arXiv:1703.09992v4 [cs.IT] 10 Oct 2018 or lcrcladCmue niern,Uiest fCampin [email protected] of E-mails: University Brazil, Campinas–SP, Engineering, Computer and Electrical Technische Systems, E-mails: gerhard.fettweis Germany, meik.doerpinghaus, Communications Dresden, Mobile Dresden, sit¨at Chair Vodafone tan fULCde o nos utperetransmissions multiple endorse not co does timing URLLC tight of the straint However, retransm fourth- packets. which received In procedure, erroneously layer. request repeat b physical obtained automatic is the hybrid reliability (4G), on requirement networks These mobile challenge generation ms). massive 1 a of pose delay end-to-end (e.g., latency eerhwti h rgam Tet2 atesi o I for Partnership wireless”. - “fast - “Twenty20 03ZZ0505B programme contract under the within Research tteIE LBCM igpr,Dc 07[1]. pr been 2017 has Dec. work Singapore, th this GLOBECOM, in which IEEE material the after The at notice, accessible. be without longer transferred no be may Copyright tpoie eil omncto rmwr htcan that framework communication Firs flexible [3]. networks a wireless provides of capacity it and reliability the xrml ihrlaiiy(.. rm ro ae of rates error frame (e.g., or reliability automation require high which industrial extremely coordination, communication critical vehicular latency for mission communications low as [2] ultra-reliable scenarios such of application (URLLC) context emerging the with in cope to challenges blt,prle aigcanl,utarlal o lat low ultra-reliable channels, munications. fading parallel ability, n ihdt ae nra ellrnetworks. cellular real reli in high rates achieve apply data to we MCo high of Additionally, and potential measurements gain. the channel illustrate field SNR thereby real to of framework maximal-r terms analytical and our in sid combining, receiver also and selection the optimal combining, decoding, comparing at (3) MCo joint and for namely gain; algorithms SNR combining block-fa of M suboptimal terms parallel of in exac improvement for SCo performance the probability over the finding outage quantifying (1) (2) the insights channels; contributions: of new main single-connecti gain with our giving coding compared are incorporated, as These MCo are (SCo). of efficiency, SNR potentials spectral the the the into links, and of loss, signal-to-noise number outage high path the quantify sim which of at we a in framework terms establish (SNR), work, analytical we in so, accurate this doing MCo yet For In throughput. of and destination. performance probability to communication mu source provides the it from as enhanced networks, links and mobile fifth-generation transmissions in reliable rates enabling for strategy .C .Sno ih swt h eateto Communication of Department the with is with Filho Santos are S. 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NTRODUCTION lrctWl,PiipShl,Mi D¨orpinghaus, Meik Schulz, Philipp Wolf, Albrecht } @tu-dresden.de. { lrctwl,philipp.schulz2, albrecht.wolf, p.br. s(UNICAMP), as svrinmay version is sne npart in esented nycom- ency ,Sho of School s, ebr IEEE Member, nnovation” ability Univer- 10 the y ltiple ratio ding data tion. atio and vity ple, tly, the and Co − the an its n- e, 9 s s . t eadn h oetaso C scmae ihsingle- with compared as open MCo remain of (SCo): questions connectivity diversity-multiplexin potentials fundamental a the some regarding such be, sing same intuitive a may the within However tradeoff delivered of be slot. case, copies best time multiple the differen the in that use can, to information such can routes frequencies, architectures multiple carrier MCo via Secondly, multiplexing destination. for diversity trade esicuecasclmlil-nu utpeotu (M multiple-output multiple-input classical include nels large-sc combat to appropriate fading. betw more distances proves large exist, me where hundred antennas [4], even or macrodiversity ineffective. ten Thus, than distance then ters. greater correlation be the proves can but fading diversity used, large-scale frequency be frequency the can diversity of that Spatial independent so almost shadowin band, by is created Shadowing is effects. which fading, suitabl late t large-scale be tight combating suited not the might well microdiversity satisfying However, is while constraints. diversity, fading, [ small-scale frequency Microdiversity combat to slot. and time desirable spatial single indepe most a including over in is (diversity) channels it redundantly fading data dent URLLC, same to the respect transmit communica With multiple links. via tion connected simultaneously are users Multi-Connectivity A. contributions. our and approach our for. outline look then we and answers the providing thereby problem, shed the that closed-fo (SNR) on signal-to-noise-ratio im- asymptotic, high More at corresponding probabilit expressions setup. obtain system outage we investigated the each portantly, of for throughput derive we and expressions questions, integral-form referred the exact answer To (SC). decoding selectio combining or joint diversity-combi (MRC) standard combining by maximal-ratio consider at namely provided we methods, latter, is algorithms consider- the former by combining For The (JD). questions suboptimal side. above and receiver the the optimal answer both we ing work, this In )Hwtoesvnsvr ihtelvlo h agtmet- target the of level the with vary savings those much How how 3) probability, outage (fixed) target a much Given how 2) efficiency, spectral (fixed) target a Given 1) salse rnilst banidpnetfdn chan- fading independent obtain to principles Established where architecture system any to refers concept MCo The algorithms, combining the MCo, describe briefly we Next, n ehr Fettweis, Gerhard and , i n ihtenme fcnetosadtopology? and connections of number the with and same ric a achieving while SNR? saved high at be throughput can power same transmit a achieving SNR? while high at saved probability be outage can power transmit ebr IEEE Member, elw IEEE Fellow, , IMO) for e for s light ning ncy een ale 4], rm n- le o n g y g 1 - - t 2 systems [5] (spatial microdiversity) with space-time block well understood [4]. However, in the aforementioned studies, coding, and distributed MIMO systems [5] (spatial macro- only linear (suboptimal) combining schemes, namely SC and diversity), suitable for single-frequency networks [6]. Alter- MRC, have been considered. In particular, the JD scheme, natively, 4G concepts such as carrier aggregation (CA) [7] which is optimum, remains open for investigation. Herein we and dual connectivity (DC) [8] have been introduced to help to fill this gap. make use of multiple so-called component carriers (frequency Deriving the outage probability of JD for parallel fading microdiversity), provided adjacent channels and sufficient RF channels has been recognized as a highly challenging problem. bandwidth transceivers can realize frequency diversity with a An important result is to evaluate the diversity-multiplexing single antenna. According to [9], the small-scale fading of tradeoff (DMT) of fading channels. The DMT states that two signals is approximately uncorrelated if their frequencies by doubling the SNR, we get both a decrease in outage d(r) are separated at least by the coherence bandwidth, which probability by the factor of 2− , yielding an increase in is confirmed, for instance, by measurement results in [10]. reliability, and an increase in throughput of r bits per channel The techniques of CA and DC also support non-collocated use, i.e., the DMT describes the slope and the pre-log factor deployments (frequency macrodiversity). of the outage probability and throughput, respectively, at infinite SNR. This concept was first proposed by Zheng and B. Combining Algorithms Tse for MIMO channels [15]. The corresponding results for The system reliability strongly depends on the combining parallel fading channels can be found in [11]. For finite- algorithm used at the receiver side, regardless of the diversity SNR the DMT of MIMO channels was proposed in [16] method. Combining algorithms merge the information received under correlated fading. However, the DMT analysis does from multiple inputs (diversity branches) into a single unified not fully characterize the outage probability, and thus is not output. The goal is to make use of the redundant information suitable to address the fundamental questions formulated at the received from the multiple inputs. There are various combining beginning of the Introduction. In [17], a tight upper and lower algorithms known in literature [11], many of which merge the bound on the outage probability based on the outage exponent received inputs at the symbol level, e.g., analysis is given but it involves heavy computational efforts 1) Selection Combining, where the best input in terms of as the results include the incomplete Gamma function and received SNR is selected, while all other inputs are Meijer’s G-function. In addition, neither the DMT analysis in discarded, and [11], [15] nor the outage exponent analysis in [17] considers 2) Maximal-Ratio Combining, where all received inputs are macrodiversity. weighted by their respective SNRs and are coherently added. D. Asymptotic Outage Analysis In contrast to SC and MRC, which combine inputs already at We want to find a good performance indicator to evaluate the symbol level, the principle of the reliability of MCo in light of URLLC applications. In fact, 3) Joint Decoding is to combine the inputs at the decoder we aim to ultimately derive simple and insightful closed-form level, e.g., by iteratively exchanging information be- expressions that can be easily used to assess or optimize prac- tween decoders of each branch. tical MCo deployments. To this end, an asymptotic analysis In addition to the different combining levels, JD differs fun- turns out to be a strong candidate, as it offers a simple yet damentally from SC and MRC in that the encoders at the in-depth characterization of the system performance’s general transmitter may produce different channel codewords based trend. In the literature, the asymptotic outage probability is on a joint codebook, while SC and MRC combine received given depending on the so-called coding gain GC and diversity inputs from the same channel codeword. gain d (see, e.g., [18]), as

