Joint Data and Active User Detection for Grant-free FTN-NOMA in Dynamic Networks
Weijie Yuan, Nan Wu, Jinhong Yuan, Derrick Wing Kwan Ng, Lajos Hanzo
aa r X i v : . [ ee ss . SP ] F e b Joint Data and Active User Detection for Grant-freeFTN-NOMA in Dynamic Networks
Weijie Yuan , Nan Wu , Jinhong Yuan , Derrick Wing Kwan Ng , and Lajos Hanzo School of Electrical Engineering and Telecommunications, University of New South Wales, NSW 2052, Australia School of Information and Electronics, Beijing Institute of Technology, Beijing 100081, China School of Electronics and Computer Science, University of Southampton, SO17 1BJ, UKEmail: { weijie.yuan, j.yuan, w.k.ng } @unsw.edu.au, [email protected], [email protected] Abstract —Both faster than Nyquist (FTN) signaling and non-orthogonal multiple access (NOMA) are promising next generationwireless communications techniques as a benefit of their capabilityof improving the system’s spectral efficiency. This paper considersan uplink system that combines the advantages of FTN andNOMA. Consequently, an improved spectral efficiency is achievedby deliberately introducing both inter-symbol interference (ISI) andinter-user interference (IUI). More specifically, we propose a grant-free transmission scheme to reduce the signaling overhead andtransmission latency of the considered NOMA system. To distinguishthe active and inactive users, we develop a novel message passingreceiver that jointly estimates the channel state, detects the useractivity, and performs decoding. We conclude by quantifying thesignificant spectral efficiency gain achieved by our amalgamatedFTN-NOMA scheme compared to the orthogonal transmissionsystem, which is up to 87.5%.
I. I
NTRODUCTION
Wireless communications has played an increasingly impor-tant role in modern digital economics. The rapid developmentof communication technologies has fueled the roll-out of theInternet-of-things (IoT) in 5G wireless systems, which requiresto accommodate a massive number of devices [1]. Unfortunately,current techniques can only support a limited number of activedevices concurrently [2]. Thus, new techniques supporting mas-sive connectivity are sought.Recent investigations on non-orthogonal multiple access(NOMA) show that by introducing controllable interference, mul-tiple users can share the same orthogonal radio resources, whichallows a communication system to support more users relyingon the same amounts of resource elements as OMA [3], [4]. Toaddress the demand for further increasing the spectral efficiency,Faster-than-Nyquist (FTN) signaling, proposed by Mazo in 1970s[5], has attracted substantial interests, since it can transmit at asymbol-rate beyond the Nyquist rate.Naturally, gleaning further gains is expected by additionallymultiplexing the FTN users employing the popular NOMA prin-
This work is supported the Australia Research Council Discovery Project(DP190101363) and Linkage Projects (LP 160100708 and LP170101196). D. W.K. Ng is supported by funding from the UNSW Digital Grid Futures Institute,UNSW, Sydney, under a cross-disciplinary fund scheme and by the AustralianResearch Council’s Discovery Project (DP190101363). L. Hanzo would like toacknowledge the financial support of the Engineering and Physical SciencesResearch Council projects EP/Noo4558/1, EP/PO34284/1, COALESCE, of theRoyal Society’s Global Challenges Research Fund Grant as well as of theEuropean Research Council’s Advanced Fellow Grant QuantCom. ciples. As a result, the co-existence of inter-user interference (IUI)as well as inter-symbol interference (ISI) in the FTN-NOMAsystems leads to a receiver complexity that grows exponentiallywith both the number of ISI taps and that of the users. To addressthis issue, several authors have developed reduced-complexityreceivers for FTN signaling [6], [7] and NOMA [8], [9]. Asa further challenge, the large number of users to be supportedby next-generation systems makes the conventional grant-basedaccess control impractical owing to the associated excessivecommunication overhead and signaling latency [10]. To tacklethis problem, grant-free schemes dispensing with “handshaking”have received considerable attention in NOMA scenarios. Forinstance, the authors of [11] and [12] assumed the user activityto be static and designed factor graph based receivers for simul-taneously solving the channel estimation as well as data and useractivity detection problem of NOMA systems. Nevertheless, theuser activity in networks fluctuates over time in practice. Someactive users may become inactive in the next few time slots,while several sleeping users may become active. Therefore, theidentification of user activity in real time is desired.In this paper, we intrinsically amalgamate FTN signalingwith the grant-free NOMA concept and consider a dynamicenvironment, where both the channel and the user activity aretime varying. By employing the autoregressive (AR) model of[13] for approximating the correlated noise samples imposed byFTN signaling, we construct a factor graph and propose a bespokeexpectation maximization (EM) - message passing algorithm(MPA) for iteratively estimating both the user activity as well asthe channel coefficients, and for detecting the data symbols. Sinceall messages defined over the factor graph and the solutions ofthe M-step of the EM are obtained in closed forms, the proposedreceiver only has a linearly increasing complexity with respect tothe number of users. Our simulation results show that the FTN-NOMA system relying on the proposed receiver significantlyimproves the spectral efficiency and reliably distinguishes theactive/inactive users.
