Physical Layer Authentication for Non-coherent Massive SIMO-Based Industrial IoT Communications
Zhifang Gu, He Chen, Pingping Xu, Yonghui Li, Branka Vucetic
aa r X i v : . [ ee ss . SP ] J a n Physical Layer Authentication for Non-coherent MassiveSIMO-Based Industrial IoT Communications
Zhifang Gu ∗ , He Chen † , Pingping Xu ∗ , Yonghui Li ‡ , and Branka Vucetic ‡∗ National Mobile Communications Research Laboratory, Southeast University, Nanjing, China † Department of Information Engineering, The Chinese University of Hong Kong, Hong Kong, China ‡ School of Electrical and Information Engineering, The University of Sydney, Sydney, Australia ∗ { zhifang gu, xpp } @seu.edu.cn, † [email protected], ‡ { yonghui.li, branka.vucetic } @sydney.edu.au Abstract —Achieving ultra-reliable, low-latency and securecommunications is essential for realizing the industrial Internetof Things (IIoT). Non-coherent massive multiple-input multiple-output (MIMO) has recently been proposed as a promisingmethodology to fulfill ultra-reliable and low-latency require-ments. In addition, physical layer authentication (PLA) technol-ogy is particularly suitable for IIoT communications thanks toits low-latency attribute. A PLA method for non-coherent mas-sive single-input multiple-output (SIMO) IIoT communicationsystems is proposed in this paper. Specifically, we first determinethe optimal embedding of the authentication information (tag) inthe message information. We then optimize the power allocationbetween message and tag signal to characterize the trade-offbetween message and tag error performance. Numerical resultsshow that the proposed PLA is more accurate then traditionalmethods adopting the uniform tag when the communicationreliability remains at the same level. The proposed PLA methodcan be effectively applied to the non-coherent system.
I. I
NTRODUCTION
The fast development of the Internet of Things (IoT)has promoted innovations in many fields. The applicationof the IoT in the industrial sector, referred to as industrialIoT (IIoT), has recently attracted tremendous attention fromresearchers and engineers owing to its ability to improvethe efficiency and productivity of industry. Compared withtraditional industrial networks mainly based on wired cables,wireless communications are more suitable for the IIoT dueto low maintenance expenditure, flexible deployment, andhigher long-term reliability [1]. However, ultra-reliable, low-latency and secure requirements of the IIoT represent mainchallenges for wireless design [2] [3]. Wireless channels sufferfrom path-loss, shadowing, fading and interference, thus it isvery challenging to design wireless networks to achieve theultra-reliable transmission [4]. Moreover, the broadcast char-acteristic of wireless channels makes the IIoT systems morevulnerable to attacks [5]. Non-coherent massive multiple-input multiple-output (MIMO) has recently been proposedas a promising methodology to meet ultra-reliable and low-latency requirements of IIoT communications [6], which usesmultiple receive antennas to reduce the effects of fading anduncorrelated noise in wireless channels and boost systemreliability. Besides, non-coherent massive MIMO uses energy-based modulation to achieve low latency by avoiding channelestimation and by applying fast non-coherent detection [7] [8].Two security services have been commonly considered in the IIoT, including integrity and authenticity, which are essential inIIoT systems. Message authentication code (MAC) is a preva-lent mechanism to provide these two services. Conventionalsystems realize message authentication by attaching a MACto the message and this authentication process is completedabove the physical layer [9], e.g., transport layer security(TLS) protocol in the transport layer and Wi-Fi protectedaccess II (WPA2) protocol in the network layer. However,these conventional mechanisms may not be able to meet thestringent low-latency requirement of IIoT communications.Because short packet transmission is one of the characteristicsof the IIoT, the transmission overhead for the MAC can belarge and excessive in the short packet transmission withsmall payload, occurring relatively large delay. In addition, theauthentication process can only be completed after the data hasbeen transferred to upper layers, which leads to low efficiency.Therefore, we aim to propose a message authentication methodat the physical layer which can also reduce transmissionoverhead.Physical layer security, according to its implementationmethod, can be divided into two categories. The first categoryis based on the information-theoretic approach, which was pro-posed by Shannon [10] and further developed by Wyner withthe wiretap channel model [11]. This kind of approach onlyguarantees data confidentiality by preventing eavesdropperfrom