Physical Layer Security in MIMO Backscatter Wireless Systems
Backscatter wireless communication is an emerging technique widely used in low-cost and low-power wireless systems, especially in passive radio frequency identification (RFID) systems. Recently, the requirement of high data rates, data reliability, and security drives the development of RFID systems, which motivates our investigation on the physical layer security of a multiple-input multiple-output (MIMO) RFID system. In this paper, we propose a noise-injection precoding strategy to safeguard the system security with the resource-constrained nature of the backscatter system taken into consideration. We first consider a multi-antenna RFID tag case and investigate the secrecy rate maximization (SRM) problem by jointly optimizing the energy supply power and the precoding matrix of the injected artificial noise at the RFID reader. We exploit the alternating optimization method and the sequential parametric convex approximation method, respectively, to tackle the non-convex SRM problem and show an interesting fact that the two methods are actually equivalent for our SRM problem with the convergence of a Karush-Kuhn-Tucker (KKT) point. To facilitate the practical implementation for resource-constrained RFID devices, we propose a fast algorithm based on projected gradient. We also consider a single-antenna RFID tag case and develop a low-complexity algorithm which yields the global optimal solution. Simulation results show the superiority of our proposed algorithms in terms of the secrecy rate and computational complexity.
Backscatter wireless communication, remarkable for its low energy consumption and low product cost, is an emerging technology which is widely used[1, 2, 3, 4]. One of its most prominent applications is in radio frequency identification (RFID) systems. RFID enables identification from a distance, thereby facilitating the handling of manufactured goods and materials[1, 2]. By employing backscatter modulation to send back the data and on-tag power harvesting to supply the power, RFID is promoted by its longevity, efficacy, and energy efficiency. It is believed that RFID will become one of the most crucial techniques to realize the Internet of Things (IoT), which allows objects to be sensed and creates more efficient interactions between the physical world and computer-based systems.
As a contactless technology in a short range, RFID systems are expected to fulfill the aim of reducing handling time despite the augment of data stored in RFID tags[6, 7]. Concerning this higher expectation of the data rate and data reliability for novel RFID applications, the implementation of the multiple-input multiple-output (MIMO) scheme appears to be effective and promising[8, 9], which attracts considerable research interests[10, 11, 12, 13, 14]. It is shown in  that adopting multiple antennas can extend the coverage of backscatter RFID systems and improve system capacity under the spatial multiplexing configuration. Furthermore, the authors in [11, 12] show that multi-antenna techniques can significantly improve the data reliability of the RFID system. The space-time coding scheme is explored in [13, 14] for MIMO RFID backscatter systems. Apart from the above analytical studies, several real experiments concerning multi-antenna RFID tags have also been conducted[15, 16, 17]. The measurement results in  show that read range can be improved when multiple antennas, instead of a single antenna, are equipped at the RFID tag. In addition, the authors in  propose a method for channel measurements in MIMO RFID systems. The authors in  showcase two multi-antenna techniques for RFID tags operating at 5.8 GHz. The MIMO scheme has been extensively investigated and recognized as an efficient approach to further extending the information-carrying ability of RFID .
Due to the widespread deployment of RFID tags, the privacy concern for users, such as clandestine physical tracking and personal information protecting, also makes a great challenge to the design of RFID systems, because the transmission is vulnerable to eavesdropping due to the broadcast nature of backscatter communication [18, 19, 20]. Most of previous works concerning RFID security issues mainly focus on lightweight cryptography such as in[20, 21, 22]. However, they still have some restrictions on the secret key generation and distribution from eavesdropping and practical limitations in terms of size, cost, and computation[22, 23]. Fortunately, in recent years physical layer security (PLS), as an alternative or complement to cryptography, has drawn considerable attention in strengthening the security of wireless communications since perfect secrecy is provided. The theoretical basis for PLS approaches lies in the notion of the secrecy capacity/rate, which was pioneered by Wyner in . Since then, wealth of relevant research has achieved a significant success in the security of conventional wireless communication systems[25, 26, 27, 28, 29, 30, 31, 32]. The main idea of PLS approaches, in addition to exploiting the randomness inherent to wireless channels, is to manually construct equivalent channels via signal design and power allocation such that the superiority of the equivalent legitimate channel to the equivalent wiretap channel can be established. One of promising approaches is to send an artificially generated noise to deteriorate the channel condition of eavesdroppers. This conception of applying artificial noise (AN) to enhance the secure transmission is first introduced in , and it is further studied in [29, 30, 31, 32]. These studies have also been generalized to the cooperative relay system[34, 35, 36, 33].