Notations:
The superscript ( · ) T and ( · ) ∗ denote the transposeand inverse operations, respectively; G ( m x , V x ) denotes theGaussian distribution of variable x having a mean vector of m x and covariance matrix of V x ; J ( · ) denotes the zeroth-orderBessel function of the first kind; diag { a } denotes a diagonalmatrix with the diagonal elements a ; E [ · ] denotes the expectationperator; ∝ represents both sides of the equation are multiplica-tively connected to a constant; ∂ denotes the partial derivativeoperator. II. S YSTEM M ODEL
The NOMA uplink is considered, where K simultaneous userstransmit their information to the BS relying on J orthogonalresource elements, with J < K and ρ = KJ representing thenormalized user-load. The coded bit stream corresponding to the k th user is first mapped to a sequence of data symbols and thenspread over J resource-slots using a low-density signature (LDS) x [ n ] k = [ x [ n ] k, , ..., x [ n ] k,J ] T , where x [ n ] k,j denotes the symbol of user k occupying the j th resource element at time instant n . To employFTN signaling in the NOMA system, the transmitted sequences, x [ n ] k of different users pass through a shaping filter q ( t ) , havinga symbol period of τ T , yielding s k ( t ) = X n x [ n ] k q ( t − nτ T ) , (1)where τ is the FTN packing factor [6] and s k ( t ) =[ s k, ( t ) , ..., s k,J ( t )] T . The transmitted signals of all users aremultiplexed over J resource elements and passed through a time-variant fading channel h k ( t ) = [ h k, ( t ) , ..., h k,J ( t )] T . Assumingperfect synchronization between the BS and the users, the signalreceived at the BS obeys: y ( t ) = K X k =1 diag { h k ( t ) } s k ( t ) + w ( t ) , (2)where y ( t ) and w ( t ) are both J -dimensional vectors with the j th entries being the received signal and noise at the j th resourceelement, respectively. We now introduce a binary variable λ [ n ] k = { , } denoting the user-activity, where λ [ n ] k = 1 represents anactive user and 0 an inactive one. Then after processing by amatched filter q ∗ ( − t ) , the discrete time model for the receivedsignal is given by r [ n ] j = K X k =1 λ [ n ] k h [ n ] k,j L X i = − L g i x [ n − i ] k,j + ω [ n ] j , (3)where g i denotes the FTN signaling-induced ISI tap and L is thelength of the taps. Note that in FTN signaling, the noise samplesof different time slots are correlated, which imposes challengeon the receiver design. As a remedy, we employ an AR modelto approximate the colored noise samples. In practice, the ARmodel with an AR order of n b is used for approximating thenoise sample ω [ n ] j , given by ω [ n ] j ≈ e [ n ] j + n b X i =1 b i ω [ n − i ] j , (4)where e [ n ] j is a random Gaussian impairment having zero-meanas well as variance σ e and b i is the AR coefficient. Based onthe known coefficients { g i } , the parameters { b i } and σ e can beestimated by solving the Yule-Walker equations [14]. We assume that for all users, the same shaping filter q ( t ) and packing factor τ are employed. III. T HE P ROPOSED L OW - COMPLEXITY R ECEIVER D ESIGN
From a statistical inference perspective, inferring channel co-efficients, and all users’ information bits from the received signalsamples is equivalent to determining the a posteriori distributionsof the corresponding variables.