out d P˜ = GC Γ¯ − , C. Related Work · Recently, research on URLLC is emerging considerably, where Γ¯ is the average received SNR.
In this work, we assume focusing on the analysis of micro- and macrodiversity and that the gain of MCo over SCo is based on the transmission of its impact on reliability. In [12], MCo solutions that utilize identical information over N parallel block-fading channels. micro- as well as macrodiversity were evaluated in system This setup can be exploited by all the combining algorithms simulations to illustrate how the signal-to-interference-plus- described beforehand. All three combining algorithms can noise ratio and the outage probability can be improved. In [13], achieve the maximum diversity gain [11], i.e., d = N, the impairments of correlated fading were evaluated and whereas SCo has no diversity gain, i.e., d = 1. But the the trade-offs between power consumption, link usage, and combining algorithms differ with respect to the coding gain. outage probability were given. In another work, multi-radio The coding gains of SC and MRC have been studied in various access-technology architectures were compared regarding their contexts [19]. However, the exact coding gain of JD has latency, which is significantly improved by MCo techniques been unknown so far, since the derivation of the exact outage [14]. For other works on MCo for URLLC, see [3] and the probability in closed form is very difficult. Only few bounds references therein. The major underlying concepts of MCo are known, e.g., a lower bound is given in [11, Ch. 9.1.3] based solutions, namely micro- and macrodiversity, have been exten- on rate allocation to the individual fading channels, and lower sively studied, and their effects on the outage probability are and upper bounds based on an outage exponent analysis [17]. 3

Both works reveal some drawbacks. The lower bound in [11, quantify the performance improvement of JD over SC and Ch. 9.1.3] offers a simple closed-form solution but is not MRC in terms of the SNR gain as tight, whereas the lower and upper bounds in [17] are tight A1(Rc) but involve heavy computational efforts as the results include GJD SC = , and , N the incomplete Gamma function and Meijer’s G-function. The AN (Rc) exact solution of the coding gain of JD remains unknown and 1 A1(Rc) GJD MRC = p , , N N likewise the asymptotic outage probability. √N! · AN (Rc) More recently, we have evaluated the packet error rate respectively. These results reveal thatp the SNR gain of JD over of MCo by real link-level Monte-Carlo simulations in Wire- SC and MRC increases at a rate of around 3(N 1)/N dB less LAN [20], from which we concluded that i) the asymptotic with respect to the target spectral efficiency (i.e.,− per source packet error rate is a good metric to evaluate the reliability of sample/channel symbol), while being insensitive to the target MCo for URLLC, ii) the asymptotic outage probability serves outage probability. as a good benchmark to evaluate practical implementations, Finally, we apply our analysis to real field channel mea- and iii) the asymptotic outage probability is suitable for link- surements and thereby illustrate the potential of MCo in actual level abstraction. cellular networks to achieve high reliability and throughput.

E. Main Contributions of this Work F. Notation and Terminology A convenient way to quantify the performance gain of MCo The upper- and lowercase letters are used to denote random over SCo is to evaluate the required transmit power to achieve variables (RVs) and their realizations, respectively, unless a given outage probability and a given spectral efficiency. stated otherwise. The alphabet set of a RV X with realization x is denoted by , and its cardinality, by . The probability Eventually, we are interested in the transmit power reduction X |X | of MCo with respect to SCo, which we refer to as SNR mass function (pmf) and probability density function (pdf) gain. Based on the SNR gain we can answer the fundamental of the discrete and continuous RV X is denoted by pX (x) questions formulated at the beginning of the Introduction. To and fX (x), respectively. The pmf and pdf are simply denoted this end, we derive a remarkably simple analytical description by p(x) and f(x), respectively, whenever this notation is ˜out unambiguous. Also, Xn and xn represent vectors containing of the asymptotic outage probability PJD,N for JD depending a temporal sequence of and with length , respectively. on the number of links N, the spectral efficiency Rc, the X x n We use t to denote the time index and i to denote a source average transmit SNRs per link Pi/N0, and the path losses η A A − , for , as index. We define = i i as an indexed series di i 1, 2, ..., N S { | ∈ S} ∈{ } of random vectors, and A = Ai i as an indexed S {A | ∈ S} ˜out AN (Rc) series of RVs. In general, a set contains elements a( ), PJD,N = N · η as in A = a1,a2, ..., a A . We define one particular set: i=1(Pi/N0)di− = 1, 2,{ ..., N . For| two|} function f(x) and g(x), the N { } where Q notation f(x) = Θ(g(x)), means that k1g(x) f(x) k g(x), k > 0, k > 0, x , x > x . We≤ denote the≤ 2 ∃ 1 ∃ 2 ∃ 0 ∀ 0 N Rc probability of an event as Pr[ ], the mutual information AN (Rc) = ( 1) 1 2 eN ( Rc ln(2)) , − − · − as I( ; ), the entropy as HE( ), theE convolution as , the binary · · · ∗ with e ( ) being the exponential sum function. The
exponent logarithm as ld( ), and the natural logarithm as ln( ). N · · N of the· SNR is the diversity gain and the Nth root of the N inverse numerator, i.e., GC,JD = 1/ AN (Rc), is the coding II. SYSTEM MODEL gain. To the best of our knowledge, the exact coding gain of p A. Multi-Connectivity System Model JD for parallel block-fading channels has been unknown so far. Our approach concentrates on the asymptotic solution, which We consider a MCo cellular network consisting of a core network and N base stations (BS , i ) communicating to is an accurate performance indicator for the operational region i ∀ ∈ N of URLLC applications, as we demonstrate by numerical a single user equipment (UE). The core network coordinates examples. Based on the asymptotic outage probability, we the data transmissions to the UE. To ease the notation, only one derive the SNR gain of MCo over SCo as link per BS is included in the system model (macrodiversity). However, a single BS can also establish multiple connections N N η to the UE (microdiversity). Connections between the core − A1(Rc) 1 i=1 di GMCo SCo = . network and each BS are realized by backhaul links, and , N η N A (R ) N N 1 q d− N c (P out) − Q 1 connections between each BS and UE, by wireless links. p q By assumption all N wireless links are orthogonal, i.e., This result reveals that the SNR gain of MCo over SCo we consider parallel block-fading channels. The achievable increases at a rate of around 3(N 1)/N dB with respect to transmission rate over the ith wireless link depends on its − the target spectral efficiency (i.e., per source sample/channel capacity Ci(Γi) and, thus, on its received SNR Γi. In the symbol) and decreases at a rate of 4.3(N 1)/N 1/P out dB system model, we distinguish between down- and uplink as with respect to the target outage probability.− In addition,· we follows: 4

n n BS 2: X2 BS 2: Y2 C2(Γ 2) C2(Γ 2)

CN (Γ N ) C1(Γ 1) Core CN (Γ N ) C1(Γ 1) Core UE: Network UE: n n n k k n k n n n Network n Y , Y , ..., Y → Sˆ n S → X , n S → X , X , ..., X n n n n BS N : XN 1 2 N BS 1: X1 1 BS N : YN 1 2 N BS 1: Y1 Y , Y , ..., Y Xn Xn 1 2 N 2 , ..., N → Sˆk

(a) (b) Fig. 1: System model with a single UE, N base stations, and a core network for (a) the downlink and (b) the uplink.

