I Introduction and Background
Backscatter communication (BSC) technology, comprising lowcost tags, without any bulkier radio frequency (RF) chain components, has gained significant recent attention owing to its potential in realizing the sustainable and pervasive ultralowpower networking [2]. The key merit of BSC, not requiring any signal modulation, amplification, or retransmission by the tags, is that it shifts the high cost and large formfactor constraints to the reader side, leading to the tagsize miniaturization, which is the basic need of numerous smart networking applications [3]. Despite these potential merits, the widespread utility of BSC is limited by shorter readrange [4] and lower achievable data rates [5]
. Further, since the tags are lightweight, passive, chipless, and batteryfree devices that do not have their own radio circuitry to process incoming signals or estimate the channel response, multiple antennas at the reader are required to separate out the backscattered signals from multiple tags by exploiting spatial multiplexing to enhance data rate and BSC reliability. Also, this multiantenna reader can implement energy beamforming (EB) during the carrier transmission to significantly improve BSC range. Therefore, to enable efficient BSC from multiple tags, there is a need for investigating the novel jointlyoptimal transmit (TX) and receive (RX) beamforming at the multiantenna reader and backscattering designs at the tags.
Ia StateoftheArt
BSC is based on the decoding of backscattered information signals at the reader as received from the multiple lowpower tags. These tags communicate their information to the reader by respectively modulating their load impedances to control the strength, phase, frequency, or any other characteristics of the carrier signal(s) as received and reflected back to the reader. Depending on the energy constraints of the tags, BSC models can be divided into three groups: (a) passive [6], (b) semipassive [7], and (c) active [8]. Though both passive and semipassive tags depend on the carrier signal excitation from the reader, the latter are also equipped with an internal power source to enable better reliability and longer range of accessibility. Whereas, the active tags are batterypowered and can broadcast their own signal, thereby achieving much longer high link quality read range at the cost of bulkier size and higher maintenance requirements. Similarly, based on the network configuration, three main types of BSC models are:

Monostatic: With carrier emitter and backscattered signal reader being the same entity, this model can share antennas for transmission to and reception from tags [9].

Bistatic: Here, the emitter and reader are geographicallyseparated two different entities [4]. This model can help in achieving a longer range.

Ambient: Widely investigated model where emitter is an uncontrollable source and the dedicated reader decodes the resulting backscattered information from the tags [8].
As a consequence, monostatic configurations are cheaper because they require relatively smaller number of antenna elements due to their sharing in fullduplex settings. In contrast, the bistatic architectures ones can achieve longer readrange at the expense of combined higher antenna count for emission and reading purposes due to the geographicseparation of emitter and reader. As shown in Fig. 1, we investigate a monostatic BSC system with multiple singleantenna semipassive tags and a multiantenna reader working in fullduplex mode [9, 10]. Henceforth, each antenna element at the reader is used for both carrier emission and backscattered signal reception [11]. This adopted configuration with a large antenna array at reader can maximize the BSC range, while meeting the desired rate requirements of tags, by exploiting the array gains during the downlink carrier transmission to the multiple tags, and uplink multiplexing gains during backscattered signals reception at the reader. Also, this setting is one of the most practical ones because it moves the computationalcomplexity and formfactor constraints from the lowpower tags to a relativelypowerful reader. Recent, experimental results [12, 13] have corroborated this fact that coverage range can be significantly improved up to a few hundred meters by exploiting array gains at reader. However, these gains in the multipleinputmultipleoutput (MIMO) readerassisted BSC can be strongly enhanced by optimally designing the underlying transceiver (TRX).
