is an N x 1 vector.As the reader does not need to distinguish Leveraging the differential of aggregated responses, empty/singleton/collision slots,the tags do not need to transmit we improve the performance of P-MTI from O(N)to multiple physical layer symbols in each time slot,which can O(K log(N/K)).The compressive sensing based information further reduce the transmission time. reconstruction allows us to extract the missing tag events directly from the differential of aggregated physical layer C.Compressive sensing based recovery symbols,thereby significantly reducing the required total time slots compared with the existing schemes.Besides,unlike We notice that P-MTI needs N measurements per mon- existing schemes which need multiple physical symbols per itoring operation.In practical scenarios,one may need to slot,P-MTI needs only one symbol per time slot.Therefore, frequently run the missing tag identification protocol to ensure the performance gain of such joint optimization is promising a timely report of missing tag events. As a matter of fact,it suffices to compute the differential D.Enhancement against noisy measurement of aggregated responses between two consecutive instances The above analysis ignores the noise in measurements. to achieve continuous monitoring.The rationale is straight- Wireless channel is mostly error-prone,subjected to various forward:if a response from a tag is detected while no factors,such as interference,quantization,etc [8.Channel dy- more response is detected later from the tag,then the tag is namics may also introduce noise to the measurements.Without probably missing.We therefore compute the differential of robustness enhancement against noise,a detection system may the aggregated responses between two consecutive instances result in unfavorable false alarms over noisy channels [14].In at time t andt-△，ya=yt-yt-△，and infer the dynamics the following,we enhance P-MTI's robustness against noise of the tags.According to Eg.(2),we have based on the theory of stable recovery [9].Incorporating the noise,Eq.(4)can be rewritten as follows. y△=yt-yt-△=Azt-Azt-△ (3) We refer the differential of z similarly by ZA =Zt-Zt-A. y△=Aza+e, (6) Then,from Eq.(③)，we have where e denotes the error due to noise.Then the optimization y△=AzA. (4) problem with relaxed constraints for recovery of z can be written as follows. Ideally,the zero entries in ZA imply that corresponding tags are present,while the non-zero entries indicate that the tags are missing.In practical RFID systems,the number of missing Minimize:ZAle tags K is typically much smaller than the number of tags under Subject to:AZA -yallea <e, (7) monitoring N during a short monitoring interval,i.e.,N where the magnitude e is bounded by e,i.e.,llelle<e.The K>0 [16,28].In other words,z is K-sparse,meaning that theory of stable recovery [9]tells us that the solution Z to the there are at most K non-zero entries in ZA [12].According convex optimization problem (7)is a good approximation of to the theory of compressive sensing,the K'-sparse vector zA ZA,and IZ-Zl2 ce where c denotes a small constant. can be accurately recovered with only M=O(K log(N/K)) In other words.a small error in y only slightly influences measurements,by solving the following convex optimization reconstruction of zA.In order to identify missing tags,P- problem [12]. MTI only needs to distinguish the zero and non-zero entries in zA.The noise may disturb zA and render the zero entries Minimize:ZAlle in zA non-zero (yet remaining small),affacting the detection Subject to:AZA=yA, (5) accuracy.As the error is well bounded by the noise in practice, where denotes (norm.l) if the noise is small we can use a threshold a to accurately classify the contaminated signals with high probabilities.In Intuitively,the objective function of minimizingle in- particular,we define the detection function f(zA:)for tag i corporates the fact that zA is sparse,while the constraint as follows function is self-explanatory.Therefore,we can refer to convex optimization solvers (e.g.,CVX [2],E1-Magic [6])to compute Present,if ilez<0 ZA.In our implementation,we use the CVX solver [2]based f(z△i)= (8) Absent, otherwise. on the interior-point algorithm [7].It has been reported that M =3K4K N measurements suffice to recover the If the magnitude of zAi is smaller than 6,then tag i is K'-sparse vector [17].We set the number of measurements present;otherwise,we say tag i is absent.When the channel as M=CKMax log(N)in practice,where KMax represents condition dramatically deteriorates,the RFID reader cannot the estimated maximum number of missing tags and C is a always accurately identify the missing tags.In such scenarios, constant.Our experiment results show that M measurements the reader may use the basic P-MTI to identify the missing of physical layer symbols are sufficient to reconstruct the K-tags,and monitor the channel H.If the channel becomes sparse vector of zA.When the number of missing tags exceeds reasonably good and stable,P-MTI can switch back to monitor KMax,our approach can identify at least KMax missing tags,the differential and identify the missing tags.The reader which allows P-MTI to effectively adjust Kmax(Section V). may also increase transmission power to increase the signalis an N ×1 vector. As the reader does not need to distinguish empty/singleton/collision slots, the tags