Noncoherent Multiuser Massive SIMO for Low-Latency Industrial IoT Communications
11 Noncoherent Multiuser Massive SIMO forLow-Latency Industrial IoT Communications
Zheng Dong ∗ , He Chen ∗ , Jian-Kang Zhang † , and Branka Vucetic ∗∗ The University of Sydney, Australia, emails: { zheng.dong, he.chen, branka.vucetic } @sydney.edu.au † McMaster University, Canada, email:[email protected]
Abstract —In this paper, we consider a multiuser massivesingle-input multiple-output (SIMO) enabled Industrial Internetof Things (IIoT) communication system. To reduce the latencyand overhead caused by channel estimation, we assume that onlythe large-scale fading coefficients are available. We employ anoncoherent maximum-likelihood (ML) detector at the receiverside which does not need the instantaneous channel state infor-mation (CSI). For such a massive SIMO system, we present anew design framework to assure that each transmitted signalmatrix can be uniquely determined in the noise-free case and bereliably estimated in noisy cases. The key idea is to utilize a newconcept called the uniquely decomposable constellation group(UDCG) based on the practically used quadrature amplitudemodulation (QAM) constellation. To improve the average errorperformance when the antenna array size is scaled up, wepropose a max-min Kullback-Leibler (KL) distance design bycarrying out optimization over the transmitted power and thesub-constellation assignment. Finally, simulation results showthat the proposed design outperforms significantly the existingmax-min Euclidean distance based method in terms of errorperformance. Moreover, our proposed approach also has a bettererror performance than the conventional coherent zero-forcing(ZF) receiver with orthogonal training for cell edge users.
I. I
NTRODUCTION
The fourth stage of industrial revolution, also termed In-dustry 4.0 or Industrial Internet of Things (IIoT), is a newvision that in general refers to the introduction of advancedInternet technologies in industrial control and manufacturing,with the aim of significantly boosting the flexibility, versatility,usability and efficiency of future smart factories [1]. There is ageneral consensus that providing powerful and pervasive con-nectivity between machines, people and objects in industrialenvironments will be essential to realize this new vision.Connectivity in factories has until now been dominatedby wired systems, which has been preferred for its real-timecapability and reliability. Wireless systems are mainly appliedin noncritical use cases such as monitoring of conditions,which poses comparatively low communication performancerequirements since communication failure will not lead to seri-ous accidents (e.g., economic losses and safety problems) [2].The main reason behind this is that stringent performance interms of relibility and latency required by critical use cases areseveral orders of magnitude better than what is achievable bytodays wireless technologies [3]. On the other hand, there areincreasing necessity for the use of wireless solutions in criticaluse cases because it reduces cost for large-scale deployment,enables flexible communication needed in smart production,and opens new fields of application, such as the control of moving objects together with their simultaneous localizationand allocation. As such, there is an urgent need to developultra-reliable low-latency wireless communications for criticalIIoT use cases. Recently, massive multiple-input multipleoutput (MIMO) technology, which refers to the usage of alarge number of antennas in wireless systems to harness thespatial diversity, has been regarded indispensable for providingultrahigh reliability [4]. In light of this, how to develop low-latency massive MIMO systems has become one of the mostimportant research problems in the field.Motivated by the above facts, we consider an uplink mul-tiuser massive single-input multiple-output (SIMO) enabledIIoT communication system. We note that the channel esti-mation is a major obstacle to reduce the transmission latencydue to the following two reasons: 1) the estimation of channelstate information (CSI) is commonly done by transmittingsome known orthogonal pilot symbols, the minimum numberof which is no less than the number of transmitter antennas [5].This will cause significant delay when the user