Noncoherent and Non-orthogonal Massive SIMO for Critical Industrial IoT Communications

Transcription

1 onoherent and on-orthogonal Massive SIMO for Critial Industrial IoT Communiations He Chen, Zheng Dong, and Branka Vueti The University of Sydney, Australia, {he.hen, zheng.dong, arxiv: v1 [s.it] 5 Mar 019 Abstrat Towards the realization of ultra-reliable low-lateny wireless ommuniations required in ritial industrial Internet of Things IIoT appliations, this paper presents a new nonoherent and non-orthogonal massive single-input multiple-output SIMO framework, in whih a large number of antennas are deployed to provide high spatial diversity so as to enable ultra-reliable ommuniations, and nonoherent transmission and non-orthogonal multiple aess tehniques are applied to effetively redue the lateny at the physial and data link layers, respetively. A two-user IIoT system is onsidered to elaborate the key design priniple of our framework, in whih two ontrolled nodes Cs transmit their data to a ommon managing node M on the same time-frequeny resoure blok, and the M implements the nonoherent maximum likelihood detetor to reover the transmitted symbols of both Cs from the reeived sum signal. We analyze the error performane of the onsidered system and then minimize the system error probability by jointly designing the onstellations of both Cs. Simulation results show that our design has lower error probability than existing designs. I. ITRODUCTIO The Internet of Things IoT, aiming to reate a smart world by onneting everyday objets and surrounding environments to the Internet, is expeted to pervade all aspets of our daily lives and fundamentally alter the way we interat with our physial environment [1]. The appliations of IoT in industrial setors, termed the Industrial IoT IIoT or Industrial Internet, has attrated tremendous attention from governments, aademia and industry, for its substantial potential to transform various industry vertials suh as eletriity, transportation, healthare, and manufaturing [], [3]. As defined by General Eletri GE, the IIoT refers to the network of a multitude of industrial devies onneted by ommuniations tehnologies that results in systems that an monitor, ollet, exhange, analyze, and deliver valuable new insights like never before [3]. From this definition, we note that ommuniation tehnologies play a ritial role in realizing the vision of the IIoT. Critial industrial use ases normally involve real-time losed-loop ontrol, where a failure of ommuniation may lead to serious eonomi losses and safety aidents [4]. Suh appliations pose stringent performane requirements on the industrial ommuniation networks, with high reliability of paket error rate down to 10 9 and ultra-low lateny at the level of sub-miroseond [5]. These strit requirements are far beyond what latest wireless tehnologies an provide, and thus have been satisfied by applying wired network infrastruture [4]. evertheless, wireless ommuniations have several benefits over the urrently-used wired infrastruture: low deployment and maintenane ost, easier deployment in senarios where ables are diffiult to deploy, and high longterm reliability by avoiding the wear and tear issues [6]. There is an emerging onsensus that developing ultra-reliable low-lateny URLL wireless is essential to fully unlok the potential of the IIoT. In wireless ommuniations, diversity tehniques have been used as the main measures to boost system reliability [7]. Among various diversity tehniques, spatial diversity whih is ahieved by equipping the transmitter and/or the reeiver multiple antennas, is partiularly appealing for realizing URLL wireless sine it does not need extra resoures in time or spetrum domain for high reliability. Considering the ultra-high reliability required by ritial IIoT use ases, deploying a massive number of antennas at the transmitter and/or the reeiver has been regarded as one of the most promising tehnologies for URLL wireless [8]. This tehnology is generally referred to as massive multiple-input multiple-output MIMO. In this paper, we term it massive single-input multiple-output SIMO when only a single antenna is equipped at the transmitter side. On the other hand, ahieving low lateny down to the sub-milliseond level in wireless ommuniations is highly hallenging. This involve a departure from the underlying theoretial priniples of wireless ommuniations Today s wireless ommuniation networks have been built to maximize data rates and network apaity with lateny suited to human pereption i.e., at the level of tens of milliseonds [5]. Realizing this several orders of magnitude redution will require signifiant lateny dedution from various layers of the protool staks. Industrial networks are typially based on redued protool staks. As suh, reduing the