A. Factor Graph Representation
By stacking all transmitted symbols, received samples, channeltaps, user states and noise samples into vectors, i.e., x , r , h , λ , and ω , the joint a posteriori distribution is written as p ( x , h , λ , ω | r ) . For an unknown variable z , we aim for derivingits marginal distribution p ( z | r ) and estimating it via the maximum a posteriori (MAP) estimators formulated as z = arg max z p ( z | r ) . (5)Nevertheless, direct marginalization is usually intractable due tothe associated high-dimensional integration. As an alternative, thefactor graph framework is capable of circumventing this problemby exploiting the conditional independence of variables giventhe observations. Exploiting the fact that x , h , λ , and ω areindependent of each other, p ( x , h , λ , ω | r ) can be factorized as p ( x , h , λ , ω | r ) ∝ p ( x ) · p ( h ) · p ( λ ) · p ( ω ) · p ( r | x , h , λ , ω ) . Since the transmitted symbols of different users at differentinstants are independent, p ( x ) can be fully factorized as p ( x ) = Y k,j,n p ( x [ n ] k,j ) , (6)where p ( x [ n ] k,j ) is determined based on the log-likelihood ratios(LLRs) output by the channel decoder.For time-varying channel taps h , it is convenient to characterize h [ n ] k,j by a Gauss-Markov model h [ n ] k,j = αh [ n − k,j + ε [ n ] , where thecoefficient α obeys the zero-order Bessel function of the first kind[15] α = E [ h [ n ] k,j ( h [ n ] k,j ) ∗ ] = J (2 πf D τ T ) , (7)and ε [ n ] is a zero-mean Gaussian distributed variable with vari-ance − | α | . Consequently, p ( h ) is expressed as p ( h ) = Y k,j p ( h k,j ) Y n p ( h [ n ] k,j | h [ n − k,j ) . (8)In a dynamically fluctuating environment, the evolution of theuser-activity state λ can be modeled by a Markov chain, where thecurrent activities of the users depend on the states of the previoustime instant. Hence, the distribution p ( λ ) can be factorized as p ( λ ) = K Y k =1 p ( λ k ) · Y n p ( λ [ n ] k | λ [ n − k ) . (9)Depending on the previous state of user k , the state transitionfunction p ( λ [ n ] k | λ [ n − k ) has different expressions. Given the user-birth probability of p [ n ] b k and the mortality probability of p [ n ] m k ,he transition probability of user-activity state p ( λ [ n ] k | λ [ n − k ) isexpressed as p ( λ [ n ] k | λ [ n − k ) = ( ( p [ n ] b k ) − λ [ n − k (1 − p [ n ] m k ) λ [ n − k λ [ n ] k = 1 , (1 − p [ n ] b k ) − λ [ n − k ( p [ n ] m k ) λ [ n − k λ [ n ] k = 0 . (10)Assuming that Λ denotes the average number of users becomingactive at a time instant, we set p [ n ] b k = Λ /K as the birth probabilityof a user. For the mortality probability, establishing an accuratemodel requires a large amount of data, but this is beyond thescope of the paper. Hence we employ a fair scheme assumingthat p [ n ] m k = 0 . .Based on the AR model (4) of the noise sample ω [ n ] j , we canfactorize p ( ω ) as p ( ω ) = Y j p ( ω