Noncoherent and Non-orthogonal Massive SIMO for Critical Industrial IoT Communications
NNoncoherent and Non-orthogonal Massive SIMOfor Critical Industrial IoT Communications
He Chen, Zheng Dong, and Branka VuceticThe University of Sydney, Australia, Email: { he.chen, zheng.dong, branka.vucetic } @sydney.edu.au Abstract —Towards the realization of ultra-reliable low-latencywireless communications required in critical industrial Internet ofThings (IIoT) applications, this paper presents a new noncoherentand non-orthogonal massive single-input multiple-output (SIMO)framework, in which a large number of antennas are deployed toprovide high spatial diversity so as to enable ultra-reliable com-munications, and noncoherent transmission and non-orthogonalmultiple access techniques are applied to effectively reduce thelatency at the physical and data link layers, respectively. Atwo-user IIoT system is considered to elaborate the key designprinciple of our framework, in which two controlled nodes (CNs)transmit their data to a common managing node (MN) on thesame time-frequency resource block, and the MN implementsthe noncoherent maximum likelihood detector to recover thetransmitted symbols of both CNs from the received sum signal.We analyze the error performance of the considered system andthen minimize the system error probability by jointly designingthe constellations of both CNs. Simulation results show that ourdesign has lower error probability than existing designs.
I. I
NTRODUCTION
The Internet of Things (IoT), aiming to create a smart worldby connecting everyday objects and surrounding environmentsto the Internet, is expected to pervade all aspects of ourdaily lives and fundamentally alter the way we interact withour physical environment [1]. The applications of IoT inindustrial sectors, termed the Industrial IoT (IIoT) or IndustrialInternet, has attracted tremendous attention from governments,academia and industry, for its substantial potential to transformvarious industry verticals such as electricity, transportation,healthcare, and manufacturing [2], [3]. As defined by GeneralElectric (GE), the IIoT refers to “the network of a multitude ofindustrial devices connected by communications technologiesthat results in systems that can monitor, collect, exchange, an-alyze, and deliver valuable new insights like never before” [3].From this definition, we note that communication technologiesplay a critical role in realizing the vision of the IIoT.Critical industrial use cases normally involve real-timeclosed-loop control, where a failure of communication maylead to serious economic losses and safety accidents [4].Such applications pose stringent performance requirements onthe industrial communication networks, with high reliabilityof packet error rate down to − and ultra-low latency atthe level of sub-microsecond [5]. These strict requirementsare far beyond what latest wireless technologies can provide,and thus have been satisfied by applying wired networkinfrastructure [4]. Nevertheless, wireless communications haveseveral benefits over the currently-used wired infrastructure:low deployment and maintenance cost, easier deployment in scenarios where cables are difficult to deploy, and high long-term reliability by avoiding the wear and tear issues [6].There is an emerging consensus that developing ultra-reliablelow-latency (URLL) wireless is essential to fully unlock thepotential of the IIoT.In wireless communications, diversity techniques have beenused as the main measures to boost system reliability [7].Among various diversity techniques, spatial diversity whichis achieved by equipping the transmitter and/or the receivermultiple antennas, is particularly appealing for realizing URLLwireless since it does not need extra resources in time or spec-trum domain for high reliability. Considering the ultra-highreliability required by critical IIoT use cases, deploying a mas-sive number of antennas at the transmitter and/or the receiverhas been regarded as one of the most promising technologiesfor URLL wireless [8]. This technology is generally referredto as massive multiple-input multiple-output (MIMO). In thispaper, we term it massive single-input multiple-output (SIMO)when only a single antenna is equipped at the transmitter side.On the other hand, achieving low latency down to thesub-millisecond level in wireless communications is highlychallenging. This involve a departure from the underlyingtheoretical principles of wireless communications—Today’swireless communication networks have been built to maximizedata rates and network capacity with latency suited to humanperception (i.e., at the level of tens of milliseconds) [5].Realizing