Joint channel training and passive beamforming design for intelligent reflecting surface-aided LoRa systems

1.   Introduction

Long Range (LoRa) is one of the most widely deployed low-power wide-area network (LPWAN) techniques for Internet-of-Things (IoT) in the world, whose signal modulation is based on a patented chirp spread spectrum ^{[1,2,3,4,5,6,7,8,9,10,11,12,13]}. Though LoRa is known to strike a good balance between coverage and throughput with low energy consumption, its performance may still be limited in real-world fading environments due to severe path loss ^[14]. To address this issue and to improve the performance of LoRa over fading channels, a single-input multiple-output (SIMO) LoRa system has been studied in ^[1]. However, deploying multiple antennas is often costly and practically challenging due to limited hardware cost, size, and complexity.

Recently, an intelligent reflecting surface (IRS), composed of a number of passive and reconfigurable reflective units, has emerged as a cost-, energy-, complexity-efficient alternative to antenna arrays, which effectively reflects incident signals such that a propagation environment becomes better suited for signal reception ^{[15,16,17,18,19,20,21,22,23,24,25,26]}. It also has significant potentials for LoRa in performance improvement; e.g., in ^[14], the IRS has been used as a means to carry additional information via the differential phase shift keying (DPSK) modulation such that a higher data rate than the conventional LoRa system can be achieved.

The novelty of our work lies that different from ^[14], in this paper, we consider deploying the IRS to improve the link reliability of the LoRa system. On top of this, the novel contributions of our paper are highlighted as follows:

● For the considered IRS-aided LoRa system, it is imperative to acquire and utilize channel state information (CSI), which is rather a challenging task. Accordingly, we propose the optimal training design for accurate CSI acquisition with LoRa modulation, which has never been reported in the literature.

● Also, considering the errors in the acquired CSI, we derive a closed-form expression of effective signal-to-noise ratio (SNR), based on which we optimize the passive beamforming (i.e., reflection pattern) for the IRS.

● Furthermore, various simulation results in realistic propagation environments are presented to validate the superiority and effectiveness of the proposed IRS-aided LoRa system. From the simulation results, useful engineering insights are drawn, which have never been provided in the literature.

2.   Signal model

As shown in Figure 1, we consider an IRS-aided LoRa system where an IRS with $L$ reflecting elements is deployed to assist the transmission of LoRa signals from a transmitter (e.g., IoT device or sensor) to a receiver (e.g., gateway or base station), each with a single antenna. ^* In LoRa, the transmitted signal takes the form of an up chirp ^[1,28]:

^*Our work for a single antenna can be readily extended to the case with multiple antennas, such that the proposed scheme is applied to each pair of a transmit antenna and a receive antenna.

where $P > 0$ is the transmission power and $m \in \{0, 1, \cdots, M-1\}$ denotes a modulation symbol. Let $h_{0}$, $f_{i}$, and $g_{i}$ denote complex-valued channel coefficients of the links from the transmitter to the receiver, from the transmitter to the IRS, and from the IRS to the receiver, respectively. The received signal at the receiver is given by

where $0 \leq \theta_{i} < 2 \pi$ is the phase shift of the $i$th IRS element, and $w(n)$ is the received noise with zero mean and variance $\sigma^{2}$. Also, $M \triangleq 2^{\rm SF}$, where ${\rm SF} \in \{ 7, \cdots, 12 \}$ denotes the spreading factor. Note that to properly adjust $\{ \theta_{i} (n) \}$ for better signal reflection, one first need to acquire the CSI, i.e., $h_{0}$ and $\{ f_{i} g_{i} \}$, which will be elaborated in the next section.

3.   Optimal training design for LoRa channel estimation

In LoRa, every transmitted frame (or packet) is initiated with a priori known preamble (or training) signal ^[28], based on which the CSI can be estimated. Typically, the base up chirp (i.e., $\{ x_{0} (n) \}_{n = 0}^{M-1}$) is chosen as the preamble signal. The received preamble signal, denoted by ${\mathbf{y}} \triangleq \left[y(0), \cdots, y(M-1) \right]^{\mathsf T}$, can be written as

where

Assuming that $M \geq L + 1$ and ${\mathbf{X}}_{{\mathbf{\theta}}}$ is of full-rank, the least squares channel estimate can be obtained as

for which the estimation mean square error (MSE) is given as $\mathbb{E} \big[\big\| \boldsymbol{h} - \hat{\boldsymbol{h}} \big\|^{2} \big] = \sigma^{2} {\rm tr} \big[\left({\mathbf{X}}_{{\mathbf{\theta}}}^{\mathsf H} {\mathbf{X}}_{{\mathbf{\theta}}} \right)^{-1} \big]$ ^[29].

