Exploiting joint sparsity in compressed sensingbased RFID
 Martin Mayer^{1, 2}Email author,
 Gabor Hannak^{1} and
 Norbert Goertz^{1}
https://doi.org/10.1186/s136390160025y
© Mayer et al. 2016
Received: 21 December 2015
Accepted: 23 February 2016
Published: 21 April 2016
Abstract
We propose a novel scheme to improve compressed sensing (CS)based radio frequency identification (RFID) by exploiting multiple measurement vectors. Multiple measurement vectors are obtained by employing multiple receive antennas at the reader or by separation into real and imaginary parts. Our problem formulation renders the corresponding signal vectors jointly sparse, which in turn enables the utilization of CS. Moreover, the joint sparsity is exploited by an appropriate algorithm.
We formulate the multiple measurement vector problem in CSbased RFID and demonstrate how a joint recovery of the signal vectors strongly improves the identification speed and noise robustness. The key insight is as follows: Multiple measurement vectors allow to shorten the CS measurement phase, which translates to shortened tag responses in RFID. Furthermore, the new approach enables robust signal support estimation and no longer requires prior knowledge of the number of activated tags.
Keywords
Compressed sensing Approximate message passing Joint sparsity Multiple measurement vectors Backscatter communication Multiple access1 Introduction
In radio frequency identification (RFID), a reader device interrogates tags for identification. A large branch of RFID deals with the identification of a multitude of tags that may identify, e.g., products in a store, parts on a conveyor belt, or items in a warehouse. Predominantly, passive tags that are powered by the field emitted by the reader are employed. Such tags are cheap and can be produced in high volumes, which has made RFID a ubiquitous technology. An overview is provided in [1, 2].
Reducing the identification time and promoting quick identification of many tags has been a major research field in recent years. The key problem in customary protocols arises from collisions during interrogation: If several tags respond simultaneously, their responses superimpose at the reader and cause collisions, resulting in loss of data. The widely adopted EPCglobal standard [3] employs a collision avoidance protocol called frame slotted ALOHA (FSA)—a summary of collision avoidance schemes is provided by [4, 5]. These protocols separate the tag responses in the time domain. The authors in [6–8] improve FSA by performing collision recovery, which is accomplished by separating tag responses in the inphase and quadraturephase plane and by employing multiple receive antennas at the reader to resolve multiple collisions.
The compressed sensing (CS)based identification protocols [9–14], on the other hand, cope with simultaneously responding tags and exploit collisions. This bears several advantages over FSAbased schemes, as reported in, e.g., [9, 14]. In particular, CS enables a quicker identification and provides an increased noise robustness. In this work, we demonstrate that CSbased schemes can be improved significantly if joint sparsity is exploited. To the best of the authors’ knowledge, this is truly novel in the realm of CSbased RFID, denoted as CSRFID in the sequel. We discuss how multiple receive antennas at the reader and the separation into real and imaginary parts lead to a CS problem with inherent joint sparsity.
We employ the Bayesian structured signal approximate message passing (BASSAMP) algorithm [15] in order to exploit the joint sparsity. While the number of activated tags is assumed to be known or has to be estimated in an additional step in the current stateoftheart protocols [9–14], we demonstrate how our approach eliminates the need for such prior knowledge by exploiting a novel signal support estimation scheme. Note that we do not present a new protocol but an improvement of the CS formulation that is applicable to all present protocols [9–14]. The benefits of our novel formulation comprise a strongly increased identification throughput, an increased noise robustness, and an implicit estimation of the number of activated tags.