However, there is only little work to study the security of backscatter systems from the perspective of PLS. In , a physical layer noise injection scheme is proposed to strengthen the security of backscatter wireless systems under a single-input single-output (SISO) system setting where all terminals employ a single antenna, and it is shown that the proposed approach yields significant performance gains. To the best of our knowledge, no work has been done on the PLS of a MIMO backscatter system, even though the MIMO technique is promising in enhancing the security due to extra spatial degrees of freedom provided by multiple antennas. Note that the PLS approaches for the conventional MIMO system cannot be directly used into the MIMO backscatter system due to the following two reasons. For one thing, the channel model of the MIMO backscatter system is quite different from the conventional one and usually is modeled as the so-called dyadic backscatter channel , which makes it hard to formulate the considered security problem. For another thing, the passive backscatter system is typically resource-constrained and thus requires low-complexity algorithms in practice. Particularly, the trade-off between secrecy performance and computational complexity should be taken into account in the algorithm design of the MIMO backscatter system.
Based on the above observations, in this paper we focus on solving the security issues of a MIMO RFID system from the perspective of PLS, wherein we take full account of the resource-constrained nature of RFID devices. The novelty and main contributions of this paper can be summarized as follows:
The MIMO backscatter wireless communication is studied from the perspective of PLS for the first time and a noise-injection precoding strategy is proposed to strengthen the security of the system.
The alternating optimization (AO) method and the sequential parametric convex approximation (SPCA) method are, respectively, invoked to tackle the non-convex secrecy rate maximization (SRM) problem. Furthermore, we show an interesting fact that the two methods are actually equivalent for our problem.
Particularly, a custom-designed algorithm based on projected gradient (PG) is proposed for fast implementation, which is especially beneficial to the resource-constrained RFID device.
As a complement, the case where the tag has only a single antenna is studied, and the global optimal solution can be obtained by one-dimensional search. Moreover, a nullspace AN design in this case is further proposed, and it is shown in the simulations that this scheme obtains the secrecy rate which is close to the optimal one and enjoys an extremely low computational complexity.
The rest of this paper is organized as follows: In Section II, we present the system model and develop the formulation for the achievable secrecy rate of a MIMO RFID backscatter system. In Section III, we focus on the SRM problem with a multi-antenna tag, and the equivalence of the AO and SPCA methods is analyzed. In Section IV, we propose a fast algorithm to efficiently solve the SRM problem with a multi-antenna tag. The case where the tag equips with only a single antenna is studied in Section V. Numerical simulations and analysis for the proposed schemes and algorithms are presented in Section VI before the conclusions drawn in Section VII.
Notations: , , and represent the transpose, Hermitian transpose, determinant and trace of a matrix , respectively. denotes an identity matrix. means that is a Hermitian positive semidefinite matrix. and denote the expectation and conjugate, respectively. means that for every . denotes a diagonal matrix with diagonal elements taken from vector . denotes a column vector whose the -th element is 1 and 0 elsewhere. means that is a random vector following a complex circular Gaussian distribution with mean and covariance . represents the Hadamard product.
Ii Secure MIMO Backscatter System Model
Consider a RFID system consisting of a multi-antenna RFID reader with transmitting antennas and receiving antennas, a RFID tag with antennas, and a passive eavesdropper with receiving antennas as shown in Fig. 1. For notational simplicity, we use the terms “reader” and “tag” as commonly used in RFID systems hereinafter.