number islarge; 2) different from most conventional communicationsystems which commonly have very long data packets, thecontrol/data packets in IIoT applications are typically veryshort. Thus, the overhead caused by channel estimation isnon-negligible and will reduce the information rate signif-icantly [6]. In our design, to avoid the overhead and la-tency caused by channel estimation, we adopt a noncoherentmaximum-likelihood (ML) receiver, where no instantaneousCSI is needed at all the nodes. The proposed new transmissionframework is based on the concept called uniquely decompos-able constellation group (UDCG) with QAM constellations.Simulation results show that our proposed design outperformsthe currently available max-min Euclidean distance basedmethod. Our contributions can be summarized as follows: • In our design, by jointly considering two time slots,constellations with phase information (e.g., QAM) canbe used by all the users [7], [8], while existing workson single-user or multiuser noncoherent massive SIMOconsidered only one time slot, the information can onlybe modulated on the amplitudes of the input signals,resulting in a low spectral efficiency [9]–[11]. • We present an explicit construction of UDCG for any number of users. Unlike design in [12] which is based onthe trellis coding structure, whose received constellationis very complicated and hard to decode, the received con-stellation in our design has a simple geometric structure. a r X i v : . [ c s . I T ] M a r II. T HE S YSTEM M ODEL AND N ONCOHERENT M AXIMUM -L IKELIHOOD (ML) D
ETECTOR
A. The System Model and Noncoherent ML Detector
We consider a massive SIMO system consisting of K single-antenna users transmitting simultaneously to the BSwith M ( M (cid:29) K ) receiving antennas on the same time-frequency grid. By using a discrete-time complex baseband-equivalent model, the received signal at the antenna arrayof BS in the t -th time slot y t = [ y ,t , . . . , y M,t ] T can beexpressed by y t = Hx t + ξ t , where x t = [ x ,t , . . . , x K,t ] T represents the transmittedsignals from all the K users, ξ t is an additive circularly-symmetric complex Gaussian (CSCG) noise vector with co-variance σ I M . We let H = GD / denote the M × K complex channel matrix between the receiver antenna arrayand all the users, where G characterizes the small-scale fadingcaused by local scattering while D = diag { β , · · · , β K } , β k > captures the propagation loss due to distance andshadowing. All the entries of G are assumed to be i.i.d. com-plex Gaussian distributed with zero mean and unit variance.The channel coefficients are assumed to be in block fadingwhich are quasi-static in the current block and change to otherindependent values in the next block with a channel coherencetime T c ≥ K . We consider a space-time block modulation(STBM) [8] scheme over T time slots and the received signalvectors can be stacked together into a matrix given by Y T = HX T + Ξ T , (1)where Y T = [ y , . . . , y T ] , X T = [ x , . . . , x T ] and Ξ T =[ ξ , · · · , ξ T ] . Assumption 1:
Throughout this paper, we adopt the follow-ing assumptions:1) The small scale channel fading matrix G is completelyunknown to the BS and all the users, while the largescale fading matrix D is available to all the nodes;2) The transmitted signals are subject to an instantaneousaverage power constraint : E {| x k,t | } ≤ P k , k =1 , . . . , K , t = 1 , . . . , T . For convenience, we assumethat the users are labeled such that P β ≤ . . . ≤ P K β K .We consider a noncoherent ML detector which is optimalfor uniformly distributed discrete input signals in terms oferror probability. First, we note that (1) can be reformulatedby Y HT = X HT D / G H + Ξ HT . Then, the vectorized versionof the received signal can be written as y = vec( Y HT ) = ( I M ⊗ X HT D / )vec( G H ) + vec( Ξ HT ) . As all the entries of G and Ξ are i.i.d. CSCG distributed, weimmediately have E [ y ] = , and the covariance matrix of y can be calculated by R y | X T = E [ yy H ] = I M ⊗ ( X HT DX T + σ I T ) . Then, the conditional distribution of the received signal y at BS for any transmitted signal matrix X T is given by p ( y | X T ) = 1 π KM det( R y | X T ) exp (cid:16) − y H R − y | X T y (cid:17) . Note that our design can be directly extended to the case with peak powerconstraint.
