IoT.To improve the multiple access performance of LoRa satellite IoT，based on the orthogonality of LoRa symbols in the fractional domain，this paper proposes a low complexity Orthogonal LoRa Multiple Access (OLMA) algorithm for multiple LoRa users occupying the same frequency bandwidth.The algorithm introduces the address code to divide the fractional bandwidth into multiple parts，and the OLMA users with different address codes occupy different parts to transmit the information code，thus avoiding mutual interference caused by collisions in the same frequency bandwidth.The multiple access capability of OLMA can be flexibly configured only by simply adjusting the length of the address code according to the actual application requirements of data transmission.Theoretical analysis and simulation results show that the OLMA algorithm can greatly improve the multiple access capability and the total transmission bit rate of LoRa IoT without changing the existing LoRa modulation parameters and process.

Keywords: LoRa; satellite Internet of Things; multiple access;fractional domain;linear frequency modulation

I.INTRODUCTION

With the rapid development of smart city，smart home，environmental monitoring and other applications，the Internet of Things (IoT) has become a typical application scenario in 5G system [1，2].The low-power wide-area network (LPWAN) [3] proposed in recent years aims to provide services for IoT applications with long distance and low power consumption.However，due to the limitations of space，environment and other factors，the traditional IoT system on the ground has insufficient coverage capability.Therefore，the satellite IoT stands out for its advantages of wide coverage and not limited by geographical location in remote areas[4-6].

As one of the typical technologies of LPWAN，LoRa[7]has received a lot of commercial attention because it is able to carry out long-distance transmission and easy to build independent networks.It has played an important role in environmental monitoring，smart grid and smart city[8].In recent years，academia has set off a research upsurge of this technology in new application scenarios such as satellites IoT[9].Reference[10]explores the feasibility of satellite IoT from the perspectives of constellation structure，spectrum allocation and network compatibility of Low Earth Orbit (LEO) satellites.Reference [11] designs a spatial LoRa wireless sensor network to meet the topological requirements of satellite applications.Reference[12-14] put forward several different physical layer waveform designs including Symmetry Chirp Spread Spectrum (SCSS)，Asymmetry Chirp Signal (ACS)，and folded Chirp-Rate Shift Keying(FCRSK)according to the communication requirements of LEO satellite IoT.A Symmetry Chirp with Multiple Chirp Rates(SC-MCR) waveform and time domain multiplexing scheme for multi-user system is designed [15].The above documents have made great achievements in the physical layer waveform design of LoRa satellite IoT.However，the multiple access technology is still one of the bottlenecks of LoRa satellite IoT due to its wide coverage and large number of users.In view of the importance of multiple access technology in satellite IoT，the new multiple access mode of the LoRa satellite IoT needs to be further studied.

Multiple access technology in satellite communication system can be divided into non-contention access mode and contention access mode [16].Due to the abrupt nature and small amount of data of the satellite IoT，the use of non-contention access to assign fixed resources to each terminal in advance will cause waste of resources and inflexible allocation of resources.Therefore，the contention access mode，commonly referred to as Random Access(RA)technology，is more suitable for the satellite IoT [17].Currently，LoRa IoT adopts an ALOHA-based protocol for medium access [18].The packet collision will inevitably occur due to the use of competition to preempt resources，which will reduce the transmission performance of LoRa.Therefore，reducing the impact of packet collisions is the key to the competitive multiple access of LoRa satellite IoT.Reference [19] studies CSMA to replace ALOHA access mechanism to improve LoRa scalability，[20]solves the collision problem by overlaying a new low-overhead scheduler on the current scheduling mechanism，and [21] improves the collision problem by interference cancellation technology.Reference[22]proposes the Interleaved Chirp Spreading LoRa(ICS-LoRa)as a parallel logical network coexisting with the nominal LoRa network to enhance the LoRa network capacity.Unfortunately，compared with the existing LoRa modulation technology，all of these studies have increased the complexity due to the introduction of new technologies.