this several orders of magnitude reduction willrequire significant latency deduction from various layers ofthe protocol stacks. Industrial networks are typically based onreduced protocol stacks. As such, reducing the latency of thephysical and the data link layers is of great importance [9]. Atthe physical layer, considering the fact the data packet (e.g.,a sensor data or a control command) in industrial networks isgenerally very short, shortening the physical layer overheadsis an effective method to reduce the latency. There has recentlybeen a line of research focusing on the design of noncoherentsingle-user massive SIMO systems so as to reduce the channelestimation overhead at the physical layer [8], [10]–[12], inwhich different modulators and detectors were designed andanalyzed, and time-division multiple access (TDMA) wasimplicitly assumed to be adopted at the data link layer.At the data link layer, an effective measures to achievelow latency is to implement non-orthogonal multiple access(NOMA) to replace the currently-used orthogonal TDMA. InNOMA, multiple transmitters are allowed to transmit simul-taneously on the same time-frequency block so as to reduce a r X i v : . [ c s . I T ] M a r he cycle time, which is defined as the minimum time neededfor all the controlled nodes (CNs) to communicate to theirmanaging node (MN) once, and has been the widely-usedlatency measure for industrial control systems [6]. To thisend, references [13] and [14] have recently proposed to jointlyoptimize the modulation constellations of multiple users inmassive SIMO systems to ensure that the symbols transmittedby multiple users at the same time are as distinguishable aspossible at the receiver side. In these designs, the minimumEuclidean distance (MED) design criterion was adopted, whichaims to maximize the minimum distance between signal pointson the sum (composite) constellation at the receiver side.However, as shown in our previous work [12], for the single-user case, the MED design criterion is obviously suboptimalin terms of system error performance, and adopting the MEDdesign criterion may lead to considerable performance loss.To achieve higher reliability, in [12] we developed a symbol-error-rate-minimization (SERM) design criterion for single-user noncoherent massive SIMO systems. Nevertheless, how toextend the SERM design criterion to address the constellationdesign for noncoherent multiuser massive SIMO with NOMAis, to the best knowledge of the authors, still an open problemin the literature.As the first effort to fill the aforementioned gap, in thispaper we consider the constellation design problem for two-user noncoherent and non-orthogonal massive SIMO systemsusing the SERM design criterion, in which two single-antennaCNs transmit to a common MN equipped with a large numberof antennas at the same time. In doing this, we constrain ourdesign to the case where the constellations of the two CNs aresuperimposed in a nested manner at the MN side. The MNadopts the optimal noncoherent maximum likelihood (ML)detector to decode the transmitted symbols of both CNs fromthe received sum signal. We derive closed-form expression forthe system SER (SSER) of the considered system, which is de-fined as the probability that the symbols transmitted by the twoCNs are not both decoded correctly. We formulate an SSERminimization problem to jointly optimize the constellations ofthe two CNs, while subject to their individual average powerconstraints. The formulated problem is a complex multi-ratiofractional programming (FP) problem, which is in general NP-hard and thus is difficult to resolve [15]. Motivated by this, wesimplify the problem to a max-min FP problem by resortingto an asymptotic analysis for the regime that the number ofantennas at the MN goes to infinity. We resolve the simplifiedproblem and attain its optimal solution in closed-form, whichserves as the asymptotically optimal solution to the originalproblem. Simulation results are provided to demonstrate thatour design is superior to the existing designs adopting theMED design criterion.II. S YSTEM M ODEL