lateny of the physial and the data link layers is of great importane [9]. At the physial layer, onsidering the fat the data paket e.g., a sensor data or a ontrol ommand in industrial networks is generally very short, shortening the physial layer overheads is an effetive method to redue the lateny. There has reently been a line of researh fousing on the design of nonoherent single-user massive SIMO systems so as to redue the hannel estimation overhead at the physial layer [8], [10] [1], in whih different modulators and detetors were designed and analyzed, and time-division multiple aess TDMA was impliitly assumed to be adopted at the data link layer. At the data link layer, an effetive measures to ahieve low lateny is to implement non-orthogonal multiple aess OMA to replae the urrently-used orthogonal TDMA. In OMA, multiple transmitters are allowed to transmit simultaneously on the same time-frequeny blok so as to redue

2 the yle time, whih is defined as the minimum time needed for all the ontrolled nodes Cs to ommuniate to their managing node M one, and has been the widely-used lateny measure for industrial ontrol systems [6]. To this end, referenes [13] and [14] have reently proposed to jointly optimize the modulation onstellations of multiple users in massive SIMO systems to ensure that the symbols transmitted by multiple users at the same time are as distinguishable as possible at the reeiver side. In these designs, the minimum Eulidean distane MED design riterion was adopted, whih aims to maximize the minimum distane between signal points on the sum omposite onstellation at the reeiver side. However, as shown in our previous work [1], for the singleuser ase, the MED design riterion is obviously suboptimal in terms of system error performane, and adopting the MED design riterion may lead to onsiderable performane loss. To ahieve higher reliability, in [1] we developed a symbolerror-rate-minimization SERM design riterion for singleuser nonoherent massive SIMO systems. evertheless, how to extend the SERM design riterion to address the onstellation design for nonoherent multiuser massive SIMO with OMA is, to the best knowledge of the authors, still an open problem in the literature. As the first effort to fill the aforementioned gap, in this paper we onsider the onstellation design problem for twouser nonoherent and non-orthogonal massive SIMO systems using the SERM design riterion, in whih two single-antenna Cs transmit to a ommon M equipped with a large number of antennas at the same time. In doing this, we onstrain our design to the ase where the onstellations of the two Cs are superimposed in a nested manner at the M side. The M adopts the optimal nonoherent maximum likelihood ML detetor to deode the transmitted symbols of both Cs from the reeived sum signal. We derive losed-form expression for the system SER SSER of the onsidered system, whih is defined as the probability that the symbols transmitted by the two Cs are not both deoded orretly. We formulate an SSER minimization problem to jointly optimize the onstellations of the two Cs, while subjet to their individual average power onstraints. The formulated problem is a omplex multi-ratio frational programming FP problem, whih is in general Phard and thus is diffiult to resolve [15]. Motivated by this, we simplify the problem to a max-min FP problem by resorting to an asymptoti analysis for the regime that the number of antennas at the M goes to infinity. We resolve the simplified problem and attain its optimal solution in losed-form, whih serves as the asymptotially optimal solution to the original problem. Simulation results are provided to demonstrate that our design is superior to the existing designs adopting the MED design riterion. II. SYSTEM MODEL Consider the uplink senario of a wireless IIoT system, where two 1 single-antenna ontrolled nodes Cs transmit 1 ote that the onsidered system an onsist of multiple two-c pairs, whih aess the wireless medium in an orthogonal manner. their data e.g., status information to a managing node M, whih is the entral ontroller unit of the system and is equipped with antennas. To redue the system irle time, the two Cs are allowed to transmit simultaneously to the M on the same time-frequeny resoure blok. By employing a disrete-time omplex baseband-equivalent model, the reeived signal vetor y = [y 1, y,..., y ] T at the M an be written as y = Hx + ξ, 1 where x = [x 1, x ] T represents the transmitted signal vetor with x k, k = 1,, denoting the transmitted symbol of the k- th C equiprobably drawn from the respetive onstellation X k, ξ is the irularly-symmetri omplex Gaussian CSCG noise vetor with ovariane σ I, and H = GD 1/ denotes the omplex hannel matrix between the two Cs and reeiving antennas at the M. We assume that