j ) Y n p ( ω [ n ] j | ω [ n − n b ] j , ..., ω [ n − j ) , (11)where p ( ω j ) ∝ G (0 , σ e ) and p ( ω [ n ] j | ω [ n − n b ] j , ..., ω [ n − j ) ∝ exp (cid:18) − ( ω [ n ] j − b T ω [ n ] j ) σ e (cid:19) , with the notations b = [ b , ..., b n b ] T and ω [ n ] j = [ ω [ n − j , ..., ω [ n − n b ] j ] T . Furthermore, we can write theevolution model of ω [ n +1] j as ω [ n +1] j = B ω [ n ] j + b ω [ n ] j , (12)where B = (cid:20) Tn b − I n b − n b − (cid:21) and b = [1 , Tn b − ] T .Based on (3), we use a Dirac delta function δ ( · ) for repre-senting the relationship between the received signal sample andthe unknown variables. By introducing an auxiliary variable s [ n ] k,j = P Li = − L g i x [ n − i ] k,j = g T ˜ x [ n ] k,j , p ( r | x , h , λ , ω ) is factorizedas p ( r | x , h , λ , ω ) = Y j,n δ ( r [ n ] j − K X k =1 λ [ n ] k h [ n ] k,j s [ n ] k,j − ω [ n ] j ) · δ ( s [ n ] k,j − g T ˜ x [ n ] k,j ) , (13)where g and ˜ x [ n ] k,j denote the vectors [ g − L , ..., g L ] T and [ x [ n + L ] k,j , ..., x [ n − L ] k,j ] , respectively. The variable ˜ x [ n ] k,j follows asimilar evolution model as in (12), ˜ x [ n ] k,j = B ˜ x [ n − k,j + b x [ n + L ] k,j , (14)where B = (cid:20) T L I L L (cid:21) and b = [1 , T L ] T .Based on the factorizations of (6)-(13), we now have thefactorization of p ( x , h , λ , ω | r ) , which we represent by a factorgraph, as depicted in Fig. 1. On this factor graph, the factornodes denoted by squares represent the functions nodes whilethe variables are denoted by edges. The equality factor nodes ofFig. 1 represented by the symbol = are introduced for variable“cloning” to enforce the condition that a variable may only appearin a maximum of two functions. To simplify the notations, weadopt −→ µ ( x ) and ←− µ ( x ) to denote the specific messages of the The auxiliary variable is introduced to reduce the number of multiplications[9]. + (cid:38)(cid:75)(cid:68)(cid:81)(cid:81)(cid:72)(cid:79)(cid:3)(cid:39)(cid:72)(cid:70)(cid:82)(cid:71)(cid:72)(cid:85) b , (cid:59) (cid:78) (cid:61) (cid:75) (cid:74) (cid:88) B , (cid:75) (cid:74) x , (cid:75) (cid:74) x = B + b , (cid:75) (cid:74) (cid:88) , (cid:59) (cid:78) (cid:61) (cid:75) (cid:74) x ...... , (cid:75) (cid:74) x ... (cid:52) g , (cid:75) (cid:74) (cid:83) (cid:59) (cid:78) (cid:13)(cid:44) (cid:61)(cid:59) (cid:78) (cid:13)(cid:44) (cid:61)(cid:59) (cid:78) (cid:13)(cid:44) (cid:61)(cid:59) (cid:78) (cid:13)(cid:44) (cid:61)(cid:59) (cid:78) (cid:13)(cid:44)(cid:13) (cid:17) (cid:61) (a) Subgraph for the multiuser detection and decoding part. (cid:1089) (cid:79) [ ] nk (cid:79) (cid:16) (cid:5) [ 1] nk (cid:48)(cid:88)(cid:79)(cid:87)(cid:76)(cid:88)(cid:86)(cid:72)(cid:85) (cid:3) (cid:39)(cid:72)(cid:87)(cid:72)(cid:70)(cid:87)(cid:76)(cid:82)(cid:81) (cid:3) (cid:68)(cid:81)(cid:71) (cid:3) (cid:39)(cid:72)(cid:70)(cid:82)(cid:71)(cid:76)(cid:81)(cid:74) (cid:51)(cid:68)(cid:85)(cid:87)(cid:41)(cid:76)(cid:74)(cid:17)(cid:3)(cid:21)(cid:11)(cid:69)(cid:12) (cid:73) [ ] nk (cid:1089) (cid:73) (cid:14) [ 1] nk (cid:79) (cid:14) [ 1] nk (cid:1089) (cid:856)(cid:856)(cid:856) (cid:79) [ ], nk j (cid:4) [ ],1 nk h (cid:16) (cid:5) [ 1],1 nk h (cid:92) [ ],1 nk (cid:1089) [ ], nk j h (cid:16) [ 1], nk j h (cid:92) [ ], nk j (cid:1089) (cid:856)(cid:856)(cid:856) (cid:79) (cid:5) [ ] nk (cid:79) [ ],1 nk (cid:45) (cid:3)(cid:85)(cid:72)(cid:86)(cid:82)(cid:88)(cid:85)(cid:70)(cid:72)(cid:86) (cid:117) (cid:117) (cid:1089) [ ],1 nk h (cid:92) (cid:14) [ 1],1 nk [ ],1 nk h (cid:5) [ ],1 nk h [ ],1 nk s [ ], nk j s (cid:856)(cid:856)(cid:856)(cid:856)(cid:856)(cid:856) (cid:1085) [ ], nk j r [ ]1, nj r [ ], nK j r (cid:1089) [ ] nj (cid:90) (cid:58) [ ] nj (cid:90) [ ] nj B (cid:1085) [ ] jn r (cid:1089) (cid:14) [ 1] nj (cid:90) b (cid:1085) (cid:79) (cid:5) [ ], nk j (cid:79) (cid:5) [ ],1 nk (b) Factor graph representation of the FTN-NOMA system. The notations φ [ n ] k , ψ [ n ] k,j , and Ω [ n ] j represent the function p ( λ [ n ] k | λ [ n − k ) , p ( h [ n ] k,j | h [ n − k,j ) , and p ( ω [ n ] j | ω [ n − n b ] j , ..., ω [ n − j ) .Fig. 1. Factor graph for joint user activity tracking and data detection. variable x that flow in the direction and in the opposite directionof the edge. B. MPA for Message Calculations1) Multiuser Detection and Decoding Part:
Let us commencefrom the message calculations in Fig. 1(a). The intrinsic infor-mation −→ µ ( x [ n ] k,j ) is calculated based on the log-likelihood ratio(LLR) output by the channel decoder. Since x [ n ] k,j is discretelydistributed, the message passing algorithm (MPA) exhibits anexcessive complexity. To this end, we approximate −→ µ ( x [ n ] k,j ) by a Gaussian distribution using expectation propagation (EP).Assuming that the extrinsic message ←− µ ( x [ n ] k,j ) obeys the Gaussiandistribution of G ( ←− m x [ n ] k,j , ←− v x [ n ] k,j ) , we can readily obtain the meanand variance of b ( x [ n ] k,j ) by moment matching and then determinehe Gaussian belief ˜ b ( x [ n ] k,j ) ∝ G ( m x [ n ] k,j , v x [ n ] k,j ) . Hence we havethe message −→ µ ( x [ n ] k,j ) expressed as −→ µ ( x [ n ] k,j ) = ˜ b ( x [ n ] k,j ) / ←− µ ( x [ n ] k,j ) ∝ G ( −→ m x [ n ] k,j , −→ v x [ n ] k,j ) . (15)By employing the MPA rules, we can determine the mean vec-tor and covariance matrix of the message −→ µ ( ˙ x [ n − L ] k,j ) . Thereforethe message output by the multiuser detector can be assumed tobe Gaussian, formulated as −→ µ ( s [ n − L ] k,j ) ∝ G ( g T −→ m ˙ x [ n − L ] k,j , g T −→ V ˙ x [ n − L ] k,j g ) . (16)Finally, we are interested in calculating the extrinsic message ←− µ ( x [ n ] k,j ) , whose mean and variance obey ←− m x [ n ] k,j = b T (cid:16) ←− m ˜ x [ n − L ] k,j − B −→ m ˙˜ x [ n − L − k,j (cid:17) , (17) ←− v x [ n ] k,j = b T (cid:16) ←− V ˜ x [ n − L ] k,j + B −→ V ˙˜ x [ n − L − k,j B T (cid:17) b , (18)respectively.