a) Downlink: The downlink system model, as illustrated the received SNR Γi is given by in Fig. 1a, has one binary memoryless source, denoted as 1 γi [S(t)] , with the k-sample sequence being represented in fΓ (γi)= exp , for γi 0, (1) t∞=1 i Γ¯ −Γ¯ ≥ vector form as Sk = [S(1),S(2), ..., S(k)]. When appropri- i  i  ate, for simplicity, we shall drop the temporal index of the with the average SNR Γ¯i being obtained as Γ¯i = (Pi/N0) η · sequence, denoting the source merely as S. By assumption di− , where Pi is the transmit power per channel, di is the S takes values in a binary set = 0, 1 with uniform distance between BSi and the UE, and η is the path loss probabilities, i.e., Pr[S = 0] = PrB[S =1]=0{ } .5. Therefore, exponent. The channel state information is assumed to be k the entropy of the sequence is 1/k H(S ) = H(S)=1. At exclusively known at the receiver. the core network, the source sequence· Sk is encoded to the transmit sequences Xn, i . The ith transmit sequence III. PRELIMINARIES AND PROBLEM STATEMENT i ∀ ∈ N is forwarded over the backhaul link to BSi. At each BS the The problem investigated in this work can be formulated transmit sequence is then sent to the UE over parallel block- Yn Yn as follows: the received channel codewords 1 , ..., N (cf. fading channels. The decoder at the UE retrieves the source received sequences in Section II-A) must comprise sufficient sequence Sk from the received sequences Yn, i . Sk i ∀ ∈ N information such that the source sequence can be per- b) Uplink: The uplink system model, as illustrated in fectly reconstructed. At the core network, for the ith link, Fig. 1b, is similar to the downlink, except that the source se- a channel code maps the source sequence Sk to a channel Xn quence is originated from the UE, and the received sequences codeword i (cf. transmitted sequence in Section II-A) with Yn i , i , are decoded at the core network to retrieve the an spectral efficiency Ri,c, measured in source samples per source∀ sequence∈ N Sk. Similar to the downlink, the ith transmit channel input symbol, associated with the modulation scheme Xn sequence i is sent from the UE to the ith BS over parallel Ri,M = ld(M), measured in bits per channel input symbol block-fading channels. for a cardinality M of the channel input symbol alphabet; and In this work, the down- and uplink system models can be the channel code rate Ri,cod, measured in source samples per treated interchangeably, so that all further results are applicable bit; i.e., R c = R M R cod. In many parts of this work, i, i, · i, to both system models. for simplicity, we shall assume Ri,c = Rc = k/n, i. As mentioned before, the transmitter has no knowledge∀ of the CSI and chooses a fixed spectral efficiency. If and only if the instantaneous capacity of the parallel fading channels support B. Link Model the spectral efficiency, the error probability at the receiver side can be made arbitrarily small [21]. As discussed in the Introduction, micro- and macrodiversity can be achieved by spatial or frequency separation of the channels. In both cases, it is reasonable to assume that the A. Channel Capacity channel fading is approximately uncorrelated, which we do in The instantaneous channel capacity of the MCo system this study. Furthermore, to cope with the low latency constraint model depends on the combining algorithm. For SC and MRC in URLLC, we consider relatively short encoded sequences. the analysis is based on the gain attained by receiving the As a result, the length of an encoded sequence is less than same channel codeword via multiple links, i.e., the channel Xn Xn Xn or equal to the length of a fading block of a block Rayleigh codewords are identical: 1 = ... = N = . On the other fading. Moreover, the signals can be transmitted from or to hand, for JD the channel codewords transmitted over all links different BSs, which leads to individual average SNR values. may be different but based on a joint codebook. The resulting Xn As argued, we can assume that the sequences i , i , instantaneous channel capacities are given as follows: are transmitted (in up- and downlink) over independent∀ chan∈ N- Selection Combining: The instantaneous received SNRs nels undergoing block Rayleigh fading and additive white of each channel are assumed to be known by the receiver. Gaussian noise (AWGN) with mean power N0. The pdf of The received sequence of the subchannel with the maximum 5

Rc Rc φ(Γ1) SNR is selected as combiner output, and the other received = Pr 0 Γ1 < 2 1, 0 Γ2 < 2 − 1, ..., sequences are discarded, so that [11] ≤ − ≤ − h Rc φ(Γ1) φ(Γ2) ... φ(ΓN−1) 0 ΓN < 2 − − − − 1 (8) Xn Yn Xn Yn ≤ − CSC (Γ1, ..., ΓN ) = max (I( ; 1 ), ..., I( ; N )) − R −φ γ −φ γ −...−φ γ 2Rc 1 2Rc φ(γ1) 1 2 c ( 1) ( 2) (iN−1) 1 − − − = φ (max (Γ1, ..., ΓN )) (2) = ... γ =0 γ =0 γN =0 is the complex AWGN channel capacity of SC. Here and Z 1 Z 2 Z f(γ )f(γ )...f(γ )dγ ...dγ dγ . (9) throughout the text, φ(x) = ld(1 + x) represents the instanta- 1 2 N N 2 1 neous complex AWGN channel capacity. The steps are justified as follows: in (6) we substitute the JD Maximal-ratio combining: As for SC, the instantaneous channel capacity in (4) into the outage probability expression received SNRs of each subchannel is assumed to be known. in (5); in (7) the sum constraint is separated into individual 1 The received sequences are scaled to their corresponding SNRs constraints; in (8) the bounds are transformed with φ− (y)= and coherently added, so that [11] 2y 1; in (9) the probability of outage is established in integral − n n n form with the assumption that the received SNRs Γi, i , CMRC (Γ1, ..., ΓN )= I(X ; Y , ..., Y ) ∀ ∈ N 1 L are independent. The pdf f(γi) is given in (1). Although the N outage expression in (9) cannot be solved in closed form, a = φ Γi (3) i=1 simple asymptotic solution can be derived at high SNR as   X R R −φ γ is the complex AWGN channel capacity of MRC. 2 c 1 2 c ( 1) 1 out − − Joint Decoding: In contrast to SC and MRC, JD pro- PJD,N ... ≈ γ1=0 γ2=0 cesses each received sequence individually until decoding. The Z − Z − − − 2Rc φ(γ1) φ(γ2) ... φ(γN−1) 1 achieved complex AWGN channel capacity is [21] − C (Γ , ..., Γ )= I(Xn; Yn)+ ... + I(Xn ; Yn ) ZγN =0 JD 1 N 1 1 N N 1 N N dγN ...dγ2dγ1 (10) = φ (Γi) . (4) ¯ i=1 i=1 Γi A (R ) X =Q N c where (11) B. Outage Formulation N ¯ i=1 Γi The outage probability is an important concept in fading N Rc (12) AN (Rc) = (Q1) 1 2 eN ( Rc ln(2)) . channels, which provides a way to characterize the per- − − · − N 1 xn − formance of communication systems in non-ergodic fading Here, eN (x) = n=0 n! is the exponential sum
function. scenarios. As N is finite, the parallel fading channel is non- For more details, we refer to the derivations in Appendix A. P N ergodic, i.e., N is not large enough to average over channel The Nth root of the inverse numerator GC,JD =1/ AN (Rc) is commonly termed the coding gain [18]. To our best variations. If the achieved instantaneous channel capacity is p less than the spectral efficiency, an outage event occurs [11]. knowledge, the coding gain of JD was unknown so far and Thus, the outage probability is given by constitutes an important original contribution of this work. In contrast to the asymptotic solution in (11), a lower bound out Pj,N = Pr [Cj (Γ1, ..., ΓN ) < Rc] (5) for (9) is given by [11, Ch. 9.1.3] out N for j SC, MRC, JD . The outage probability for SC and P [Pr[0 φ(Γ) < Rc/N]] ∈ { } JD,N ≥ ≤ MRC are well studied, whereas it is very difficult to derive a N A (R /N) tractable exact formula of the outage probability of JD. = 1 exp 1 c , (13) − − Γ¯    ¯ ¯ ¯ Rc/N IV. OUTAGE PROBABILITIES for Γ1 = ... = ΓN = Γ and A1(Rc/N)=2 1. Note that the lower bound is based on the assumption that− an outage In this section, we establish the exact outage probability event occurs if the channel capacity of at least one fading of JD in integral form. More importantly, we derive in closed channel cannot support the spectral efficiency R /N, i.e., each form a corresponding asymptotic expression for the high-SNR c fading channel is allocated an equal share of the information. regime. In addition, we reproduce known bounds on the JD In [17], based on the outage exponent analysis, a lower and outage probability given in [11], [17], as well as the exact and an upper bound are given by asymptotic outage probabilities of SC and MRC [19], [22], out,lower which we require later on for comparison. P = a exp N φ Γ¯ Rc/N E Γ¯ ≥ JD,N − 1,1 ¯ E (Γ)¯ out  +E1,0 Γ + 0
/N + o(
N) ,
PJD,N out,upper ¯ Rc ¯ A. Joint Decoding  PJD = b exp N φ Γ /N E1,1 Γ ≤ ,N
− 
¯ E (Γ)¯ The outage probability for JD can be calculated as follows: +E1,0 Γ + 0
/N + o(
N)
 (14) out  out,upper PJD,N = Pr [0 φ(Γ1)+ φ(Γ2)+ ... + φ(ΓN ) < Rc] (6) where aand b are constants, with a
b. P out,lower and P
≤ ≤ JD,N JD,N = Pr [0 φ(Γ1) < Rc, 0 φ(Γ2) < Rc φ(Γ1), ..., are refereed to as the lower and upper outage exponents, ≤ ≤ − ¯ ¯ 0 φ(ΓN ) < Rc φ(Γ1) φ(Γ2) ... φ(ΓN 1)] respectively. The exact reliability functions E1,1 Γ , E1,0 Γ , ≤ − − − − − and E Γ¯ are given in [17]. According to [17], the outage (7) 0