all the entries of G are i.i.d. CSCG distributed with unit variane to haraterize the loal sattering fading, D = diag{β 1, β } β k > 0 is a diagonal matrix whih aptures the large-sale propagation loss due to the distane and shadowing effet. We also let h n = [h 1,n, h,n ] T denote the n-th olumn of H. To further redue the system yle time, we assume that no instantaneous hannel estimation is performed. As suh, G is ompletely unknown and the nonoherent detetion is adopted at the M to reover the transmitted signals from the two Cs. evertheless, the matrix D is assumed to be available at the M sine it hanges muh slower and thus an be estimated with muh lower overhead ompared with the estimation of instantaneous hannel oeffiients [16]. A. onoherent Maximum-Likelihood Detetor For the onsidered nonoherent multiuser SIMO system with uniform inputs, it is known that the nonoherent ML deoder is optimal in the sense that it minimizes average probability of error of the reeived sum signal at the M [17]. To proeed, we note that 1 an be rewritten as y = GD 1/ x + ξ. As all the entries of G and ξ are i.i.d. Gaussian, we immediately have E[y] = 0. By noting y T = x T D 1/ G T + ξ T, and with the help of [18], we have y = vey T = I x T D 1/ veg T + ξ. Then, the ovariane matrix of y an be given by { [I R y x = E{yy H } = E x T D 1/ veg T + ξ ] [ I x T D 1/ veg T + ξ ] } H = x T Dx + σ I = x H Dx + σ I = xi, where x is the suffiient statisti of the input signal, whih is defined as x = x H Dx + σ = k=1 β k x k + σ. 3

3 The probability density funtion PDF of the reeived signal y at the M onditioned on the input signal x an thus be given by fy x = 1 π x exp y. 4 x The nonoherent ML detetor aims to estimate the transmitted information by arrying out the following optimization problem: Combining 4 and 5, we have ˆx = arg min x ln fy x. 5 y ˆx = arg min x + ln x. 6 x We an observe from 6 that the phase information of the input signal is lost. As suh, we an only modulate the information to be transmitted on the power of the transmitted signal i.e., x k in the onsidered system, whih is termed energy-based modulation in [10], [14]. ote that we hereafter use energy and power interhangeably as the symbol duration of the onsidered system is fixed. We define the nonnegative onstellation of eah C as a olletion of the power of the transmitted symbols. For notation simpliity, we assume that both Cs use the same M-ary onstellation. We then use X k = {s k,i } M i=1 to denote the onstellation of the k-th C, k = 1,, or equivalently x k X k. We assume that eah C is subjet to an individual average power onstraint given by M s k,i/m P k, k = 1,, 7 i=1 where P k is the average power onstraint of the k-th C. For the sake of notation later, we further define the onstellation set A k = {a k,i } M i=1 = {β ks k,i } M i=1. The power onstraint in 7 is then equivalent to M a k,i/m β k P k, k = 1,. 8 i=1 Motivated by the fat that uniform onstellations is preferred in most pratial ommuniation systems, we onsider that all A k s are uniform onstellations. We then an express the onstellation set A k of the k-th C as A k = {m δ k +q k } m=0. The individual average power onstraint an be simplified as q k + M 1 δ k β k P k, k = 1,. 9 Without loss of generality, we assume that β 1 P 1 β P. We then an set δ 1 δ. As we an see from 4, the PDF of the reeived signal onditioned on the input signal, fy x, is ompletely haraterized by the suffiient hannel statisti funtion x. Furthermore, x involves the summation of elements drawn from the sets A k, k = 1,. To formally model this, we define It is worth mentioning that our design framework an be extended to the ase with all Cs using distint orders of modulation, where a more ompliated notation system is required. 1 3 δ δ Fig. 1. Illustration of the nested sum onstellation of two nonnegative uniform onstellations of order 4 with q 1 = q = 0, where the sum of the first and seond onstellations produes the third onstellation. We also note that the sum onstellation is uniquely determined by the three distanes δ 1 and δ. { the sum onstellation B = k=1 a k : a k A k }. To ensure that in the noise-free ase, the reeiver an always distinguish all the transmitted symbols one any sum signal b, b B, is reeived, we require that the set B must be uniquely fatorable [19], whih is denoted by B = A 1 A and is formally defined as: Definition 1: The set B is uniquely fatorable if and only if B = k=1 A k = M. That is, for b = k=1 a k and b = k=1 a k, the equality b = b is equivalent to a 1, a = a 1, a. In other words, we require the term x defined in 3 to have a one-to-one orrespondene with the transmitted signal vetor x. Then, the transmitted signal of eah C an be uniquely determined if the sum signal an be orretly deteted. With the aid of the uniquely fatorable