Exploiting Constructive Interference for Backscatter Communication Systems

It is noted that, most of the existing works assume that the DLI is always harmful, and thus focus on how to avoid the effect of interference for BackCom systems. However, there are several natural questions that arise: Is interference necessarily destructive in all cases? Is it possible to blaze a trail to turn the destructive interference into a constructive one? These problems have received little attention, and there have been only a few research efforts paid to answer the above questions. The constructive interference (CI) was achieved among multiple cooperative tags in centralized scenarios [24] and decentralized scenarios [25], respectively, in which several tags backscatter the same packet synchronously, which makes it possible that backscattered signals can be constructively combined, leading to an increased signal strength at the reader. Besides, a BackCom system with the symbol level precoding (SLP) technique was proposed in [26] to convert the multi-user interference into the constructive region of the desired data symbol by designing the transmit beamforming on the symbol level.

Although the CI was considered in [24] and [25] among different tags in BackCom systems, which means that CI can be found among multiple weak signals, they ignore the real protagonist, namely, the strong interference from the DL, since compared with the mutual interference among tags, the DLI is more harmful. In [26], the multi-user interference in broadcasting channels was considered to be converted into the CI, where the tag signal detection was not taken into account and there was actually no DLI. In this regard, we take up the baton in this paper to probe how to convert the DLI into a helpful interference in enhancing the backscattered signal strength, which is in contrast to most of the existing works that aim at canceling the DLI [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 21, 20, 19, 22]. Besides, for more visualization, Table I briefly summarizes the differences between this paper and the mentioned works. To the best of our knowledge, this is the first attempt to harness the DLI for the backscattered signal for BackCom systems. Specifically, we consider a multi-tag BackCom system with a dedicated power source (PS) supplying energy to the tags, and a multi-antenna reader to enhance the reception efficiency.

In a nutshell, the contributions of this paper are summarized as follows:

• •

  To evaluate whether the DLI is constructive or not, the detection error probabilities (DEPs) under the cases with and without the DL are evaluated. To make our analysis tractable, a Kullback-Leibler divergence (KLD)-oriented approach is proposed, where the KLDs under these two cases are requisitioned. By doing so, the CI can be obtained. However, the mutual interference among multiple tags also hinders the exploitation of the CI, and we consider tag selection to avoid this problem. It is worth pointing out that the KLD analysis of the DLI is new, and not presently available in existing works.

• •

  Our objective is to maximize the received signal-to-noise ratio (SNR) under the two different CI requirements (i.e., the consensual and evolved CIs) with the single- and multi-antenna PSs, respectively, via beamforming design and tag selection, which leads to two different optimization problems in each scenario. Moreover, the formulated problems are non-convex and hard to solve. In order to circumvent this difficulty, we decompose these problems based on the alternating optimization method. Then, the corresponding sub-problems for beamforming design are respectively solved by proposing two different algorithms with successive convex approximation (SCA), and the sub-problems for tag selection are solved via greedy search.

• •

  In order to gain more insight into the CI, we consider a special case with one single antenna at the reader. In this case, we are able to derive the closed-form expressions for the feasible region of the input SNR and the channel angle, where the evolved CI exists. This result is new and unique in the sense of its potential to build up the CI in practice.

• •

  Simulation results show that significant performance gain can always be achieved with the proposed algorithms compared with the benchmark algorithms in terms of the received SNR. Besides, the CI region is demonstrated for the case with multiple antennas at the reader, and the derived angle for the case with single antenna at the reader is also confirmed by simulation results. Moreover, we also demonstrate from the derived angle that it is more likely to build up the CI for the relatively large DEP tolerance or large path loss.

The remainder of the paper is organized as follows: In Section II, we present the system model and analyze the DEPs for two different cases. In Section III, we formulate two optimization problems, and two different greedy-based algorithms with SCA are proposed to solve them. Besides, a special case is also considered for more insight in Section III. Section IV gives the simulation results, and Section V concludes this paper.