understanding the information, but other security serviceslike data integrity and authenticity are not considered. Thesecond category is based on the signal and channel features,which aims to provide authenticity and data integrity. In theIIoT scenario, active attacks (e.g., modification, masqueradeor replay attack) are much more harmful than passive attacks(e.g., eavesdropping) [12]. Therefore, we consider physicallayer authentication (PLA) from the perspective of the secondcategory. Existing PLA methods have two forms: passive andactive [13]. Passive PLA utilizes the intrinsic features ofcommunication systems to authenticate the transmitter, suchas radio signal strength indicator, channel state information(CSI) and radio frequency fingerprints [5]. These featureswere thoroughly analyzed in [14] with a theoretical modeland experimental validation. The results of [14] revealed thatthe intrinsic features are not reliable in practical scenarios dueto the device mobility, wireless fading channels and indistin-guishable RF fingerprints. In contrast, active PLA refers to theethods in which the transmitter sends additional information(normally referred to as tag) for authentication at the physicallayer. Active PLA features embedding a tag in the messageinformation and does not take extra time to transmit the tag.Thanks to its potential to meet the low-latency requirement,active PLA has advantages over conventional authenticationmethods sending a message and its tag separately [15].The key issue of implementing active PLA is how to embeda tag in message information at the physical layer. Severalmethods dealing with this issue have been published. Thetag was added as noise in [16]: different additional angleoffsets to normal quadrature phase shift keying (QPSK) indi-cate different tag bits. 4/16 hierarchical quadrature amplitudemodulation (QAM) was applied in [17] to transmit a messageand its tag simultaneously, where a 4-QAM tag constellation issuperimposed on a 4-QAM message constellation. Challengeresponse PLA was introduced in [18] and the authenticationinformation was embedded during a “challenge and response”process. Although these active PLA methods transmit a tagand a message at the same time, they all need to send separatepilots for channel estimation to acquire the instantaneous CSI.In [19], a tag is embedded in the original pilot to form anew pilot, and the tag detection is completed by a correlationoperation. Then the new pilot signal is used to estimate CSI formessage recovery. Note that these existing active PLA methodsare not suitable for non-coherent massive MIMO-based IIoTsystems, in which no estimation of the instantaneous CSI isperformed. To our best knowledge, how to perform active PLAfor non-coherent systems is still an open problem.As the first attempt to fill this gap, we focus on designingan active PLA mechanism for non-coherent massive single-input multiple-output (SIMO) IIoT systems. As elaborated in[20], non-negative pulse amplitude modulation (PAM) is afavorable scheme for the considered system. In the system,the variance of the received signal power increases as thetransmitted signal amplitude increases. In this context, thetag embedding constellation pattern is not necessary to beuniform as in the existing methods, but needs to be optimizedaccording to the message constellation. The tag embeddingdesign becomes a nontrivial problem as the increased tagsignal power reduces the error rate of tag while increasingthe error rate of message, leading to the error performancetrade-off between message and tag. In this paper, we manageto find an optimal 1-bit tag embedding design based on agiven message constellation. Then for a fixed average systempower, we attain the optimal power allocation of message andtag signals to characterize the trade-off between message andtag error performance, which can provide useful insights forpractical system design.II. S
YSTEM M ODEL
In this paper, we consider the PLA in a massive SIMO-based IIoT communication system, where multiple sensorstransmit data to a controller with N antennas. Each sensorhas one single antenna and these sensors send data to thecontroller with time-division multiple access (TDMA) manner. (cid:48)(cid:82)(cid:71)(cid:88)(cid:79)(cid:68)(cid:87)(cid:82)(cid:85)(cid:3)(cid:20)(cid:48)(cid:82)(cid:71)(cid:88)(cid:79)(cid:68)(cid:87)(cid:82)(cid:85)(cid:3)(cid:21)(cid:43)(cid:68)(cid:86)(cid:75) (cid:1854)(cid:1863) (cid:1839) (cid:1865) (cid:1872) (cid:1876) (cid:48)(cid:82)(cid:71)(cid:88)(cid:79)(cid:68)(cid:87)(cid:82)(cid:85)(cid:3)(cid:20)(cid:48)(cid:82)(cid:71)(cid:88)(cid:79)(cid:68)(cid:87)(cid:82)(cid:85)(cid:3)(cid:21)(cid:43)(cid:68)(cid:86)(cid:75) (cid:1854)(cid:1863) (cid:1839) (cid:1865) (cid:1872) (cid:1876) Fig. 1. Diagram of the PLA at the transmitter side.