The noncoherent ML detector aims to estimate the trans-mitted information carrying matrix from the received sig-nal vector y by solving the optimization problem: (cid:98) X T =arg max X T ln p ( y | X T ) , or equivalently, (cid:98) X T = arg min X T y H R − y | X T y + log det( R y | X T ) . (2)From (2), we can observe that the detector relies on thesufficient statistic of the transmitted signal matrix: R y | X T = I ⊗ ( X HT DX T + σ I T ) . The detailed discussion is given inthe following subsection. B. Unique Identification of the Transmitted Signal Matrix
In this section, we first identify what conditions the trans-mitted signal matrix must satisfy to enable the unique identi-fication of the transmitted signal matrix X T . We can observefrom (2) that, to achieve reliable communication between allthe users and BS in the considered massive SIMO system, thereceiver must be able to uniquely determine each transmittedsignal matrix X T once R = X HT DX T has been received,which can be formally stated as follows: Proposition 1:
Any reliable communications for the mul-tiuser massive SIMO system described in (1) require that,for the transmitted signal matrix selected from M K × T ⊆ C K × T , if and only if there exist any two signal matrices X T , (cid:101) X T ∈ M K × T satisfying X HT DX T = (cid:101) X HT D (cid:101) X T , thenwe have X T = (cid:101) X T . (cid:4) The proof is omitted due to space limitation. Inspired byProposition 1, to aid our system design, we introduce thefollowing concept called uniquely-factorable multiuser space-time modulation (UF-MUSTM) as follows:
Definition 1:
A multiuser space-time modulation codebook S K × T ⊆ C K × T is said to form a UF-MUSTM codebook if forany pair of codewords S , (cid:101) S ∈ S K × T satisfying S H S = (cid:101) S H (cid:101) S ,we have S = (cid:101) S . (cid:4) Definition 1 motivates us to design a UF-MUSTM codebookfor the considered noncoherent massive SIMO, which will begiven in Sec. III. Therefore, our primary task in the rest of thispaper is to propose a new method for the systematic designof such UF-MUSTM S K × T .We note that, most of the existing space-time code designsconsider point-to-point MIMO systems, where all the trans-mitting antennas are connected to the same transmitter. As aresult, the transmitted information carrying signals are acces-sible by all the antennas where unitary space-time code designcan be employed [7], [8]. However, in our considered MUSTMnoncoherent massive SIMO system, the signals transmittedfrom different users are not allowed to fully collaborate, whichgreatly limits the codebook design. Moreover, the performanceanalysis for non-unitary codeword of MUSTM is very chal-lenging as shown in [7].III. T HE M ULTIUSER M ASSIVE
SIMO S
YSTEM WITH N ONCOHERENT
ML R
ECEIVER
In this section, we present a UF-MUSTM framework witha slot-by-slot ML detection receiver. We find that when the number of receiver antennas increases, the pairwise error prob-ability (PEP) between the two codewords will be dominated bythe Kullback-Leibler (KL) distance between them. Motivatedby this fact, a max-min KL distance design is proposed byperforming optimization on sub-constellation assignment andpower allocation among all the users.