6

probability differs for Rc/N < Cergodic and Rc/N Cergodic, a) Identical received average SNRs: For Γ¯ = ... = ≥ 1 where Cergodic = limN CJD(Γ1, ..., ΓN )/N = E [φ(1 + Γ)] Γ¯N = Γ¯ the pdf of the total received SNR is given by [19, is the ergodic capacity.→∞ The derivations of (14) are mainly (2.30)] based on large deviations theory and Meijer’s -function [23], G (N 1) [24]. For more details on the outage exponent analysis, please γMRC− γMRC fΓ (γMRC)= exp . (19) refer to [17]. MRC (N 1)! Γ¯N − Γ¯ − · In contrast to the bounds in (13) and (14), our asymptotic   The outage probability can be then calculated as follows [19, solution in (11) offers a simple, yet accurate solution at high (2.31)-(2.33)]: SNR. The outage exponent analysis in (14) can achieve tight out bounds on the outage probability, but the calculations involve P = Pr [0 φ(ΓMRC) < Rc] (20) MRC,N ≤ the incomplete Gamma function and Meijer’s G-function, 2Rc 1 which makes any further analytical derivations on the SNR − = f(γMRC)dγMRC (21) gain, DMT, and throughput rather involved. As [17] did not ZγMRC=0 explicitly considered the properties of the asymptotic outage A (R ) (i 1) A1(Rc) N ( 1 c /Γ¯) − probability, the exact solution of the coding gain remained =1 exp . − − Γ¯ i=1 (i 1)! unknown. The lower bound in (13) is a simple solution,   X − (22) but not tight, as we show later on. On the other hand, our asymptotic solution in (11) offers a remarkable simple The steps can be justified similarly to (15)-(16). The closed asymptotical description of the outage probability at high SNR. form of the integral in (21) is given in [19, (2.33)]. An Especially for URLLC, high-SNR results are well suited, as asymptotic solution can be derived at high SNR [22, (16)] 5 as we are interested in frame error rates below 10− . Note that, more generally than (13) and (14), our solution also allows 1 A (R ) N P out 1 c , (23) for different average SNRs, which is of practical relevance, if MRC,N ≈ N! Γ¯ the signals are transmitted from or to different BSs. We detail   N this comparison via numerical examples in Section IX. from which GC,MRC = √N!/A1(Rc) is the coding gain of MRC. b) Different received average SNRs: For Γ¯ = ... = Γ¯ 1 6 6 N B. Linear Combining the pdf of the total received SNR is given by [25, Proposi- tion 3.1] The outage probability for SC can be derived as follows [19, N (2.42)]: N 2 γMRC fΓ (γMRC)= Γ¯ − exp MRC i=1 i − Γ¯  i  out X N PSC,N = Pr [0 φ(max (Γ1, ..., ΓN )) < Rc] (15) 1 ≤ j=1 . (24) Rc ¯ ¯ N 2 1 × j=i Γi Γj − 6 − = f(γi)dγi (16) Y i=1 The outage probability can be then calculated as follows: Zγi=0 YN A (R ) = 1 exp 1 c , (17) out ¯ PMRC,N = Pr [0 φ(ΓMRC) < Rc] (25) i=1 − − Γi ≤    Rc Y 2 1 N − ¯N 2 γMRC Rc = Γ − exp with A1(Rc)=2 1. The steps are justified as follows: i=1 i − Γ¯ − ZγMRC=0  i  in (15) we substitute the SC channel capacity in (2) into N X 1 the outage probability expression in (5); (16) follows similar j=1 ¯ ¯ dγMRC (26) × j=i Γi Γj arguments as in (6), (8), and (9), respectively, and the multiple 6 − YN N 1 A1(Rc) integral can be rewritten as the product of single integrals, = Γ¯ − 1 exp i=1 i − − Γ¯ since the integral domain is normal and the SNRs are inde-   i  pendent; (17) is the closed-form solution of the integral in X N 1 j=1 ¯ ¯ . (27) (16). An asymptotic solution at high SNR can be derived × j=i Γi Γj 6 − by using the MacLaurin series of the exponential function Y The steps can be justified similarly to (15)-(17). No further exp( x ) 1 x for x 0, giving [19, (2.43)] − i ≈ − i i → simplification based on high SNRs can be achieved for (27). N However, the asymptotic solution of identical received average out (A1(Rc)) PSC,N , (18) SNRs in (23) gives an upper bound at high SNR for (27) by N N ≈ Γ¯i ¯N ¯ i=1 replacing Γ with i=1 Γi yielding Q N from which GC,SC =1/A1(Rc) is the coding gain of SC. Q 1 (A (R )) P out . 1 c . (28) To calculate the outage probability of MRC we define an MRC,N N! N ¯ i=1 Γi auxiliary RV, namely, the total received SNR, as ΓMRC = N i=1 Γi. For more details on this upper bound,Q we refer to the deriva- We have to distinguish between two cases: tions in Appendix B. P 7