An attacker is within the range of this wireless communicationsystem, who can receive the signal from sensors and sendmalicious signals to the controller. To meet the low-latencyrequirement of the IIoT, we adopt non-negative PAM atthe transmitter and non-coherent maximum likelihood (ML)detector at the receiver [6]–[8]. Since only statistics of thechannel are needed in this method, the channel estimationprocess is not required and the authentication can be executedfaster. To realize PLA, the transmitter embeds the tag signal inthe message signal at the physical layer, as illustrated in Fig. 1.MAC is denoted by M , which is generated by message bits b and secret key k with a hash function, and it is given by M = hash( b, k ) . (1)The modulated signal of b and M are denoted by m and t ,which are termed message signal and tag signal, respectively.The transmitted signal x is then determined by x = p | m | + | t | , (2)where m ∈ M = { m i | i = 1 , · · · , L m } and t ∈ T = { t i,j | i = 1 , · · · , L m ; j = 1 , · · · , L t } . L m and L t arethe number of constellation points of message signal andtag signal, respectively. The transmitted signal that involvesmessage signal m i is denoted as x i . The average power ofmessage signal E m and the average power of tag signal E t are constrained by the total average system power E tot E m + E t ≤ E tot , (3)where E m = L m L m P i =1 | m i | and E t = L m L t L m P i =1 L t P j =1 | t i,j | . Thereceived signal y at the controller in the considered massiveSIMO system can be represented by y = h x + n , (4)where h = [ h , · · · , h N ] T is the SIMO channel vector and n = [ n , · · · , n N ] T is the noise vector between the sensorand controller. We assume that h is a circularly symmetriccomplex Gaussian random vector, specifically, each element of h is independently and identically distributed with zero meanand unit variance (i.e., Rayleigh fading). Another assumptionis that n is a circularly symmetric complex Gaussian randomvector which is independent of h . The mean vector of n is azero vector and its covariance matrix is σ I N . σ is assumedto be known and I N is an N -dimensional unit matrix.Two steps are required in the signal detection at the receiverside. First, following the ML detection rule, the estimatedmessage signal ˆ m is obtained by solving the problem ˆ m = arg max m ∈M f ( y | m ) , (5)here f ( y | m ) is the probability density function (PDF) of y conditioned on m . Second, the estimated tag signal ˆ t isobtained based on ˆ m with the ML rule, ˆ t = arg max t ∈T f ( y | ˆ m, t ) , (6)where f ( y | ˆ m, t ) is the PDF of y conditioned on ˆ m and t .After the detection, we can get the estimated message bits b ′ and the estimated MAC M ′ with the demodulation of ˆ m and ˆ t , respectively. Then a new MAC M n is calculated with b ′ and k by the same hash function, i.e., M n = hash( b ′ , k ) . Theauthentication process is completed by comparing M ′ with M n . If M ′ and M n are identical, the message can be regradedas coming from a legitimate user and not being tamperedwith. Otherwise, this message will be discarded because it isnot authenticated successfully.III. P RELIMINARIES AND D ETECTION R ULES
In this section, we first introduce preliminaries on non-negative PAM design for message constellation in the massiveSIMO system, and then present the 1-bit tag embedding designwhen the message constellation points are given.