2) Channel Estimation:
Provided that the message −→ µ ( ˙ h [ n − k,j ) is available in Gaussian form, by applying the MPA rules, themessage −→ µ (˜ h [ n ] k,j ) is expressed as −→ µ (˜ h [ n ] k,j ) ∝ G (cid:16) −→ m ˜ h [ n ] k,j , −→ v ˜ h [ n ] k,j (cid:17) ∝ (cid:16) α −→ m ˙ h [ n − k,j , | α | ( −→ v ˙ h [ n − k,j − (cid:17) . (19)Then, we are able to derive the message −→ µ ( ˙ h [ n ] k,j ) , given by −→ v ˙ h [ n ] k, = −→ v ˜ h [ n ] k, + ←− v h [ n ] k, −→ v ˜ h [ n ] k, ←− v h [ n ] k, , (20) −→ m ˙ h [ n ] k, = −→ v ˙ h [ n ] k, −→ m ˜ h [ n ] k, −→ v ˜ h [ n ] k, + ←− m h [ n ] k, ←− v h [ n ] k, . (21)Next, we consider the process of colored noise. Since theAR process given by (4) is causal, the messages are onlypropagated forward along the arrow’s direction. Provided that themeans of the noise parameters are , we can readily derive thecorresponding messages as follows −→ v ω [ n ] j = b T −→ V ω [ n ] j b , (22) −→ V ω [ n +1] j = B −→ V ω [ n ] j B T + b −→ v ω [ n ] j b T . (23)
3) User-activity Idenfitication:
For the discrete random vari-able representing a user-activity state at time instant n − , wehave ˙ λ [ n − k , and the message −→ µ ( ˙ λ [ n − k ) is the belief of user k ’s state at time instant n − , which is fully characterized bythe probability of λ [ n − k = 1 , i.e., −→ p ˙ λ [ n − k . Therefore, passingon the user-activity probability −→ p ˙ λ [ n − k instead of the messagecan simplify the expressions. We arrive at the forward message −→ p ( λ [ n ] k ) expressed as −→ p λ [ n ] k = (1 − p [ n ] m k ) −→ p ˙ λ [ n ] k + p [ n ] b k (1 − −→ p ˙ λ [ n ] k ) . (24) The equality node of Fig. 1 is equivalent to the product ofmessages. Therefore the message updating concerning ˙ λ [ n ] k isderived as −→ p ˙ λ [ n ] k = −→ p λ [ n ] k ←− p ˙ λ [ n ] k, − −→ p λ [ n ] k − ←− p ˙ λ [ n ] k, . (25)The message −→ p λ [ n ] k,j forwarded to the multiplier node can beobtained similarly.Note that the above message calculations depend on the as-sumption of having known backward messages gleaned from themultiplier node. According to the update rules of the conventionalMPA, the messages derived at the multiplier node × areunable to provide Gaussian form messages. Hence, we invokethe expectation maximization algorithm for the multiplier node. C. Modified EM Algorithm for × Node
Without loss of generality, we consider the multipliernode connected with r [ n ] k,j and the joint distribution p ( λ [ n ] k,j , s [ n ] k,j , h [ n ] k,j | r [ n ] k,j ) . We first define λ [ n ] k,j as the unknownparameter and { h [ n ] k,j , s [ n ] k,j , r [ n ] k,j } as the complete data setassociated with incomplete data r [ n ] k,j and latent variables h [ n ] k,j , s [ n ] k,j . Assuming that the beliefs b ( s [ n ] k,j ) and b ( h [ n ] k,j ) are available,the expectation of the complete-data log augmented density iscalculated as q ( λ [ n ] k,j ) = − Z Z (cid:16) ←− m r [ n ] k,j − λ [ n ] k,j s [ n ] k,j h [ n ] k,j (cid:17) ←− v r [ n ] k,j · b ( s [ n ] k,j ) b ( h [ n ] k,j ) d h [ n ] s,j d h [ n ] k,j + ln −→ µ ( λ [ n ] k,j ) + C, (26)where C is a constant that is irrelevant to λ nk,j . It can beobserved that (26) is a concave function and the estimate ˆ λ [ n ] k,j is given by the solution of ∂q ( λ [ n ] k,j ) ∂λ [ n ] k,j = 0 . However, note that(26) only considers the j th resource element, while the useractivity applies to all radio resources. The maximization shouldbe performed by obtaining the necessary information from all J resource elements, i.e., with respect to the variable ˙ λ nk . To thisend, the multiplier node will feed back the message ←− µ ( λ [ n ] k,j ) to the equality node of Fig. 1(b). Having q ( λ [ n ] k,j ) in hand, ←− µ ( λ [ n ] k,j ) is calculated as exp h q ( λ [ n ] k,j ) i / −→ µ ( λ [ n ] k,j ) . Similar to theforward message −→ p ( λ [ n ] k ) , we can simply