C. Single-Connectivity respectively. In addition, we give the asymptotic throughput In addition, the exact and asymptotic outage probability of of SCo (e.g., from N =1 in (35)): SCo (e.g., N = 1 for (17) and (18)) are given as a baseline out out TSCo B ld P Γ+1¯ (1 P ) in bit/s. (37) by ≈ −
out A1(Rc) P =1 exp (29) VI. MULTI-CONNECTIVITY GAIN SCo − − Γ¯   A (R ) In this section, we quantify the performance gain of MCo 1 c , (30) ≈ Γ¯ over SCo in terms of transmit power reduction. For MCo we consider the optimal combining algorithm, i.e., JD. To ensure a from which GC SCo =1/A (Rc) is the coding gain of SCo. , 1 fair comparison between different setups, we equally allocate V. THROUGHPUT the total transmit power PT to all channels, such that Pi = PT/N, i . However, this assumption is non-essential, The throughput captures how much information is received and other∀ system∈ N setups can be evaluated from our formulas at the destination on average per transmission, depending on with some effort. the SNR. To capture this, following the standard approach in We assume a target (fixed) spectral efficiency Rc (i.e., a the literature [4], we define the throughput T as the product of certain throughput has to be guaranteed) and target (fixed) the bandwidth B, spectral efficiency Rc, and the non-outage out out outage probability P , and evaluate the required total average probability (1 P ), i.e., ¯ out − SNR Γ = σ( )(P ) in the high-SNR regime. The SNR gain out · T = BRc(1 P ) in bit/s. (31) is defined as the ratio of the required average SNR between − SCo and MCo. To evaluate (31), we require the achieved spectral efficiency Rc A reformulation of (11) yields for the different combining algorithms (j JD, SC, MRC ) ∈ { out } for a given number of links N, outage probability P , and P N A (R ) 1 out T N c (38) average received SNRs Γ¯ . σJD(P )= = N out , N N0 r P N N η Now, by reformulating (11) so as to express the achieved i=1 di− spectral efficiency in terms of a given outage probability, we q where σ (P out) is the required total averageQ SNR for JD. The obtain a high-SNR asymptotic expression for the throughput JD required total average SNR for SCo based on the reformulation of JD as of (30) is N 1 out out TJD,N BA− P Γ¯i (1 P ) in bit/s. (32) ≈ N i=1 − out PT A1(Rc) 1   σSCo(P )= = out η . (39) 1 Y N0 P d1− Here, AN− ( ) is the inverse function of AN ( ). Unfortunately, · 1 · the inverse function A− ( ) does not have a closed-form The SNR gain is then given as the ratio between the required N · solution. However, we can give a good approximation. At average SNRs for SCo and JD as high SNR, the achieved spectral efficiency for a given outage out σSCo(P ) probability is Rc 1. In this case, the inverse function can be G = ≫ MCo,SCo out given by use of an approximation of the asymptotic Lamber σJD(P ) N N η W function [26] by − A1(Rc) 1 i=1 di N = . 1 out N 1 N η ¯ N A (R ) N out N 1 q d− AN− P Γi = Rc − [ln(ζ) ln(ln(ζ))] , N c (P ) − Q 1 i=1 ≈ ln(2) −  Y  (33) p q (40) where Based on (40), we can answer the fundamental questions formulated at the beginning of the Introduction, i.e., how much N−1 out N ¯ transmit power can be saved by MCo as compared with SCo (N 1)!P i=1 Γi ζ = − . (34) depending on the number of links N, the spectral efficiency q N 1 η − Q (corresponding to the throughput T BR ), the path loss d− For more details, we refer to the derivations in Appendix C. c i for i , and the outage probability≈ P out. The asymptotic throughputs for SC and MRC can be given ∈ N based on (18) and (28) as VII. JOINT DECODING VS.LINEAR COMBINING N N out ¯ TSC,N B ld P Γi +1 In this section, we evaluate the performance improvement ≈ r i=1 ! Y of JD over SC and MRC. All combining algorithms for MCo (1 P out) in bit/s, (35) × − are superior to SCo, since the multiple diversity branches are N exploited, i.e., the diversity gain is N. However, there exists a N out ¯ TMRC,N B ld N! P Γi +1 difference of the outage probabilities governed by the coding ≈ r · i=1 ! Y gains GC,( ) of each combining algorithm ((11), (18), and (1 P out) in bit/s, (36) · × − (23)). Based on these equations, the performance improvement 8 of JD over SC and MRC can be quantified in terms of the SNR value decomposition, the MIMO fading channel can also be gain, which is the ratio of the coding gains, transformed into a parallel fading channel in the space domain [11]. Thus, it is reasonable to relate the DMT with the SNR GC,JD A1(Rc) N GJD,SC = = > √N!, and (41) gain. G N C,SC AN (Rc) For JD, the diversity gain is a function of the multiplexing GC,JD 1 A1(Rc) gain given by GJD MRC = =p > 1, (42) , N N GC,MRC √N! · AN (Rc) dJD(r)=N r, r [0,N]. (48) respectively. In Lemma 3 (see Appendixp D) we prove that the − ∈ SNR gain of JD over MRC is strictly larger than one, which The DMT for SC and MRC is given by implies that the SNR gain of JD over SC is strictly larger than N√ dj (r)=N (1 r), r [0, 1], (49) N!. · − ∈ for j SC,MRC . For more details, we refer to the VIII. SNR GAIN VS.DIVERSITY-MULTIPLEXING derivations∈ { in Appendix} E. TRADEOFF The DMT of JD based on the lower bound in (13) given In this section we relate the SNR gain to the DMT analysis. in [11, Ch. 9.1.3] is aligned with our results. It is not Note that the SNR gain in (40) can be separated into two surprising that JD outperforms SC and MRC in terms of the parts, one depending on the spectral efficiency and the other multiplexing gain. Both SC and MSC perform a non-invertible depending on the outage probability. Both parts are influenced linear transform on the received signal vector, collapsing the by the number of links. As we can easily see, with a decreasing dimension from N to one. It is obvious that diversity can outage probability, the SNR gain increases, scaled by the be maintained with SC and MRC, but both will suffer with power of (N 1)/N, i.e., respect to JD when the goal is to achieve multiplexing gain. − At full diversity (d = N, r = 0) (i.e., fixed spectral ∂10 log (G ) dB N 1 1 JD 10 MCo,SCo efficiency), the SNR gain increases by a factor proportional to out = 4.3 − out dB. (43) ∂P − N P (N 1)/N with a decreasing outage probability (cf. (43)). The The dependency of the SNR gain in terms of the spectral term−(N 1)/N is the relative maximum diversity gain of MCo efficiency cannot be seen that easily. However, similarly as − (with JD) and SCo. At full multiplexing (dJD = 0, r = N) for the throughput (cf. (72)) we can simplify the SNR gain (i.e., fixed outage probability), the SNR gain increases by a for sufficiently high spectral efficiencies, i.e., Rc 1, as factor proportional to (N 1)/N with a increasing spectral ≫ − N−1 efficiency (cf. (45)). Similar to the full diversity, the term Rc N N (N 1)! 2 GMCo,SCo N− 1 N−1 (N 1)/N is the relative maximum multiplexing gain of ≈ − N N − s(ln(2)) N Rc MCo (with JD) and SCo. In summary, both performance N N η improvements are governed by the relation of the maximum d− 1 i=1 i diversity and maximum multiplexing gains of MCo (with JD) η . (44) × N N 1 q d− (P out) − Q 1 and SCo. Based on the DMT analysis, one can give the slope of the q Now, we can see that with increasing spectral efficiency, the SNR gain in the spectral efficiency and outage probability, but SNR gain increases, scaled by the factor (N 1)/N, i.e., − not the SNR gain itself. Furthermore, the slope in the spectral efficiency is merely valid for sufficiently high values, as we ∂10 log10 (GMCo,SCo) dB N 1 3 − dB. (45) discuss in the next section. ∂Rc ≈ N A similar analysis for the SNR gain of JD over SC and MRC in IX. NUMERICAL RESULTS (41) and (42), respectively, leads to the same result as in (45) with respect to the spectral efficiency, while being insensitive In this section, we illustrate and discuss the exact, asymp- to the target outage probability. totic, and the lower bound outage probabilities of JD as well as We see that in (43) and (45) there is a factor of (N 1)/N. the exact and asymptotic throughput of JD. Furthermore, we This factor can be related to the DMT analysis, as we− show illustrate and discuss the corresponding SNR gain. We equally in the following. In the context of MIMO systems [15], it is allocate the total transmit power PT to all channels, such that P = PT/N, i . Furthermore, we normalize all distances proven that for a multiplexing gain i ∀ ∈ N to one. We define the average system transmit SNR as PT/N0. R (Γ)¯ r = lim c , (46) Fig. 2a depicts the outage probability of JD (Monte-Carlo Γ¯ ld NΓ¯ →∞ simulation of (9), our asymptote in (11), and the existing lower bound in (13)) versus the average system transmit the diversity gain d will not exceed
SNR PT/N0. For comparison, we include the SCo outage ld P out (r, Γ)¯ probability in (29). We show results for N 2, 3, 5 and ,N ∈ { } d(r)= lim · , (47) a constant spectral efficiency of Rc = 0.5. We can observe − Γ¯ ld NΓ¯ →∞ the following: (i) the asymptote is very tight at medium and for Γ¯1 = ... = Γ¯N = Γ¯, i.e., all distances
are normalized to high SNR; (ii) with every additional link the diversity gain unit, and average system SNR NΓ¯. By applying a singular dJD(r) increases by one with constant spectral efficiency, i.e., 9

9 0 ·10 10 1.5 Approx. asymptote Numerical solution −2 ,N 10 SCo out JD P , bit/s 1 N = 5 ,N −4 N 10 = 2 JD T

N = 3 −6 N = 3 10 0.5

Asymptote N = 2 Outage Probability, N = 5 Throughput, −8 Monte-Carlo 10 Lower bound SCo − 0 5 0 5 10 15 20 25 10 20 30 40 50 60 70

Average system transmit SNR, PT/N 0, dB Average system transmit SNR, PT/N 0, dB

(a) (b)

Fig. 2: (a) JD outage probability for Monte-Carlo simulation, asymptote, and lower bound, with N ∈ {2, 3, 5} and, Rc = 0.5, and (b) JD throughput Monte-Carlo simulation, approximated asymptote with N ∈ {2, 3, 5}, B = 20 MHz, and P out = 10−3. The outage probability and throughput of SCo are depicted for comparison.

the multiplexing gain is r =0; and (iii) the SNR offset of the in (40)— versus the spectral efficiency Rc (corresponding lower bound increases with the number of links. At this point, to the throughput T BRc). We show results with N we would like to clarify our assumptions on the SNR range. 2, 3, 4 number of links≈ and an outage probability of P out ∈ { 3 } 5 ∈ It is noteworthy that, even though our outage analysis is based 10− , 10− . The following can be observed: (i) the SNR on high SNR, it leads to accurate results in the low-to-medium {gain increases} with the number of links, spectral efficiency, SNR region as well. In Fig. 2a, for instance, the asymptotic and decreasing outage probability; (ii) a decrease in outage outage probability with 5 links is already tight for an outage probability manifests itself as a vertical shift of the respective out 3 N 1 probability of PJD,5 = 10− . The corresponding average SNR gain, i.e., from (43) we have ∆GMCo,SCo =2 10 N− dB 3 5 · system transmit SNR is then PT/N0 = 5 dB. That means for an outage probability shift from 10− to 10− ; and (iii) for that the average transmit SNR per link is Pi/N0 = 2 dB, sufficiently high spectral efficiencies, the SNR gain increases which falls in the low-to-medium SNR region. − by 3(N 1)/N dB per source sample/channel symbol (cf. Remark: The lower and upper bounds based on the outage (45)). − exponent analysis in (14) are tight, cf. [17, Fig. 3 - Fig. 7], Fig. 3b depicts the SNR gain of JD —GJD,( ) given in but to achieve tight bounds for the entire SNR range heavy (41) and (42)— over SC and MRC, respectively,· versus the computation efforts are required. Especially for the class of spectral efficiency Rc. The following can be observed: (i) the URRLC applications, the low-SNR range is not of interest, as SNR gain of JD is greater than one, as proven in Lemma 3; the region of the required outage probabilities is at medium (ii) the SNR gain of JD increases with N and Rc; (iii) the to high SNR. SNR gain of JD with respect to MRC differs from the SNR Fig. 2b depicts the throughput for JD (numerical solution gain of JD with respect to SC by 1/ N√N!; (iv) for very low of (31), our asymptote in (32) with the approximation of spectral efficiencies the SNR gain of JD over MRC vanishes; the asymptotic inverse function in (33)) versus the average and (iv) for sufficiently high spectral efficiencies, the SNR system transmit SNR PT/N0. For comparison, we illustrate the gain increase by 3(N 1)/N dB per source sample/channel SCo throughput in (37). We show results for N 2, 3, 5 , symbol (cf. (45)). − out ∈ { 3 } B = 20 MHz, and an outage probability of P = 10− . The Note that the range of practical spectral efficiencies is within following can be observed: (i) the approximated asymptote is Rc [0.5, 4.6]¯ (e.g., 1/2 channel code rate and BPSK, or very tight at high SNR; and (ii) for increasing SNR, the JD 2/3∈channel code rate and 128-QAM). However, we illustrate throughput increases asymptotically with N 20 Mbit/s per spectral efficiency up to Rc = 25, in order to show the 3 dB, whereas the SCo throughput increases· asymptotically asymptotic slope of the SNR gain. with 1 20 Mbit/s per 3 dB. In the· following, we show numerical results for the SNR X. CELLULAR FIELD TRIALFORTHE UPLINK out 3 5 gain with P 10− , 10− and N 2, 3, 4 . As shown In [27], measurements were carried out in a field trial ∈{ } ∈{ } in Fig. 2a, the asymptotic outage probability is very tight testbed in Dresden (Germany) downtown. In this section we within this range, i.e., the following numerical results based make use of this measurement data to show the potential on the asymptotic outage probability barely differ from the of MCo in a real cellular network. In the field trial, the numerical results based on the exact outage probability. uplink was considered. Next we shortly introduce the field Fig. 3a depicts the SNR gain of MCo over SCo —GMCo,SCo trial setup and then present the empirical outage probability 10