A. Preliminaries on Message Constellation Design
We now consider only a message signal m ∈ M istransmitted. The problem (5) has been solved in [8], whichshowed that the ML detection problem can be solved bya quantization operation. More specifically, the quantizationoperation is described as [8] ˆ m = m , if || y || N < B ; m i , if B i − ≤ || y || N ≤ B i , i = 2 , · · · , L m − m L m , if || y || N > B L m − , (7)where B i is the optimal decision threshold between m i and m i +1 . The threshold B i can be represented by B i = A i A i +1 ln A i +1 A i A i +1 − A i , i = 1 , ..., L m − , (8)where A i = | m i | + σ . Based on this optimal decision rule,the correct detection probability of i -th symbol m i , denotedby P c,i , is determined by [8] P c,i = G (cid:16) NB i A i (cid:17) , if i = 1; G (cid:16) NB i A i (cid:17) − G (cid:16) NB i − A i (cid:17) , if i = 2 , · · · , L m − − G (cid:16) NB Lm − A i (cid:17) , if i = L m , (9)where G ( z ) is the cumulative distribution function (CDF) ofa complex Chi-squared distribution variable Z , given by G ( z ) = 1 − e − z N − X L =0 z L L ! , z > . (10) Note that time stamps or session identifiers are required to resist replayattacks.
When each message symbol is selected from M with equalprobability, the average message symbol error rate (SER),denoted by P e , can be calculated as follows: P e = 1 − L m L m X i =1 P c,i . (11)By minimizing P e under the constraint that average messagepower is not greater than E m , the asymptotically optimal non-negative PAM constellation design for massive SIMO systemscan be represented as follows [8]: (cid:8) , σ ( R − , σ ( R − , · · · , σ ( R L m − − (cid:9) , (12)where R ( R > ) is obtained by solving the equation L m − X j =0 R j = L m (cid:18) E m σ + 1 (cid:19) . (13)The message signal-to-noise ratio (SNR), denoted by γ m , isdefined as γ m = E m σ . The constellation points are describedfrom the perspective of power since there is a specific cor-respondence between power and amplitude in non-negativePAM. Take L m = 4 as an example, the message constellationpoints described with A i and the corresponding decisionthresholds B i are shown in Fig. 2(a). B. Tag Embedding Design
We are now ready to elaborate the proposed tag embeddingscheme and the corresponding detection rule. One-bit embed-ding is considered in this paper, i.e., L t = 2 . When the i -thmessage symbol is transmitted, the power of the embeddedsymbol E i,j has two possible values according to two differenttag bits, E i,j = ( | m i | + | t i, | , if j = 1 (tag bit is 0); | m i | + | t i, | , if j = 2 (tag bit is 1) . (14)Note that the power of tag signal t i, and t i, are variablesdepending on message signal m i , which is the key idea ofthe proposed “Message-based Tag Modulation”. In existingmethods that use uniform tag embedding, t i, and t i, are con-stants for different message signal m i . The reason we do notuse uniform tag embedding is that the message constellationpoints in our method contain the relationship of geometricseries as shown in Fig. 2(a). If t i, and t i, are high power tagsignals, they may be suitable for signal points A and A , butwill not work well for A and A , and vice versa. Due to the ! " $ % ! % " % (a) Message constellation design and decision thresholds !,! !," ",! "," $,! $," % ! % " % % $ & ! ’ & "’ & (b) Tag Embedding design and updated decision thresholds.Fig. 2. Signal design and decision thresholds. ffect of the embedding tag, the message decision thresholds(8) need to be updated. We also take L m = 4 as an example toshow the tag embedding design and the corresponding decisionthresholds in Fig. 2(b). Since the message constellation pointsthat are close to each other are more error-prone, we use thenearest two constellation points to calculate the new messagedecision threshold B ′ i by (8), B ′ i = A i, A i +1 , ln A i +1 , A i, A i +1 , − A i, , i = 1 , · · · , L m − , (15)where A i,j = E i,j + σ . Using (7) and (15), the messagesymbol can be estimated, which is the first step of thedetection. Based on the result of estimated message symbol,tag symbol is detected subsequently with the following rule, ˆ t = ( t i, (tag bit is 0) , if || y || N ≤ C i ; t i, (tag bit is 1) , if || y || N > C i ; (16)where C i is the optimal decision threshold to decide which tagbit is embedded in message signal m i . Since (8) also followsthe general form of non-coherent ML decision threshold, C i can be expressed as C i = A i, A i, ln A i, A i, A i, − A i, , i = 1 , , · · · , L m . (17)Note that the instantaneous CSI is not required in messagedetector (7) and tag detector (16). As such, successive interfer-ence cancellation (SIC) detection widely used in existing PLAmethods is no longer applicable in this method. According tothe constellation design results of Section III-A, the optimalpower of first constellation point is zero. Therefore, we canset the power of the tag signal to zero when tag bit is , i.e., | t i, | = 0 (for i = 1 , · · · , L m ).IV. E RROR P ERFORMANCE A NALYSIS AND O PTIMIZATION
In this section, the message SER and tag SER are analyzedfirst as the performance metrics of the proposed PLA method.Then the optimization problem of signal design is formulatedand solved according to the specific system requirements.