Outline: In Section 2, we briefly summarize CS and introduce the concept of multiple measurement vectors and joint sparsity. Section 3 gives an overview of CSRFID and discusses our novel contributions. Section 4 explains the origins of joint sparsity in CSRFID and highlights the advantages of its exploitation. In Section 5, the channel model and channel coefficient distribution are introduced, and the BASSAMP algorithm is defined for the RFID scenario. Section 6 deals with the estimation of the signal support and the number of activated tags. Numerical results are provided and discussed in Section 7, and we conclude in Section 8.
Notation: Boldface letters such as A and a denote matrices and vectors, respectively. The bth column of a matrix A is denoted a _{ b }, while the nth entry of the bth column is denoted a _{ n,b }. The superscript (·)^{T} denotes the transposition of a matrix or vector, and (·)^{H} denotes the conjugate transpose. The vectorization of an M×N matrix into a column vector is denoted \(\mathbf {A}(:) \equiv \left [\mathbf {a}_{1}^{{\mathrm {T}}},...,\mathbf {a}_{N}^{{\mathrm {T}}}\right ]^{{\mathrm {T}}}\).
The N×N _{ B } allone matrix is denoted \(\mathbf {1}_{N\times N_{B}}\). The Frobenius norm of a matrix A is denoted \(\\mathbf {A}\_{F}=\sqrt {\text {trace}(\mathbf {A} \mathbf {A}^{\mathrm {H}})}\). Calligraphic letters such as \(\mathcal {S}\) denote sets.
The cardinality of a set \(\mathcal {S}\) is denoted by \(\mathcal {S}\). Random variables, vectors, and matrices are written in sans serif font as x, x, and X, respectively, while realizations thereof are written in serif as x, x, and X.
2 Compressed sensing and joint sparsity
where \(\mathbf {y} \in \mathbb {C}^{M}\) is the measurement vector, \(\mathbf {A} \in \mathbb {R}^{M \times N}\) is the fixed sensing matrix, and \(\mathbf {w} \in \mathbb {C}^{M}\) is additive measurement noise. If the signal vector x features only K≪N nonzero entries, it is said to be Ksparse.
measurements, with a small constant c. Most importantly for CSRFID, this holds for Rademacher distributed sensing matrices where the entries are picked from the set {−1,1} with equal probability.
While many recovery algorithms aiming at solving (1) for x have been proposed in literature [18, 19], we utilize the versatile approximate message passing (AMP) framework that was introduced in [21–23]. These algorithms enable efficient recovery with low computational complexity, while maintaining excellent recovery performance. Note that the CSRFID schemes presented in [9–11] utilize computationally demanding convex optimization algorithms, while the schemes in [12–14] employ an efficient AMP algorithm that iteratively solves the least absolute shrinkage and selection operator (LASSO) [24]. In this work, we utilize a powerful extension of the AMP algorithm—termed BASSAMP and introduced in [15]—that allows to leverage prior knowledge and joint sparsity. A detailed specification follows in Section 5.
where \(\mathcal {S}_{\mathbf {x}_{b}}\) contains the indices of the nonzero entries in x _{ b }. Aside from having the same support, all vectors are K sparse with \(\mathcal {S}_{\mathbf {x}}=K\ll N\).
where \(\mathbf {y}_{b} \in \mathbb {C}^{M}\), \(\mathbf {A}^{\!(b)}\in \mathbb {R}^{M \times N}\), and \(\mathbf {w}_{b} \in \mathbb {C}^{M}\). Let us collect the data blocks in matrices: \(\mathbf {Y} = [\mathbf {y}_{1},...,\mathbf {y}_{b},..., \mathbf {y}_{{N_{B}}\phantom {\dot {i}\!}}]\), \(\mathbf {X} = [\mathbf {x}_{1},...,\mathbf {x}_{b},..., \mathbf {x}_{{N_{B}}\phantom {\dot {i}\!}}]\), and \(\mathbf {W} = [\mathbf {w}_{1},...,\mathbf {w}_{b},..., \mathbf {w}_{{N_{B}}\phantom {\dot {i}\!}}]\).
If all N _{ B } sensing matrices are identical, i.e., \(\mathbf {A} \equiv \mathbf {A}^{\!(b)}, \forall b\in \mathcal {B}\), (4) can be rewritten as Y=A X+W. This is the relevant case for our approach.
3 CSRFID: Related work and novel contributions
The emerging field of CS triggered a shift of paradigm in signal processing and digital communications that also sparked new ideas in the field of RFID. In this work, we focus on protocols where a single reader identifies a multitude of tags. It is investigated how multiple receive antennas at the reader improve the performance.
We assume commonly used passive RFID tags that employ backscatter modulation to convey information back to the reader [25]. Before going into details about the CSRFID protocols presented in [9–14], let us first discuss what those schemes have in common.
3.1 CSRFID problem formulation