A. The KL Distance between the Transmitted Space-TimeModulation Codewords
In practice, the computational complexity of the optimalnoncoherent ML detector described in (2) can be prohibitivelyhigh when the size of X T is large. To reduce the receivercomplexity, our main idea is to use a small block size forthe ML receiver. If only one time slot is involved in theML detector in (2), i.e., we consider T = 1 , the corre-lation matrix R = X HT DX T degenerate into a real scalar x H Dx = (cid:80) Kk =1 β k | x k, | , where the phase information ofthe transmitted symbols is lost and information bits from allthe users can only be modulated on the amplitudes of thetransmitted symbols. However, such a design typically has avery low spectral efficiency [9]–[11]. To improve the spectrumefficiency by allowing constellation with phase information betransmitted by all the users, we need at least two time slots [7],[8]. Motivated by the above observation, we consider the casewith T = 2 , where the transmitted signals from the first andsecond time slots are represented by X = [ x , x ] . We alsodenote R y | X = I M ⊗ R , in which R = X H DX + σ I = (cid:20) a cc ∗ b (cid:21) , (3)where a = x H Dx + σ , b = x H Dx + σ , and c = x H Dx ,with ab > | c | . By (3), we immediately have R − = 1 ab − | c | (cid:20) b − c − c ∗ a (cid:21) . (4)By inserting (3), (4) into(2), the ML receiver can be given by: (cid:98) X = arg min X y H R − y | X y + log det( R y | X )= arg min X a (cid:107) y (cid:107) + b (cid:107) y (cid:107) − (cid:60) ( c y H y ) ab − | c | + M ln (cid:0) ab − | c | (cid:1) , (5)where y and y are the received signal vectors in the firstand second time slots, respectively. It can be observed that thediagonal entries in (3) are a = x H Dx = (cid:80) Kk =1 β k | x k, | and b = x H Dx = (cid:80) Kk =1 β k | x k, | , in which the phase informa-tion is lost, while the off-diagonal term is c = x H Dx = (cid:80) Kk =1 β k x ∗ k, x k, = (cid:80) Kk =1 β k | x k, || x k, | exp (cid:0) j arg( x k, ) − j arg( x k, ) (cid:1) , suggesting that we can transmit a known refer-ence signal vector x in the first time slot and then transmit theinformation bearing signal vector x to enable a “differential-like” transmission. The exact PEP is extremely hard to evaluatefor the matrix X given above [7]. Moreover, the exactexpression for the pairwise error probability (PEP) do not seemto be tractable for optimization. Inspired by the Chernoff-SteinLemma, when the number of receiver antennas M goes toinfinity, the PEP will goes to zero exponentially where the exponent determined by the KL distance [8]. Hence, in thispaper, we propose to maximize the minimum KL distancebetween the conditional distributions of the received signalscorresponding to different input signals.We now calculate the KL distance between the received sig-nals induced by the transmitted signals matrices X = [ x , x ] and (cid:101) X = [˜ x , ˜ x ] , which is also the expectation of thelikelihood function between two received signals vectors. Inessence, the likelihood function between the received signalvectors corresponding to the two transmitted signals conver-gence in probability to the KL-distance with the increaseof the number of receiver antennas. More specifically, theKL-distance between the received signal corresponding totransmitted matrix X and (cid:101) X is given by: D ( M )KL ( X || (cid:101) X ) = E f ( y | X ) (cid:20) ln (cid:16) f ( y | (cid:101) X ) f ( y | X ) (cid:17)(cid:21) = E f ( y | X ) (cid:20) ln (cid:16) det( R y | X )det( R y | (cid:101) X ) (cid:17) + (cid:16) y H R − y | (cid:101) X y − y H R − y | X y (cid:17)(cid:21) = E f ( y | X ) (cid:20) tr (cid:16)(cid:0) R − y | (cid:101) X − R − y | X (cid:1) yy H (cid:17)(cid:21) + ln (cid:16) det( R y | X )det( R y | (cid:101) X ) (cid:17) = tr (cid:16)(cid:0) R − y | (cid:101) X − R − y | X (cid:1) R y | X (cid:17) +ln (cid:16) det( R y | X )det( R y | (cid:101) X ) (cid:17) = M D KL ( X || (cid:101) X ) , in which D KL ( X || (cid:101) X ) = tr (cid:2) ( X H DX + σ I )( (cid:101) X H D (cid:101) X + σ I ) − (cid:3) − ln (cid:104) det (cid:0) ( X H DX + σ I )( (cid:101) X H D (cid:101) X + σ I ) − (cid:1)(cid:105) − . (7)We can observe that D KL ( X || (cid:101) X ) is actually the KL-distanceif there is only one receiver antenna. Due to the assumptionof the independence of channel coefficients, and the KLdistance with M antennas D ( M )KL ( X || (cid:101) X ) is M times of D KL ( X || (cid:101) X ) . B. Multiuser Space-Time Modulation witn QAM Division
The main objective of this subsection is to propose anew QAM division based MUSTM design framework forthe considered massive SIMO system. This design is basedon the uniquely decomposable constellation group (UDCG)originally proposed in [13], building upon the commonly usedspectrally efficient QAM signaling. Now, we introduce thedefinition of UDCG as follows:
Definition 2:
A group of constellations {X k } Kk =1 form aUDCG, denoted by (cid:8) (cid:80) Kk =1 x k : x k ∈ X k (cid:9) = (cid:93) Kk =1 X k = X (cid:93) . . . (cid:93) X K , if there exist two groups of x k , ˜ x k ∈ X k for k = 1 , · · · , K such that (cid:80) Kk =1 x k = (cid:80) Kk =1 ˜ x k , then we have x k = ˜ x k for k = 1 , · · · , K . (cid:4) As QAM constellation is commonly used in modern digitalcommunications, which has a simple geometric structure, wenow give the following construction of UDCG based on QAMconstellation. For simplicity, we consider that each user isusing the 4-QAM constellation . The case with general QAM constellation is much more complicated andwill be left as a future work.
