A Low-Complexity Demodulation for Oversampled LoRa Signal

This paper deals with a method of demodulation for oversampled LoRa signal. The usual maximum likelihood (ML) based demodulation method for LoRa chirp spread spectrum (CSS) waveform is dedicated to signals sampled at Nyquist rate, whereas considering oversampled signals may improve the performance of the LoRa demodulation process. In this respect, when an oversampling rate (OSR) 2 is assumed, the method suggested in this paper consists in applying two demodulation processes to the even and odd samples of the oversampled LoRa signal, and then combining the results. This principle is then generalized to any OSR, and we show that the complexity of the method is low since it only involves discrete Fourier transforms (DFT). Moreover, a performance analysis in terms of symbol and bit error rate (SER and BER) is presented considering both additive white Gaussian noise (AWGN) and Rayleigh channel models. Simulations show the relevance of the method and the performance analysis as a gain of 3 dB is achieved by the demodulation at OSR 2 compared with OSR 1.


I. INTRODUCTION
In the recent years, the Internet of things (IoT) technologies, among which the low power wide area network (LPWA) solutions, have enabled the connectivity of a constantly growing number of devices, offering digital transformation across industry verticals [1]. Among the LPWA technologies, LoRa is one of the most deployed and studied by both academic and industrial researchers, mainly because it offers the advantage of an easy deployment.
The LoRa waveform is originally based on the chirp spread spectrum (CSS) modulation, described by Winkler in [2]. More generally, the physical layer (PHY) of LoRa is developed by Semtech, and the system has been promoted by the LoRa Alliance, that specifies the LoRaWAN open protocol.
The PHY layer of LoRa as well as related signal processing algorithms and performance analyzes have been extensively studied in the literature. Thus, the LoRa waveform and its main properties are described by the authors of [3], [4]. In [5]- [9], theoretical symbol and bit error rate (SER and BER) expressions and approximations are provided for different channel models. However, these performance analyzes are limited to LoRa signals sampled at Nyquist rate, because the LoRa demodulation is inherently designed to process signals sampled at oversampling rate (OSR) 1 (i.e. Nyquist rate). For this reason, only few papers like [10] deal with demodulation of oversampled Manuscript submitted Vincent Savaux is with IRT b<>com, Rennes, France (corresponding author: e-mail: vincent.savaux@b-com.com) LoRa signals, whereas it is a straightforward strategy for performance improvement. In [10] a discrete Fourier transform (DFT) of the size of the oversampled signal is applied, and the resulting spectrum is processed to take advantage of the gain of energy brought by the oversampling. Despite this method requires a low computational cost, the processing in the frequency domain leads to an increase of the observed noise level, making the technique relevant only for OSR strictly larger than 2.
In this paper, we suggest a new demodulation method for oversampled LoRa signals combining both good performance and low complexity. The principle of the method considering OSR 2, consists in splitting the oversampled signal into two signals sampled at Nyquist rate, one containing the even samples, and the other one containing the odd samples of the oversampled LoRa signal. Then, dedicated DFT are applied to each of the signals, and the resulting responses are combined to apply the usual LoRa demodulation process. It is shown that such principle can be extended to any OSR as well. The complexity of the suggested technique is limited as it only involves to apply a number of DFT equal to the OSR. Furthermore, the theoretical achievable SER and BER performance is derived considering both additive white Gaussian noise (AWGN) and Rayleigh channel models, and it is validated through simulations. Thus, a gain of 3 dB is achieved by the suggested demodulation at OSR 2 compared with the usual demodulation at Nyquist rate.
The rest of the paper is organized as follows: Section II presents a background on LoRa modulation and demodulation. In Section III we introduce the new demodulation method for oversampled LoRa signal, as well as the corresponding performance analysis. Simulations results show the relevance of the technique in Section IV, and Section V concludes this paper.
where u(.) indicates the Heaviside step function. We assume a synchronized reception (see [11]- [14] for synchronization in LoRa), and we consider that the propagation channel can be modeled as a one-tap block fading channel h, therefore, for any n = 0, 1, ..N ′ − 1, the received signal r[n] is given by where w[n], n = 0, 1, .., N ′ −1, are the independent and identically distributed (iid) samples of the complex additive white Gaussian noise (AWGN) such that w[n] ∼ CN (0, σ 2 ), where σ 2 = E{|w[n]| 2 }. In the following, we assume that channel h is either an AWGN channel (i.e. h = 1) or a Rayleigh channel (i.e. h =∼ CN (0, σ 2 h )). In any case, we define the signal-to-noise ratio as The basic principle of the LoRa demodulation process consists in estimating the symbol index m from the observation r[n], by means of the maximum likelihood (ML) estimator since it is optimal in condition of synchronized reception [5], [12], [13]. For OSR r o = 1, the LoRa demodulation process is based on three mains steps: the "dechirp" operation, consisting in removing the quadratic component from the received LoRa symbol: Then, a discrete Fourier transform (DFT) is applied: where where |Y [k]| 2 is called the periodogram of y[n]. The symbol error rate (SER) performance corresponding to the LoRa demodulation has been largely studied in from [6]- [8]. Thus, given the channel gain h, it can be expressed as From (8), the SER performance of the LoRa signal in AWGN channel is simply obtained through P awgn s = P s|h=1 , and the SER of the LoRa signal over Rayleigh channel, denoted by P ray s , is given by averaging P s|h weighted by The BER P b is, in turn, obtained from the SER through P b = 2 SF −1 2 SF −1 P s . Besides the performance analysis, we also gives the complexity of the LoRa demodulation in terms of complex multiplications: we deduce from (4), (5), and (6) that these operations require N , N log(N ), and N operations, respectively.
In [10], the demodulation of LoRa has been extended to any OSR r o > 1. Similar to the case OSR r o = 1, the "dechirp" operation becomes where n = 0, 1, .., N ′ − 1. A DFT of size N ′ is then applied to y[n] in (11). The resulting spectrum Y [k], k = 0, 1, .., N ′ − 1, highlights two peaks at the positions k = m and k = m − N + N ′ , and with amplitudes . As a consequence, it is suggested in [10] to sum these peaks through It must be noticed thatỸ The SER performance of the LoRa demodulation proposed in [10], for any OSR r > 1 is given by where the term 2σ 2 in the denominator within the exponential comes from can be deduced from (14) that an OSR r o = 2 leads to the same SER as in (8)