We intend to identify K tags that are activated by the reader (i.e., are in read range).

After a query from the reader, all K tags respond simultaneously with a signature sequence.

Activated tag k responds with signature \(\mathbf {s}_{a_{k}}\), where a _{ k }∈{1,...,N} is the signature index, and there are N possible signatures in total.

Each signature entails M realvalued (ASK) symbols, i.e., \(\mathbf {s}_{a_{k}} \in \{b_{0},b_{1}\}^{M}\) (the two amplitudes of backscatter modulation).

The N signature sequences form the columns of the signature matrix S=[s _{1},...,s _{ N }]∈{b _{0},b _{1}}^{ M×N }. The signature sequences are generated pseudorandomly, each with a certain seed [26]. The possible seeds are known to the reader such that it can construct S.
where the nonzero entries of \(\mathbf {x}\in \mathbb {C}^{N}\) store the complexvalued channel coefficients and dictate which columns in S are selected, and with \(\mathbf {w}\in \mathbb {C}^{M}\) being additive Gaussian measurement noise with i.i.d. entries \(\mathsf {w}_{m} \sim \mathcal {CN}(0,\sigma _{\mathsf {w}}^{2})\). In our application, x is a sparse vector with K≪N nonzero entries, i.e., there are much fewer signatures (or tags to be identified) than exist in total. Our goal is to recover x from z knowing S, because the locations of the nonzero entries in x tell us which signatures have been chosen by the tags. This information is used to directly identify the tags [12, 14] or to establish a handshake mechanism in order to read out the tag information in an additional step [9–11, 13].
3.2 CSRFID overview
Let us give an overview of the individual protocols. Most tag identification protocols can be separated into two subsequent phases:
Tag acquisition refers to the process of obtaining information about the activated tags in order to communicate with them. Data readout refers to the process of obtaining the data (payload) of the acquired tags. For example, the widely employed FSA protocol [3] schedules the activated tags to respond during time slots in a frame, thereby trying to avoid collisions in the acquisition phase. The tags respond with a 16bit pseudorandom sequence called RN16 [3]. The reader acquires the uncorrupted RN16 sequences from collision free slots, which enables a successful handshake mechanism with the corresponding tags. After acquisition, the data that identifies the tags is read out in a sequential manner (tag by tag). Let us discuss how these phases are handled by CSRFID protocols. Buzz: This CSbased scheme was introduced in [9].
During the acquisition phase, the tags respond simultaneously with pseudorandom sequences that are seeded by the tag’s temporary identifier, which is a 16bit random number (i.e., the RN16 number a tag would have picked for FSA). This is formulated as a CS measurement (5). Because the total number of possible identifiers (signatures in S) is N=2^{16}, the CS measurement (5) features a very large sensing matrix S that renders an efficient recovery of x infeasible. The scale of the problem is reduced by hashing the identifiers into buckets [9] and eliminating the buckets that contain no energy, thereby strongly reducing the number of possible signatures (and, consequently, N). However, this requires knowledge of the number of activated tags K that has to be estimated in a prior step. An improved scale reduction that utilizes a gradient algorithm was introduced in [10]. Another improvement that does not require arbitrary restrictions of the huge initial identifier space was proposed in [11].
In the data readout phase, the tags respond simultaneously as well. Based on the temporary identifier (that is now known to the reader), each tag generates a random sequence of bits. If a bit is ‘1’, the tag transmits its data, whereas it is silent if a bit is ‘0’. This results in a rateless code; the superposition of the randomly encoded tag responses can be decoded by a belief propagation decoder, for details, see [9]. CSF: In [12], we introduced a scheme to quickly identify tags in applications with fixed inventory, e.g., a book store with N books, and K≪N books are brought to the checkout to be identified. Each of the N tags features a unique signature (identifier) that is not based on a random number.
During the acquisition phase, the tags respond simultaneously with their signature sequence—this process is cast as a CS measurement (5). Recovering x from S yields complete identification because each signature corresponds to a unique item of the inventory, no data readout phase is required for identification.
Another novelty proposed in [12] was the utilization and investigation of an AMP recovery algorithm that enables efficient iterative recovery of largescale CS problems. CSR: In [13, 14], a flexible alternative to CSF was introduced that allows for arbitrary inventory sizes.
During the acquisition phase, the tags respond simultaneously with a signature sequence that is randomly chosen from a pool of N possible signatures (N is now a design parameter). Recovering x form S yields the estimated set of assigned signatures. A scale reduction as in Buzz is not required.
In the data readout phase, these signatures are enquired and the corresponding tags that recognize their signature transmit their data. In [13, 14], the data is read out in a sequential manner (tag by tag). However, it is also possible to employ the rateless code scheme from Buzz [9].
In [14], the optimal choices of signature length M and signature pool size N based on the number of tags K have been discussed.
How RFID protocols handle the two phases of tag identification
Phase 1  Phase 2  