do not need to transmit multiple physical layer symbols in each time slot, which can further reduce the transmission time. C. Compressive sensing based recovery We notice that P-MTI needs N measurements per mon￾itoring operation. In practical scenarios, one may need to frequently run the missing tag identification protocol to ensure a timely report of missing tag events. As a matter of fact, it suffices to compute the differential of aggregated responses between two consecutive instances to achieve continuous monitoring. The rationale is straight￾forward: if a response from a tag is detected while no more response is detected later from the tag, then the tag is probably missing. We therefore compute the differential of the aggregated responses between two consecutive instances at time t and t − ∆, y∆ = yt − yt−∆, and infer the dynamics of the tags. According to Eq.(2), we have y∆ = yt − yt−∆ = Azt − Azt−∆. (3) We refer the differential of z similarly by z∆ = zt − zt−∆. Then, from Eq.(3), we have y∆ = Az∆. (4) Ideally, the zero entries in z∆ imply that corresponding tags are present, while the non-zero entries indicate that the tags are missing. In practical RFID systems, the number of missing tags K is typically much smaller than the number of tags under monitoring N during a short monitoring interval, i.e., N  K ≥ 0 [16, 28]. In other words, z∆ is K-sparse, meaning that there are at most K non-zero entries in z∆ [12]. According to the theory of compressive sensing, the K-sparse vector z∆ can be accurately recovered with only M = O(K log(N/K)) measurements, by solving the following convex optimization problem [12]. Minimize: kz∆k`1 Subject to: Az∆ = y∆, (5) where k · k`p denotes `p-norm, i.e., kxk`p , ( Pi=N i=1 |xi | p ) 1/p . Intuitively, the objective function of minimizing kz∆k`1 in￾corporates the fact that z∆ is sparse, while the constraint function is self-explanatory. Therefore, we can refer to convex optimization solvers (e.g., CVX [2], `1-Magic [6]) to compute z∆. In our implementation, we use the CVX solver [2] based on the interior-point algorithm [7]. It has been reported that M = 3K ∼ 4K  N measurements suffice to recover the K-sparse vector [17]. We set the number of measurements as M = CKMax log(N) in practice, where KMax represents the estimated maximum number of missing tags and C is a constant. Our experiment results show that M measurements of physical layer symbols are sufficient to reconstruct the K￾sparse vector of z∆. When the number of missing tags exceeds KMax, our approach can identify at least KMax missing tags, which allows P-MTI to effectively adjust KMax (Section V). Leveraging the differential of aggregated responses, we improve the performance of P-MTI from O(N) to O(K log(N/K)). The compressive sensing based information reconstruction allows us to extract the missing tag events directly from the differential of aggregated physical layer symbols, thereby significantly reducing the required total time slots compared with the existing schemes. Besides, unlike existing schemes which need multiple physical symbols per slot, P-MTI needs only one symbol per time slot. Therefore, the performance gain of such joint optimization is promising. D. Enhancement against noisy measurement The above analysis ignores the noise in measurements. Wireless channel is mostly error-prone, subjected to various factors, such as interference, quantization, etc [8]. Channel dy￾namics may also introduce noise to the measurements. Without robustness enhancement against noise, a detection system may result in unfavorable false alarms over noisy channels [14]. In the following, we enhance P-MTI’s robustness against noise based on the theory of stable recovery [9]. Incorporating the noise, Eq.(4) can be rewritten as follows. y∆ = Az∆ + e, (6) where e denotes the error due to noise. Then the optimization problem with relaxed constraints for recovery of z can be written as follows. Minimize: kz∆k`1 Subject to: kAz∆ − y∆k`2 ≤ , (7) where the magnitude e is bounded by , i.e., kek`2 ≤ . The theory of stable recovery [9] tells us that the solution ˆz∆ to the convex optimization problem (7) is a good approximation of z∆, and kz∆ − ˆz∆k2 ≤ c where c denotes a small constant. In other words, a small error in y∆ only slightly influences reconstruction of z∆. In order to identify missing tags, P￾MTI only needs to distinguish the zero and non-zero entries in z∆. The noise may disturb z∆ and render the zero entries in z∆ non-zero (yet remaining small), affacting the detection accuracy. As the error is well bounded by the noise in practice, if the noise is small we can use a threshold θ to accurately classify the contaminated signals with high probabilities. In particular, we define the detection function f(z∆i) for tag i as follows. f(z∆i) = ( Present, if kz∆ik`2 < θ Absent, otherwise. (8) If the magnitude of z∆i is smaller than θ, then tag i is present; otherwise, we say tag i is absent. When the channel condition dramatically deteriorates, the RFID reader cannot always accurately identify the missing tags. In such scenarios, the reader may use the basic P-MTI to identify the missing tags, and monitor the channel H. If the channel becomes reasonably good and stable, P-MTI can switch back to monitor the differential and identify the missing tags. The reader may also increase transmission power to increase the signal