Noting that the tagstoreader backscatter uplink channel is coupled to the readertotags downlink one, novel higher order modulation schemes have been investigated in [9] for the monostatic MIMOBSC settings. Whereas, a frequencymodulated continuouswave BSC system with monostatic reader, whose one antenna was dedicated for transmission and remaining for the reception of backscattered signals, was studied in [14] to precisely determine the number and position of active tags. On similar lines, considering a multiantenna power beacon assisted bistatic BSC model, robust inference algorithms, not requiring any channel state or statistics information, were proposed in [15]
to detect the sensing values of multiple single antenna backscatter sensors at a multiantenna reader by constructing Bayesian networks and using expectation maximization principle. Pairwise error probability and diversity order achieved by the orthogonal space time block codes over the dyadic backscatter channel (i.e., monostatic BSC system with multipleantennas at the reader for transmission and reception from a multiantenna tag) were derived in
[9] and [16]. Authors in [17] designed a data detection algorithm for an ambient BSC system where differential encoding was adopted at the tag to eliminate the necessity of channel estimation (CE) in minimizing the underlying sum bit error rate (BER) performance. The asymptotic outage performance of an adaptive ambient BSC scheme with Maximum Ratio Combining (MRC) at the multiantenna reader was analyzed in [18] to demonstrate its superiority over the traditional nonadaptive scheme. Adopting the BSC model with multiple antennas the reader, authors in [19] first presented maximum likelihood (ML) based optimal combiner for simultaneously recovering the signals from emitter and tag. Then, they also investigated the relative performance of the suboptimal linear combiners (MRC, Zero Forcing (ZF) and Minimum MeanSquared Error (MMSE)) and successive interference cancellation (SIC) based combiners, where MMSESIC combiner was shown to achieve the nearML detection performance. In [20], a dyadic backscatter channel between multiantenna tag and reader was studied to quantify the impact of underlying pinhole diversity and the RF tag’s scattering aperture on enhancing the achievable BER performance and tag operating range. Authors in [21] noted that if separate reader transmitter and receiver antennas are used in conjunction with multiple RF tag antennas, the envelope correlation between the forward and backscatter links can be significantly reduced to enhance the BER performance. Furthermore, investigating the optimal detection threshold for ambient BSC in [22], it was found that an increasing array size can yield larger gains in BER at low signaltonoise (SNR), with lower returns in high SNR regime.On different lines, with the goal of optimizing harvested energy among tags, suboptimal EB designs for monostatic multiantenna reader were investigated in [23]. More recently, a leastsquaresestimator for the BSC channels between a multiantenna reader and singleantenna tag was proposed in [10]. Based on that a linear MMSE based channel estimator was designed in [24] to come up with an optimal energy allocation scheme maximizing underlying singletag BSC performance while optimally selecting number of orthogonal pilots for CE.
IB Notations Used
The vectors and matrices are respectively denoted by boldface lowercase and capital letters.
, , and respectively denote the Hermitian transpose, transpose, and conjugate of matrix . and respectively represent zero, allones, and identity matrices. stands for th element of matrix , represents for th column of , and stands for th element of vector . With and respectively being the trace and rank of matrix , and respectively represent Frobenius norm of a complex matrix and absolute value of a complex scalar. is used to denote a square diagonal matrix with ’s elements in its main diagonal and for representing the vectorization of matrix into a column vector. and represent the inverse and squareroot, respectively, of a square matrix , whereas means that is positive semidefinite and operator represents the Hadamard product of two matrices. Expectation is defined using andrepresents the principal eigenvector corresponding to maximum eigenvalue
of a Hermitian matrix . Lastly, with , and respectively denoting the real and complex number sets,denotes the complex Gaussian distribution with mean
and covariance .Ii Motivation and Significance
This section first discusses the novel aspects of this work targeted towards addressing an important existing research gap along with the potential scope of the proposed designs. In the latter half, we summarize the main contributions of this paper.
Iia Novelty and Scope
Since the information sources in BSC, i.e., tags, do not have their own RF chains for communication, the two key roles of the reader are: (a) carrier transmission to excite the tags in the downlink, and (b) efficient detection of the received backscattered signals in the uplink. Therefore, new TRX designs are needed because the requirements of RX design for the uplink involving effective detection of the backscattered information signals at the reader as received from the multiple tags are different from those of the TX beamforming in the downlink involving singlegroup multicastingbased carrier transmission. Furthermore, the underlying nonconvex optimization problem is more challenging than in conventional wireless networks because the corresponding backscattered throughput definition involves product or cascaded channels. Also, the resourcelimitations of tags put additional constraints on the precoder and combiner designs.