Consider the uplink scenario of a wireless IIoT system,where two single-antenna controlled nodes (CNs) transmit Note that the considered system can consist of multiple two-CN pairs,which access the wireless medium in an orthogonal manner. their data (e.g., status information) to a managing node (MN),which is the central controller unit of the system and isequipped with N ( N (cid:29) ) antennas. To reduce the systemcircle time, the two CNs are allowed to transmit simultane-ously to the MN on the same time-frequency resource block.By employing a discrete-time complex baseband-equivalentmodel, the received signal vector y = [ y , y , . . . , y N ] T atthe MN can be written as y = Hx + ξ , (1)where x = [ x , x ] T represents the transmitted signal vectorwith x k , k = 1 , , denoting the transmitted symbol of the k -th CN equiprobably drawn from the respective constellation X k , ξ is the circularly-symmetric complex Gaussian (CSCG)noise vector with covariance σ I N , and H = GD / denotesthe N × complex channel matrix between the two CNsand N receiving antennas at the MN. We assume that all theentries of G are i.i.d. CSCG distributed with unit varianceto characterize the local scattering fading, D = diag { β , β } ( β k > ) is a diagonal matrix which captures the large-scalepropagation loss due to the distance and shadowing effect.We also let h n = [ h ,n , h ,n ] T denote the n -th column of H .To further reduce the system cycle time, we assume that noinstantaneous channel estimation is performed. As such, G iscompletely unknown and the noncoherent detection is adoptedat the MN to recover the transmitted signals from the two CNs.Nevertheless, the matrix D is assumed to be available at theMN since it changes much slower and thus can be estimatedwith much lower overhead compared with the estimation ofinstantaneous channel coefficients [16]. A. Noncoherent Maximum-Likelihood Detector
For the considered noncoherent multiuser SIMO systemwith uniform inputs, it is known that the noncoherent MLdecoder is optimal in the sense that it minimizes averageprobability of error of the received sum signal at the MN[17]. To proceed, we note that (1) can be rewritten as y = GD / x + ξ . As all the entries of G and ξ arei.i.d. Gaussian, we immediately have E [ y ] = . By noting y T = x T D / G T + ξ T , and with the help of [18], we have y = vec( y T ) = ( I N ⊗ x T D / )vec( G T ) + ξ . (2)Then, the covariance matrix of y can be given by R y | x = E { yy H } = E (cid:110)(cid:2)(cid:0) I N ⊗ x T D / )vec( G T ) + ξ (cid:3)(cid:2) ( I N ⊗ x T D / )vec( G T ) + ξ (cid:3) H (cid:111) = ( x T Dx ∗ + σ ) I N = ( x H Dx + σ ) I N = c ( x ) I N , where c ( x ) is the sufficient statistic of the input signal, whichis defined as c ( x ) = x H Dx + σ = (cid:88) k =1 β k | x k | + σ . (3)he probability density function (PDF) of the received signal y at the MN conditioned on the input signal x can thus begiven by f ( y | x ) = 1 π N c N ( x ) exp (cid:16) − (cid:107) y (cid:107) c ( x ) (cid:17) . (4)The noncoherent ML detector aims to estimate the transmittedinformation by carrying out the following optimization prob-lem: ˆ x = arg min x ln f ( y | x ) . (5)Combining (4) and (5), we have ˆ x = arg min x (cid:107) y (cid:107) c ( x ) + N ln c ( x ) . (6)We can observe from (6) that the phase information of theinput signal is lost. As such, we can only modulate theinformation to be transmitted on the power of the transmittedsignal (i.e., | x k | ) in the considered system, which is termedenergy-based modulation in [10], [14]. Note that we hereafteruse energy and power interchangeably as the symbol durationof the considered system is fixed. We define the (nonnegative)constellation of each CN as a collection of the power of thetransmitted symbols. For notation simplicity, we assume thatboth CNs use the same M -ary constellation . We then use X k = { s k,i } Mi =1 to denote the constellation of the k -th CN, k = 1 , , or equivalently | x k | ∈ X k . We assume that eachCN is subject to an individual average power constraint givenby (cid:88) Mi =1 s k,i /M ≤ P k , k = 1 , , (7)where P k is the average power constraint of the k -th CN. Forthe sake of notation later, we further define the constellationset A k = { a k,i } Mi =1 = { β k s k,i } Mi =1 . The power constraint in(7) is then equivalent to (cid:88) Mi =1 a k,i /M ≤ β k P k , k = 1 , . (8)Motivated by the fact that uniform constellations is preferredin most practical communication systems, we consider thatall A k ’s are uniform constellations. We then can express theconstellation set A k of the k -th CN as A k = { m ¯ δ k + q k } M − m =0 .The individual average power constraint can be simplified as q k + M −