The goal of the training design is to minimize the channel estimation MSE by optimizing the reflection pattern of the IRS as

It is known that an optimal solution to the above problem satisfies ${\mathbf{X}}_{{\mathbf{\theta}}}^{\mathsf H} {\mathbf{X}}_{{\mathbf{\theta}}} = \frac{P}{M} {\mathbf{I}}$ ^[29]. From (4), this condition is equivalent to^†

^†To fulfill this optimality condition (i.e., for a proper training), it is required that $M \geq L+1$ (i.e., the number of data measurements for training needs to be larger than that of channel coefficients to be estimated). In LoRa, this requirement is usually met even for a large number of IRS elements (i.e., large value of $L$) due to a long symbol duration (i.e., large value of $M$) supported by LoRa. Specifically, LoRa supports six different SF values (from 7 to 12) ^[5,28,33], and thus, the value of $M$ in LoRa ranges from $2^{7} = 128$ to $2^{12} = 4096$. In practice, therefore, the value of $M$ (in the order of hundreds or thousands) is usually much larger than that of $L$ (in the order of tens or hundreds).

implying that the optimality is achieved when the channel estimation errors become uncorrelated with each other. In general, there exist infinitely many solutions to problem (P1) fulfilling (6). One of the simplest choices is given by the primitive $M$th roots of unity as

4.   Passive beamforming design for LoRa signal reflection

Subsequent to the channel estimation, we now consider the design of the IRS's passive beamforming utilizing the acquired CSI to effectively reflect the incident LoRa signals. Considering the imperfectness of the estimated CSI, the received signal in (2) can be represented as

where $\tilde{h}_{i} \triangleq h_{i} - \hat{h}_{i}$, $i = 0, 1, \cdots, L$. Note that the first term is the desired signal component, whereas the second term is an additional noise induced by the channel estimation errors. Therefore, from (8), we define the effective SNR as

To enhance the signal detection performance, we aim at maximizing the effective SNR in (9) for each $n$ by optimizing the reflection coefficients of the IRS as follows:

where we drop the index $n$ for simplicity. From the well-known triangle inequality $\left| \hat{h}_{0} + \sum_{i = 1}^{L} \hat{h}_{i} e^{ j \theta_{i} } \right| \leq \sum_{i = 0}^{L} \left| \hat{h}_{i} \right|$, the solution to problem (P2) can be obtained by achieving the equality as

It is rather intriguing to note from (11) that the IRS's reflection pattern is fixed when reflecting the LoRa signals, whereas it varies over time when estimating the channels.

5.   Simulation results

In simulations, we consider an IRS-aided LoRa system with ${\rm SF} = 12$. The bandwidth and noise power spectral density are set to 500 kHz^‡ and $- 174$ dBm/Hz, ^§ respectively, and thus, the noise variance is $\sigma^{2} = -137.01$ dBm. Also, $h_{0}$, $\{ f_{i} \}$, and $\{ g_{i} \}$ are modeled as the Rician fading channels with the Rician factor set to $2$ and the average path losses given by $d_{0}^{- \beta}$, $d_{f}^{- \beta}$, and $d_{g}^{- \beta}$, respectively, where $d_{0}$, $d_{f}$, and $d_{g}$ denote the distances between the transmitter and the receiver, between the transmitter and the IRS, and between the IRS and the receiver, respectively, and $\beta$ is the path loss exponent. Unless stated otherwise, we use $\beta = 3.2$, $d_{f} = 2$ m, $d_{g} = 3$ km, ^¶ and $d_{0} = \sqrt{d_{f}^{2}+d_{g}^{2}}$ throughout the simulations.

^‡As specified in the standards ^[5,28,33], LoRa supports three BW settings (125,250 or 500 kHz) and six different SF values (from 7 to 12).