The basic idea of realizing the RFID backscatter wireless communication is as follows: First, the reader transmits a continuous carrier wave (CW) to power up the tag. Then, the passive tag reflects back the CW signal by changing the different impedance loads of its antennas according to its stored secret information such as identification data. This procedure can be regarded as a backscatter modulation, where the secret information, to be transmitted from the tag to the reader, is modulated on the reflected CW signal by the tag. Finally, the reader extracts the desired information by estimating and decoding the echoed back signal after the backscatter modulation. During the whole backscatter procedure, the reader continuously transmits a CW signal to power up the tag and concurrently receives the echoed back signal modulated by the tag, namely, it works in a full-duplex mode. Thus, the received self-interference or leaked signal directly from its transmitter to receiver needs to be canceled. Whereas in practice, it is difficult for the reader to perform this cancellation without any signal leakage. Thereby in this work, we use as in  to denote the self-interference attenuation factor which reflects the capacity for the reader to cancel the interference from the transmitted signal in the received signal. Based on the procedure above, under the assumption that all the channels undergo slow frequency-flat fading, the received signal at the reader is given by111This signal model is first proposed in , wherein self-interference is not considered and only spatial domain is involved. The model in  is further generalized to the space-time coding model in [4, 13, 14] where temporal domain is also taken into account. In this paper, we focus on transmit optimization against eavesdropping only in spatial domain and thus adopt the signal model similar as that in .
where is the signal transmitted by the reader222Since the reader receives its transmitted signals from both the backscatter and its self-interference channels at different time instants, we adopt in (1) to denote the signal received directly from the reader’s self-interference channel and distinguish it from the signal received from the backscatter channel, i.e., . A similar notation will be also used in (6). with the total power constraint . , , and represent the channel from the reader to the tag, the channel from the tag to the reader, and the self-interference channel at the reader, respectively. is the additive white Gaussian noise (AWGN) at the reader. is the tag’s information signaling matrix which represents the backscatter gain at the tag. The structure of describes the backscatter modulation by the tag, during which the RF tag absorbs and scatters radio signals by its antennas. This signaling matrix takes several different forms depending upon the physical implementation of the modulation circuitry and RF tag antennas. In this paper, we consider a common scenario employed in [4, 10] that the signaling matrix takes the form of a diagonal matrix, given by
We assume that the elements in vector are i.i.d. and .
In this paper, we propose a noise-injection precoding scheme to create additional interference at the eavesdropper and thus to strengthen the security of the system. The transmit vector takes the structure , where is the CW signal to provide the tag with the power supply and we consider the conventional case as in uniform query333There exist two different query methods widely adopted in the existing literature: the conventional uniform query [4, 10, 13, 38] and the newly developed unitary query adopted in the design of space-time coding for MIMO Backscatter RFID[14, 39]. In contrast to the diversity analysis in space-time coding where the message is coded over both space and time, here we focus on the transmit design against eavesdropping only in spatial domain and thus adopt conventional uniform query. that the reader transmits a constant CW with the transmit power equally allocated among all its transmit antennas, i.e., , where denotes the power allocated to the CW signal. is an AN vector, generated by the reader and injected into the transmitted CW signal, to interfere with the eavesdropper. Let denote the spatial covariance matrix of the AN. The total power constraint at the reader now changes to
For the considered MIMO RFID channel, the reader uses the noisy observation of to estimate the unknown signal with the CW signal . We can rewrite as
For notational simplification, let and . We can rewrite (1) as
where , following from the setting in , is the attenuation factor which denotes how successful the reader is in canceling the backscattered AN. Note that the CW signal , received directly from the self-interference channel at the reader, has been removed from (5) due to the fact that the standardized CW signal is commonly known and can be eliminated by the reader.
Similar to the reader, the received signal at the eavesdropper is given as
where and are the tag-eavesdropper and reader-eavesdropper channel matrices, respectively. is the AWGN at the eavesdropper. is the backscattered AN signal modulated by the tag’s information signal, while is the injected noise term received directly from the reader-eavesdropper channel. It should be noted that, unlike the reader, the eavesdropper cannot perform AN attenuation due to the absence of the prior knowledge about the random AN signal transmitted by the reader. However, the CW signal received directly from the reader-eavesdropper channel can be eliminated by the eavesdropper due to the fact that the standardized CW signal is commonly known. That is why this signal term does not appear in (6).
The achievable secrecy rate for a given secure transmission scheme determines the performance limit of PLS. Unfortunately, the exact expression of the secrecy rate here is difficult to obtain due to the non-Gaussian distribution of the combined signal and AN terms and received at the reader and the eavesdropper, respectively. However, following the similar method adopted in , we regard these terms as interference and obtain an approximation of the achievable secrecy rate given by 
where the covariance matrices of the interference and noise are given by
From (7) and (8), to maximize the achievable secrecy rate in (7a) under the total power constraint in (3), the power allocation between the AN and the CW as well as the covariance matrix of AN needs to be carefully designed since the jamming signal from the reader degrades the performance of both the reader itself and the eavesdropper. The case where the tag has multiple antennas will be studied in the next two sections, while the scenario where a single antenna is employed at the tag will be investigated in Section V.