12 ¯ δ k ≤ β k P k , k = 1 , . (9)Without loss of generality, we assume that β P ≤ β P . Wethen can set ¯ δ ≤ ¯ δ .As we can see from (4), the PDF of the received sig-nal conditioned on the input signal, f ( y | x ) , is completelycharacterized by the sufficient channel statistic function c ( x ) .Furthermore, c ( x ) involves the summation of elements drawnfrom the sets A k , k = 1 , . To formally model this, we define It is worth mentioning that our design framework can be extended tothe case with all CNs using distinct orders of modulation, where a morecomplicated notation system is required. 𝛿 (cid:2869) 𝛿 (cid:2870) Fig. 1. Illustration of the nested sum constellation of two nonnegative uniformconstellations of order 4 with q = q = 0 , where the sum of the first andsecond constellations produces the third constellation. We also note that thesum constellation is uniquely determined by the three distances δ and δ . the sum constellation B = (cid:110)(cid:80) k =1 a k : a k ∈ A k (cid:111) . To ensurethat in the noise-free case, the receiver can always distinguishall the transmitted symbols once any sum signal b , ∀ b ∈ B , isreceived, we require that the set B must be uniquely factorable [19], which is denoted by B = A (cid:93)A and is formally definedas: Definition 1:
The set B is uniquely factorable if and onlyif |B| = (cid:81) k =1 |A k | = M . That is, for b = (cid:80) k =1 a k and b (cid:48) = (cid:80) k =1 a (cid:48) k , the equality b = b (cid:48) is equivalent to ( a , a ) =( a (cid:48) , a (cid:48) ) . (cid:4) In other words, we require the term c ( x ) defined in (3)to have a one-to-one correspondence with the transmittedsignal vector x . Then, the transmitted signal of each CN canbe uniquely determined if the sum signal can be correctlydetected. With the aid of the uniquely factorable propertybetween the sum constellation and the separate constellationused by each CN, the optimization problem (6) to be solvedby the noncoherent ML detector can be simplified into thedetection of the received sum signal as: ˆ c = arg min c ∈C (cid:107) y (cid:107) c + N ln c, (10)where C = { c (cid:96) } M (cid:96) =1 = { b (cid:96) + σ } M (cid:96) =1 .As an initial effort, in this paper we constrain our designframework to the scenario where the signal constellations ofthe two CNs are superimposed in a nested manner over the air.That is, the distance between the two end points of the smallerconstellation is less than the distance between the adjacentpoints of the larger constellation. Mathematically, we have ¯ δ > ( M − δ . To facilitate the understanding, we illustratethe process of a nested summation of two nonnegative uniformconstellations of the same order 4 in Fig. 1. We can see fromthis figure that the nested summation of the constellationssignificantly reduce the minimum Euclidean distance of thesum constellation at the receiver side. Fortunately, the resultantperformance loss can be effectively compensated by the largenumber of antennas equipped at the MN.We observe from Fig. 1 that we can define a new notation δ , as the difference between the minimum Euclidean distanceof constellation A (i.e., ¯ δ ) and the Euclidean distance of thetwo end points on the constellation A . We also let ¯ δ = δ .We will later show that using δ k instead of ¯ δ k can simplifythe presentation of the optimization problem. With this newefinition, the resultant sum constellation of both CNs can becompletely characterized by { δ k } k =1 and { q k } k =1 . Specif-ically, { b (cid:96) } M (cid:96) =1 and { c (cid:96) } M (cid:96) =1 are both nonnegative weightedsum of { δ k } k =1 and { q k } k =1 . That is, given the modulationsize M , { δ k } k =1 and { q k } k =1 , we can readily enumerate theexpressions of { c (cid:96) } M (cid:96) =1 . In the meanwhile, the constellationsof the two CNs, A and A , can also be determined. Hereafter, { δ k } k =1 and { q k } k =1 are the key parameters to be optimizedin this paper. Furthermore, by applying the mathematicalinduction, the average power constraints of both CNs givenin (9) can be further expanded as q + M − δ ≤ β P , (11) q + q + M −
12 [( M − δ + δ ] ≤ β P . (12)III. E RROR P ERFORMANCE A NALYSIS AND P ROBLEM F ORMULATION
A. Optimal Decision Regions and Error Performance
We subsequently derive the optimal decision regions of (cid:107) y (cid:107) in the non-coherent ML detector for a given group ofconstellations {A k } k =1 (i.e., the set C is given). Without lossof generality, we consider that all the elements of the set C are arranged in an ascending order such that c (cid:96) < c (cid:96) +1 for (cid:96) = 1 , , . . . , M − . We now resolve the optimizationproblem of the adopted noncoherent ML detector given in(10) and attain the following theorem on the optimal decisionregions of (cid:107) y (cid:107) : Theorem 1:
The optimal decision regions of (cid:107) y (cid:107) for theadopted non-coherent ML detector can be written as ˆ c = c , if (cid:107) y (cid:107) N ≤ d ; c (cid:96) , if d (cid:96) − < (cid:107) y (cid:107) N ≤ d (cid:96) , (cid:96) = 2 , . . . , M − c |B| , if (cid:107) y (cid:107) N > d M − , (13)where d (cid:96) = c (cid:96) +1 µ (cid:16) c (cid:96) +1 c (cid:96) (cid:17) with µ ( x ) = ln xx − . (cid:4) The proof is omitted due to space limitation.
