Outage probability computation in multi-backscatter systems with multi-modes of operation

In this article, we study the outage performance of an ambient multi-backscatter system supported by non-orthogonal multiple access (NOMA) downlink transmission, where legacy users receive information either directly from the base station (BS) or from multiple backscatter devices (BDs) as well. Specifically, we characterize the network connections between the BS and network users into three modes of operation. In the first mode, the BS communicates with the users via the BDs by exploiting NOMA. In the second mode, the BS communicates directly with the users by employing selection combining (SC) beamforming. In the third mode, the users receive signals from both the BS and BDs simultaneously, by using maximal ratio combining (MRC) to combine the BS and BDs links at the users. Consequently, we derive the exact outage expressions of each mode and utilize Monte Carlo simulations to validate the individual exact expressions.


INTRODUCTION
The rapid deployment of internet-of-things (IoTs) in diverse systems such as energy systems (Smart Grid), healthcare systems, industrial systems, and transportation systems, has seen a tremendous push into researching ways of powering these billions of devices in a sustainable and environmentally friendly way [1], [2]. Ambient backscatter communication (AmBC) has been proposed by the research community as a solution to address this unique challenge [3]. AmBCs can free up IoT devices from needing batteries as backscatter devices (BDs) harvest energy from traditional radio frequency (RF) transmitters such as television and radio transmitters to transmit information to nearby IoT devices [2]- [13]. Yang et al. in [4], the authors note that the harvested energy from ambient RF transmitters is adequate to provide power to high-rate battery-less sensors. Therefore, AmBC can be used in IoT environments where it's not convenient to replace or maintain sensor batteries, e.g., in wide-body area networks (WBANs) [14], [15]. However, due to the broadcast nature of ambient RF transmitters, AmBC tend to suffer from direct-link interference from the RF transmitter [4]- [6]. Several methods such as the design of maximum-likelihood (ML) detectors, and interference cancellation techniques, have been proposed to eliminate the direct-link interference, see [4], [6] and references therein for more details.
In this study, we propose utilizing non-orthogonal multiple access (NOMA) technology to address this problem, owing to its ability to simultaneously serve multiple wireless users over the same radio resource via linear superposition coding at the transmitter and successive interference cancellation (SIC) at the user's receiver [7], [16]- [18]. Therefore, integrating NOMA with AmBC networks shows great promise for future  [21]. However, in this article, we focus on the contribution of NOMA to AmBCs performance. Our main contributions are: Firstly, we derive exact outage expressions for three practical AmBC link cases. In the first case, the base station (BS) communicates with the users via the BDs. In the second case, the BS communicates directly with the users.
In the third case, the users receive signals from both the BS and BDs simultaneously; Secondly, from these expressions, we analyze the role of channel vectors, power allocation coefficients and backscatter reflection coefficient on outage probability. All expressions are confirmed using Monte Carlo simulations. The paper is organized is being as. In section 2, we describe the proposed system model and signal characteristics at the users. Thereafter, in section 3, we derive exact outage probability (OP) expressions depending on the mode of operation. In section 4, we explain results, followed by the summary in section 5.

RECEIVED SIGNALS AT TWO USERS
In this paper, we study a downlink wireless system to allow the base station (BS) serving two multiple antennas users (user U 1 and user U 2 ) under assistance of multiple backscatter devices (BDs), shown in Figure  1. In some practical case, the second user U 2 might connect with the BS directly.
BS broadcasts the information x BS to multi-backscatter (BD n ), n = 1, 2, . . . N , which is given by: where x 1 and x 2 are the messages with unit power transmitted to U 1 and U 2 , respectively. Further, P S is the total transmit power of the BS with power allocation parameters a 1 and a 2 , respectively, with a 1 > a 2 and a 2 + a 1 = 1.
The maximal-ratio combining (MRC) is used with beamforming vector to achieve optimal transmission scheme as in [22], [23] with ∥w i ∥ = 1: where ∥·∥ denotes the Euclidean norm of a matrix.
The BD the BS signal to U 1 and U 2 with its own message s, where E |s| 2 = 1. Thus, U 1 and U 2 receive one type of signal: the backscatter link signal from the BD. The received signals at U 1 and U 2 can thus be written as: where β is a complex reflection coefficient used to normalize, h 1 and h 2 are the 1 × M channel vector of BD − U 1 link and 1 × K channel vector of BD − U 2 link, n U1 and n U2 are the additive white Gaussian noise (AWGN) with zero mean and variance matrix σ 2 U1 I M and σ 2 U2 I K , respectively. Hence, the signal x 1 is detected successfully and x 2 as interference. Then, the signal-interference to ratio at U 1 is given by: Thus, the instantaneous SINR at U 2 to detect x 1 and the instantaneous SNR at U 2 to detect x 2 is given respectively as: 3. OUTAGE PROBABILITY COMPUTATION OF DIFFERENT MODES 3.1. Scheme 1: outage probability for backscatter-non-orthogonal multiple access system without direct link In this study, we assume independent Rayleigh-distributed random variables (RVs) for all the channels.
The OP of U 1 is: where γ 1 = 2 R1 − 1 with R 1 being the target rate at U 1 to detect x 1 . Theorem 1. The exact OP expression of U 1 is calculated as: where (9) is based on the condition of a 1 > γ 1 a 2 Proof: ❒ ISSN: 2302-9285 Based on [24], (3.471.9). We can rewrite (10) as: Next, the OP of U 2 is given by: where γ 2 = 2 R2 − 1 with R 2 being the target rate at U 2 to detect x 2 , θ 2 = γ2 ρa2 . Similarly with solving OP I