property between the sum onstellation and the separate onstellation used by eah C, the optimization problem 6 to be solved by the nonoherent ML detetor an be simplified into the detetion of the reeived sum signal as: ĉ = arg min C y + ln, 10 where C = { l } M = {b l + σ } M. As an initial effort, in this paper we onstrain our design framework to the senario where the signal onstellations of the two Cs are superimposed in a nested manner over the air. That is, the distane between the two end points of the smaller onstellation is less than the distane between the adjaent points of the larger onstellation. Mathematially, we have δ > M 1 δ 1. To failitate the understanding, we illustrate the proess of a nested summation of two nonnegative uniform onstellations of the same order 4 in Fig. 1. We an see from this figure that the nested summation of the onstellations signifiantly redue the minimum Eulidean distane of the sum onstellation at the reeiver side. Fortunately, the resultant performane loss an be effetively ompensated by the large number of antennas equipped at the M. We observe from Fig. 1 that we an define a new notation δ, as the differene between the minimum Eulidean distane of onstellation A i.e., δ and the Eulidean distane of the two end points on the onstellation A 1. We also let δ 1 = δ 1. We will later show that using δ k instead of δ k an simplify the presentation of the optimization problem. With this new

4 definition, the resultant sum onstellation of both Cs an be ompletely haraterized by {δ k } k=1 and {q k} k=1. Speifially, {b l } M and { l} M are both nonnegative weighted sum of {δ k } k=1 and {q k} k=1. That is, given the modulation size M, {δ k } k=1 and {q k} k=1, we an readily enumerate the expressions of { l } M. In the meanwhile, the onstellations of the two Cs, A 1 and A, an also be determined. Hereafter, {δ k } k=1 and {q k} k=1 are the key parameters to be optimized in this paper. Furthermore, by applying the mathematial indution, the average power onstraints of both Cs given in 9 an be further expanded as q 1 + M 1 δ 1 β 1 P 1, 11 q 1 + q + M 1 [M 1 δ 1 + δ ] β P. 1 III. ERROR PERFORMACE AALYSIS AD PROBLEM FORMULATIO A. Optimal Deision Regions and Error Performane We subsequently derive the optimal deision regions of y in the non-oherent ML detetor for a given group of onstellations {A k } k=1 i.e., the set C is given. Without loss of generality, we onsider that all the elements of the set C are arranged in an asending order suh that l < l+1 for l = 1,,..., M 1. We now resolve the optimization problem of the adopted nonoherent ML detetor given in 10 and attain the following theorem on the optimal deision regions of y : Theorem 1: The optimal deision regions of y for the adopted non-oherent ML detetor an be written as 1, ĉ = l, B, if y where d l = l+1 µ l+1 d 1; if d l 1 < y d l, l =,..., M 1; if y > d M 1, 13 l with µx = ln x x 1. The proof is omitted due to spae limitation. Remark 1: In Theorem 1, we have simplified the nonoherent ML detetor into an average reeived power-based detetor. Speifially, the M only needs to the estimate the y average power of the reeived signal i.e., to detet the sum signal. Then, the respetive signal transmitted by eah C an be uniquely determined by using the one-to-one orrespondene between and x. We now analyze the suessful transmission probability of the signal vetor x l. Reall that x l and l have one-to-one orrespondene. Denote by y x l the reeived signal at the M when x l is transmitted by the Cs. Aording to Theorem 1, the suessful transmission probability of the signal vetor x l, denoted by P,l, an be written as Pr yx1 d 1, if l = 1; P,l = Pr d l 1 < yx l d l, if l M 1; yxm Pr > d M 1, if l = M. 14 In this paper, we onsider the senario that the M needs to ollet both Cs information orretly so as to make a further deision. In this ase, the M will laim an error if the sum signal as a whole is deoded erroneously. We define the probability of suh an error as the system symbol error rate SSER. Reall that the transmitted signals of both Cs are drawn from their respetive onstellations with the same probability. We thus an express the SSER as P e = 1 1 M M P,l. 15 To proeed, we note that the random variable yx l l follows a Chi-squared distribution and its umulative distribution funtion CDF is given by 1 x m G x = 1 exp x, x > m! m=0 We an further simplify 14 as follows Pr yx1 1 1 µ 1, if l = 1; Pr µ l l 1 < yx l l l+1 l µ l+1 l, P,l = if l M 1; yxm M Pr > µ, if l = M. M 1 M G 1 µ 1, if l = 1; G l+1 = l µ l+1 l G µ l l 1, if l M 1; M 1 G µ, if l = M. M Substituting 18 into 15 and making neessary manipulations, we an obtain a losed-form expression for the SSER as follows where P e = 1 M 1 M F l+1 l, 19 F t = 1 + G µ t G tµ t 0 is defined for notation simpliity.