LoRa signal adopts non-stationary frequency shift chirp modulation and its instantaneous frequency varies linearly with time.As a time-frequency analysis tool，F(xiàn)ractional Fourier Transform(FRFT)[23-25]is very suitable for analyzing chirp signal because it has both time domain and frequency domain information of a signal.

In this paper，to improve the multiple access performance of LoRa satellite IoT，a novel low-complexity Orthogonal LoRa Multiple Access(OLMA)algorithm for multiple LoRa users occupying the same frequency bandwidth is proposed.

The main novelty and contributions of this paper are as follows.Without occupying additional timefrequency resources，based on the orthogonality of LoRa symbols in the fractional domain，the proposed OLMA algorithm can make multiple LoRa users realize orthogonal multiple access in the fractional domain by adding address code to the LoRa symbols，so as to avoid mutual interference between multiple LoRa users caused by collisions in the same frequency bandwidth.The multiple access capability of OLMA can be flexibly configured by adjusting the length of the address code according to the actual application requirements of data transmission.The multiple access capability and the total transmission bit rate of LoRa satellite IoT can be greatly improved.In addition，the proposed OLMA algorithm has the same complexity as the existing LoRa modulation technology since it does not change the existing LoRa modulation parameters and process.

This paper is organized as follows.In Section II，we present the LoRa model and analyze the characteristics of LoRa symbol in the fractional domain.In Section III，the orthogonal LoRa multiple access algorithm and its implementation are proposed.In Section IV，simulation results are presented.In Section V，we draw the conclusions and put forward the focus of future research.

II.LORA MODEL AND FRFT

2.1 LoRa Model

LoRa modulation is a special Linear Frequency Modulation(LFM)technology.The chirp signal is used as the carrier，and the initial frequency is used to carry the information bits of LoRa symbol.A LoRa symbol can be expressed as Eq.(1)in the time domain[26].whereBis the bandwidth of LoRa，SFis the spreading factor，Kis the cyclic shift value corresponding to the transmission bits，Tis the period of LoRa symbol andT= 2SF/B，andT0is the time of frequency hopping andT0=(2SF-K)/B.Assume thatf0is the initial frequency of LoRa symbol which is determined byK，andf0=KB/2SF.The time-frequency relationship of LoRa symbol is shown in Figure 1.

Figure 1. The time-frequency relationship of LoRa symbol.

Without considering all kinds of interference and noise，the LoRa symbol is undersampled at the receiving end.The sampling frequency isB，so the sampling interval is Δt= 1/B.Then the discrete sequence of lengthN=T/Δt= 2SFis expressed as Eq.(2).

Whennis integer，exp(j2πn) = 1.Sos(n)can be simplified as

It can be seen from Eq.(3) that undersampling makes two segments of LoRa symbol into one segment，which makes the energy of LoRa symbol more concentrated.

The matching downchirp signal is

Then the dechirped symbolr(n)can be obtained by

It can be seen from Eq.(5) that dechirping makes the LoRa LFM signal into a sinusoidal signal，which makes it easy to demodulate in the frequency domain.

In summary，after undersampling and dechirping，the original LoRa symbol of the two-segment LFM turns into a single sinusoidal signal.The informationKtransmitted by LoRa symbol is carried by the frequency of the single sinusoidal signal.Therefore，to demodulateK，there is no need to recover the original LoRa symbol，but only the frequency of the single sinusoidal signal.

The frequency of the single sinusoidal signal can be obtained by FFT.

The estimated value ofKcan be obtained through the spectral peak search as Eq.(7).

Convert the decimal ?Kto binary，and then the information bits are demodulated accurately.