Lemma 1: The UDCG with multilevel 4-QAM constella-tions : The K -ary square QAM constellation Q = (cid:8) [ ± ( m − ) ± j ( n − )] d : m, n = 1 , . . . , K − (cid:9) , with d being theminimum Euclidean distance between the constellation points,can be uniquely decomposed into the sum of K multilevel 4-QAM sub-constellations {X k } Kk =1 denoted by Q = (cid:93) Kk =1 X k ,where X k = (cid:8) ( ± ± j ) × k − d (cid:9) for k = 1 , . . . , K . (cid:4) With the help of UDCG, we are now ready to propose aQAM division based UF-MUSTM for the considered massiveSIMO system with a noncoherent ML receiver given in (5).The structure of each transmitted signal matrix is given by X = [ x , x ] = D − / ΠS , in which S = [ s , s ] = √ p √ p s √ p √ p s ... ... √ p K √ p K s K . (8)In our design, the diagonal matrix D − / is used to com-pensate for the large scale fading between different users.The vector p = [ p , . . . , p K ] is introduced to adjust therelative transmitting power between all the users and s =[ s , . . . , s K ] is the information carrying vector. We let s k ∈ X k where X k constitute a UDCG with sum-QAM constellation Q such that Q = (cid:93) Kk =1 X k as defined in Lemma 1 and E [ | s k | ] = E k d , with E k = 2 k − , k = 1 , . . . , K . Thematrix Π = [ e π (1) , . . . , e π ( K ) ] T is a permutation matrix,where e k denotes a standard basis column vector of length K with 1 in the k -th position and 0 in other positions. π : { , . . . , K } → { , . . . , K } is a permutation over K elements characterized by (cid:18) . . . Kπ (1) π (2) . . . π ( K ) (cid:19) . Wealso let π − : { , . . . , K } → { , . . . , K } be a permutationsuch that π − ( π ( k )) = k for k = 1 , . . . , K . From the abovedefinition, we immediately have Π T Π = I K .For transmitted signal matrices X , we have the followingdesired properties: Proposition 2:
Consider X = D − / ΠS and (cid:101) X = D − / Π (cid:101) S , where S and (cid:101) S belong to S K × as describedin Definition 1. If X H DX = (cid:101) X H D (cid:101) X , then we have X = (cid:101) X . (cid:4) C. User-constellation Assignment and Power Allocation forthe Noncoherent ML Detector
We consider the user-constellation assignment π and powerallocation p for the noncoherent ML detector of design. Forour design given in (8), we have X H DX + σ I = (cid:20) s H s + σ s H s s H s s H s + σ (cid:21) = (cid:20) a cc ∗ b (cid:21) , (cid:101) X H D (cid:101) X + σ I = (cid:20) s H s + σ s H ˜ s ˜ s H s ˜ s H ˜ s + σ (cid:21) = (cid:20) a ˜ c ˜ c ∗ ˜ b (cid:21) . where a = K (cid:88) k =1 p k + σ , c = K (cid:88) k =1 s k , ˜ c = K (cid:88) k =1 ˜ s k ,b = ˜ b = K (cid:88) k =1 p k | s k | + σ = K (cid:88) k =1 E k d p k + σ , (9)in which s k , ˜ s k ∈ X k , ˜ c, c ∈ Q = (cid:93) Kk =1 X k , and ab > | c | .From (9), we can find that X H DX + σ I and (cid:101) X H D (cid:101) X + σ I are independent of the permutation function π , but aredetermined the power allocation vector p = [ p , . . . , p K ] T ,and the information carrying vectors s = [ s , . . . , s K ] T and ˜ s = [˜ s , . . . , ˜ s K ] T . Now, with the help of (9), we have det (cid:104) ( X H DX + σ I )( (cid:101) X H D (cid:101) X + σ I ) − ) (cid:105) = ab − | c | a ˜ b − | ˜ c | , tr (cid:104) ( X H DX + σ I )( (cid:101) X H D (cid:101) X + σ I ) − (cid:105) = 1 a ˜ b − | ˜ c | tr (cid:40) (cid:20) a cc ∗ b (cid:21) (cid:20) ˜ b − ˜ c − ˜ c ∗ a (cid:21) (cid:41) = ab + a ˜ b − c ˜ c ∗ − c ∗ ˜ ca ˜ b − | ˜ c | . As a consequence, (7) can be reformulated by D KL ( X || (cid:101) X ) = 2 ab − c ˜ c ∗ − c ∗ ˜ cab − | ˜ c | − ln (cid:16) ab − | c | ab − | ˜ c | (cid:17) − ab − | c | ab − | ˜ c | − ln (cid:16) ab − | c | ab − | ˜ c | (cid:17) − | c − ˜ c | ab − | ˜ c | . Recall that, the power constraint in Assumption 1 is E {| x k,t | } ≤ P k , k = 1 , . . . , K, t = 1 , . That is, for the firstand second time slots, we have E {| x k, | } = p π ( k ) β k ≤ P k ,and E {| x k, | } = p π ( k ) E π ( k ) d β k ≤ P k . The above powerconstraints are equivalent to P π − ( k ) β π − ( k ) ≤ p k ≤ P π − ( k ) β π − ( k ) E k d , ∀ k. For the considered massive SIMO when M is large, thePEP will goes to zero exponentially where the exponentdetermined by the KL distance [8]. Since we have one-to-one correspondence between c , { s k } Kk =1 , and ˜ c , { ˜ s k } Kk =1 , wenow aim to solve the following optimization problem: Problem 1:
Find the optimal power control coefficients { p k } Kk =1 and permutation π , such that: max { p k } Kk =1 ,π min { c, ˜ c } D KL ( X || (cid:101) X )= ab − | c | ab − | ˜ c | − ln (cid:16) ab − | c | ab − | ˜ c | (cid:17) − (cid:124) (cid:123)(cid:122) (cid:125) T + | c − ˜ c | ab − | ˜ c | (cid:124) (cid:123)(cid:122) (cid:125) T (11a) s . t . a = K (cid:88) k =1 p k + σ , b = K (cid:88) k =1 p k E k d + σ , (11b) c = K (cid:88) k =1 s k , ˜ c = K (cid:88) k =1 ˜ s k , (11c) P π − ( k ) β π − ( k ) ≤ p k ≤ P π − ( k ) β π − ( k ) E k d , ∀ k. (11d) (cid:4) We can observe that (11) is a max-min optimization problemwhere the objective function can be divided into two parts: T ≥ and T ≥ .We first consider the inner optimization problem on { c, ˜ c } .It can be observed that the minimum of T = 0 is attainedwhen ab −| c | ab −| ˜ c | = 1 , or equivalently | c | = | ˜ c | . Also, theminimum value of T is attained when c and ˜ c are the nearestneighboring points and | ˜ c | is minimized, e.g., the minimalvalue of T can be obtained simultaneously when c = (1+ j ) d and ˜ c = (1 − j ) d . In this case, we have | c | = | ˜ c | and hence T = 0 and T = d ab − d / = abd − , where the objectivefunction in (11) is a monotonically decreasing against abd . Wenote that abd = ( (cid:80) Kk =1 1 p k + σ )( (cid:80) Kk =1 p k E k + σ d ) , and henceproblem (11) can be reformulated by: min { p k } Kk =1 ,π (cid:16) K (cid:88) k =1 p k + σ (cid:17)(cid:16) K (cid:88) k =1 p k E k + σ d (cid:17) (12a) s . t . P π − ( k ) β π − ( k ) ≤ p k ≤ P π − ( k ) β π − ( k ) E k d , ∀ k. (12b)The optimization on Problem (12) can be carried out byfirst fixing π to find the optimal value of p , and then performfurther optimization on π . To that end, we know from (12b)that, for any given π , the feasible range of d is