80 40 N = 2 N = 2 N = 3 N = 3 N = 4 N = 4 P out = 10 −3 30 MRC , dB 60

out −5 , dB SC

P = 10 ) · SCo ( , , JD MCo G 20 G 40

SNR gain, 10 SNR gain, 20

0 0 5 10 15 20 25 0 5 10 15 20 25

Spectral efficiency, Rc, source samples/channel symbol Spectral efficiency, Rc, source samples/channel symbol

(a) (b) Fig. 3: (a) SNR gain of MCo (with JD) over SCo, with N ∈ {2, 3, 4} and P out ∈ {10−3, 10−5} and (b) SNR gains of JD over SC and MRC, with N ∈ {2, 3, 4}. and throughput cumulative distribution functions (CDFs) for can be assessed with (11), (17) and (27) for JD, SC, and MRC, the measurement data. Our results elaborate on the following respectively, for each measurement. Similarly, the throughput points: (i) the performance improvement of MCo over SCo is given by (32), (35), and (36) for JD, SC, and MRC, and (ii) the performance gain of JD in comparison to SC and respectively. MRC. Fig. 4c and Fig. 4d depict the empirical outage probability CDF and the empirical throughput CDF, respectively. We show results for N 2, 3 number of links, a spectral efficiency A. Field Trial Setup ∈ { } out 5 of Rc =1 (Fig. 4c), and an outage probability of P = 10− The field trial testbed, deployed in Dresden downtown, is (Fig. 4d). The following can be observed: (i) MCo is much depicted in Fig. 4a. In total, 16 BSs located on five sites with superior to SCo and (ii) JD outperforms SC and MRC, the up to six-fold sectorization were used for the measurements. performance gain increasing with the number of links from During the field trial, two UEs were moved on a measurement source to destination. bus in 5 m distance while transmitting on the same time and frequency resources employing one dipole antenna each. C. Discussion The superimposed signal is jointly received by all BSs, which took snapshots of 80 ms (corresponding to 80 transmit time Based on the field trial setup, we can conclude that MCo intervals) every 10 s. In total, about 1900 such measurements can achieve a substantial performance improvement in real were taken in order to observe a large number of different cellular networks. The measurement data at hand documents transmission scenarios. In Fig. 4b the measured average SNR that multiple relatively high average SNR values frequently occur, for which combining algorithms are particularly bene- Γ¯m values for around 1000 measurements observed at all BSs i ficial. From the uplink measurement data we can also draw and locations are shown, where m denotes the measurement conclusions for the downlink. As argued in Section II-B, the number and i the BS index, i Hbf 0°, Hbf 60°, ... . The two largest average SNRs measured∈ { at any BS for} each statistical properties of the link model are identical for the up- measurement are depicted in the upper part of the figure. An and downlinks. Thus, it is reasonable to assume that multiple interesting result is that multiple relatively high average SNR relatively high average SNR values frequently occur in the values of two different BSs are observed at each location of the downlink as well. Based on this, MCo can also achieve low UEs. Since combining algorithms are particularly beneficial in outage probabilities and high throughput in the downlink of scenarios with multiple relatively high average SNR values, real cellular networks. this result indicates that cooperation among BSs can provide a much more reliable data transmission, as confirmed next. XI. CONCLUSION For more details on this field trial setup, please refer to [27]. In this work, we answer fundamental questions regard- ing the performance improvement of multi-connectivity over single-connectivity. For doing so, we derive the exact and B. Empirical CDFs for Outage Probability and Throughput asymptotic outage probabilities for optimal and suboptimal ¯m With the measured average SNR Γi in Fig. 4b we can receiver algorithms, namely joint decoding, selection combin- generate an empirical outage probability and throughput CDF. ing, and maximal-ratio combining. We evaluate and show the For each measurement we consider the N strongest links, i.e., tremendous transmit power reduction of MCo over SCo, by ¯m the largest measured average SNRs Γi . The outage probability analytically deriving the corresponding SNR gain. Our results 11

largest 30 2nd largest Mitte 180 ◦ ◦ , dB

Mitte 60 m i ◦ ¯ Sued 300 Γ Sued 180 ◦ 20 Sued 60 ◦ FFP 30 ◦ Lenne 300 ◦ Lenne 180 ◦ Lenne 60 ◦ Hbf 300 ◦ 10 Hbf 240 ◦ Hbf 180 ◦ Hbf 120 ◦ ◦

Hbf 60 Measured average SNR, ◦ Hbf 0 0 200 400 600 800 Measurement number, m

(a) (b) 1 1 JD JD SC SC 0.8 MRC 0.8 MRC SCo N = 3

0.6 0.6

SCo N = 2 0.4 N = 2 0.4

. 0 2 0.2 Empirical throughput CDF N = 3 Empirical outage probability CDF 0 −10 −8 −6 −4 −2 0 0 10 10 10 10 10 10 10 4 10 5 10 6 10 7 10 8 out Outage probability, P· ,N Throughput, T·,N

(c) (d) ¯m Fig. 4: (a) Testbed deployment, (b) measured average SNR Γi achieved at all BSs of the testbed during the complete field trial [27], (c) empirical outage out −5 probability CDF (N ∈ {2, 3}, Rc = 1), and (d) empirical throughput CDF (N ∈ {2, 3}, P = 10 ) for JD, SC, and MRC. reveal that the SNR gain increases with the number of links ACKNOWLEDGMENTS and the spectral efficiency, while decreasing with the outage probability. The authors wish to acknowledge the helpful discussions ¨ In addition, we compare the optimal combining algorithm, with David Ohmann and Nick Schwarzenberg. i.e, joint decoding, with the suboptimal combining algorithms, i.e., selection combining and maximal-ratio combining. Again, APPENDIX A we quantify the performance gain in terms of the SNR reduc- ASYMPTOTIC JOINT DECODING OUTAGE PROBABILITY tion. Our results reveal that JD becomes more advantageous in terms of the SNR gain as the spectral efficiency and the The asymptotic outage probability can be obtained as number of links increases. − Finally, we have applied our analytical framework to real 2Rc 1 2Rc φ(γ1) 1 out − − cellular networks. Based on the measurement data recorded PJD,N = ... γ1=0 γ2=0 in a field trial, we have evaluated the achievable performance −Z − Z − − 2Rc φ(γ1) φ(γ2) ... φ(γN−1) 1 improvement by the use of multi-connectivity. The measure- − 1 γ1 ¯ exp ¯ ment data documents that multiple relatively high average γ =0 Γ1 − Γ1 Z N   SNR values frequently occur, in which case multi-connectivity 1 γ2 1 γN proves particularly beneficial. exp ... exp dγN ...dγ2dγ1 × Γ¯2 − Γ¯2 Γ¯N − Γ¯N     (50) − − − − − 2Rc 1 2Rc φ(γ1) 1 2Rc φ(γ1) φ(γ2) ... φ(γN−1) 1 − − − ... ≈ Zγ1=0 Zγ2=0 ZγN =0 12