2.2 FRFT Spectrum of LoRa Modulation Symbol

FRFT is a generalized form of Fourier transform[27].The basis function of Fourier transform is in the form of trigonometric signal，but the basis function of FRFT is in the form of LFM signal.The FRFT spectrum of LFM signal at a specific angle is an impulse function，which is the theoretic basis for analyzing the FRFT spectrum of LoRa symbols.The FRFT of a generalized functionx(t)is defined as

whereαis the transforming angle of FRFT，andKα(u，t) is the FRFT kernel function，which is expressed as

The inverse FRFT(IFRFT)with respect to angleαis the FRFT at angle-α.

where the superscript*represents the complex conjugate.

To obtain the FRFT spectrum of LoRa symbol[28]，put Eq.(1)into Eq.(8).

whereA=Aαejπu2cotα.Let+ cotα=0，which meansα=-arccot()，and Eq.(11) can be simplified as

Letk=then the Eq.(12)can be expressed as Eq.(13).

It can be seen that the FRFT spectrum of LoRa symbol atαis the superposition of two non-overlapping sinc functions，and theαis the optimal transformation angle of FRFT.The energy of LoRa symbol is concentrated in two narrow-bands centered atk=Kandk=K -2SF.The distance between the two centers is 2SF.TakeSF= 7，the FRFT spectrum of LoRa symbol whenKis 87 is shown in Figure 2.

Figure 2. The FRFT spectrum of LoRa symbol.

For LoRa receiver，since the FRFT spectrum takes 2SFas the period whenBis the sampling frequency，the FRFT spectrum of LoRa symbol becomes the sum of two sinc functions centered atk=Kand only a single spectral peak is formed.

III.THEPROPOSEDORTHOGONAL LORA MULTIPLE ACCESS ALGORITHM

3.1 The Orthogonal LoRa Multiple Access Algorithm Principle

In general，a multiple access algorithm can be implemented through different domains，such as time domain，frequency domain，and code domain.Its core is to avoid mutual interference between multiple users.

In this paper the proposed OLMA algorithm is designed based on the fractional domain.The energy convergence and narrowband characteristics of LoRa symbol in the fractional domain are the basis of the OLMA algorithm.

To realize the orthogonal multiple access of LoRa users in the fractional domain，the fractional bandwidth is divided into multiple small parts and each part only transmits the information bits of a single LoRa user respectively.Therefore，for LoRa users with the same frequency bandwidth，the orthogonal multiple access in the fractional domain can be realized，thereby reducing collisions.

When the spreading factor of LoRa isSF，the number of points of the conventional LoRa symbol in the fractional domain will be 2SF.The number of bits transmitted by the conventional LoRa symbol is

Assume that the fractional bandwidth is evenly divided intoPparts，so there are 2SF/Ppoints in each part.Letlais the number of address bits andla=log2P，so the number of information bits transmitted by each LoRa symbol corresponding to each part can be expressed as

Therefore，the total number of information bits transmitted by all the OLMA users can be expressed as

Compared with Eq.(14)，under the same frequency bandwidth，the total number of information bits transmitted by all OLMA users isltotal/l=2la·(1-la/l)times that of conventional LoRa.

Letl= 7，la= 2 ，thenlm= 5.The fractional bandwidth division is shown in Figure 3.

Figure 3. The bandwidth division in the fractional domain.

The number of OLMA users without collision in the fractional domain is determined byla.Aslaincreases，the number of accessible OLMA users increases.Although the number of information bits transmitted by each OLMA user decreases，the total number of information bits transmitted by all OLMA users increases.Therefore，lacan make a trade-off between the multiple access capability of LoRa IoT and the bit rate of a single user.It is necessary to choose the appropriatelaaccording to the actual application requirements of data transmission.In summary，the proposed OLMA algorithm can simultaneously demodulate LoRa symbols of multiple users under the same frequency bandwidth，which improves the multiple access capability and spectrum efficiency of the LoRa Satellite IoT.

3.2 Derivation of Orthogonal LoRa Multiple Access Algorithm

Although the principle of the proposed OLMA algorithm is based on the orthogonality of LoRa symbols in the fractional domain，the algorithm can be realized in the time domain.