Throughout the paper, scalars, vectors, and matrices are denoted by lowercase, boldface lowercase, and boldface uppercase letters, respectively. $|\cdot|$ and $||\cdot||$ denote the absolute value of a complex scalar and the $l_{2}$-norm of a vector, respectively. $(\cdot)^{H}$, det$(\cdot)$, Rank$(\cdot)$, Tr$(\cdot)$, $||\cdot||_{*}$, $||\cdot||_{1}$, and $||\cdot||_{2}$ represent the Hermitian, the determinant, the rank, the trace, the nuclear norm, the $\mathcal{L}_{1}$ norm, and the spectral norm of the matrix, respectively. $\mathcal{CN}(\mu,\sigma^{2})$ is the circularly symmetric complex Gaussian (CSCG) distribution with mean $\mu$ and variance $\sigma^{2}$. $\lambda^{\max}(\cdot)$ and $\lambda^{\min}(\cdot)$ denote the maximum and minimum eigenvalues of the matrix, respectively. $\bm{I}_{N}$ represents the $N\times N$ identity matrix. $W_{0}(\cdot)$ and $W_{-1}(\cdot)$ denote the principal branch and the bottom branch of the Lambert $W$ function, respectively.

We consider a BackCom system with single input multiple output (SIMO), as shown in Fig. 1, with a single-antenna PS, $K$ single-antenna tags ($k\in\mathcal{K}=\{1,2,\cdots,K\}$), and a reader with $M$ antennas ($m\in\mathcal{M}=\{1,2,\cdots,M\}$). In particular, all tags are powered by the PS so that each tag can backscatter its information to the reader. The channel coefficients of the PS-tag $k$ link, the PS-reader link, and the tag $k$-reader link are respectively given by $h_{\text{ST}}^{k}\in\mathbb{C}^{1\times 1}$, $\bm{h}_{\text{SR}}\in\mathbb{C}^{M\times 1}$, and $\bm{h}_{\text{TR}}^{k}\in\mathbb{C}^{M\times 1}$. Assume that channels between the transmitter and the receiver follow a block-fading based model, namely, $h_{\text{ST}}^{k}$, $\bm{h}_{\text{SR}}$, and $\bm{h}_{\text{TR}}^{k}$ remain constant during backscattering in one time interval, but vary independently in different intervals. It is further assumed that the PS transmits $N$ CSCG signals $s(n)$, $n\in\mathcal{N}=\{1,2,\cdots,N\}$ with the transmitted power $\sigma_{s}^{2}$ in one time interval [27]. Moreover, $s(n)$ for $n=\{1,2,\cdots,N\}$ are independent and identically distributed. The signal received at the $k$-th tag can be expressed as

where $z_{k}(n)$ denotes the noise at $k$-th tag. Then, the $k$-th tag backscatters the signal $r_{k}(n)$ and transmits its own binary signal $B_{k}$. Note that the noise backscattered by each tag can be ignored in the backscattering process due to its poor signal strength. Consequently, the backscattered signal by the $k$-th tag can be expressed as

where $\alpha_{k}\in[0,1]$ represents the attenuation inside the tag, which is achieved by adjusting the load impedance at the port of the antenna, and also depends on the structure mode of the antenna [28].

In order to avoid the mutual interference among tags, only one tag can be selected to backscatter its information at one slot. Noting that the signal received by the reader is not only from the selected tag (e.g., the $k$-th tag), but also from the PS. Thus, the signal received by the reader can be expressed as

where $\bm{w}(n)\in\mathbb{C}^{M\times 1}$ denotes the noise vector of the reader and is assumed to be an i.i.d. CSCG random vector with $\bm{w}(n)\sim\mathcal{CN}(0,\sigma_{w}^{2}\bm{I}_{M})$. $\sigma_{w}^{2}$ denotes the noise power. By using the receive beamforming, the received signal at the reader can be expressed as

where $\bm{v}_{k}\in\mathbb{C}^{M\times 1}$ denotes the receive beamforming vector at the reader.