k+1 2x 1 1 γ1 1 γ2 1 γN n − 1 1 ... 1 dγN ...dγ2dγ1 k n n k (ln(1 + γK+1)) Γ¯ − Γ¯ Γ¯ − Γ¯ Γ¯ − Γ¯ ( 1) x − k 1 1 2 2 N N × k=0 − k (k + 1) (ln(2))       (51)   γK+1=0 X (60) − 2Rc 1 2Rc φ(γ1) 1 1 − − K 1 1 ... = ( 1)K 2x 1 2x − ( 1)n xn+1 (ln(2))n+1 ¯ ¯ ¯ n=0 ≈ Γ1Γ2...ΓN γ1=0 γ2=0 − − − − n! − −Z − −Z  2Rc φ(γ1) φ(γ2) ... φ(γN−1) 1 n n X1 − ( 1)k (61) dγN ...dγ2dγ1 (52) × k=0 − k k +1 ZγN =0 X    1 AN (Rc) /(n + 1) = . (53) K+1 x Γ¯ Γ¯ ...Γ¯ = ( 1) (1 2 eK+1 ( x ln(2))) . (62) 1 2 N −| −{z · }− The steps are justified as follows: (50) is the substitution of the pdf f(γi) given in (1) into (9); (51) MacLaurin series The steps are justified as follows: (58) is our inductive for exponential function exp( xi) 1 xi for xi 0, hypothesis; for (59) we have used the binomial formula; (62) (52) expanding the resulting product− ≈ as −(1 x ) →1 for we have used the following i − i ≈ xi 0; (53) is proven in Lemma 1 and the assumption that → Q n n 1 1 n n +1 the received SNRs are independently distributed, thus we can ( 1)k = ( 1)k interchange the integral bounds. k=0 − k k +1 n +1 k=0 − k +1 X   X   Lemma 1. For any N N 1 , ∈ \{ } − − − − (63) 2x 1 2x φ(γN ) 1 2x φ(γN ) ... φ(γ2) 1 − − − A (x)= ... 1 n+1 n +1 n +1 n +1 N = − ( 1)k + ZγN =0 ZγN−1=0 Zγ1=0 n +1 k=1 − k 0 − 0        dγ1...dγN 1dγN (54) X − Pn+1( 1)k n+1 =(1 1)n+1=0 N x k=0 − ( k ) − = ( 1) (1 2 eN ( x ln(2))) . (55) − − · − | {z } (64) N 1 yn Here, e (y)= − is the exponential sum function. 1 N n=0 n! = (65) n +1 Proof. Base case:PIf N =2, then (54) is − 2x 1 2x φ(γ2) 1 and carried out some algebraic manipulations. Equation (62) − − A2(x)= 1dγ1dγ2 corresponds to (55) for N = K +1. So, the theorem holds γ2=0 γ1=0 Z x Z for N = K + 1. Hence, by the principle of mathematical 2 1 x − 2 N = 1 dγ induction, the theorem holds for all N 1 . 1+ γ − 2 ∈ \{ } Zγ2=0  2  =2x (x ln(2) 1)+1, (56) · − APPENDIX B which is (55) for N =2. So, the theorem holds for N =2. LEMMA 2 Inductive hypothesis: Suppose the theorem holds for all values of up to some . N K 2 Lemma 2. For any N N, R > 0, and RVs Γ R , i Inductive step: Let ≥ , then (54) is c i + N = K +1 = 1, ..., N , ∈ ∈ ∀ ∈ 2x 1 N { } − A (x)= N K+1 N γ =0 1 (A1(Rc)) K+1 Pr 0 Γ A (Rc) , (66) − Z − − − i 1 N 2x φ(γK+1) 1 2x φ(γK+1) ... φ(γ2 ) 1 ≤ i=1 ≤ ≤ N! Γ¯ − −   i=1 i ... 1dγ1...dγK X γK =0 γ1=0 Rc Q1 ¯ γi ¯ Z Z where A (Rc)=2 1 and pdf f (γ )= /Γi exp( /Γi) , 1 − Γi i − AK (x φ(γK+1)) i . − ∀ ∈ N d|γK+1 {z }(57) 2x 1 x Proof. Let us define a N-fold simplex as − 2 K 1 = ( 1)K + − ( 1)K+n+1 − 1+ γ n=0 − γK+1=0  K+1 RN Z X N := (x1, ..., xN ) : xi 0, 1 n n S ∈ ≥ (x φ(γK+1)) (ln(2)) dγK+1 (58)  × n! − N x  Γ¯ixi A1(Rc) . (67) 2 1 x K 1 i=1 ≤ − K 2 − K+n+1 1 = ( 1) + ( 1) X  − 1+ γK+1 n=0 − n! ZγK+1=0  The geometric simplex volume is [28] n X n k n n k k (ln(2)) ( 1) x − φ(γK+1) dγK+1 × k=0 − k N    (59) 1 (A1(Rc)) X Vol ( N ) := dxN ...dx1 = N . (68) K 1 S ··· N! Γ¯ K x − K+n+1 1 n i=1 i = ( 1) γK+1 +2 ( 1) (ln(2)) Z N Z n=0 n! S − − Q X 13

Now, the left hand side of Lemma 2 can be rewritten as APPENDIX D LEMMA 3 N Pr 0 Γi A1(Rc) Lemma 3. For any N N 1 and x> 0, ≤ i=1 ≤ ∈ \{ }   N X N XN (x) = (A1(x)) >N! AN (x)= YN (x), (78) = Pr 0 Γ¯iXi A1(Rc) (69) · ≤ i=1 ≤   where AN (x) is given in Lemma 1. XN = exp( xi)dxN ...dx1 (70) Proof. For x =0 we have X (0) = 0 and Y (0) = 0 in (78). ··· i=1 − N N Z N Z Next, show that the slope of XN (x) is larger than the slope S Y of YN (x) for x> 0 and thus XN (x) > YN (x), x 0, N. dxN ...dx1. (71) ≥ ∀ ≤ ··· d d x N Z N Z X (x)= (2 1) S dx N dx − x x N 1 The steps can be justified as follows: in (69) and (70) we intro- = N2 ln(2) (2 h 1) − , i (79) Γi ¯ − duce the RV Xi = /Γi, with pdf fXi (xi) = exp( xi) , i d d N 1 1 − ∀ ∈ N x − n ; in (71) the exponential function can be upper bounded by YN (x)= N!( 1) 1 2 ( x ln(2)) N dx dx − − n=0 n! − exp( x) 1 for x 0; (71) is the geometric simplex volume h  (80)i − ≤ ≥ X in (68). N 1 1 = N! ln(2)2x( 1)N+1 − ( 1)n (x ln(2))n − n=0 − n! N 1 n 1 X n 1 APPENDIX C + − ( 1) (x ln(2)) − (81) n=1 − (n 1)! INVERSE FUNCTION −  X − PN 2( 1)n 1 (x ln(2))n − n=0 − n! For Rc 1, the term of the exponential sum function ≫ | x 2N {z 1 }N 1 with the highest order in (12) is dominant. Thus, we can = N! ln(2)2 ( 1) (x ln(2)) − (82) − (N 1)! approximate AN (Rc = x) y = f(x) to x N −1 ≈ = N2 ln(2) (x ln(2)) − . (83) N 1 x (N 1) (ln(2)) − x a y =2 x − =2 x b. (72) We have to show that (N 1)! x N 1 N 1 − (2 1) − > (x ln(2)) − for x> 0. (84) We can reformulate (72) as follows − Since both sides in (84) are equal for x = 0 and have the x x a y x 1 a y same exponent, it is sufficient to show 2 x = , 2 a = , b ⇔ a a b r d d x ln(2) x ln(2) ln(2) y (2x 1) > (x ln(2)) (85) exp = a , (73) dx − dx ⇔ a a a b x   r ln(2)2 > ln(2). (86) ln(2) y z exp(z)= a , (86) holds for x> 0. ⇔ a b r az a ln(2) y x = = W a , (74) APPENDIX E ⇔ ln(2) ln(2) a b  r  DERIVATION OF DIVERSITY-MULTIPLEXING TRADEOFF where W ( ) is the Lambert W function, i.e., z = Based on the outage probability analysis considered before 1 · g− (z exp(z)) = W (z exp(z)) is the inverse function of the diversity gains for JD, SC, and MRC are given by g(z) = z exp(z), for z 1. For z e, the Lambert W ¯ ¯ ≥ − ≥ ld AN (r, Γ) N ld Γ function is bounded by [26, Theorem 2.7] dJD(r)= lim − (87) − Γ¯ ld NΓ¯ →∞

ln(ln(z)) ld NΓ¯ W (z) = ln(z) ln(ln(z))+Θ . (75) = lim r
− ln(z) − Γ¯ ld NΓ¯   →∞ 
(N 1) 1 ¯ − (N 1) Finally, with (74) and (75) we can give an approximation of ld (N 1)! r ld(
NΓ) (ln(2)) − the inverse function as + − ld N
Γ¯
1 N 1 N ld Γ¯ AN− (AN )= Rc − [ln(ζ) ln(ln(ζ))] , (76)
(88) ≈ ln(2) − − ld NΓ¯
 = N r, (89) where −
N ld A1(r, Γ)¯ N ld Γ¯ N−1 dSC(r)= lim − (90) (N 1)!AN − Γ¯ ld NΓ¯ ζ = − . (77) →∞

p N 1 ld Γ¯ ld Γ¯ − = N lim r
(91) − Γ¯ ld NΓ¯ − ld NΓ¯ →∞ 



14

= N (1 r), and (92) [8] D. S. Michalopoulos, I. Viering, and L. Du, “User-plane multi- · − ¯ ¯ connectivity aspects in 5G,” in Proc. IEEE 23rd International Confer- N ld A1(r, Γ) ld (N!) N ld Γ ence on Telecommunications (ICT), Thessaloniki, Greece, May 2016, dMRC(r)= lim − − − Γ¯ ld NΓ¯ pp. 1–5. →∞