The OLMA algorithm divides thelbits of LoRa symbol into two parts.One part haslabits，which is called address code corresponding to the specific part of the fractional bandwidth.Another part haslmbits，which is called information code corresponding to the information bits transmitted by OLMA users.The relationship of address code and information code is shown in Figure 4.

Figure 4. The relationship of address code and information code.

LetKdbe the cyclic shift value of LoRa symbol corresponding to the OLMA userd.Since the initial frequency of LoRa symbol is determined by the cyclic shift valueKcorresponding to theSFtransmission bits of LoRa symbol，the relationship betweenKdand address code and information code is the key to the OLMA algorithm.The relationship can be expressed as Eq.(17).wherebiis thei-th information bit of LoRa symbol，d ∈{0，1，...，D -1}corresponding to the address code of different OLMA users，D= 2lawhich represents the number of accessible users of the OLMA algorithm without collision in the fractional domain.From Eq.(17)，it can be seen that with the difference ofd，KdhasDnon-overlapping value ranges，and the following conclusion can be drawn for any two OLMA users.

It can be seen from Eq.(18)that theKdof any LoRa symbol of different OLMA users is different.Therefore，in the frequency domain，sincewhich is the initial frequency of LoRa symbol of the OLMA userd，thef0dalso hasDnon-overlapping parts.Whenlais 2，the time-frequency relationship of LoRa symbols of different OLMA users is shown in Figure 5.

The bits transmitted by LoRa symbols of OLMA users can be expressed as Table 1

Table 1. Address code allocation.

In the uplink，the received waveform is the superposition of LoRa symbols of multiple OLMA users，and it can be expressed as

wheresd(t)is the LoRa symbol of the OLMA userd.

whereT0dis the frequency hopping time of LoRa symbol of the OLMA userd.

After undersampling，Eq.(19) can be rewritten as

Comparing Eq.(21)with Eq.(3)，it can be seen thatsd(n) has the same expression form as the conventional LoRa symbols(n)，sosd(n)also has the same single spectral peak as the conventional LoRa symbol in the fractional domain.Furthermore，since the LoRa symbols of different OLMA users have orthogonality in the fractional domain，so the demodulation of thesd(n) of each OLMA user can be similar to the demodulation of conventional LoRa symbol.

Letla= 2，lm= 5 ，the fractional spectrum ofsOLMA(n) is shown in Figure 6.It can be seen that there are four spectral peaks corresponding to different OLMA users in the fractional domain，which is consistent with the theoretical analysis of the aforementioned OLMA algorithm.

Figure 5. The time-frequency relationship when la is 2.

Figure 6. The fractional spectrum when la is 2.

Since thesd(n) of different OLMA users is superimposed in the time domain，it cannot be separated directly.The demodulation process ofsOLMA(n) is described as follows.

MultiplysOLMA(n) with the matching downchirp signald(n)given in Eq.(3).

Therefore，rOLMA(n)is the superposition ofrd(n).

Considering the linear property of FFT，we can obtain

Convert decimalKdto binary.

wherebijrepresents thej-th demodulated information bit of thei-th OLMA user.The address code and the information code are independent and easy to separate.By matching the address code to the corresponding OLMA user，the information bits of each OLMA user can be obtained correctly.

From the above descriptions of the principle and implementation of OLMA，it can be seen that compared with the conventional LoRa symbol，the LoRa symbol of OLMA user is only different in the transmitted information bits，and its parameters have not changed，such as spreading factor，bandwidth，period，power，etc.In addition，the implementation processes of LoRa symbol modulation，synchronization，and demodulation have not changed at all.Therefore，the proposed OLMA algorithm has the same lowcomplexity as the conventional LoRa algorithm，and it can be compatible with the physical layer digital signal processing process of the conventional LoRa terminal while improving the multiple access capability of LoRa IoT.