The existing works [11, 14, 12, 13, 15, 9, 16, 19, 23, 10, 24, 22, 25] on multiantenna reader supported BSC did not focus on utilizing reader’s efficacy in designing smart signal processing techniques to overcome the radio limitations of tags by jointly exploiting the array and multiplexing gains. To the best of our knowledge, the joint TRX design for the multiantenna reader has not been investigated in the literature yet. Also, the backscattering coefficient (BC) optimization at the tags for maximizing the sumbackscatteredthroughput is missing in the existing stateofthe art on the multitag BSC systems. Recently, a few BC design policies were investigated in [25] for maximizing the average harvested power due to the retrodirective beamforming at multiantenna energy transmitter based on the backscattered signals from the multiple single antenna tags. But [25] ignored the possibility of uplink backscattered information transfer, and only focused on the downlink energy transfer.
In this work, we have presented novel design insights for both TRX and BC optimization. Specifically, new solutions for the individuallyoptimal designs and asymptoticallyglobaloptimal jointdesigns are proposed along with an efficient lowcomplexity iterative algorithmic implementation. These designs can meet the basic requirement of extending the BSC range and coverage by imposing the nontrivial smart signal processing at multiantenna reader. Significance of the proposed designs is corroborated by the fact that they can yield substantial gains without relying on any assistance from the resourceconstrained tags in solving the underlying nonconvex sumbackscatteredthroughput maximization problem. Our optimal designs are targeted for serving applications with the overall BSC systemcentric goal, rather than individual taglevel, where the besteffort delivery is desired to maximize the aggregate throughput. Practical utility of these designs targeted for monostatic BSC can be easily extended for addressing the needs of other BSC models. Also, we discourse how the proposed optimization techniques can be used for solving the nonconvex throughput maximization problems in wireless powered communication networks (WPCN). Thus, this investigation, providing designs for achieving longer readrange and higher backscatteredthroughput, enables widespread applicability of BSC technology in ultralowpower emergingradio networks for lastmile connectivity and InternetofThings networking.
IiB Key Contributions and Paper Organization
Fivefold contribution of this work is summarized below.

A novel optimization framework has been investigated for maximizing the sumbackscatteredthroughput from multiple singleantenna tags in a monostatic BSC setting. It involves: (i) smart allocation of reader’s resources by optimally designing the TRX, and (ii) maximizing the benefit of tags cooperation by optimally designing their BC. The corresponding basic building blocks and problem definition addressed are presented in Section III.

Next, the asymptoticallyoptimal joint designs are derived in Sections VA and VB for the high and low SNR regimes, respectively. We show that both these jointlyoptimal designs, which can be efficiently obtained, provide key novel design insights. Using these results as performance bounds, a lowcomplexity iterativealgorithm is outlined in Section VC to obtain a nearoptimal design.

To corroborate the practical utility of the proposed designs, in Section VI we discuss their extension to address the requirements of application networks like WPCN, bistatic and ambient BSC models with imperfect channel state information (CSI) and multiantenna tags.

Detailed numerical investigation is carried out in Section VII to validate the analytical claims, present key optimal design insights, and quantify the performance gains over the conventional designs. There other than comparing the efficacy of individuallyoptimal designs, we have shown that on an average sumthroughput gains can be achieved by the proposed joint TRX and BC design over the relevant benchmarks [27, 19].
Throughout this paper, the main outcomes have been highlighted as remarks and Section VIII concludes this work with the keynotes and possible future research extensions.
Iii Problem Definition
We start with briefly describing the adopted system model and network architecture, followed up by the BSC and semipassive tag models. Later, we present the expression for the achievable backscatteredthroughput at reader from each tag.
Iiia System Model and Network Architecture
We consider a multitag monostatic BSC system comprising singleantenna semipassive tags, and one fullduplex reader equipped with antennas which is responsible for simultaneous carrier transmission and backscattered signal decoding. Hereinafter, the th tag is denoted by with and the reader is denoted by . We assume that these tags are randomly deployed in a square field of length meters (m), with being at its center as shown in Fig. 1. To enable fullduplex operation [11], each of the antennas at can transmit a carrier signal to the tags while concurrently receiving the backscattered signals.