Note that this framework of multi-tag selection for the Backcom system is beneficial for interference-free transmission and can efficiently combat the double-fading channel, which can be well used in radio frequency identification (RFID) [29] for logistics management, wireless body area networks (WBANs) [30] for health monitoring and data sensing, etc.

It is assumed that the BackCom system operates in the time-division multiplexing (TDD) mode, and thus the channel state information (CSI) can be obtained by the training-based channel estimation at the uplink transmission [31]. Specifically, the reader is designed to transmit two sets of pilots in two training phases [32]. In the first phase, the $k$-th tag switches its impedance into an initial matched state, and then it receives the pilot signals from the reader as excitation and does not backscatter any signal. Thus, the PS can estimate the CSI for the DL (i.e., $\bm{h}_{\text{SR}}$) via channel reciprocity. In the second phase, the $k$-th tag switches its impedance into a fixed and known backscatter state to represent a symbol $B_{k}^{0}$ after receiving the training signals in the first phase. Thus, the PS can estimate the composite channel $\bm{h}_{\text{SR}}+B_{k}^{0}h_{\text{ST}}^{k}\bm{h}_{\text{TR}}^{k}$ via the training pilots. Then, for the given $B_{k}^{0}$, the CSI of the backscattering link (BL) (i.e., $h_{\text{ST}}^{k}\bm{h}_{\text{TR}}^{k}$) can be obtained at the PS by subtracting the estimated the DL channel component from the estimated composite channel.

In this subsection, we give some analysis for the received signal when the DLI is available or not. Specifically, when the DL is available, the received signal of the reader can be rewritten with a binary hypothesis. In particular, the null hypothesis ($\mathcal{H}_{0}$) is that the $k$-tag does not backscatter the incident signal and the alternative hypothesis ($\mathcal{H}_{1}$) is that the $k$-tag backscatters the incident signal. In this regard, we have

where $\bm{h}_{0}=\bm{h}_{\text{SR}}$, $\bm{h}_{1}^{k}=\bm{h}_{\text{SR}}+\bm{h}_{\text{STR}}^{k}$, and $\bm{h}_{\text{STR}}^{k}=\alpha_{k}h_{\text{ST}}^{k}\bm{h}_{\text{TR}}^{k}$.

Thus, the distributions of the received signal $y_{k}(n)$ can be expressed as

where the variances under $\mathcal{H}_{0}$ and $\mathcal{H}_{1}$ can be expressed as

Accordingly, the received signal $\bm{y}_{k}$ at the reader in one time interval obeys the following distributions

where the covariance matrices under $\mathcal{H}_{0}$ and $\mathcal{H}_{1}$ can be expressed as $\textbf{C}_{i}^{k}=\delta_{i}^{k}\bm{I}_{N},i\in\{0,1\}.$

Let $p_{0}(\bm{y}_{k})$ and $p_{1}(\bm{y}_{k})$ denote the likelihood functions of the received signal of the reader under $\mathcal{H}_{0}$ and $\mathcal{H}_{1}$, respectively. Based on (8), we have

As a counterpart, assuming the DL is unavailable, the received signal at the reader can be expressed as

Accordingly, the received signal $\bm{\bar{y}}_{k}$ in one time interval obeys the following distributions

where $\bar{\textbf{C}}_{i}^{k}=\bar{\delta}_{i}^{k}\bm{I}_{N}$ are the covariance matrices, and

where $\bm{\bar{h}}_{0}=\bm{0}$, and $\bm{\bar{h}}_{1}^{k}=\bm{h}_{\text{STR}}^{k}$.

In this case, the likelihood functions of the received signal of the reader under $\mathcal{H}_{0}$ and $\mathcal{H}_{1}$, respectively, can be expressed as

In BackCom systems, the goal for the reader is to distinguish two different hypotheses $\mathcal{H}_{0}$ or $\mathcal{H}_{1}$ by applying a specific decision rule. In general, the priori probabilities of hypotheses $\mathcal{H}_{0}$ and $\mathcal{H}_{1}$ are assumed to be equal. As such, the detection error probability (DEP) of the reader is defined as in [33, 34]

where $\mathcal{D}_{1}$ and $\mathcal{D}_{0}$ are the binary decisions that infer whether backscattering transmission is present or not, respectively. ${\text{Pr}}(\mathcal{D}_{1}|{\mathcal{H}}_{0})$ and ${\text{Pr}}(\mathcal{D}_{0}|{\mathcal{H}}_{1})$ denote the false alarm and miss detection probabilities, respectively.