^§This corresponds to the thermal noise floor for 1 Hz bandwidth at room temperature (20℃) ^[34].

^¶It has been studied in ^[8] and ^[30,31,32] that the coverage of LoRa ranges from 100m to 30km in different scenarios. Therefore, it appears feasible for the LoRa modulation to cover a communication range of 3 km even with the passive IRS.

In Figure 2, the channel estimation MSE of the proposed training design is shown versus the transmission power $P$ when $L = 16$ (left sub-figure) and versus the number $L$ of reflecting elements at the IRS when $P = - 75$ dBm (right sub-figure). In Figure 2, for comparison, we also present the performance of the following baseline schemes:

● Baseline Scheme I: A training scheme based on a directional reflection pattern with phase shifts randomly chosen such that $\theta_{i} (n) \in [- \delta, \delta]$, $\forall i, n$. We set $\delta = 0.035$ rad.

● Baseline Scheme II: A training scheme based on an on-off reflection pattern (i.e., $\theta_{i} (n) \in \{ 0, 1\}$, $\forall i, n$) such that reflecting elements are sequentially turned on one by one.

● Baseline Scheme III: A training scheme based on an alternating optimization of the reflection coefficients $\theta_{i} (n)$, $\forall i, n$.

From Figure 2, it can be seen that the proposed scheme considerably outperforms the baseline schemes. The channel estimation performance of all the schemes improves as $P$ increases (resp. $L$ decreases), because the impact of noise decreases (resp. the number of channel coefficients to be estimated decreases).

In Figure 3, the bit error rate (BER) performance of the following IRS-aided LoRa systems is compared:

● IRS-aided LoRa I: The proposed passive beamforming design with CSI estimated by the proposed training design.

● IRS-aided LoRa II: The IRS-aided LoRa system with the proposed passive beamforming using perfect CSI.

● IRS-aided LoRa III: The IRS-aided LoRa system with an omnidirectional passive beamforming (i.e., $\theta_{i} = \frac{2 \pi i}{L}$, $\forall i$) using no CSI.

● Conventional LoRa: The conventional LoRa system without the IRS.

In the IRS-aided LoRa systems Ⅰ and Ⅱ, coherent detection is employed to evaluate the BER performance with estimated CSI and perfect CSI, respectively; whereas, in the IRS-aided LoRa system Ⅲ and the conventional LoRa system, non-coherent detection is adopted.

In Figure 3, the BERs of the various LoRa systems are shown versus $P$ (left sub-figure) and versus $d_{g}$ when $P = 0$ dBm (right sub-figure) for different values of $L$, where for the IRS-aided system Ⅰ, the CSI is estimated with $P = 0$ dBm. From Figure 3, it can be observed that the IRS-aided LoRa system Ⅰ yields almost the same performance as the IRS-aided LoRa system Ⅱ and performs much better than the other systems, thereby validating the effectiveness of the proposed passive beamforming design as well as demonstrating the feasibility of coverage extension of the LoRa system with the aid of IRS. The reason for such superiority is that in the proposed scheme, the channel training and IRS beamforming are jointly optimized such that the IRS beamforming compensates for the CSI errors in the channel training, whereas the other schemes suffer from their suboptimalities. Particularly, the performance improvement of the IRS-aided LoRa system Ⅰ or Ⅱ becomes more pronounced as $L$ increases, but that of the IRS-aided LoRa system Ⅲ is marginal due to the lack of CSI.

6.   Conclusions

The optimal training scheme minimizing the channel estimation MSE and the optimal passive beamforming maximizing the effective SNR were derived for the IRS-aided LoRa system, which significantly enhanced the BER performance over the conventional LoRa system.

Use of AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

This research was supported by the National Research Foundation of Korea(NRF) grant funded by the Korea government(MSIT) (No. 2022R1A4A1033830).

Conflict of interest

All authors declare no conflicts of interest in this paper. The manuscript was written through the contributions of all authors. All authors have approved the final version of the manuscript.

Prof. Jae-Mo Kang is the Guest Editor of special issue "Advances of Artificial Intelligence-based mathematical modeling optimization in engineering applications" for AIMS Mathematics. Prof. Jae-Mo Kang was not involved in the editorial review and the decision to publish this article.