The multiantenna adopts linear precoding and assigns each a dedicated precoding vector . We denote by the vector of independent and identically distributed (i.i.d.) symbols as simultaneously transmitted by . Hence, the complex baseband transmitted signal from is given by , and we assume that there exists a total power budget to support this transmission. The resulting modulated reflected data symbols as simultaneously backscattered from the tags are respectively spatially separated by with the aid of linear decoding vectors as denoted by . Here combiner is used for decoding ’s message. This restriction on TRX designs to be linear has not only been considered to address lowpowerconstraints of BSC, but also because for , these designs are nearlyoptimal [28].
IiiB Adopted BSC and Tag Models
In contrast to the practical challenges in implementing the fullduplex operation in conventional communication systems involving modulated information signals, the unmodulated carrier leakage in monostatic fullduplex BSC systems can be efficiently suppressed [11]. Further, we consider semipassive tags [7] that utilize the RF signals from for backscattering their information and are also equipped with an internal power source or battery to support their low power onboard operations. Thus, they do not have to wait for having enough harvested energy, thereby reducing their overall access delay [8]. However, note that this battery is only used for powering the tag’s circuitry to set the desired modulation or BC and for regular operations like sensing. Also, these benefits of longer BSC range and higher rate due to an ontag battery suffer from few problems like extra weight, larger size, higher cost, and batterylife constraints.
For implementing the backscattering operation, we consider that each modulates the carrier received from via a complex baseband signal denoted by [4]. Here, the loadindependent constant is related to the antenna structure of the th tag and the loadcontrolled reflection coefficient switches between the distinct values to implement the desired tag modulation [29]. Without the loss of generality, to produce impedance values realizable with passive components, we assume that the effective signal from each tag satisfies because the scaling factor corresponding to the magnitude of the ’s complex baseband signal can be included in its reflection coefficient or BC definition [6]. The higher values of reflect increasing amounts of the incident RF power back to which thus result in higher backscattered signal strength and thereby maximizing the overall readrange of . Whereas, the lower value of BC for a tag implies that its backscattering to causes lesser interference for the other tags.
The to wireless reciprocalchannel is denoted by an vector . Here, parameter represents average channel power gain incorporating the fading gain and propagation loss over to or to link. Although we have considered i.i.d. fading coefficients for all to channels due to sufficient antenna separation at reader [16, 9, 24], the proposed designs in this work can also be used for the BSC settings with dependent and not necessarily identically distributed fading scenarios. In this paper we assume that this perfect CSI for each is available at to investigate the best achievable performance. However, our proposed designs can be extended to imperfect CSI cases as discussed in Section VIA2 and their robustness under inaccuracy in CSI is also demonstrated in Section VII.
Therefore, on using these models, the baseband received signal at is expressed as:
(1) 
where for each are i.i.d. symbols and
is the zeromean Additive White Gaussian Noise (AWGN) vector with independent entries having variance
.IiiC BackscatteredThroughput at
We note that the backscattered noise strength due to the AWGN power is practically negligible [6, 7, 8, 9, 4, 10] in comparison to the corresponding carrier reflection strength due to the signal power . So, ignoring this backscattered noise, which in comparison to the excitation power gets practically lost during backscattering from tags, the received signal available for information decoding at , as obtained using the definition (1), is:
(2) 
where the vector represents the received zeromean AWGN at and is the noise power spectral density. Applying the linear detection at , the received signal can be separated into streams by multiplying it with detection matrix and the corresponding decoded information signal is:
(3) 
As each of the streams can be decoded independently, the complexity of the above linear receiver is on the order of , where denotes the cardinality of the finite alphabet set of for each . Thus, with , the th element of , to be used for decoding the backscattered message of , is:
(4) 
Therefore, using (IIIC) and the backscattered message definition from Section IIIB, the resulting signaltointerferenceplusnoiseratio (SINR) at from each is given by:
(5) 
where denotes the effective transmit SNR at as realized due to the carrier transmission from , which itself on ignoring the backscattered noise can be defined as:
(6) 
Thus, the backscatteredthroughput for at is given by:
(7) 
From the above throughput definition which has been extensively used in existing multitag BSC investigations, we notice that the key difference from the throughput in conventional networks is the existence of the product or cascaded channels definition and additional BC parameters.
Lastly, the resulting sumbackscatteredthroughput , which is the systemlevel performance metric as maximized in the current multitag monostatic BSC setting, is given by:
(8) 
Next we use the above sumthroughput definition for carrying out the desired optimization of TRX and BC designs.