Taking the case with the DL as an example, for an optimal detector at the reader, the optimal DEP ($\xi_{k}^{*}$) for detecting the backscattered signal by the $k$-th tag can be obtained as [34, 35]

where $\mathcal{V}_{T}(p_{0},p_{1})$ denotes the total variation between $p_{0}(\bm{y}_{k})$ and $p_{1}(\bm{y}_{k})$.

Similarly, the optimal DEP ($\zeta_{k}^{*}$) of the reader for detecting the backscattered signal by the $k$-th tag under the case without the DL can be obtained as

where $\mathcal{V}_{T}(\bar{p}_{0},\bar{p}_{1})$ denotes the total variation between $\bar{p}_{0}(\bm{\bar{y}}_{k})$ and $\bar{p}_{1}(\bm{\bar{y}}_{k})$.

To evaluate the enhancement of the BackCom transmission with the assistance of the DL, an optimization problem for maximizing the received SNR at the reader is formulated by the joint optimization for the receive beamforming design and the tag selection, which is given by

where $\beta_{k}$ is the tag selection factor, i.e., $\beta_{k}=1$ denotes the $k$-th tag is selected, otherwise $\beta_{k}=0$. $\gamma=\frac{\sigma_{s}^{2}}{\sigma_{w}^{2}}$ denotes the input SNR. $\xi^{\max}$ and $\zeta^{\max}$ denote the tolerated threshold of the DEPs with and without the DL for the BackCom system. For problem (17), $C_{1}$ is the tag selection constraint. $C_{2}$ denotes the receive beamforming constraint. $C_{3}$ and $C_{4}$ are the corresponding DEP constraints, which guarantee that the DEPs with and without the DL are not greater than the corresponding tolerable DEPs, respectively. It is noted that, by setting $\xi^{\max}\leq\zeta^{\max}$, the constructive DLI can be obtained, which is defined as the consensual CI.

As can be seen, problem (17) is challenging to solve directly due to the mixed-integer objective and the non-smooth constraints. To make it more treatable, assuming the $k$-th tag has been selected, the subproblem for beamforming design can be transformed as

Unfortunately, there are still many obstacles in solving problem (18), one of which is that the total variation metrics in $C_{3-1}$ and $C_{4-1}$ are unwieldy and intractable for products of probability measures [36]. To deal with that, the Bretagnolle–Huber bound [37] is adopted, i.e.,

or

where $D_{\text{KL}}^{k}(p_{0}||p_{1})$ and $D_{\text{KL}}^{k}(p_{1}||p_{0})$ denote the KLD from $p_{1}(\bm{y}_{k})$ to $p_{0}(\bm{y}_{k})$ and the KLD from $p_{0}(\bm{y}_{k})$ to $p_{1}(\bm{y}_{k})$, respectively. Specifically, the expression of $D_{\text{KL}}^{k}(p_{0}||p_{1})$ can be written as¹¹1It is noted that the Gaussian-distributed signal input is restrictively assumed in this paper, which is commonly used for analysis of communication systems due to its mathematical tractability. The impact of practical modulation type and order on the exploration of constructive interference will be another work, and is left for future work.

Based on (15) and (19), we have the lower bound related to the KLD for $\xi_{k}^{*}$ under the case with the DL, which satisfies

Similarly, the KLD of the corresponding likelihood functions under different hypotheses without the DL can be given by

Correspondingly, the lower bound of the DEP related to the KLD in the case without the DL can be expressed as

Note that the maximum value of problem (17) can be achieved when $\xi_{k}^{*}$ and $\zeta_{k}^{*}$ are equal to their lower bounds, respectively, in (22) and (24). Thus, $C_{3-1}$ and $C_{4-1}$ can be rewritten by, respectively,

or equivalently

Thus, problem (18) can be transformed into the following problem, i.e.,

However, it is difficult to solve problem (27) due to the tight coupled relationship in $C_{3-2}$ and $C_{4-2}$. To deal with it, we propose Theorem 1.