pass on the normalizedprobability ←− p λ [ n ] k,j , which is used to calculate ←− p λ [ n ] k,j − , then ←− p λ [ n ] k, and finally −→ p ˙ λ [ n ] k .To obtain the beliefs of s [ n ] k,j and h [ n ] k,j , we apply the conceptof EM again that s [ n ] k,j becomes an unknown parameter and h [ n ] k,j remains the latent variable. In this way, the belief of s [ n ] k,j isupdated as follows: b ( s [ n ] k,j ) ∝ −→ µ ( s [ n ] k,j ) · exp (cid:16) Z b ( h [ n ] k,j ) ln p ( r [ n ] k,j , s [ n ] k,j | h [ n ] k,j , ˆ λ [ n ] k,j ) d h [ n ] k,j (cid:17) , (27)here λ [ n ] k,j is replaced by the estimate ˆ λ [ n ] k,j obtained from themaximization of (26), expressed as ˆ λ [ n ] k,j = ←− m r [ n ] k,j m s [ n ] k,j m h [ n ] k,j + ←− v r [ n ] k,j (1 − −→ p λ [ n ] k,j )( | m s [ n ] k,j | + v s [ n ] k,j )( | m h [ n ] k,j | + v h [ n ] k,j ) . (28)Since we have b ( s [ n ] k,j ) = −→ µ ( s [ n ] k,j ) · ←− µ ( s [ n ] k,j ) , it is natural todefine the second term on the right-hand side of (27) as ←− µ ( s [ n ] k,j ) .Assuming that b ( h [ n ] k,j ) , it follows that G ( m h [ n ] k,j , v h [ n ] k,j ) , ←− µ ( s [ n ] k,j ) can be modeled by a Gaussian distribution with a mean andvariance of ←− m s [ n ] k,j = ←− m r [ n ] k,j m h [ n ] k,j | m h [ n ] k,j | + v h [ n ] k,j , (29) ←− v s [ n ] k,j = ←− v r [ n ] k,j | m h [ n ] k,j | + v h [ n ] k,j . (30)Consequently the belief b ( s [ n ] k,j ) is readily obtained. By exchang-ing the roles of s [ n ] k,j and h [ n ] k,j , we have the updating rules of themessage ←− µ ( h [ n ] k,j ) and of the belief b ( h [ n ] k,j ) . After obtaining thebeliefs b ( s [ n ] k,j ) and b ( h [ n ] k,j ) , we can now determine q ( λ [ n ] k,j ) in thenext iteration following (26).Finally, we can obtain the forward message −→ µ ( r [ n ] k,j ) . Sincethe variables λ [ n ] k,j , s [ n ] k,j , and h [ n ] k,j are independent, the momentsof r [ n ] k,j are given by the product of the moments of the abovethree variables. Consequently, we have −→ m r [ n ] k,j =ˆ λ nk,j m h [ n ] k,j m h [ n ] s,j , (31) −→ v r [ n ] k,j =ˆ λ nk,j (cid:0) (1 − ˆ λ nk,j ) | m h [ n ] k,j | | m s [ n ] k,j | + | m h [ n ] k,j | v s [ n ] k,j + | m s [ n ] k,j v h [ n ] k,j + v h [ n ] k,j v s [ n ] k,j | (cid:1) . (32)Above, we have obtained −→ p ˙ λ [ n ] k and b ( h [ n ] k,j ) . Then we canproceed by comparing −→ p ˙ λ [ n ] k to a specific threshold for decidingwhether the user k is active, while the estimate of the channeltap h [ n ] k,j is given by ˆ h [ n ] k,j = E h h b ( h [ n ] k,j ) i = m h [ n ] k,j .It is observed that in the proposed algorithm, the messagesare represented only by a few parameters and the integration issimplified to additions and multiplications, which dramaticallyreduces the receiver’s complexity. Explicitly, the complexity onlyincreases linearly with the number of active users |K + | , whichshows the superiority of the proposed algorithm in massiveconnectivity scenarios.IV. S IMULATION R ESULTS
This section presents our simulation results. A rate-1/2 length- , low-density parity-check (LDPC) code is adopted. At totalof K = 180 users are supported by J = 120 resource elementsin our NOMA system, leading to the normalized user-load of ρ =150% . Quadrature Phase Shift Keying (QPSK) is used for bit-to-symbol mapping. For each user, 5 sequences of symbols having alength of , are transmitted. The data symbols correspondingto different users are shaped by the raised root cosine filter having E b /N (dB) -4 -3 -2 -1 BE R Genie-aidedMPA-APPEM-MPALS-AMP-MPA