5 B. Problem Formulation We are now ready to formulate a SSER minimization problem for the onsidered system, in wihh we optimize the onstellations of both Cs i.e., {δ k } k=1 and {q k} k=1 while onsidering the individual average power onstraint of eah C. Mathematially, we have P1 min P e = 1 M 1 {δ k } k=1,{q k} M F k=1 l+1 l, 1 s.t. δ k 0, q k 0, 11, 1, where we reall that { l } M are nonnegative weighted sum of {δ k } k=1 and {q k} k=1. We an see that P1 is a multiratio frational programming FP problem. More speifially, it is a sum-of-funtions-of-ratio problem, whih is generally P-hard [15]. We now try to simplify P1 by investigating the harateristis of its objetive funtion and onstraints. We first arrive at the following lemma regarding the optimal value of {q k } k=1 : Lemma 1: The optimal values of {q k } k=1 in P1 are q 1 = q = 0. The proof is omitted due to spae limitation. Remark : Lemma 1 indiates that all the optimal onstellations must inlude the origin. This result an be understood intuitively as follows: When not all the onstellations used by the Cs inlude zero, the resultant sum onstellation will not inlude the zero. In this ase, we an always move the most left-side onstellation point of the sum onstellation to the origin to further redue the SSER without violating the average power onstraints of the Cs. As suh, all the optimal onstellations used by Cs should inlude the origin. Applying Lemma 1, we redue P1 to the following optimization problem P1.1 min P e = 1 M 1 l+1 δ 1,δ M F, s.t. δ k 0, M 1 l M 1 δ 1 β 1 P 1, 3 [M 1 δ 1 + δ ] β P 4 where { l } M are nonnegative weighted sum of {δ k} k=1 only. Though we have simplified the original P1 by removing half of the variables to be optimized, the new P1.1 is still diffiult to resolve due to the ompliated struture of the objetive funtion. To the best knowledge of the authors, only a stationary point loal optimality of P1.1 an be effiiently ahieved by applying the latest quadrati transform algorithm developed in [15]. Motivated by this issue, in the subsequent setion we will study the asymptoti ase with the number of antennas at the M i.e., approahing infinity so as to attain the asymptotially optimal solution to P1.1, i.e., asymptotially optimal onstellation design for the onsidered IIoT system. IV. ASYMPTOTICALLY OPTIMAL COSTELLATIO DESIG In this setion, we perform the asymptoti analysis of the SSER for the regime that the number of antennas equipped at the M goes to infinity i.e.,, so as to further simplify the objetive funtion of P1.1. In our previous work [1], we have onduted similar asymptoti analysis for a single-user nonoherent massive SIMO system. By following a similar proedure, we attain that both the upper and lower bounds of the SSER P e are monotonially dereasing } M 1 { funtions of the term min l+1 l. We omit the detailed derivation for brevity and refer interested readers to [1, Theorem 3] and its proof for details. It is worth mentioning that the asymptoti expression has shown to be very tight and an approah its exat ounterpart when is moderately large [1]. The nie feature identified in the asymptoti analysis indiates that minimizing the SSER is equivalent to } M 1 { maximizing the term min l+1 l. Mathematially, we an simplify P1.1 to the following problem P max min {δ k } k=1 { } M 1 l+1 l, s.t. 3, 4, 5 whih is a max-min-ratio problem. In fat, P an be diretly resolved by applying the quadrati transform algorithm developed in [15]. After a areful observation at the ratios in the objetive funtion of P, we find that we an further simplify P due to the following two important observations: Observation 1: By realling the definitions of { l } M and {δ k } k=1, we notie that for any l, the differene between l+1 l is always equal to