Tag acquisition  Data readout  
FSA  Schedule responses into time slots (avoid collisions)  Sequential (list) 
Buzz  Concurrent responses, CS measurement (exploit collisions)  Simultaneous (rateless code) 
CSF  Concurrent responses, CS measurement (exploit collisions)  Not required 
CSR  Concurrent responses, CS measurement (exploit collisions)  Sequential or simultaneous 
3.3 Novel contributions

We identify the origins of joint sparsity in CSRFID and provide a mathematical problem formulation. In particular, a reader with multiple receive antennas features multiple jointly sparse signals, and a separation into real and imaginary parts doubles their number.

We adapt BASSAMP—an algorithm used for CS recovery of jointly sparse signals—to the RFID problem formulation. This adaptation involves the calculation of several functions—used in the algorithm—for a specific channel coefficient prior distribution. We perform a rigorous calculation for a dyadic channel model, where the individual channels are Gaussian distributed, and finally provide closed form expressions for those functions. This allows for a straightforward implementation of the algorithm.

We propose a relaxation of the channel prior distribution in order to obtain simpler functions and to further reduce the computational complexity (reader side) of the implementation. This relaxation is validated by experiments.

We introduce a novel approach for robust signal support estimation that utilizes joint sparsity. This enables implicit estimation of the number of activated tags K. FSA requires K for an optimal choice of the frame size, Buzz requires K to reduce the scale of the CS problem, and CSF and CSR require K to determine the ideal signature length for optimal identification throughput. With our proposed approach, K can be implicitly estimated during CS recovery.

We compare CSRFID to a FSAbased collision recovery scheme and show the superior performance of our proposed approach. It is investigated how the number of receive antennas at the reader influences the performance.
4 Exploiting joint sparsity in CSRFID

The same mean squared error (MSE) performance as standard AMP can be achieved by a significantly reduced number of CS measurements M. Consequently, shorter signature sequences (length M) can be employed for tag acquisition.

This increases the acquisition throughput and reduces the jitter sensitivity [14] (jitter refers to link frequency deviations among tags). The noise robustness is improved as well. Furthermore, passive tags require less energy during the acquisition phase.