U1
can be achieved OP I U2 as: 3.2. Scheme 2: outage probability of directed link by selection combining mode In the MRT scheme, to maximize the signal-to-interference noise (SINR) at U 1 and U 2 , BS uses a beamforming transmit vector w r = h † r ∥hr∥ , r ∈ {SU 1 , SU 2 } to signal x BS before sending w r to users, h SU1 and h SU2 are the 1 × M and 1 × K channel vector of BS to U 1 link and BS to U 2 link, respectively. The BS transmits signals directly to two users as y r = ∥h r w r ∥ x BS + n r , r ∈ {SU 1 , SU 2 }.
We have SINR at U 1 as: and SINR and the successive interference cancellation (SIC) at U 2 as γ SU2,1 = ∥hSU 2 ∥ 2 a1ρ and U 2 as Finally, the instantaneous SINRs at U 1 and U 2 based on selection combining, can be written as: The OP of U 1 is given by: Ω h 0 t −m dt and ∆ (x) = (a 1 − xa 2 ) ρβ 2 . Using [24], (3.471.9) and after some transform. F γ1,1 (x) and F γ SU 2 (x) can be express respectively as: Substituting (18) and (19) into (16), the close-form of OP of OP II U1 is obtained as: Next, the OP of U 2 is: Similarly, OP II U2 can be achieved as:

Scheme 3: outage probability of directed link by maximal ratio combining mode
In this scheme, the backscattering link signals and direct link signals are combined by maximal ratio combining (MRC) at the two users. Hence, after using MRC the received SINR two users are given by: Then, the OP at U 1 can be calculated as: It is very difficult to obtain the close-form solution of the integral of (24). Therefore, we use the approach in [25] by making an I-step staircase approximation to the actual triangular integral region in (24) with sufficiently large I to obtain: Thus, the OP at U 2 is given by: ❒ ISSN: 2302-9285 where θ 2 = γ2 ρa2 . Υ 1 can be calculated as: where γ (x, y) is the lower incomplete Gamma function [24], (3.351.1). And Υ 2 is written as: But it is very difficult to obtain the close-form solution of Υ 2 , to do so, we use Gaussian-Chebyshev quadrature [26], (25.4.38), to get: Substituting (26) and (28) into (25), OP III U2 is given by: where ξ l = cos 2l−1 2L π and Λ (x) = θ2

NUMERICAL RESULTS
In this section, we set and Ω SU1 = d −τ SU1 and τ is the path-loss exponent setting to be τ = 2. In particular, main parameters can be seen in Table 1. In addition, the Gauss-Chebyshev parameter is selected as L = 100 to yield a close approximation.   Figure 3, we also observe that the users experience different outage performances depending on the complex reflection coefficient β. The best performances are achieved by scheme 3, with the worst performance obtained by user 2 in scheme 1. Figure 4 plots the relationship between OP and γ 1 = γ 1 (dB), varying β = 0.3, and β = 0.7. From Figure 4, we observe that some network users performance curves approach an outage probability ceiling. Generally, the outage performance reduces with increase in γ 1 = γ 1 (dB). The best performances are achieved by scheme 3, with the worst performance obtained by scheme 1.  Figure 5, we note that several OP curves approach an OP floor with increasing a 2 , while other OP curves converge at 0 when a 2 = 1. Figure  6 plots the relationship between OP and N = M = K, with ρ = 5 (dB). From Figure 5, we observe that maximal ratio combining delivers the best outage performance with the OP for backscatter-NOMA system performing the worst. In both Figures, user 2 in scheme 1 had the worst OP performance, due to the weak backscatter device signals that it recovers, therefore, it can easily experience outage in any condition. With user 1 in scheme 3 achieving the best performance due to the signals from the very near multi-backscattering devices and the further away base station direct link being combined by maximal ratio combining at its receiver.

CONCLUSION
In this paper, we provided the outage probability (OP) analysis of an ambient multi-backscatter system with multi-modes of operation. We derived exact solutions of OP for different users depending on the mode of operation. We observe from the simulation results, that the direct link between the base station and users affects the OP. In future work, we will consider ergodic capacity of the system.