Remark 1:
In Theorem 1, we have simplified the non-coherent ML detector into an average received power-baseddetector. Specifically, the MN only needs to the estimate theaverage power of the received signal (i.e., (cid:107) y (cid:107) N ) to detectthe sum signal c . Then, the respective signal transmitted byeach CN can be uniquely determined by using the one-to-onecorrespondence between c and x . (cid:4) We now analyze the successful transmission probability ofthe signal vector x (cid:96) . Recall that x (cid:96) and c (cid:96) have one-to-onecorrespondence. Denote by (cid:107) y ( x (cid:96) ) (cid:107) the received signal at theMN when x (cid:96) is transmitted by the CNs. According to Theorem 1, the successful transmission probability of the signal vector x (cid:96) , denoted by P c,(cid:96) , can be written as P c,(cid:96) = Pr (cid:16) (cid:107) y ( x ) (cid:107) N ≤ d (cid:17) , if (cid:96) = 1;Pr (cid:16) d (cid:96) − < (cid:107) y ( x (cid:96) ) (cid:107) N ≤ d (cid:96) (cid:17) , if 2 ≤ (cid:96) ≤ M − (cid:18) (cid:107) y ( x M ) (cid:107) N > d M − (cid:19) , if (cid:96) = M . (14)In this paper, we consider the scenario that the MN needsto collect both CNs’ information correctly so as to make afurther decision. In this case, the MN will claim an error ifthe sum signal as a whole is decoded erroneously. We definethe probability of such an error as the system symbol errorrate (SSER). Recall that the transmitted signals of both CNsare drawn from their respective constellations with the sameprobability. We thus can express the SSER as P e = 1 − M M (cid:88) (cid:96) =1 P c,(cid:96) . (15)To proceed, we note that the random variable (cid:107) y ( x (cid:96) ) (cid:107) c (cid:96) fol-lows a Chi-squared distribution and its cumulative distributionfunction (CDF) is given by G ( x ) = 1 − exp ( − x ) N − (cid:88) m =0 x m m ! , x > . (16)We can further simplify (14) as follows P c,(cid:96) = Pr (cid:16) (cid:107) y ( x ) (cid:107) c ≤ N c c µ (cid:16) c c (cid:17)(cid:17) , if (cid:96) = 1;Pr (cid:16) N µ (cid:16) c (cid:96) c (cid:96) − (cid:17) < (cid:107) y ( x (cid:96) ) (cid:107) c (cid:96) ≤ N c (cid:96) +1 c (cid:96) µ (cid:16) c (cid:96) +1 c (cid:96) (cid:17)(cid:17) , if 2 ≤ (cid:96) ≤ M − (cid:18) (cid:107) y ( x M ) (cid:107) c M > N µ (cid:16) c M c M − (cid:17)(cid:19) , if (cid:96) = M . (17) = G (cid:16) N c c µ (cid:16) c c (cid:17)(cid:17) , if (cid:96) = 1; G (cid:16) N c (cid:96) +1 c (cid:96) µ (cid:16) c (cid:96) +1 c (cid:96) (cid:17)(cid:17) − G (cid:16) N µ (cid:16) c (cid:96) c (cid:96) − (cid:17)(cid:17) , if 2 ≤ (cid:96) ≤ M − − G (cid:16) N µ (cid:16) c M c M − (cid:17)(cid:17) , if (cid:96) = M . (18)Substituting (18) into (15) and making necessary manipu-lations, we can obtain a closed-form expression for the SSERas follows P e = 1 M M − (cid:88) (cid:96) =1 F (cid:18) c (cid:96) +1 c (cid:96) (cid:19) , (19)where F ( t ) = 1 + G ( N µ ( t )) − G ( N tµ ( t )) (20)is defined for notation simplicity. . Problem Formulation We are now ready to formulate a SSER minimizationproblem for the considered system, in wihch we optimize theconstellations of both CNs (i.e., { δ k } k =1 and { q k } k =1 ) whileconsidering the individual average power constraint of eachCN. Mathematically, we have ( P1 ) min { δ k } k =1 , { q k } k =1 P e = 1 M M − (cid:88) (cid:96) =1 F (cid:18) c (cid:96) +1 c (cid:96) (cid:19) , (21) s . t . δ k ≥ , q k ≥ , (11) , (12) , (22)where we recall that { c (cid:96) } M (cid:96) =1 are nonnegative weighted sumof { δ k } k =1 and { q k } k =1 . We can see that ( P1 ) is a multi-ratio fractional programming (FP) problem. More specifically,it is a sum-of-functions-of-ratio problem, which is generallyNP-hard [15].We now try to simplify ( P1 ) by investigating the character-istics of its objective function and constraints. We first arrive atthe 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 ∗ = q ∗ = 0 .The proof is omitted due to space limitation. Remark 2:
Lemma 1 indicates that all the optimal constel-lations must include the origin. This result can be understoodintuitively as follows: When not all the constellations usedby the CNs include zero, the resultant sum constellation willnot include the zero. In this case, we can always move themost left-side constellation point of the sum constellation tothe origin to further reduce the SSER without violating theaverage power constraints of the CNs. As such, all the optimalconstellations used by CNs should include the origin. (cid:4)
Applying Lemma 1, we reduce ( P1 ) to the followingoptimization problem ( P1 . ) min δ ,δ P e = 1 M (cid:88) M − (cid:96) =1 F (cid:18) c (cid:96) +1 c (cid:96) (cid:19) , s . t . δ k ≥ , M − δ ≤ β P , (23) M −
12 [( M − δ + δ ] ≤ β P (24)where { c (cid:96) } M (cid:96) =1 are nonnegative weighted sum of { δ k } k =1 only .Though we have simplified the original ( P1 ) by removinghalf of the variables to be optimized, the new ( P1 . ) isstill difficult to resolve due to the complicated structure ofthe objective function. To the best knowledge of the authors,only a stationary point (local optimality) of ( P1 . ) can beefficiently achieved by applying the latest quadratic transformalgorithm developed in [15]. Motivated by this issue, in thesubsequent section we will study the asymptotic case with thenumber of antennas at the MN (i.e., N ) approaching infinity soas to attain the asymptotically optimal solution to ( P1 . ) , i.e.,asymptotically optimal constellation design for the consideredIIoT system. IV. A SYMPTOTICALLY O PTIMAL C ONSTELLATION D ESIGN
In this section, we perform the asymptotic analysis of theSSER for the regime that the number of antennas equippedat the MN goes to infinity (i.e., N → ∞ ), so as to furthersimplify the objective function of ( P1 . ) . In our previouswork [12], we have conducted similar asymptotic analysis fora single-user noncoherent massive SIMO system. By follow-ing a similar procedure, we attain that both the upper andlower bounds of the SSER P e are monotonically decreasingfunctions of the term min (cid:110) c (cid:96) +1 c (cid:96) (cid:111) M − (cid:96) =1 . We omit the detailedderivation for brevity and refer interested readers to [12,Theorem 3] and its proof for details. It is worth mentioningthat the asymptotic expression has shown to be very tightand can approach its exact counterpart when N is moder-ately large [12]. The nice feature identified in the asymptoticanalysis indicates that minimizing the SSER is equivalent tomaximizing the term min (cid:110) c (cid:96) +1 c (cid:96) (cid:111) M − (cid:96) =1 . Mathematically, wecan simplify ( P1 . ) to the following problem ( P2 ) max { δ k } k =1 min (cid:26) c (cid:96) +1 c (cid:96) (cid:27) M − (cid:96) =1 , s . t . (23) , (24) , (25)which is a max-min-ratio problem. In fact, ( P2 ) can bedirectly resolved by applying the quadratic transform algo-rithm developed in [15]. After a careful observation at theratios in the objective function of ( P2 ) , we find that wecan further simplify ( P2 ) due to the following two importantobservations: • Observation 1 : By recalling the definitions of { c (cid:96) } M (cid:96) =1 and { δ k } k =1 , we notice that for any (cid:96) , the differencebetween c (cid:96) +1 − c (cid:96) is always equal to one of the δ k ’s. Bythis observation, we divide all the M − ratios in theobjective function of ( P2 ) into two groups, with the k thgroup being denoted by (cid:110) c (cid:96) + δ k c (cid:96) (cid:111) (cid:96) ∈{ ,...,M | c (cid:96) +1 − c (cid:96) = δ k } .Note that the number of ratios in each group can bedifferent. • Observation 2 : For a given δ k , the larger the c (cid:96) , thesmaller the ratio c (cid:96) + δ k c (cid:96) . As such, the minimal ratio in the k th group is achieved when (cid:96) equals