Support estimation (location of nonzero entries) can be improved significantly by combining soft information from multiple vectors, see Section 6. This leads to fewer identification cycles [14] (i.e., fewer repetitions of the acquisition phase) and quicker identification. Furthermore, the number of activated tags K can be estimated implicitly.
jointly sparse vectors.
This reformulation is necessary in order to obtain an appropriate sensing matrix that is compatible with the employed recovery algorithm. Signature matrix S comprises entries from the set {b _{0},b _{1}}. Assuming signature sequences (columns in S) where b _{0} and b _{1} are equally likely, all sequences have the same mean. Consequently, the reformulation features a sensing matrix \(\mathbf {A}\in \{\overline {b},\overline {b}\}^{M\times N}\) with zero mean columns and with \(\overline {b}=b_{1}b_{0}/2\), i.e., Rademacher distributed up to a constant factor. This renders A an appropriate sensing matrix for CS recovery that satisfies (2). In order to apply BASSAMP, we have to specify the functions used in Algorithm 1 (see below). Note that we use the original algorithm from [15] without algorithmic changes but demonstrate how the utilized functions have to be specified for our application case.
5 Adaptation of BASSAMP for RFID
The BASSAMP algorithm—introduced in [15] and depicted in Algorithm 1—aims at recovering X from Y in (10). It utilizes the knowledge of A, the signal prior, and the joint sparsity structure.
Let us briefly summarize how Algorithm 1 works.
where the noise \(\widetilde {\mathsf {w}}_{n,b}\) accounts for the measurement noise and the undersampling noise. It is assumed to be Gaussian distributed as \(\widetilde {\mathsf {w}}_{n,b} \sim \mathcal {N}(0,\beta _{b})\). This assumption is satisfied in the asymptotic case (M,N→∞ while \(\frac {M}{N}=const.\)) and approximately satisfied in finite but high dimensions. The decoupled measurement (12) refers to line 5 of Algorithm 1. The effective noise variance β _{ b } is estimated in line 6, and the current estimate for x _{ b } is computed in line 7 using the minimum mean squared error (MMSE) estimator function F(·;·,·) that will be defined later. A residual is computed in line 8. Above points subsume the Bayesian approximate message passing (BAMP) [21–23] iteration that is executed for all N _{ B } blocks. In each BAMP iteration, the signal vector x _{ b } is newly estimated. The energy of the residual decreases over iterations, and so does the effective noise variance β _{ b } — the MMSE estimator F(·;·,·) acts as a denoiser. In line 9, the extrinsic group update function U _{ G }(·,·,·) enforces the joint sparsity structure. This is done via binary latent variables that indicate whether a signal entry is zero or nonzero; in a probabilistic manner, the prior probability for a specific signal entry to be zero is updated. The likelihood ratios that are generated by the extrinsic group update function are converted into new prior probabilities in line 10. After several iterations of BASSAMP, a consensus emerges. For a detailed derivation, we refer the interested reader to [15].
The algorithm assumes independently distributed signal entries for the BAMP iteration, where the prior distribution of the nth entry of the bth signal vector is denoted as \(f_{\mathsf {x}_{n,b}\phantom {\dot {i}\!}}(x_{n,b})\). This prior plays a major role in the computation of the functions F(·;·,·), F ^{′}(·;·,·), and U _{ G }(·,·,·). Therefore, let us specify the signal prior for the RFID scenario, which is essentially dictated by the channel model.
5.1 Channel model and distribution
where h is a placeholder for the real part \(\mathsf {h}_{r,k}^{\mathfrak {(R)}}\) or imaginary part \(\mathsf {h}_{r,k}^{\mathfrak {(I)}}\) of the total channel (13).
5.2 Sparsity enforcing signal prior
where γ _{ n,b } is the probability that the nth signal entry of the bth vector is zero. If the number of activated tags, K, is known a priori, the initial value computes as \(\gamma _{n,b}=1\frac {K}{N}\). In BASSAMP, this probability is adapted in each iteration, and it is sufficient to initialize it with a very coarse assumption of the number of activated tags; details follow in Section 6.
5.3 Specification of functions
The conditional expectation (22) yields the MMSE estimate of x _{ n,b } given the decoupled measurement \(u_{n,b}=x_{n,b} +\widetilde {w}_{n,b}\), where \(\widetilde {\mathsf {w}}_{n,b}\sim \mathcal {N}(0,\beta _{b})\); for details, consider [15, 28]. Note that in Algorithm 1 (line 7), (22) is applied separately on the vector components u _{ n,b } of the vector input u _{ b }.
A detailed explanation of functions (34) and (35) can be found in [15].
5.4 Specification of functions—Gaussian relaxation
The prior update (35) stays the same.
To justify this approximation of the channel PDF, we conducted numerical experiments that suggest that the MSE performance is hardly affected, see Section 7.
5.5 Choice of parameters
The choice of the channel model and the forward channel variance σ ^{(f)} ^{2} and backward channel variance σ ^{(b)} ^{2} depends on the location of the reader antennas, the environment (scatterers and reflectors), and the effective read range. The variances describe the strength of the spatial fading of the forward and backward link, respectively. In order to estimate them, one would have to measure the forward and backward links separately for many prospective tag positions.
In practice, one would rather measure the total channel (13), or avoid measuring the channels and determining their distribution entirely. The AMP framework allows to perform prior estimation during recovery, i.e., the recovery algorithm can be adapted to estimate the prior over iterations. One such algorithmic extension was proposed in [30], and it was shown that the performance degradation due to unknown prior is negligible in practice.
6 Support estimation