Iv Sum Backscattered Throughout Maximization
Here we first mathematically formulate the joint optimization problem in Section IVA and discuss its salient features. Next, after discussing the reasons for nonconvexity of the problem, we present the individual optimization schemes for obtaining the optimal TX precoding, RX beamforming, and BC designs in Sections IVB, IVC, and IVD, respectively.
Iva Mathematical Optimization Formulation
The joint reader’s TRX and tags’ BC design to maximize the achievable sumbackscatteredthroughput at , as defined in (8), can be mathematically formulated as below:
where is the available transmit power budget at , and respectively the practicallyrealizable [6] lower and upper bounds on BC for each tag . All the computations for obtaining the jointlyoptimal solution of are performed at , which then sets its TRX to the optimal one and instructs the tags to set their respective BC accordingly. Here, the battery energy consumption at semipassive tags in setting their respective BC as per ’s instruction is negligible in comparison to their regular operations [7].
We notice that although has convex constraints, it is in general a nonconvex optimization problem because its nonconcave objective includes coupled terms involving the product of optimization variables, i.e., precoders , combiners , and BC for each . Despite the nonconvexity of joint optimization problem , we here reveal some novel features of the underlying individual optimizations that can yield the globaloptimal solution for one of them while keeping the other two fixed. In other words, we decouple this problem into individual optimizations and then try to solve them separately by exploiting the reduced dimensionality of the underlying problem. We next discuss the individual optimizations, onebyone, starting with the TX precoder optimization at during the downlink carrier transmission.
IvB Optimal Transmit Precoding Design at
The proposed method for obtaining the optimal TX beamforming vectors for each at can be divided into two parts. In the first part, we discourse the relationship between precoder designs for the different tags in the form of Lemma 1. Thereafter proving the concavity of the equivalent semidefinite relaxation (SDR) [30] for the precoder design optimization problem, the randomization process [31] is used to ensure desired implicit rankone constraint on the matrix solution. The implementation steps are provided in the form of Algorithm 1.
Lemma 1
The optimal precoder designs for the tags that maximize the resulting sumbackscatteredthroughput are identical. In other words, for each .
Proof:
The proof is given in Appendix A.
Lemma 1 actually implies that transmits with same precoder for all the tags, i.e., multicasting is optimal TX design. This is due to the fact that carrier transmission from is just to effectively excite (powerup) the tags, and this excitation can be made most efficient when the TX precoder aligns with the strongest eigenmode of the matrix . A similar observation was made in context of the precoder designs for efficient downlink energy transfer in WPCN [33, 34].
Subsequently, with the above result, the sumbackscatteredthroughput can be rewritten below as a function of the common precoding vector , satisfying , RX beamforming matrix and BC vector for tags:
(9) 
where backscatteredthroughput as function of , received SINR as a function of , and transmit SNR as a function of for are respectively defined as:
(10a)  
(10b)  
(10c) 
However, since is still nonconcave in , we next show that by using an equivalent SDR with matrix definition satisfying rankone constraint, we can resolve this issue.
Lemma 2
The sumbackscatteredthroughput, for a given combiner design for and BC vector for the tags, is a concave function of the matrix variable .
Proof:
Please refer to Appendix B for details.
independent uniformly distributed random vectors
on , where both these vectors are in size.Following Lemmas 1 and 2 along with definition (B.1), the TX beamforming optimization for a given combiner and BC design can be formulated as the optimization problem below:
which on ignoring is a convex problem with objective function to be maximized being concave in (cf. Lemma 2) and constraints being convex. So, although the optimal solution of can be obtained using any standard convex optimization toolbox, like the CVX MATLAB package [36], there lie two challenges. First, the objective function does not satisfy the Disciplined Convex Programming (DCP) rule set for using the CVX toolbox [36, 32] because each summation includes the ratio of linear functions of . Second issue is that the optimal as obtained after solving the SDR needs to implicitly satisfy the rankone constraint .