A. Error Performance Analysis
According to the assumptions of h and n in Section II, y isalso a circularly symmetric complex Gaussian random vector.The mean vector of y is a zero vector, and its covariancematrix can be written as E h yy H i = E (cid:20)(cid:16) h p | m | + | t | + n (cid:17) (cid:16) h p | m | + | t | + n (cid:17) H (cid:21) = (cid:0) | m | + | t | + σ (cid:1) I N . (18) Define a new random variable Z ′ Z ′ = || y || | m | + | t | + σ , (19)which follows complex Chi-squared distribution. When m istransmitted with embedding tag t , , according to (7) and (15), the message signal can be correctly detected if || y || N < B ′ .This condition is equivalent to || y || | m | + | t , | + σ < N B ′ | m | + | t , | + σ = N B ′ A , . (20)Let P cm,i denote the average correct message detection prob-ability of x i . Then P cm, can be derived as P cm, = G (cid:18) N B ′ A , (cid:19) . (21)Similarly, P cm,i can be given by P cm,i =
12 2 P j =1 G (cid:16) NB ′ i A i,j (cid:17) , if i = 1;
12 2 P j =1 h G (cid:16) NB ′ i A i,j (cid:17) − G (cid:16) NB ′ i − A i,j (cid:17)i , if i = 2 , · · · , L m −
12 2 P j =1 h − G (cid:16) NB ′ Lm − A i,j (cid:17)i , if i = L m . (22) The average message SER P em can be calculated as P em = 1 − L m L m X i =1 P cm,i . (23) The assumption is made that the message symbols are selectedfrom constellation points collection with equal probability.Due to the uniformity characteristic of hash functions, dif-ferent tag bits can also be considered appearing with equalprobability [15].We now consider the error performance of the tag signal.Suitable hash functions exhibit “avalanche effect”, i.e., theoutput changes significantly if the input alters slightly [15].The authentication fails when the message estimation hasonly one bit error, which indicates that the recalculated MACwill be meaningless if the message parts have errors. Asa result, we consider the tag SER under the condition thatthe message symbol is detected correctly. It should be notedthat the message SER can be controlled to be low enough(e.g. less than − ) during the embedding design process.Therefore, in this paper, the tag correct rate and error rateare calculated from the perspective of conditional probability.When the embedded symbol x i is transmitted, the tag correctdetection rate under the condition that the message symbol m i is correctly obtained can be determined by (16), which is P ct,i = 12 (cid:20) G (cid:18) NC i A i, (cid:19) + 1 − G (cid:18) NC i A i, (cid:19)(cid:21) . (24) To simplify the expression, we define the proportionalvariables r i ( i = 1 , · · · , L m ) r i = A i, A i, , < r i < R. (25) Therefore, the average tag SER, denoted by P et , can berepresented by P et = 1 L m L m X i =1 (1 − P ct,i )= 12 L m L m X i =1 [1 + G ( Nu ( r i )) − G ( Nv ( r i ))] , (26) here u ( r i ) = ln r i r i − and v ( r i ) = r i ln r i r i − . B. Trade-off Characterization
One primary purpose of this paper is to design an optimalembedding scheme to minimize the tag SER subject to the totalaverage power constraint. Meanwhile, the system reliabilityshould meet the certain requirement, i.e., P em < δ , where δ is the message SER requirement threshold. This optimizationproblem is formulated as follows min { A i,j } j =1 , i =1 , ··· ,Lm P et (27a) s . t . E t + E m ≤ E tot , (27b) P em ≤ δ. (27c)This problem is non-trivial because of multiple optimizationvariables and the complex structure of constraint functions.To simplify this problem, we use two steps to solve it. First,we find the optimal tag embedding scheme when the messageconstellation is fixed (i.e., E m is given). Second, we search forthe optimal allocated power of message E m that