Iii Multi-Antenna Tag
In this section, we consider the case where the tag equips with multiple antennas. The considered problems are first formulated as matrix optimization problems in Section III-A, and then the AO and SPCA methods are introduced to solve the SRM problem in Section III-B and Section III-C, respectively. The equivalence of the two methods in the considered problem will be illustrated in Section III-D.
Iii-a Problem Formulation
To facilitate analysis, we first recast the secrecy rate in (7a) as
where the feasible set is defined as
As discussed in [33, 29], a simplified and special design of the general AN scheme in (10) is the so-called nullspace AN scheme where the transmitted AN lies in the nullspace of the legitimate user’s channel. In our MIMO backscatter model, from (8a) we know that the reader receives two altered copies of its transmitted AN from the backscatter channel and the self-interference channel, respectively. When both the AN copies cannot be perfectly eliminated at the same time and the number of transmit antennas at the reader is adequate to perform the nullspace AN precoding, namely or , the AN transmitted by the reader can be designed to lie in the nullspace of the reader-tag channel or the self-interference channel , respectively. To be specific, we can construct the injected AN as , and now we have
where contains all the right singular vectors of or corresponding to zero singular values, and is an or positive semidefinite matrix to be optimized, respectively. Mathematically, the nullspace SRM problem is formulated as follows:
The SRM problem in (10) is non-convex and difficult to tackle due to the non-concave property of the term in the objective function in (9). The AO and SPCA methods are the two methods widely exploited in tackling non-convex matrix optimization problems [30, 40, 41], and it will be interesting to see in the sequel that the two methods are actually equivalent under our SRM problem. Therefore, our approach is to reformulate the non-concave term to a concave one by exploiting the AO and SPCA methods. The two methods will be, respectively, introduced in the next two subsections.
Iii-B AO Method for SRM
The main idea of the AO method is to exploit the coordinate-wise convexity property of the non-convex problem where optimization over two subsets of variables is non-convex, but optimization with respect to (w.r.t.) one while fixing the other is convex. To re-express the general SRM problem in (10) as a form that can be tackled by the AO method, we first introduce the following lemma.
From Lemma 1, one can easily see that the non-concave term can be changed to a linear (and thus concave) term w.r.t. the original optimization variable by adding an auxiliary variable. By applying Lemma 1 to the objective function in (9) via setting and , we have the following equivalent formulation of problem (10):
Note that we have dropped the constant in the objective of problem (15) for simplicity. The equivalent problem in (15) is non-convex w.r.t. . However, it is not hard to see that problem (15) is convex w.r.t. either or while fixing the other. We can exploit this coordinate-wise convexity property to use the AO method to solve the problem. To be specific, let denote the solution obtained at the -th AO iteration. We iteratively use the values at the -th iteration to obtain the ones at the -th iteration by alternatingly solving the following two optimization problems for ,
Problem (16b) is convex, and thus can be numerically solved. The solution to our SRM problem can be obtained by iteratively calculating (17) and solving problem (16b) until the corresponding secrecy rate fulfills the given accuracy requirement.
As a basic result of AO, the method proposed above produces non-descending objective values. More specifically, we have . In addition, we will show by Theorem 1 in Section III-D that the sequence generated by the AO method converges to a Karush-Kuhn-Tucker (KKT) point of the original problem in (10).
Iii-C SPCA Method for SRM
As an alternative way, we can also use the SPCA method to solve the SRM problem in (10). The basic idea of the SPCA method is to approximate a non-convex problem by a sequence of convex problems. In each convex problem, every non-convex constraint is replaced by an appropriate inner but convex one. Generally, the convergence rate of the SPCA method is fast. More details about SPCA can be found in [40, 41].
To apply the SPCA method, we first transform the non-concave terms in the objective function in (9) to the inner-approximated but concave ones. Note that the non-concave terms in the objective function are actually convex w.r.t. the optimization variable. Here the first-order Taylor’s series approximation can be used as a global underestimator of the convex function [43, p. 69]. More specifically, the approximation for the function w.r.t. around is given by
From the right-hand side of (18), one can see that non-concave terms can be changed to linear (and thus concave) ones by exploiting this method.