The estimated set of assigned signatures \(\widehat {\mathcal {T}}_{\mathrm {A}}\) [14] (this is the information from the acquisition phase used to communicate with the tags),

The estimated number of activated tags \(\widehat {K}\).
The schemes presented in [12–14] assumed to know the number of activated tags K, i.e., \(\widehat {K}=K\). Utilizing this knowledge, the estimated set of assigned signatures \(\widehat {\mathcal {T}}_{\mathrm {A}}\) is dictated by the K largest entries of signal recovery \(\widehat {\mathbf {x}}\). The newly employed BASSAMP algorithm allows for a robust support estimation by combining the soft information of all recovered vectors \(\widehat {\mathbf {x}}_{b}\).

In Buzz [9–11], this set represents the seeds used in the pseudo random generator for the data readout via rateless code. An erroneous set hampers decoding, and the acquisition has to be repeated.

In CSF [12], this set directly identifies the activated tags. The reader enquires the signature indices in order to confirm the identification. An erroneous set prolongs this enquiry phase and leads to a repetition of the acquisition phase.

In CSR [13, 14], the indices of the assigned signatures are used to communicate with the tags for data readout. Again, an erroneous set prolongs the enquiry phase and leads to a repetition of the acquisition phase.
For all schemes, a wrongly estimated set of assigned signatures prolongs the identification and increases the readertotag communication overhead.
Note that the support estimation is performed after executing the BASSAMP algorithm; it considers the values u _{ n,b }, γ _{ n,b }, and β _{ b } after the last iteration t. The prior probabilities in BASSAMP Algorithm 1 are initialized with a coarse assumption of K, termed K ^{0}: \(\gamma _{n,b}^{0} = 1\frac {K^{0}}{N}\).
7 Numerical results and comparison
it gives indication about the overall recovery performance.
For the subsequent numerical experiments, the stopping criteria of Algorithm 1 were chosen as ε _{tol}=10^{−5} and t _{max}=100.
7.1 Gaussian relaxation of prior
In Section 5.4, we proposed a relaxation of the prior distribution (Gaussian instead of Laplace) in order to obtain an implementation of Algorithm 1 that features a lower computational complexity and better numerical stability. To demonstrate that the performance is not significantly affected by this relaxation, we investigate the empirical phase transition curves that illustrate the recovery performance over a wide range of parameters K and M for fixed N=1 000. We chose N _{ R }=1 receive antenna.
The average success is obtained as \(\overline {S} = \frac {1}{1\,000}\sum _{r=1}^{1\,000}S_{r}\). The empirical phase transition curves are finally obtained by plotting the 0.5 contour of \(\overline {S}\) using MATLAB®;function contour.
7.2 Support estimation
In this section, we demonstrate the capabilities of the support detection scheme that was introduced in Section 6. The BASSAMP Algorithm 1 requires an initialization of the zero probabilities, which is done with a coarse initial assumption of the number of activated tags: \(\gamma _{n,b}^{0}=1\frac {K^{0}}{N}\). We now demonstrate that the initial assumption of K _{0} can indeed be very coarse.
We consider the number of correct detections (CDs) and false alarms (FAs) [32] that partition the estimated set of assigned signatures \(\widehat {\mathcal {T}}_{\mathrm {A}}\) (47) (an index in this set either refers to a CD or a FA).
7.3 Improvement of acquisition phase—perfect conditions