can minimizethe tag SER.Now we consider the first step. When E m is fixed, themessage SER P em and the tag SER P et are given by (23)and (26), respectively. Note that P em is complicated since itcontains multiple variables ( A i,j , i = 1 , · · · , L m , j = 1 or 2 ).To reduce the complexity of constraint (27c), an upper boundof message SER, P uem , can be derived to replace P em in (27c).The upper bound can be derived as follows P uem = 1 L m L m − X i =1 { − G [ Ng ( r i )] + G [ Nh ( r i )] } , (28) where g ( r i ) = R ln Rri R − r i , h ( r i ) = r i ln Rri R − r i . The proof is providedin Appendix A.Using variable substitution r i = e k i (0 < k i < ln R ) and themessage SER upper bound in (28), the optimization problem(27) can be rewritten as min { k i } Lmi =1 L m L m X i =1 n G h Nu ( e k i ) i − G h Nv ( e k i ) io (29a) s . t . L m L m X i =1 A i, ( e k i − ≤ E tot − E m , (29b) L m L m − X i =1 n − G h Ng ( e k i ) i + G h Nh ( e k i ) io ≤ δ. (29c) We can show that (36) is a convex optimization problem andthe proof is provided in Appendix B. Therefore (36) can beefficiently solved by interior-point method and the optimal tagembedding scheme can be determined when E m is given.In the second step, we consider the situation when E m isa variable, different E m results in different values of P et . Inthis case, the optimization result of (36) is a function of E m ,which is denoted by H ( E m ) . Note that when H ( E m ) achievesits minimal value, the inequality (27b) will become a equality,i.e., E m + E t = E tot . The reason is that if E tot still has asurplus, P em and P et can be further reduced by increasingmessage and tag power at the same time. Therefore, this optimization problem can be narrated as finding the optimalpower allocation between message signal and tag signal whenthe constraints are still satisfied. Define the power allocationfactor α , α = E m E tot , α ≤ α ≤ , (30)where α is the minimum factor that makes E m satisfy theconstraint P uem ≤ δ . The function H ( E m ) can be representedby H ( α ) , which indicates that different α corresponds todifferent minimized tag SER. Therefore, the new power al-location problem can be formulated as min α ≤ α ≤ H ( α ) (31a) s . t . E t ≤ (1 − α ) E tot , (31b) P ue,m ≤ δ. (31c) This is a problem of single variable with a limited range, whichcan be efficiently solved by one-dimensional search.When we set different values of message SER requirementthreshold δ for the considered system, different minimized tagSER can be obtained by solving the optimization problem(27). Therefore, we can characterize the trade-off between themessage and tag error performance through changing the valueof the message SER requirement thresholds.V. N UMERICAL R ESULTS
We carried out computer simulations to verify the theoreticalresults. In our simulations, we set N = 128 and σ = 1 .Firstly, to show the accuracy of the message and tag errorperformance analysis, the theoretical and simulation resultsof P em and P et are demonstrated in Fig. 3 and they matchwith each other. E tot is set to be large enough becauseoptimization has not yet been considered. The “Message-based Tag Modulation” scheme is compared with the “UniformEmbedding” scheme that embeds same tag power levels forall message symbols. For a fair comparison, we control the tagpower of two embedding schemes to make sure they have thesame message SER performance (i.e., less than − ) whenthe message SNR is dB. The results show that the tag SER Message SNR (dB) -6 -5 -4 -3 -2 -1 S y m bo l e rr o r r a t e Uniform Embedding, Message (Theoretical)Uniform Embedding, Message (Simulation)Uniform Embedding, Tag (Theoretical)Uniform Embedding, Tag (Simulation)Proposed Embedding, Message (Theoretical)Proposed Embedding, Message (Simulation)Proposed Embedding, Tag (Theoretical)Proposed Embedding, Tag (Simulation)