[9] A. Goldsmith, Wireless Communications. Cambridge University Press, (93)
2005. ld Γ¯ ld (N!) ld Γ¯ [10] B. Van Laethem, F. Quitin, F. Bellens, C. Oestges, and P. De Doncker, = N lim r (94) “Correlation for multi-frequency propagation in urban environments,” ¯ ¯ ¯ ¯ − Γ ld NΓ
− N ld NΓ − ld NΓ
Progress In Electromagnetics Research Letters, vol. 29, pp. 151–156, →∞   2012. = N (1 r), (95) · −


[11] D. Tse and P. Viswanath, Fundamentals of Wireless Communication. Cambridge University Press, 2005. respectively. The steps can be justified as follows: (87), (90), [12] G. Pocovi, B. Soret, M. Lauridsen, K. I. Pedersen, and P. Mogensen, and (93) are given by substituting (11), (18), (23) into (47); “Signal quality outage analysis for ultra-reliable communications in ¯ cellular networks,” in Proc. IEEE Globecom Workshops (GCW), San in (88) we use the infinite SNR properties of AN r, Γ in Diego, CA, USA, Dec. 2015, pp. 1–6. (98) and some algebraic manipulations; for infinite SNR the [13] F. Kirsten, D. hmann, M. Simsek, and G. P. Fettweis, “On the utility properties (99) and (100) hold which yields (89); in (91)
and of macro- and microdiversity for achieving high availability in wireless ¯ networks,” in Proc. IEEE International Symposium on Personal, Indoor, (94) we use the infinite SNR property of A1 r, Γ in (96) and and Mobile Radio Communications (PIMRC), Aug 2015, pp. 1723– some algebraic manipulations; for infinite SNR the properties 1728. in (99) and (100) hold which yields (92) and (95).
[14] J. J. Nielsen and P. Popovski, “Latency analysis of systems with multiple ¯ interfaces for ultra-reliable M2M communication,” in Proc. IEEE 17th Substituting (46) into (12) the constants AN (r, Γ) and International Workshop on Signal Processing Advances in Wireless its special case A1(r, Γ)¯ can be given depending on the Communications (SPAWC), Edinburgh, UK, Jul. 2016, pp. 1–6. multiplexing gain r by [15] L. Zheng and D. N. C. Tse, “Diversity and multiplexing: a fundamental tradeoff in multiple-antenna channels,” IEEE Trans. Inf. Theory, vol. 49, ¯ r ld(Γ¯) r ld(Γ¯) no. 5, pp. 1073–1096, May 2003. lim A1(r, Γ) = lim 2 1 = lim 2 , (96) [16] R. Narasimhan, “Finite-SNR diversity-multiplexing tradeoff for corre- Γ¯ Γ¯ − Γ¯ →∞ →∞ →∞ lated Rayleigh and Rician MIMO channels,” IEEE Trans. Inf. Theory, N
r ld(NΓ¯) lim AN (r, Γ)¯ = lim ( 1) +2 vol. 52, no. 9, pp. 3965–3979, Sep. 2006. Γ¯ Γ¯ − [17] B. Bai, W. Chen, K. B. Letaief, and Z. Cao, “Outage exponent: A →∞ →∞ N 1 1 n unified performance metric for parallel fading channels,” IEEE Trans. − ( 1)N+n+1 r ld(NΓ)¯ (ln(2))n (97) Inf. Theory, vol. 59, no. 3, pp. 1657–1677, March 2013. n=0 − n! [18] Z. Wang and G. B. Giannakis, “A simple and general parameterization X r ld(NΓ¯) 1
(N 1)  (N 1) quantifying performance in fading channels,” IEEE Trans. Commun., = lim 2 r ld(NΓ)¯ − (ln(2)) − , Γ¯ (N 1)! vol. 51, no. 8, pp. 1389–1398, 2003. →∞ − (98) [19] T. M. Duman and A. Ghrayeb, Coding for MIMO Communication
Systems. John Wiley & Sons, 2008. [20] N. Schwarzenberg, A. Wolf, N. Franchi, and G. Fettweis, “Quantifying where (98) can be justified with the infinite SNR properties in the gain of multi-connectivity in wireless LAN,” in Proc. IEEE European (101). Further properties for infinite SNR are: Conference on Networks and Communications (EuCNC), Ljubljana, Slovenia, Jun 2018, pp. 16–20. ld Γ¯ ld(N!) [21] E. Biglieri, J. Proakis, and S. Shamai, “Fading channels: Information- lim =1, lim =0, (99) theoretic and communications aspects,” IEEE Trans. Inf. Theory, vol. 44, Γ¯ ld NΓ¯ Γ¯ N ld NΓ¯ →∞
→∞ no. 6, pp. 2619–2692, 1998. (N 1) ld ld NΓ¯ [22] Z. Chen, J. Yuan, and B. Vucetic, “Analysis of transmit antenna lim −
=0,
(100) selection/maximal-ratio combining in Rayleigh fading channels,” IEEE Γ¯ ld NΓ¯ Trans. Veh. Technol., vol. 54, no. 4, pp. 1312–1321, 2005. →∞

(N 1) (N n) [23] I. S. Gradshteyn and I. M. Ryzhik, Table of integrals, series, and lim ld(NΓ)¯ −
lim ld(NΓ)¯ − (101) products. Academic press, 2014. Γ¯ ≫ Γ¯ [24] A. Dembo and O. Zeitouni, “Large deviations techniques and applica- →∞ →∞

tions, volume 38 of stochastic modelling and applied probability,” 2010. for n =2, ..., N. [25] M. Bibinger, “Notes on the sum and maximum of independent exponentially distributed random variables with different scale parameters,” 2013. [Online]. Available: https://arxiv.org/abs/1307.3945 REFERENCES [26] A. Hoorfar and M. Hassani, “Approximation of the Lambert W function and hyperpower function,” Research report collection, vol. 10, no. 2, [1] A. Wolf, P. Schulz, D. Ohmann,¨ M. D¨orpinghaus, and G. Fettweis, 2007. “On the gain of joint decoding for multi-connectivity,” in Proc. IEEE [27] M. Grieger, “Uplink joint detection in a realistic macro cellular envi- Globecom (GC), Singapore, Dec. 2017, pp. 1–6. ronment,” Ph.D. dissertation, TU Dresden, Germany, 2014. [2] P. Schulz, M. Matthe, H. Klessig, M. Simsek, G. Fettweis, J. Ansari, [28] R. S. Ellis, “Volume of an n-simplex by multiple integration.” Elemente S. A. Ashraf, B. Almeroth, J. Voigt, I. Riedel, A. Puschmann, der Mathematik, vol. 31, pp. 57–59, 1976. A. Mitschele-Thiel, M. Muller, T. Elste, and M. Windisch, “Latency critical IoT applications in 5G: Perspective on the design of radio interface and network architecture,” IEEE Communications Magazine, vol. 55, no. 2, pp. 70–78, Feb. 2017. [3] D. Ohmann,¨ “High reliability in wireless networks through multi- connectivity,” Ph.D. dissertation, TU Dresden, Germany, 2017. [4] A. F. Molisch, Wireless Communications. John Wiley & Sons, 2012. [5] C. Oestges and B. Clerckx, MIMO wireless communications: From real- world propagation to space-time code design. Academic Press, 2010. [6] M. Eriksson, “Dynamic single frequency networks,” IEEE Journal on Selected Areas in Communications, vol. 19, no. 10, pp. 1905–1914, Oct. 2001. [7] 3GPP J. Wannstrom. (2013, Jun.) LTE-advanced. [Online]. Available: http://www.3gpp.org/technologies/keywords-acronyms/97-lte-advanced