Fig. 2. BER performance of the proposed algorithm versus E b /N . a roll-off factor of α = 0 . and a FTN packing factor of τ = 0 . and. A Rayleigh fading channel is considered and the taps aregenerated by Jake’s model with a normalized Doppler rate of f D τ T = 0 . . The channel estimation is based on the leastsquare (LS) method and as few as 5 pilots. The parameter Λ isset to , which indicates that approximately 11% of users areactive. Again, the mortality probability was set to p [ n ] m k = 0 . .The user activity is assumed to remain static for a sequence of1,080 symbols. The threshold of . is employed for user activityidentification.In Fig. 2, we compare the proposed receiver design to someexisting benchmark algorithms in terms of its BER performance.Explicitly, the BER performance versus E b /N of the proposedalgorithm as well as of the MPA-APP and LS-AMP-MPA meth-ods are illustrated in Fig. 2. The LS-AMP-MPA is a two-stepmethod which firstly identifies the active users and then performsMPA based MUD. Since the two-stage method only providesthe estimates of user-activities for data detection, considerableperformance loss can be seen compared to the proposed EM-MPA algorithm. The MPA-APP method regards all users tobe active although most users are inactive, hence leading tocertain performance degradation. Moreover, we also include thecurve corresponding to the OMA system using MPA for channelestimation and user-activity identification (Genie-aided). A slightperformance loss is observed for the proposed algorithm due tothe ISI and IUI imposed by our FTN-NOMA system. Neverthe-less, the spectral efficiency is increased by ( ρ − ∗ /τ = 87 . ,given the same radio resources.We plot the normalized minimum mean-squared errors(NMSE) of the channel estimate based on the proposed algorithmas well as on the GA-MPA and on the full pilot based sparseBayesian learning (SBL) method in Fig. 3. Since EP can exploitthe extrinsic information from the detector when approximatingthe discrete distribution by a Gaussian one, the proposed EM-MPA algorithm outperforms the GA-MPA method using direct .5 5 5.5 6 E b /N (dB) -3 -2 -1 N M SE EM-MPAInitial estimationFull-pilot SBL
Fig. 3. NMSE of channel estimation versus E b /N .
20 40 60 80 10010 -5 -4 -3 -2 BE R EM-MPAEM-MPA-Ideal
Fig. 4. Impact of number of active users on BER performance. moment matching. It is also worth noting that the NMSE perfor-mance of the proposed algorithm has a modest performance losscompared to the full-pilot based method, which demonstrates thepowerful capability of the proposed algorithm.In Fig. 4, the impact of the number of active users on thedecoding performance is considered. We plot the BER perfor-mance versus the parameter Λ at E b /N = 5 . dB, where theperformance of the EM-MPA-Ideal relying on perfectly knownuser-activity is also shown as a performance upper bound. In fact,a higher Λ results in more active users. Hence, the IUI becomesmore severe and degrades the BER performance, which can beobserved for both the proposed method and the performanceupper bound. It is interesting to see that when Λ increases, theperformance of the proposed algorithm approaches the upperbound, because user identification becomes more accurate formore active users. In particular, in the extreme case, when all users are active, the identification error will drop to .V. C ONCLUSIONS
We have conceived a low-complexity receiver for the grant-free FTN-NOMA uplink. We considered dynamically fluctuatingenvironments, which assume that the user state and channel varywith time. A new expectation maximization - message passingalgorithm combination was proposed based on the factor graphframework for joint FTN symbol detection, channel estimationand user-activity tracking. The complexity of the proposed re-ceiver increases linearly with the number of active users, which issignificantly lower than that of the conventional message passingreceiver. Our simulation results demonstrated the efficiency ofthe proposed method in identifying active users and decoding theinformation bits whilst enhancing the bandwidth efficiency by upto 87.5%. R
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