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On the Lora Modulation for Iot: Waveform Properties and Spectral Analysis Marco Chiani, Fellow, IEEE, and Ahmed Elzanaty, Member, IEEE

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On the Lora Modulation for Iot: Waveform Properties and Spectral Analysis Marco Chiani, Fellow, IEEE, and Ahmed Elzanaty, Member, IEEE ACCEPTED FOR PUBLICATION IN IEEE INTERNET OF THINGS JOURNAL 1 On the LoRa Modulation for IoT: Waveform Properties and Spectral Analysis Marco Chiani, Fellow, IEEE, and Ahmed Elzanaty, Member, IEEE Abstract—An important modulation technique for Internet of [17]–[19]. On the other hand, the spectral characteristics of Things (IoT) is the one proposed by the LoRa allianceTM. In this LoRa have not been addressed in the literature. paper we analyze the M-ary LoRa modulation in the time and In this paper we provide a complete characterization of the frequency domains. First, we provide the signal description in the time domain, and show that LoRa is a memoryless continuous LoRa modulated signal. In particular, we start by developing phase modulation. The cross-correlation between the transmitted a mathematical model for the modulated signal in the time waveforms is determined, proving that LoRa can be considered domain. The waveforms of this M-ary modulation technique approximately an orthogonal modulation only for large M. Then, are not orthogonal, and the loss in performance with respect to we investigate the spectral characteristics of the signal modulated an orthogonal modulation is quantified by studying their cross- by random data, obtaining a closed-form expression of the spectrum in terms of Fresnel functions. Quite surprisingly, we correlation. The characterization in the frequency domain found that LoRa has both continuous and discrete spectra, with is given in terms of the power spectrum, where both the the discrete spectrum containing exactly a fraction 1/M of the continuous and discrete parts are derived. The found analytical total signal power. expressions are compared with the spectrum of LoRa obtained Index Terms—LoRa Modulation; Power spectral density, Dig- by experimental data. ital Modulation, Internet of Things The main contributions of this paper can be summarized as follows: I. INTRODUCTION • we provide the analytical expression of the signal for the The most typical IoT scenario involves devices with limited M-ary LoRa chirp modulation in the time domain (both energy, that need to be connected to the Internet via wire- continuous-time and discrete-time); less links. In this regard, Low Power Wide Area Networks • we derive the cross-correlation between the LoRa wave- (LPWANs) aim to offer low data rate communication capa- forms, and prove that the modulation is non-orthogonal; bilities over ranges of several kilometers [1]–[4]. Among the • we prove that the waveforms are asymptotically orthog- current communication systems, that proposed by the LoRa onal for increasingly large M; alliance (Low power long Range) [5] is one of the most • we derive explicit closed-form expressions of the contin- promising, with an increasing number of IoT applications, uous and discrete spectra of the LoRa signal in terms of including smart metering, smart grid, and data collection from the Fresnel functions; wireless sensor networks for environmental monitoring [6]– • we prove that the power of the discrete spectrum is [11]. Several works discuss the suitability of the LoRa com- exactly a fraction 1/M of the overall signal power; munication system when the number of IoT devices increases • we compare the analytical expression of the spectrum [12]–[15]. with experimental data from commercial LoRa devices; The modulation used by LoRa, related to Chirp Spread • we show how the analytical expressions of the spectrum Spectrum, has been originally defined by its instantaneous can be used to investigate the compliance of the LoRa arXiv:1906.04256v1 [cs.NI] 10 Jun 2019 frequency [16]. Few recent papers attempted to provide a modulation with the spectral masks regulating the out- description of the LoRa modulation in the time domain, but, as of-band emissions and the power spectral density. will be detailed below, they are not complaint with the original The provided time and spectral characterization of the LoRa LoRa signal model. The LoRa performance has been analyzed signal is an analytical tool for the system design, as it allows by simulation or by considering it as an orthogonal modulation suitable selection of the modulation parameters in order to ful- M. Chiani is with the Department of Electrical, Electronic and Information fill the given requirements. For example, our analysis clarifies Engineering “G. Marconi” (DEI), University of Bologna, Via dell’Universita` how the spreading factor, maximum frequency deviation, and 50, Cesena, Italy (e-mail: [email protected]). transmitted power determine the occupied bandwidth, shape A. Elzanaty was with the Department of Electrical, Electronic and Informa- tion Engineering, University of Bologna, 40136 Bologna, Italy. He is now with of the power spectrum and its compliance with spectrum the Computer, Electrical and Mathematical Science and Engineering Division, regulations, system spectral efficiency, total discrete spectrum King Abdullah University of Science and Technology, Thuwal 23955, Saudi power, maximum cross-correlation, and SNR penalty with Arabia (e-mail: [email protected]). This work was supported in part by MIUR under the program “Dipartimenti respect to orthogonal modulations. di Eccellenza (2018-2022),” and in part by the EU project eCircular (EIT Throughout the manuscript, we define the indicator function Climate-KIC). gT ¹tº = 1 for 0 ≤ t < T and gT ¹tº = 0 elsewhere, and indicate Copyright (c) 2019 IEEE. Personal use of this material is permitted. However, permission to use this material for any other purposes must be as u¹tº the unit step function. The Dirac’s delta is indicated obtained from the IEEE by sending a request to [email protected]. as δ¹xº, and its discrete version as δm, with δ0 = 1, δm = ACCEPTED FOR PUBLICATION IN IEEE INTERNET OF THINGS JOURNAL 2 ¯ x 0 m , 0. We also indicate with C¹xº , cos t2π/2 dt 0 f(t; a1) 8 ¯ x 2 ¹ º / f(t; a ) and S x , 0 sin t π 2 dt the Fresnel functions [20]. 2 II. LORA SIGNAL MODEL B The LoRa frequency shift chirp spread spectrum modulation has been originally described in terms of the instantaneous B ) a2 a M ; frequency reported in [16, Figure 7]. It is an M-ary digital t ( modulation, where the M possible waveforms at the output f of the modulator are chirp modulated signals over the fre- quency interval ¹ f0 − B/2; f0 + B/2º with M different initial frequencies. The data modulated signal is usually preceded by B synchronization waveforms, not considered here. For the data, a1 M the instantaneous frequency is linearly increased, and then wrapped to f0 − B/2 when it reaches the maximum frequency 0 f0 + B/2, an operation that mathematically can be seen as 0 τ2 τ1 Ts a reduction modulo B. Having the instantaneous frequency t sweeping over B does not imply that the signal bandwidth is Fig. 1: Example of the instantaneous frequency f ¹t; aº as a B, as will be discussed in Section III. function of time for two different modulating symbols a ; a 2 SF 1 2 For LoRa the parameters are chosen such that M = 2 with f0;:::; M − 1g. SF integer, and BTs = M, where Ts is the symbol interval. The bit-rate of the modulation is 1 SF SF Assuming the modulation starts at t = 0, from (1) the phase R = log M = = B b 2 SF φ¹t aº t 2 » ; T » Ts Ts 2 ; for 0 s is given by ¹ t The ratio between the chip-rate Rc = M/Ts = B and the bit- φ¹t; aº 2π f ¹τ; aº dτ 1 , rate is therefore 0 SF B B 2 Rc B 2 = 2 π a t + t − B ¹t − τaº u¹t − τaº : (3) η = = = : M 2 Ts Rb Rb SF Also, with the LoRa parameters we see from (2) that the Its reciprocal 1/η can be seen as the modulation spectral product Bτ = M−a is an integer, and can therefore be omitted efficiency in bit/s/Hz. Some values of the spectral efficiency a in the phase. are reported in Table I for M ranging from 23 to 212. Note that a factor 1/2 for the quadratic term is missing in the phase definitions reported in [17]–[19], making the A. Continuous-time description instantaneous frequency of the signal not complaint with that of LoRa. That difference also propagated in the discrete-time To describe mathematically the signal in the time domain, let version of the signals used in [18], [19], so that even the time- us start for clarity by assuming that the frequency interval over discrete analysis made there is not applicable to the LoRa which to linearly sweep the frequency is »0; B¼ as depicted signal. in Fig. 1. For the time interval t 2 »0; Ts» and a symbol a 2 f0; 1;:::; M −1g the instantaneous frequency in LoRa can thus Property 1. The LoRa modulation is a memoryless continuous be written as phase modulation with φ¹0; aº = φ¹Ts; aº. B B f ¹t; aº = a + t ¹mod Bº Proof. The initial phase is φ¹0; aº = 0. The phase at the end M Ts of the symbol interval is B B = a + t − B u ¹t − τ º 0 ≤ t < T (1) B B M T a s φ¹T ; aº = 2 π a T + T − B ¹T − τ º u¹T − τ º s s M s 2 s s a s a where a B/M is the initial frequency which depends on the M = 2 π a + − M u¹T − τ º = 0 ¹mod 2πº modulating symbol, and 2 s a a τa = Ts 1 − (2) where the last equality is due to that a + M/2 − Mu¹Ts − τaº is M always an integer.
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