Fig. 3. Comparison of SER of the proposed embedding scheme and uniformembedding scheme. .5 0.55 0.6 0.65 0.7 0.75 0.8
Power allocation factor, -5 -4 -3 -2 T ag sy m bo l e rr o r r a t e Etot=14dBEtot=15dBEtot=16dBEtot=17dB
Fig. 4. Optimization results of tag SER with different E tot . of the proposed embedding scheme decreases as the messageSNR increases. However, the tag SER of the “Uniform Em-bedding” scheme is always above . Therefore, “UniformEmbedding” scheme is not suitable for PLA in non-coherentmassive SIMO communications.Then we focus on the optimal performance by solvingoptimization problems in Section IV-B. The message SERupper bound threshold δ is set to − . The results ofoptimization problem (36) and the search results for powerallocation are provided in Fig. 4. The tag SER decreases as thetotal system average power increases and the optimal powerallocation factor can be found from the results.Finally, to understand the entire system performance, thetrade-off between P em and P et is depicted in Fig. 5. We canobserve from the trade-off curves that the tag SER decreases asthe message SER requirement threshold increases. The resultsin Fig. 5 also show that both tag SER and message SER canbe reduced when the total system average power increases.The trade-off curve presents the optimal tag SER performanceunder different system requirements for message SER, whichcan provide useful insights for practical PLA system design.VI. C ONCLUSIONS
In this paper, we proposed an active PLA mechanismfor non-coherent massive SIMO-based IIoT systems. Thispaper is the first to show that active PLA can be achievedwithout the need of pilot signal and channel estimation. Wedesigned the optimal tag embedding scheme when the messageconstellation is given. Then we solved the power allocationproblem to obtain the optimal tag SER performance. Thetrade-off curve between tag SER and message SER wasdepicted to offer a comprehensive understanding of the systemperformance. From the simulation results, we can concludethat “Message-based Tag Modulation” is necessary for theconsidered non-coherent system. Moreover, the proposed au-thentication method can meet the specific power and errorrate requirements of IIoT system. As an initial effort, the tagembedding design was limited to 1-bit tag per message symbolin this paper. We will extend our work to multiple bits tag permessage symbol for future work. -6 -5 -4 -3 Message symbol error rate -7 -6 -5 -4 -3 -2 T ag sy m bo l e rr o r r a t e Etot=14dBEtot=15dBEtot=16dBEtot=17dB
Fig. 5. Trade-off curves of tag SER and message SER with different E tot . A PPENDIX AP ROOF OF AN U PPER B OUND OF THE M ESSAGE
SERNotice that the average message SER of x is less thantransmitting p m + t , , because close constellation pointsproduce a large message SER. Let P em,i denote the averagemessage SER when m i is transmitted. P em, and its upperbound can be derived as follows P em, = 12 " P (cid:18) ˆ m = m , · · · , m L m | x = q m + t , (cid:19) + P (cid:18) ˆ m = m , · · · , m L m | x = q m + t , (cid:19) < P (cid:18) ˆ m = m , · · · , m L m | x = q m + t , (cid:19) = 1 − G (cid:18) NB ′ A , (cid:19) . (32) Similarly, the average message SER of x L m is less thantransmitting p m L m + t L m , , so the upper bound of P em,L m can be derived as follows P em,L m < P (cid:18) ˆ m = m , · · · , m L m − | x L m = q m L m + t L m , (cid:19) = G (cid:18) NB ′ L m − A Lm, (cid:19) . (33) Further, the average message SER of x i ( i = 2 , · · · , L m − also has an upper