The first of the abovementioned issues can be resolved using the recently proposed quadratic transform technique for maximizing the multiple ratio concaveconvex linear fractional programming problems [37]. Further as on ignoring , is a convex problem, the stationary point as obtained using this quadratic transformation yields the globaloptimal solution of . Hence, we can obtain the globaloptimal by solving SDR using CVX toolbox. Thereafter, the second issue can be resolved by deploying the randomization process [31] to ensure the implicit satisfaction of the rankone constraint . The detailed algorithmic steps resolving these two issues are summarized in iterative Algorithm 1. It starts with an initial precoding matrix with , which is motivated by the fact that for singletag case, the respective maximal ratio transmission (MRT) design is optimal [24]. Then, after initializing the auxiliary variable vector as in step 5, we apply the quadratic transformation as suggested in [37, Theorem 1] to each underlying SINR term in (B.1) and maximize the corresponding convex reformulation with respect to , for a given , as denoted by in step 7. Thereafter, we continue to update and optimize in an iterative fashion. Since the sumbackscatteredthroughput is concave in , this sequence of convex optimization problems converges to a stationary point of , which is also its globaloptimal solution, with nondecreasing values for the underlying objective after each iteration. When this improvement in the throughput value reduces below a certain acceptable threshold, the Algorithm 1 terminates with the globaloptimal precoding matrix . Next, for this precoding solution to satisfy the rankone constraint we deploy the randomization process [30] as given by steps 10 to 25 of Algorithm 1 which returns the optimal TX precoder . The randomization process involves generation of set of candidate weight vectors and selecting the one which yields the highest sumbackscatteredthroughput among them. Here, we have set samples as mentioned in the results section of [31] because it maintains a good tradeoff between the solution quality and complexity.
IvC Receive Beamforming or Combiner Design at Reader
For a given precoder and BC , the optimal RX beamforming problem is formulated as:
Below we outline a key result defining the optimal RX beamforming or combiner design at .
Lemma 3
For a given precoder design for and BC vector for the tags, the optimal combiner design is characterized by the Wiener or MMSE filter, as defined below:
(11) 
Proof:
Firstly, from (5) and (7) we notice that for each depends only on its own combiner . Accordingly, we can maximize the individual rates or SINRs in parallel with respect to , while satisfying their underlying normalization constraint . Further, as the in (5) can be alternatively represented as a generalized Rayleigh quotient form [38, eq. (16)], the optimal combiner for each , can be obtained as the generalized eigenvector of the matrix set with largest eigenvalue. Using it along with and Lemma 1, the optimal combiner in (11) is obtained.
IvD Backscattering Coefficient (BC) Optimization at Tags
Mathematical formulation for this case is presented below:
We would like to mention that although is a nonconvex problem, it can be solved globally, but in nonpolynomial time, using the block approximation approach [39]. Furthermore, even though the sumbackscatteredthroughput defined in (8) is nonconcave function of the BC vector , below we present a key property for the backscatteredthroughput for each , which we have exploited in designing a computationallyefficient solution methodology.
Proposition 1
For a given precoder and combiner design for , both the backscattered SINR and throughput for each tag is pseudolinear in .
Proof:
As SINR involves the ratio of two linear functions, and of , using the results of [40, Tables 5.5 and 5.6] we note that is both pseudoconvex and pseudoconcave in the BC vector . Now as functions which are both pseudoconvex and pseudoconcave are called pseudolinear [26], each is pseudolinear in . Further, since the monotonic transformations preserve pseudolinearity of a function [40], we observe that throughput is also pseudolinear in .
Here, it is worth noting that since the summation operation does not perverse pseudolinearity [26], the sumbackscatteredthroughput maximization with respect to is not a convex problem and hence does not possess globaloptimality. However, we notice that can be alternatively casted as an optimal power control problem for the sumrate maximization over the multiple interfering links [41, and references therein]. For instance, recently in [37] an application of fractionalprogramming was proposed for efficiently obtaining a stationary point for the nonconvex power control problem over the multiple interfering links. We have used that to yield an efficient lowcomplexity suboptimal design for BC vector .