By applying the approximation in (18) centering around the point to non-concave terms in the objective function in (9), we obtain (19) at the top of the next page where the superscript refers to the optimal solution obtained at the -th iteration. In each iteration we perform the approximation centering around the optimal solution obtained at the previous iteration. At the -th iteration, the original non-convex SRM problem in (10) can be locally approximated by the following convex optimization problem for ,
Like the AO method in the last subsection, the sequence of objective values produced by the SPCA method is non-descending. Moreover, we will show by Theorem 1 in the next subsection that the sequence generated by the SPCA method converges to a KKT point of the original SRM problem in (10).
As seen in Section III-A, the nullspace SRM problem in (13) is just a degenerate form of the SRM problem in (10) when the nullspace AN constraint in (12) is imposed. Thus, once the general SRM problem in (10) is solved by the AO or SPCA method, the nullspace SRM problem in (13) can be easily solved in a similar way.
Generally, the AO and SPCA methods are two totally different methods and either the AO or SPCA method individually will give a solution. However, it is interesting to see that the two methods are actually equivalent for our SRM problem in the sense that the two methods can equivalently lead to the same optimization problem. We will analyze this equivalence in the next subsection.
Iii-D Equivalence, Convergence, and Complexity Analyses
It is not hard to see that the objective function of problem (III-C) obtained by the SPCA method and (III-D) only differ in some added terms that are irrespective of our optimization variable , and thereby have no effect on the optimization result. This leads to the equivalence of the two methods for the considered SRM problem.
The main reasons for this equivalence are summarized as follows:
As for the SPCA method, the terms and in of the objective function in (9) are linear w.r.t. the optimization variable . This linear property makes the terms in the objective function convex, and thus the approximation of the first-order Taylor’s series in (18) can be applicable. The choice of this approximation in the SPCA method makes the equivalence between problem (III-C) obtained by the SPCA method and problem (22) possible.
Concerning the AO method, Lemma 1 plays an important role in this equivalence. It should be noted that the specific way to apply the AO method actually depends largely on the form of the considered problem. In our case, it is the non-concave term that makes Lemma 1 applicable, and then the AO method can be employed to lead us to problem (22).
Since we have shown that both the AO and SPCA methods for our SRM problem are equivalent to iteratively solving the convex optimization problem in (22), the two methods actually have the same convergence result and computational complexity. We first give the convergence result in the following theorem.
Both the AO and SPCA methods produce non-descending achievable secrecy rates and converge to a KKT point of the original SRM problem in (10).
The proof is given in Appendix A. \qed
At each AO or SPCA iteration, the convex optimization problem in (22) can be solved by using a general-purpose convex optimization toolbox, such as
CVX, to obtain a numerical solution.
This computational complexity can be approximated by the complexity of solving a standard semidefinite program (SDP) problem through the interior point method, though problem (22) is not a standard SDP problem.
Problem (22) has independent real and imaginary parts in the Hermitian matrix . Let denote the total number of the AO or SPCA iteration. Then the complexity cost of the AO or SPCA method is at most , while the cost reduces to at most or for the two nullspace SRM problems in (13). These computational complexity costs are usually much less when the modern SDP solver like SeDuMi in
CVX is employed.
Iv A Fast PG Algorithm for SRM with a Multi-Antenna Tag
As we have indicated, the RFID system is typically resource-constrained compared with the conventional wireless communication system. Thus, it particularly requires low-complexity algorithms in practice.
In this section, by exploiting the characteristic of the feasible set of the SRM problem in (10), we develop a fast and low-complexity algorithm based on projected gradient (PG) to replace the inefficient
CVX while solving the convex problem in (22).
Let denote the result obtained in the -th inner PG iteration at the -th outer equivalent AO or SPCA iteration. The general PG iteration is given by 
where and are positive step sizes, denotes the projection on the feasible set in (11), i.e.,
and denotes the gradient of the function in (III-D) w.r.t. at the point , which is shown in (27) at the top of the page where , , , and are defined in (28) with and defined in (17). To be specific, here is actually the conjugate derivative of a real function w.r.t. a Hermitian matrix, which falls into the field of generalized complex-valued matrix derivatives and follows from . Note that once the initial value for the PG method is chosen to be feasible, i.e. , it is clear that the sequence generated by the PG method is feasible for any fixed due to the projection operation in (25a) and the condition . By exploiting the structure of the set , the projection in (26) can be formulated in a semi-closed form, which is shown in the following theorem.