where T _{ps} is the throughput per slot [8], i.e., the number of tags acquired per slot, and the number 16 refers to the RN16 sequences utilized during acquisition. If the number of activated tags is known, the optimal choice of the frame size leads to a maximum average throughput of T _{ps}=e ^{−1}≈0.368 [33]. In [6–8], collision recovery schemes have been proposed that allow shortened frame sizes and, thus, increased throughput numbers. A reader with N _{ R } receive antennas can resolve up to 2N _{ R } collisions [8]. Assuming perfect channel knowledge and knowledge of the number of activated tags K, a reader with N _{ R }=1 receive antenna can resolve one collision and features a maximal theoretical throughput of T _{ps}=0.841, while a reader with N _{ R }=4 receive antennas achieves a maximal theoretical throughput of T _{ps}=4.479, for details, see [8].
It was shown in [12] that an optimally tuned AMP recovery algorithm requires a measurement multiplier c=2 to yield perfect recovery results in the noiseless case. In Sections 7.1 and 7.2, we observed that our proposed scheme requires fewer measurements than the legacy AMP scheme, which enables a reduction of c that results in the same recovery quality. Scrutinizing Fig. 6, we observe that M≈210 (c≈0.9) for N _{ R }=1 and M≈140 (c≈0.6) for N _{ R }=4 lead to perfect recovery (i.e., only CDs and no FAs), respectively.
By utilizing the channel statistics and the joint sparsity among the signal vectors, a strong improvement over previous approaches is observed. Furthermore, the novel approach already shows a significant improvement for a reader that employs only N _{ R }=1 receive antenna.
7.4 Improvement of acquisition phase—imperfect conditions
Note that here, the simulated bit overhead \(\beta _{\text {CS}}^{\text {(A)}}\) that may include several CS measurements (cycles) is used, whereas (53) refers to the optimal bit overhead of only one CS measurement.

4.3 times quicker than the legacy CS approach that employed AMP and a single receive antenna,

3 times quicker than FSA with collision recovery,

26 times quicker than legacy FSA without collision recovery capability, and a reader with a single receive antenna.
We emphasize again that our approach is applicable to all stateoftheart CSRFID schemes [9–14].
8 Conclusions
We proposed a novel extension to CSRFID that improves the acquisition phase of the tag identification by leveraging joint sparsity. We demonstrated how multiple receive antennas at the reader produce multiple measurement vectors and that their number can be doubled beneficially by separation into real and imaginary parts. The corresponding signal vectors are jointly sparse, i.e., they share a common support. This is exploited by the BASSAMP algorithm that we defined for a dyadic channel model.
We showed that the usage of a Gaussian prior relaxation is applicable. Simulation results suggest that an exact knowledge of the channel coefficient distribution is not required. Furthermore, the relaxation promotes a low complexity implementation of our iterative recovery algorithm.
Robust signal support estimation is facilitated by combining the soft information from multiple jointly sparse signal vectors. Support estimation is crucial for quick tag identification, as the support dictates the overhead of the readertotag communication (correct detections lead to correctly read out tags, while false alarms prolong the identification). It was shown that prior knowledge of the exact number of activated tags is not required for robust support estimation.
The main benefits of exploiting joint sparsity are the possible reduction of the sequence length (i.e., shorter tag responses) and the increased noise robustness during acquisition. This enables quicker, more reliable identification, reduces the sensitivity to jitter, and lowers the energy requirements of the tags.
Declarations
Acknowledgements
This work has been funded by the Christian Doppler Laboratory for Wireless Technologies for Sustainable Mobility, and its industrial partner Infineon Technologies. The financial support by the Federal Ministry of Economy, Family and Youth and the National Foundation for Research, Technology and Development is gratefully acknowledged.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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