A. OSR r o = 2
We first present the proposed LoRa demodulation method for OSR r o = 2 before generalizing to any OSR. The principle is illustrated in Fig. 1: the received signal r[n] sampled at OSR r o = 2 is split into two downsampled signals, one consisting in the even samples and the other one consisting in the odd samples (illustrated by the delay z −1 in Fig. 1) of the received oversampled signal. Then, specific "dechirp" and DFT processes are applied, and the resulting signals are combined to apply the usual LoRa demodulation (i.e. the ML estimator of m). In the following, the steps of the suggested LoRa demodulation are detailed.
Then, since exp(2jπpu( 2p+1 2 − N + k)) = 1 for any p and k values, and exp −jπ k N = 1 we obtain: where Then Y ′ [k] in (23) can be expressed in a matrix notation as (26) the SER of the suggested demodulation leads to From (27) we notice that the ratio 2kN |2| 2 (k+1)σ 2 within the exponential is twice larger than the ratio kN |2| 2 (k+1)σ 2 in (8). Since the larger the ratio, the lower the SER, we deduce that the suggested method at OSR r o = 2 actually outperforms the usual LoRa demodulation at OSR r o = 1. Note that the SER in AWGN and Rayleigh channel can be straightforwardly obtained as previously, i.e. P awgn s = P s|h=1 , and P ray Moreover the complexity of the suggested demodulation is only twice that of OSR r o = 1, since the processes for even and odd n samples are similar.

B. Generalizing for any OSR
We can generalizing the principle of the demodulation described in Fig. 1 Then, since exp(2jπpu( rop+1 ro − N + k)) = 1 for any p and k values, and exp −2jπ qk roN = 1 we obtain: where . Once again, we can rewrite (34) by noticing that Then Y (q,ro) [k] in (34) can be expressed in a matrix notation as where the complexity of the matrix multiplication straightforwardly reduces to that of the DFT if r o ∈ {1, 2, 4}.
For other values of r o , dedicated algorithms for matrix multiplication may be necessary, which limits the ease of implementation. Then, similarly to (26) we can combine the results from the r o branches to apply the ML estimator as follows:m Moreover, when a Rayleigh channel is considered, the channel gain is set to σ 2 h = 1. Fig. 2 shows the SER performance versus SNR (dB) of the LoRa demodulation at OSR 1 for both the even and the odd samples (corresponding to (18) and (23), respectively), compared with the suggested demodulation at OSR 2 (26). In Fig. 2-(a), the AWGN channel is considered, and the SNR varies in the range [−15, −5] dB, and in Fig.   2-(b), the Rayleigh channel is considered, and the SNR varies in the range [−23, 10] dB. We observe in both Figs.
2-(a) and (b) that the SER performance of the LoRa demodulation at OSR 1 is the same for both the even and the odd samples. Moreover, a gain of 3 dB is achieved by the suggested method at OSR 2 compared with OSR 1. This is consistent with the analysis, as we noticed that the ratio 2kN |2| 2 (k+1)σ 2 within the exponential in (27) is twice larger than the ratio kN |2| 2 (k+1)σ 2 in (8), corresponding to a 3 dB gain. Fig. 3 compares the SER performance results versus SNR (dB) of the suggested demodulation obtained through analysis ( (27) and (29)) and through simulations for SF 7, 8, and 9, and considering AWGN (a), and Rayleigh (b) channel models. In any configuration, we observe that the theoretical results match the simulations ones, therefore validating the suggested performance analysis. Furthermore, a gain of about 3dB is achieve between the performance of "SF N" compared with "SF N-1", which is consistent with the usual demodulation at OSR 1.

V. CONCLUSION
In this paper, we have introduced a new demodulation method for oversampled LoRa signal. The principle consists in splitting the received signal into downsampled signals, processing them using dedicated DFT, then combining them and finally applying a usual LoRa demodulation to the combined signals. The advantages of the suggested method lie in the improvement of the demodulation performance compared with the usual one performed at OSR 1, while keeping a low computational cost. Furthermore, a performance analysis in term of SER is developed, which is validated through simulations. Future works will investigate the impact of non-ideal reception including synchronization mismatch on the performance of the demodulation.