The detailed algorithmic implementation is outlined in Algorithm 2. It starts with an initial BC vector with all its entries being which is motivated by the fact that for highSNR regime, the optimal BC is characterized by the fullreflection mode. Then, after initializing the auxiliary variable vector as in step 5, we apply the quadratic transformation as suggested in [37, Theorem 1] to the underlying each SINR term and maximize corresponding convex reformulation with respect to , for a given , as denoted by in step 7. Thereafter, we continue to update and optimize in an iterative fashion. Since each throughput term is nondecreasing and concave in its respective SINR term, which itself is pseudolinear in , this sequence of convex problems converges to a stationary point of with nondecreasing values for the underlying objective after each iteration. When this improvement in throughput value reduces below a tolerance , the Algorithm 2 terminates with a nearoptimal BC .
V Proposed LowComplexity Optimal Designs
Using the key insights developed for the individuallyoptimal TX precoding, RX beamforming, and BC vector designs in previous section, now we focus on deriving the jointlyoptimal TRX and BC designs by simultaneously solving the original joint optimization problem (but with for each based on Lemma 1) in the three optimization variables and . We start with presenting novel asymptoticallyoptimal joint designs for the TRX at and BC at tags in both low and high SNR regimes. In this context, first a joint design for lowSNR application scenarios is proposed, followed by the other one for the highSNR regime. These two efficient lowcomplexity asymptoticallyoptimal designs shed new key design insights on the bounds for jointlyglobaloptimal solution. Thereafter, we conclude by presenting a Nelder–Mead (NM) method [26, Ex. 8.51] based lowcomplexity iterative algorithm that does not require the explicit computation of complex derivatives for the objective sumbackscatteredthroughput.
Va AsymptoticallyOptimal Design Under HighSNR Regime
First from Lemma 3 we revisit that regardless of the precoder and BC design, the optimal combiner is characterized by the MMSE filtering defined in (11). Next, we recall that under the highSNR regime, the ZFbased RX beamforming is known to be a very good approximation for the Wiener or MMSE filter [38, eq. (14)]. So, using the definition below,
(12) 
the ZF based combiner matrix is given by:
(13) 
As the RX beamforming vector has to satisfy constraint , the optimal combiner for the highSNR scenarios, as obtained from in (13), is given below:
(14) 
Here, the ZFbased RX beamforming vectors satisfy:
(15) 
Thus, with , the sumbackscatteredthroughput under highSNR regime where employs ZF based combiner , is given by:
(16) 
Next revisiting the matrix definition , the equivalent SDR for jointly optimizing the remaining variables and is formulated below as , which is followed by Lemma 4 outlining a key result to be used for solving it.
Lemma 4
is concave in with optimal being equal to for each .
Proof:
Please refer to Appendix C for the proof.
Using Lemma 4 and ignoring , we notice that , with for each tag, is a convex problem in the optimization variable . Further, since satisfies the DCP rule, the CVX toolbox can be used to obtain the optimal , as denoted by . However, for this precoding solution to satisfy the rankone constraint we need to deploy the randomization process, as discussed in Section IVB and implemented via steps 10 to 25 of Algorithm 1 while setting in step 10, to finally get the optimal precoder .
Remark 1
Under highSNR regime, optimal precoder is obtained by solving SDR with followed by randomization process. Whereas, optimal combiner follows ZF based design and all the tags are in fullreflection mode, i.e., BC vector .
VB Novel Joint Design For LowSNR Applications
Under lowSNR regime, we can use the following two approximations for simplifying :
(17a)  
(17b) 
where (17a) is owing to the fact that under lowSNR regime, the backscattered signals from all the other tags, causing interference to the tag of interest, is relatively very low in comparison to the received AWGN. Whereas, (17) is obtained using the approximation . Using these properties, the sumbackscatteredthroughput reduces to:
(18) 
where is based on the individual optimizations of combiner and BC vector respectively following MRC and fullreflection mode in this scenario. So, with above as objective and as variable, the corresponding maximization problem can be formulated as below:
From , we notice that the TX precoder design at that maximizes the sum received power at the tags also eventually yields the maximum sumbackscatteredthroughput from them. Thus, the optimal precoder, same for all tags and called TX energy beamforming (EB), is denoted by:
(19) 
where is the right singular vector of the matrix that corresponds to its maximum eigenvalue . So, the total sum received power at tags is:
(20) 
On substituting and , for each , in (11) and using Lemma 3, the optimal combiner (MMSE based design) at tag for the lowSNR regime is given by:
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