Theorem 1: $C_{3-2}$ and $C_{4-2}$ can be transformed into the following equivalent forms, i.e.,

where $F(x)=\exp(W_{0}(-\exp(-x))+x)$, $f^{\min}\triangleq\frac{D^{\min}}{N}+1$, and $g^{\min}\triangleq\frac{E^{\min}}{N}+1$.

Proof: Please see Appendix A.

Based on Theorem 1, problem (27) can be transformed into the following problem, i.e.,

However, it is still hard to solve problem (29) directly due to the non-convexity of $C_{3-3}$. To deal with this, the SCA-based method is applied [38]. In each iteration, we construct a global underestimator of the intractable term via its first-order Taylor approximation, which is given by

where $\bm{\bar{v}}_{k}(l)$ is the feasible point of $\bm{v}_{k}$ in the $l$-th iteration of the SCA.

As a result, problem (29) can be finally expressed as

Note that problem (31) is a standard convex problem, which can be solved by CVX. The detail of this algorithm is shown in Algorithm 1. Here, the computational complexity for Algorithm 1 is evaluated. Given the number of constraints, interior point methods with a vector variable $\bm{v}_{k}$ of size $M\times 1$ will take $\mathcal{O}(\sqrt{M}\ln(1/\epsilon))$ iterations, with each iteration requiring $\mathcal{O}(M^{3})$ arithmetic operations for the worst case [39], where $\epsilon$ denotes the precision of the interior point algorithm. Letting the maximum iteration number for SCA is $L$, the total computation complexity for beamforming design is $\mathcal{O}(LM^{3.5}\ln(1/\epsilon))$.

So far, we have studied the consensual CI for the BackCom system in the previous subsection. However, the relationship between $\xi_{k}^{*}$ and $\zeta_{k}^{*}$ is unknown. In this subsection, we are going to investigate another CI, i.e., evolved CI, by exploiting the following relationships between $\xi_{k}^{*}$ and $\zeta_{k}^{*}$, i.e.,

It is worth mentioning that when $\xi_{k}^{*}>\zeta_{k}^{*}$ holds, the dilemma will be that the involvement of the DL leads to a larger DEP than the case without the DL, which would be intolerable for perfectionists if the DL is required to have no negative impact on the BackCom system. On the contrary, when $\xi_{k}^{*}\leq\zeta_{k}^{*}$ holds, we can assert that introducing the DL does not destroy the detection performance compared with the case without the DL, and can even improve it.

Motivated by this observation, a new optimization problem for achieving the evolved CI can be reformulated, which is expressed as

Assuming that the $k$-th tag has been selected, problem (33) can be further rewritten as

Based on the same approach for the DEP-to-KLD transformation, serving the purpose of maximizing the objective of problem (34), the left-hand items in $\bar{C}_{3-1}$ and $\bar{C}_{4-1}$ are replaced by their lower bounds, i.e.,

Further, to guarantee the CI evolution, the lower-bound relaxation is applied, thus (35) is transformed into

Correspondingly, problem (34) can be further rewritten as

where

Remark 1: By comparing problems (27) and (37), it can be seen that the potential adverse condition that lurks in problem (27) against the construction of CI is removed in problem (37). The purpose of problem (37) is to explore whether it is always possible to create a larger KLD by using the DL compared with that without the DL, which is different from the former. For illustration, Fig. 2 is provided to portray the relationship between different KLDs. By doing so, we can effectively enhance the backscattering strength without affecting or even reducing the DEP compared with the conventional BackCom system.

Note that problem (37) can be regarded as evolving from problem (27), but the proposed algorithm for problem (27) can not suitable for problem (37) since $\bar{C}_{3-2}$ poses different challenge in solving the receive beamforming. This motivates us to design a new algorithm for problem (37).