bound due to the same reason above, P em,i < P (cid:18) ˆ m = m , · · · , m i − | x i = q m i + t i, (cid:19) + P (cid:18) ˆ m = m i +1 , · · · , m L m | x i = q m i + t i, (cid:19) = 1 − G (cid:18) NB ′ i A i, (cid:19) + G (cid:18) NB ′ i − A i, (cid:19) . (34) From the above, an upper bound of P em can be derived as em = 1 L m L m X i =1 P em,i < L m L m − X i =1 (cid:20) − G (cid:18) NB ′ i A i, (cid:19) + G (cid:18) NB ′ i A i +1 , (cid:19)(cid:21) = 1 L m L m − X i =1 { − G [ Ng ( r i )] + G [ Nh ( r i )] } , (35) where g ( r i ) = R ln Rri R − r i , h ( r i ) = r i ln Rri R − r i . This completes theproof of an upper bound of the message SER.A PPENDIX BP ROOF OF THE CONVEX OPTIMIZATION PROBLEM
The considered optimization problem is rewritten as fol-lows: min { k i } Lmi =1 L m L m X i =1 n G h Nu ( e k i ) i − G h Nv ( e k i ) io (36a) s . t . L m L m X i =1 A i, ( e k i − ≤ E tot − E m , (36b) L m L m − X i =1 n − G h Ng ( e k i ) i + G h Nh ( e k i ) io ≤ δ. (36c) Let F ( k ) = 1 + G (cid:2) N u ( e k ) (cid:3) − G (cid:2) N v ( e k ) (cid:3) and W ( k ) =1 − G (cid:2) N g ( e k ) (cid:3) + G (cid:2) N h ( e k ) (cid:3) . F ( k ) is a convex function for < k < ln R accordingto the lemma in [8]. The objective function in (36a) is asum of L m convex functions F ( k i )( i = 1 , · · · , L m ) and itsHessian matrix is a diagonal matrix, thus the Hessian matrixis positive definite and the objective function in (36a) is aconvex function. Note that e k − is a basic convex function.The left side of (36b) is a sum of L m convex functions e k i − i = 1 , · · · , L m ) and its Hessian matrix is a diagonalmatrix, so it is also a convex function. The left side of (36c)is a sum of L m − functions W ( k i )( i = 1 , · · · , L m − and its Hessian matrix is a diagonal matrix. Therefore, theoptimization problem (36) is a convex problem if W ( k ) is aconvex function for < k < ln R .Let W ( k ) = G (cid:2) N g ( e k ) (cid:3) and W ( k ) = G (cid:2) N h ( e k ) (cid:3) . Thenwe have W ′ ( k ) = W ′ ( k ) − W ′ ( k ) . The derivative of G ( z ) is f Z ( z ) = N − z N − e − z . Then we have W ′ ( k ) = f Z h Ng ( e k ) i Ng ′ ( e k )= 1( N − h Ng ( e k ) i N − e − Ng ( e k ) Ng ′ ( e k ) . (37) Since h ( e k ) = g ( e k ) e k R = g ( e k ) + ( k − ln R ) , we can obtainthat h ′ ( e k ) = R e k (cid:2) g ( e k ) + g ′ ( e k ) (cid:3) . Then we have W ′ ( k ) = f Z h Nh ( e k ) i Nh ′ ( e k )= 1( N − h Nh ( e k ) i N − · e − Nh ( e k ) Nh ′ ( e k )= 1( N − (cid:20) Ng ( e k ) e k R (cid:21) N − · e − N [ g ( e k )+( k − ln R ) ] · N (cid:26) R e k h g ( e k ) + g ′ ( e k ) i(cid:27) = W ′ ( k ) + 1( N − N N g ( e k ) N e − Ng ( e k ) . (38) Therefore, we can simplify the W ′ ( k ) as follows W ′ ( k ) = 1( N − N N g ( e k ) N e − Ng ( e k ) . (39) The second-order derivative of W ( k ) can be obtained by W ′′ ( k ) = N N +1 g ( e k ) N − e − Ng ( e k ) g ′ ( e k ) (cid:2) − g ( e k ) (cid:3) ( N − , (40) where g ′ ( e k ) = R (cid:2) e k − R + e k (ln R − k ) (cid:3) ( R − e k ) r = e k = Rr ( R − r ) (1 + ln Rr − Rr ) . (41) Since ln x − x is a monotonically decreasing function for x > , then we have Rr − Rr < . Therefore, g ′ ( e k ) < for < k < ln R . Then we know that g ( e k ) is a decreasingfunction when < k < ln R . Moreover, lim k → ln R g ( e k ) = 1 ,then − g ( e k ) < for < k < ln R . We can conclude that W ′′ ( k ) > < k < ln R ) , thus W ( k ) is a convex functionfor < k < ln R . Above all, the optimization problem (36)is a convex problem. R EFERENCES[1] M. Luvisotto, Z. Pang and D. Dzung, “Ultra High Performance WirelessControl for Critical Applications: Challenges and Directions,”
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