one of the δ k s. By this observation, we divide all the M 1 ratios in the objetive funtion of P { into two groups, with the kth group being denoted by l +δ k l }l { 1,...,M l+1 l =δ k }. ote that the number of ratios in eah group an be different. Observation : For a given δ k, the larger the l, the smaller the ratio l+δ k l. As suh, the minimal ratio in the kth group is ahieved when l equals to its maximum possible value in the set l { 1,..., M } l+1 l = δ k. We denote the maximum possible value of l in the kth group as l δk. By the above two important observations, we have suessfully redued the number of ratios in P from M 1 to. By applying the mathematial indution method to enumerate the values of lδk s for given M, we suessfully simplify P to the following problem P3 max min δ 1,δ { M M 1 δ1 + M 1 δ + σ M M 1 δ 1 + M 1 δ + σ, δ 1 } M 1 δ 1 + M 1 δ + σ s.t. 3, 4. M 1 δ 1 + M 1 δ + σ δ

6 SSER =1,β =5,σ =1 =1,β =5,σ =1 =4,β =5,σ =1 =4,β =5,σ =1 =1,β =5,σ =0.1 =1,β =5,σ =0.1 =4,β =5,σ =0.1 =4,β =5,σ = umber of M antennas Fig.. Comparison between our design and the MED design, where P 1 = P = 316 mw 5 dbm and M =. After some mathematial manipulations, we arrive at the following proposition regarding the optimal solution to P3. Proposition 1: The optimal solution to P3, denoted by δ1 and δ, is determined in the following two ases: If δ 1 β P β 1 P 1, we have δ1 = δ 1 β P /M 1, and δ = β P /M 1 δ 1 β P. If δ 1 β P > β 1 P 1, we have δ 1 = β 1 P 1 /M 1, and δ = σ + +β1p1 σ + 4 β1p1 δ = δ+σ + σ β 1 P 1 +β 1 P 1 σ + β 1 P 1 σ. + + δ+σ + σ 4 δ + δ σ. Here, δ 1 The proof is omitted due to spae limitation. Till now we have obtained the asymptotially optimal onstellations of the two Cs in the onsidered system. V. SIMULATIO RESULTS We now present simulation results to ompare the SSER performane of the proposed design and the existing design using MED riterion. In doing this, we plot the SSER urves of these two shemes versus the number of antennas equipped at the M i.e., for different values of β 1, β, and σ in Fig.. We an see from Fig. that our design is superior to the MED ounterpart for all simulated senarios, as long as is large enough. Moreover, for a given signal-to-noise ratio i.e., σ is fixed, the performane gap of our sheme over the MED one beomes lager when the value of β 1 is loser to that of β. This is beause when β is fixed, the larger β 1 gives us more spae to optimize the smaller onstellation suh that the performane gain over the MED design riterion is enlarged. VI. COCLUSIOS In this paper, we developed a new nonoheret and nonorthogonal massive SIMO framework to enable ultra-reliable low-lateny wireless needed in emerging ritial industrial Internet of Things IIoT appliations. As the first work within this framework, we have designed a two-user IIoT system, whih onsists of two single-antenna ontrolled nodes Cs and one managing node M equipped with a large number of antennas. The two Cs transmit their information to the M simultaneously on the same radio resoure, and the M applies the nonoherent maximum likelihood detetor to reover both Cs information from the reeived sum signal. We jointly optimized the onstellations of both Cs to maximize the system reliability. We managed to find the losed-form expression of the asymptotially optimal solution to the formulated problem. Simulation results demonstrated that the proposed design has better system reliability than the existing designs adopting the minimum Eulidean distane riterion. As future work, we will extend our framework to arbitrary number of users, and will implement the design on softwaredefined radio platforms to demonstrate and evaluate its performane in real environments. REFERECES [1] A. 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