In order to make problem (37) more tractable, we propose Theorem 2.

Theorem 2: $\bar{C}_{3-2}$ can be transformed into the equivalent form, i.e.,

Proof: Please see Appendix B.

Based on Theorem 1 and Theorem 2, problem (37) can be transformed into the following problem, i.e.,

Then, by introducing a new variable $\bm{W}_{k}=\bm{\hat{v}}_{k}\bm{\hat{v}}_{k}^{H}$, $\bar{C}_{3-3}$ and $\bar{C}_{4-3}$ can be transformed, respectively, into

and

where $\bm{H}_{1}^{k}=\bm{h}_{1}^{k}(\bm{h}_{1}^{k})^{H}$, $\bm{H}_{0}=\bm{h}_{0}\bm{h}_{0}^{H}$, and $\bm{\bar{H}}_{1}^{k}=\bm{\bar{h}}_{1}^{k}(\bm{\bar{h}}_{1}^{k})^{H}$.

Thus, problem (40) can be expressed as

However, problem (43) is still a non-convex problem caused by the non-convexity in $\bar{C}_{3-4}$. In general, it is difficult or even intractable to obtain the global optimal solution to such a non-convex problem. To make it more treatable, we introduce a slack variable $t_{k}$ such that $\text{Tr}(\bm{H}_{0}\bm{W}_{k})\leq t_{k}$, problem (43) can be transformed into the following problem, i.e.,

It is noted that the rank-one constraint $\bar{C}_{5}$ is an impediment to solving problem (44). Semi-definite relaxation (SDR) has been commonly adopted to tackle the rank-one constraint, which, however, may not result in a rank-one matrix. Moreover, some approximation methods, such as Gaussian randomization, may cause performance deterioration when the number of antennas is quite large [40]. To deal with this, $\bar{C}_{5}$ is equivalently transformed into the following difference-of-convex function [41], such as

where $||\bm{W}_{k}||_{*}=\sum\limits_{i}\delta_{i}(\bm{W}_{k})$ and $||\bm{W}_{k}||_{2}=\delta_{1}(\bm{W}_{k})$. It is noted that for any $\bm{X}$ belonging to the positive semi-definite matrix, $||\bm{X}||_{*}\geq||\bm{X}||_{2}=\underset{i}{\mathop{\max}}~{}\delta_{i}(\bm{X})$ is always held. The inequality can be treated as an equation if and only if $\bm{X}$ is a rank-one matrix, especially, $\delta_{1}(\bm{X})=||\bm{X}||_{2}$ and $\delta_{i}(\bm{X})=0,\forall i=2,\cdots,M.$ As a result, (45) can be efficiently met for the rank-one constraint $\bar{C}_{5}$. However, (45) is still non-convex. To deal with it, a penalty function is employed, and thus problem (44) can be equivalently transformed into the following problem, such as

where $\bar{C}_{5}$ is relaxed to a penalty term added to the objective function, and $\chi$ is a constant which penalizes the objective function for any matrix whose rank is not one.

However, problem (46) is still an intractable problem with any $\chi>0$ due to the non-convexity of the objective function. To deal with this, the SCA method is applied. Specifically, for a given $\bm{W}_{k}(j)$ in the $j$-th iteration of SCA, a convex upper bound for the penalty term can be obtained by using first-order Taylor expansion, which can be expressed as

where $\bar{\bm{W}}_{k}(j)\triangleq||\bm{W}_{k}(j)||_{2}+{\rm Tr}\left[\bar{\bm{u}}\left(\bm{W}_{k}(j)\right)\left(\bar{\bm{u}}(\bm{W}_{k}(j))\right)^{\rm H}\left(\bm{W}_{k}-\bm{W}_{k}(j)\right)\right]$ and $\bar{\bm{u}}\left(\bm{W}_{k}(j)\right)$ represents the eigenvector corresponding to the largest eigenvalue of $\bm{W}_{k}(j)$ [41].