3.3 Collision Analysis

The duty cycle of LoRa signal varies with the actual application requirements of data transmission of LoRa IoT.Assume that the duty cycle isρ，and the number of total LoRa users isQ.

For the conventional LoRa，the collision probability is

Therefore，whenQ ＞1，the collision probability of the conventional LoRa decreases asρdecreases，but the collision probability will not be zero，which will affect the BER performance.

For the OLMA，when the length of address bits isla，the collision probability is

WhenQ ≤2la，it means that each OLMA user can have a special address code，the collision probability will be zero no matter what the value ofρis.WhenQ ＞2la，comparing Eq.(25) and Eq.(24)，it can be seen that the collision probability of the OLMA is much smaller than that of the conventional LoRa.

IV.SIMULATION RESULTS

We have evaluated the performance of the proposed OLMA algorithm under Rician fading channel.

Assume that the power of OLMA user and ICSLoRa[22]user is the same as conventional LoRa user.

The simulation parameters are shown in Table 2.In order to illustrate the relationship between BER performance and SNR of the proposed OLMA algorithm，whenSFis 7 and the number of users is 8，theBER performance under differentlaandρis shown in Figure 7.

Table 2. Simulation parameters.

Figure 7. BER performance under different la and ρ.

It can be seen that no matter what valueρtakes，the BER performance gets better aslaor SNR increases，and it is worst whenla=0，which is the conventional LoRa.This is precisely because with the increase oflathe collision probability decreases.But，when SNR is greater than-3 dB，except forla=3，the BER performance no longer improves with the increase of SNR.This is precisely because when SNR＞-3 dB，with the increase of SNR，the noise is not the main factor that affects the BER compared with the collision probability.Whenla=3，the OLMA algorithm can provide fractional domain orthogonal multiple access channels，so that these 8 users will not collide and the BER performance is not affected by the duty cycle.ICS-LoRa offers one more channel than conventional LoRa，so the BER performance is similar to that ofla=1.

In order to show the relationship between BER performance and duty cycleρof the proposed OLMA algorithm，when SNR is 0 dB and the number of users is 64，the BER performance under differentlaand SF is shown in Figure 8.

Figure 8. BER performance under different la and SF.

It can be seen that asρdecreases，the BER of the OLMA algorithm at anylaorSFis improved.And under the sameρ，with the increase ofla，the BER of the OLMA algorithm at anySFis also improved.This is because since the number of access users far exceeds the number of the OLMA parallel channels，the BER performance will be greatly affected by the collision probability and the collision probability will be decreased with the decrease ofρor the increase ofla，thus the OLMA algorithm can improve the BER performance.

In summary，withlaincreasing，the OLMA can decrease the collision probability of users in the fractional domain，thus the BER performance gets better.

V.CONCLUSION

In this paper，a low-complexity OLMA algorithm based on the orthogonality of LoRa symbols in the fractional domain is proposed.The algorithm divides the fractional bandwidth of LoRa symbols into multiple parts by introducing address code and allocates each part to different users to avoid mutual interference between OLMA users.The number of accessible users without collision can be adjusted by the length of the address code.The OLMA algorithm can greatly improve the multiple access capability and the total transmission bit rate of LoRa satellite IoT.And because the algorithm does not change the existing LoRa modulation parameters and process，it has the same low-complexity as the conventional LoRa algorithm.

It should be noted that the spreading factor of the LoRa symbols used in the OLMA algorithm is the same，but the spreading factor can vary from 7 to 12.How the OLMA algorithm accommodates more users with different spreading factors is the focus of future research.

ACKNOWLEDGEMENT

The authors would like to thank the reviewers for their detailed reviews and constructive comments，which have helped improve the quality of this paper.This work was supported in part by the National Natural Science Foundation of China under Grant 61871153，in part by Science and Technology on Communication Networks Laboratory under Grant 6142104200202，and in part by Science and Technology Project of Ministry of Public Security(2019GABJC35).