to its maximum pos-sible value in the set (cid:96) ∈ (cid:8) , . . . , M (cid:12)(cid:12) c (cid:96) +1 − c (cid:96) = δ k (cid:9) .We denote the maximum possible value of (cid:96) in the k thgroup as (cid:96) δ k .By the above two important observations, we have success-fully reduced the number of ratios in ( P2 ) from M − to .By applying the mathematical induction method to enumeratethe values of c (cid:96) δk ’s for given M , we successfully simplify ( P2 ) to the following problem ( P3 ) max δ ,δ min (cid:26) M ( M − δ + ( M − δ + σ M ( M − δ + ( M − δ + σ − δ , ( M − δ + ( M − δ + σ ( M − δ + ( M − δ + σ − δ (cid:41) s . t . (23) , (24) . −7 −6 −5 −4 −3 −2 −1 Number of MN antennas (N)
SSE R Proposed, β =1, β =5, σ =1MED, β =1, β =5, σ =1Proposed, β =4, β =5, σ =1MED, β =4, β =5, σ =1Proposed, β =1, β =5, σ =0.1MED, β =1, β =5, σ =0.1Proposed, β =4, β =5, σ =0.1MED, β =4, β =5, σ =0.1 Fig. 2. Comparison between our design and the MED design, where P = P =
316 mW (25 dBm) and M = 2 . After some mathematical manipulations, we arrive at thefollowing proposition regarding the optimal solution to ( P3 ) . Proposition 1:
The optimal solution to ( P3 ) , denoted by δ ∗ and δ ∗ , is determined in the following two cases: • If ˜ δ † (2 β P ) ≤ β P , we have δ ∗ = ˜ δ † (2 β P ) / ( M − , and δ ∗ = 2 β P / ( M − − ˜ δ † (2 β P ) . • If ˜ δ † (2 β P ) > β P , we have δ ∗ = 2 β P / ( M − , and δ ∗ = − (cid:16) σ M − +2 β P (cid:17) + (cid:115)(cid:16) σ M − − β P (cid:17) + (cid:18) (2 β P β P σ
2+ 2 β P σ M − (cid:19) M − .Here, ˜ δ † (cid:16) ˜ δ (cid:17) = − (cid:16) ˜ δ + σ + σ M − (cid:17) + (cid:114)(cid:16) ˜ δ + σ + σ M − (cid:17) + ( ˜ δ δ σ ) M − .The proof is omitted due to space limitation.Till now we have obtained the asymptotically optimalconstellations of the two CNs in the considered system.V. S IMULATION R ESULTS
We now present simulation results to compare the SSERperformance of the proposed design and the existing designusing MED criterion. In doing this, we plot the SSER curvesof these two schemes versus the number of antennas equippedat the MN (i.e., N ) for different values of β , β , and σ inFig. 2. We can see from Fig. 2 that our design is superior tothe MED counterpart for all simulated scenarios, as long as N is large enough. Moreover, for a given signal-to-noise ratio(i.e., σ is fixed), the performance gap of our scheme over theMED one becomes lager when the value of β is closer to thatof β . This is because when β is fixed, the larger β gives usmore space to optimize the smaller constellation such that theperformance gain over the MED design criterion is enlarged.VI. C ONCLUSIONS
In this paper, we developed a new noncoheret and non-orthogonal massive SIMO framework to enable ultra-reliablelow-latency wireless needed in emerging critical industrial Internet of Things (IIoT) applications. As the first work withinthis framework, we have designed a two-user IIoT system,which consists of two single-antenna controlled nodes (CNs)and one managing node (MN) equipped with a large numberof antennas. The two CNs transmit their information to theMN simultaneously on the same radio resource, and theMN applies the noncoherent maximum likelihood detectorto recover both CNs’ information from the received sumsignal. We jointly optimized the constellations of both CNsto maximize the system reliability. We managed to find theclosed-form expression of the asymptotically optimal solutionto the formulated problem. Simulation results demonstratedthat the proposed design has better system reliability thanthe existing designs adopting the minimum Euclidean distancecriterion.As future work, we will extend our framework to arbitrarynumber of users, and will implement the design on software-defined radio platforms to demonstrate and evaluate its per-formance in real environments.R
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