given by d ≤ P π − k ) β π − k ) p k E k ≤ P π − k ) β π − k ) E k for k = 1 , . . . , K , orequivalently d ≤ min (cid:110) P π − k ) β π − k ) E k (cid:111) Kk =1 . By the Cauchy-Swartz inequality, we have (cid:18) K (cid:88) k =1 p k + σ (cid:19)(cid:18) K (cid:88) k =1 p k E k + σ d (cid:19) ( a ) ≥ (cid:18) K (cid:88) k =1 √ p k (cid:112) p k E k d + σ d (cid:19) = (cid:18) K (cid:88) k =1 (cid:112) E k + σ d (cid:19) , where the inequality in ( a ) holds if and only if √ p k E k / √ p k = d ,for k = 1 , . . . K . Or equivalently, the optimal power allocationis p = [ p (cid:63) , . . . , p (cid:63) ] T where p (cid:63)k = √ E k d for k = 1 , . . . , K . Ournext task is to check the power constraint on p (cid:63)k given in (12b)is violated or not. For d ≤ min (cid:110) P π − k ) β π − k ) E k (cid:111) Kk =1 , wehave p (cid:63)k = √ E k d ≤ P π − ( k ) β π − ( k ) , and p (cid:63)k E k d = √ E k d ≤ P π − ( k ) β π − ( k ) , ∀ k , where no power constraints on are vio-lated for p . Finally, the optimization problem on π can begiven by min π K (cid:88) k =1 (cid:112) E k + σ d s . t . d ≤ P π − ( k ) β π − ( k ) E k , ∀ k. Or equivalently, we aim to solve max π d s . t . d ≤ P k β k E π ( k ) , ∀ k. Before proceeding on, we establish the following lemma.
Lemma 2:
Suppose that two positive sequences { a k } Kk =1 and { b k } Kk =1 are arranged both in a nondecreasing order. If welet Π denote the set containing all the possible permutationsof , · · · , K , then, the solution to the optimization problem, max π ∈ Π min (cid:110) a k b π ( k ) (cid:111) Kk =1 , is given by π (cid:63) ( k ) = k for k =1 , · · · , K . (cid:4) The proof is omitted due to space constraint. By Lemma 2,and note that P β ≤ . . . ≤ P k β K , to maximize d , we shouldlet E π (1) ≤ . . . ≤ E π ( K ) , i.e., the average power of thesub-constellations should be in ascending order. All the abovediscussions can be summarized into the following theorem: Theorem 1:
The users are ordered such that P β ≤ P β ≤ . . . ≤ P k β K , and we denote d (cid:63) = min k (cid:8) P k β k √ E k (cid:9) Kk =1 , thethe optimal power for each user can be given by p (cid:63) =[ √ E d (cid:63) , . . . , √ E K d (cid:63) ] T . And the optimal permutation matrixis the identity matrix, i.e., Π = I K . (cid:4) IV. S
IMULATION R ESULTS AND D ISCUSSION
In this section, computer simulations are performed todemonstrate the effectiveness of our proposed design incomparison with other existing benchmarks. The small-scalefading is assumed to be the normalized Rayleigh fading. Thepath-loss L as a function of transmission distance d at antennafar-field can be approximated by
10 log L = 20 log (cid:16) λ πd (cid:17) − γ log (cid:16) dd (cid:17) − ψ, d ≥ d , where d = 100 m is the reference distance, λ = v c /f c ( f c = 3 GHz) is the wavelength of carrier, γ = 3 . is thepath-loss exponent [14]. In the above model, ψ ∼ N (0 , σ ψ ) ( σ ψ = 3 . ) is the Gaussian random shadowing attenuationresulting from blockage of objects. For the receiver, we assumethat the noise power is
10 log σ = 10 log N B w =10 log . × − = − .