Let be the eigenvalue decomposition of . Then the optimal solution to problem (26) is given by
where and are unique and take the form
with and the water-filling level chosen as the minimum nonnegative value such that .
The proof is given in Appendix B. \qed
We remark that the water-filling level in Theorem 2 can be efficiently obtained by some practical algorithms based on hypothesis testing. As for the choosing of the step sizes and in the PG method, several strategies obeying the Armijo rule [47, Section 2.3.1] can be exploited. Here, we fix the second step size as , while the backtracking line search is adopted to determine the first step size . In this way, the iteration in (25) degenerates into
The iteration in (IV) is guaranteed to converge to the global maximum for the convex optimization problem in (22), and it achieves a good balance between the convergence rate and computational complexity. The procedure of backtracking line search for choosing is listed in Algorithm 1. The parameter , and typical algorithmic parameters are , , and . Algorithm 1 is referred to as the Armijo search along the boundary of , .
We summarize our fast algorithm for SRM, combining the PG method for inner convex problem with the formerly proposed outer AO or SPCA iteration, in Algorithm 2, where and are used to denote and , respectively, for notational convenience.
The main computational complexity of Algorithm 2 lies in the multiplication, inverse, and eigenvalue decomposition of a matrix.
To facilitate the complexity comparison with
CVX given in Section III-D, here we give the complexity cost of Algorithm 2 w.r.t. only, which is , where denotes the total iteration number of the outer AO or SPCA method as before and represents the average iteration number of the inner PG method at each outer iteration.
Recalling the computational complexity of
CVX given in Section III-D, we see that the improvement in that of the fast PG algorithm is significant. This observation will also be verified by the simulations in Section VI.
Note that this fast PG algorithm can be also applied to solve the nullspace SRM problem in (13) after some modifications. From the chain rule of the gradient, the gradient of the function w.r.t. at the point is given by
V Single-Antenna Tag
In the previous sections, we have solved the SRM problem with a multi-antenna tag. In real applications, due to the resource and cost constrained property of the RFID network, currently a single antenna is usually used at the tag in the market. Thus, it is necessary to consider security issues under the scenario where the tag has a single antenna, while the reader and the eavesdropper have multiple antennas. It should be noted that previously proposed methods can still be exploited to obtain a local optimal solution for this scenario. In this section, we focus on finding a low-complexity algorithm which yields the global optimal solution to the SRM problem with a single-antenna tag under some practical assumptions.
When the tag employs a single antenna, all the channels form/to the tag reduce to vectors and we redefine . To facilitate analysis and obtain the traceable optimal solution to the SRM problem with a single-antenna tag, here we make the following two assumptions:
The eavesdropper is not aware of the noise injection scheme and thereby simply adopts maximum ratio combining (MRC) to deal with the received signal. Note that when the noise injection scheme is known by the eavesdropper, it may adaptively apply the minimum mean square error (MMSE) receiver to mitigate the jamming from the reader. The SRM problem under this scenario can be similarly tackled by the methods used in the multi-antenna tag case.
The reader transmits the AN signal in the nullspace of its self-interference channel, which is a practical assumption mainly in that the reader can equip with only one more antenna for transmitting compared with for receiving. It should be noted that the mathematical model under this assumption is similar to the one under the assumption that the reader can perfectly cancel the AN received from the self-interference channel, i.e., . Thus, for notational simplicity we only consider the latter assumed situation in this section.
respectively, where follows from the matrix inverse lemma. The SRM problem with a single-antenna tag now becomes
In the following, we will show that although problem (V) is still non-convex, its optimal solution is traceable. The basic idea to solve problem (V) is to reduce the original problem to a single-argument optimization problem, and then the optimal solution can be efficiently obtained by one-dimensional search. Before proceeding, we first give the following lemma about the rank property of the optimal AN covariance for problem (V).
The optimal AN covariance for problem (V) is rank-one.
To show the optimal is rank-one for problem (V), we first let be fixed and the optimal must satisfy
To see more clearly, the above problem can be recast as