97 dB , where the channelbandwidth B w = 20 MHz and N = k T F / is the powerspectral density of noise with k = 1 . × − J/K beingthe Boltzman’s constant, reference temperature T = 290 K(“room temperature”), and noise figure F = 6 dB. − − − − Number of BS antennas, M BE R Proposed method, N=1MED detector, N=1Proposed method, N=2MED detector, N=2Proposed method, N=3MED detector, N=3Proposed method, N=4MED detector, N=4
Fig. 1. Comparison of the proposed scheme with MED based design on theaverage BER of all users versus M , where 4-QAM are used by all the users. We first examine the error performance of the proposeddesign under the instantaneous average power constraint for Number of BS antennas, M -4 -3 -2 -1 BE R Orthogonal training, d=300mOrthogonal training, d=500mOrthogonal training, d=900mProposed method, d=300mProposed method, d=500mProposed method, d=900m
Fig. 2. The comparison between the proposed design and ZF receiver withorthogonal channel training for K = 3 users and 4 time slot. different number of users as illustrated in Fig. 1. It is assumedthat the average power upper bound is P k =316 mW (25 dBm), ∀ k . All the K users are assumed to be uniformly distributedwithin the cell of radius d . It can be observed that, with theincreased number of users, the error performance deterioratesquickly caused by the mutual interference between users.Then, more BS antennas are needed to achieve the sameaverage BER. We also compare our design with MED basedmethod proposed in [9], [15]. Since we are using two timeslots, while the MED methods only need one time slot, weassume 2-PAM constellations are used by all the users forthe MED based design. We can also find that the proposedapproach outperforms the MED based method significantly interms of BER in all the schemes.Next, we compare the error performance of the proposedmethod with the conventional zero-forcing (ZF) receiver usingorthogonal training sequence in Fig. 2. Without loss of general-ity, we consider a system with N = 3 users. For the orthogonaltraining based method, at least 4 time slots are needed and weassume that the channel coefficients are quasi-constant in thesetimes slots. As 4-QAM are used by each user for the proposedscheme, we assume that 64-QAM are assumed for the trainingbased approach in order to achieve a fair comparison. For thetraining prcocess, we assume that, a popular least-square (LS)channel estimator is employed [5]. It can be observed fromFig. 2 that, when the antenna number M is small and thechannel gain is large (i.e., the distance d is small), the trainingbased method outperforms the proposed design in term ofBER. However, when the antenna number is relatively large,the proposed design has a better error performance, especiallyat the cell edge. V. C ONCLUSION
In this paper, we have proposed a new noncoherent mul-tiuser massive SIMO design for the low-latency IIoT wireless communication applications. Assuming that the large-scalefading coefficients are known, we presented a simple andsystematic construction of the transmitted signal matrix basedon the concept of UDCG. In our design, we have useda noncoherent ML receiver over two time slots where noinstantaneous CSI is required. To improve the error perfor-mance, the minimum KL distance between the received signalscorresponding to different transmitted signal matrices wasmaximized by proper power allocation and sub-constellationassignment. Computer simulations reveal that, our methodoutperforms significantly the MED based design in termsof average BER. Our approach can also have a better errorperformance than the orthogonal training design for cell edgeusers when the array size is large.R
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