On Fading Channel Dependency Structures with a Positive Zero-Outage Capacity
aa r X i v : . [ c s . I T ] F e b On Fading Channel Dependency Structureswith a Positive Zero-Outage Capacity
Karl-Ludwig Besser,
Student Member, IEEE , Pin-Hsun Lin,
Member, IEEE , andEduard A. Jorswieck,
Fellow, IEEE
Abstract
With emerging technologies like 6G, many new applications like autonomous systems evolve whichhave strict demands on the reliability of data communications. In this work, we consider a system withmultiple slowly fading channels that constitute a diversity system. We show that the joint distributionof the fading coefficients has a tremendous impact on the outage performance of a communicationsystem. In particular, we investigate the zero-outage capacity (ZOC) and characterize the joint fadingdistributions with a strictly positive ZOC. Interestingly, the set of joint distributions, that lead to positiveZOCs, is larger than a singleton in general. We also derive expressions for the maximum ZOC withrespect to all possible joint distributions for selection combining (SC) as diversity combining techniqueat the receiver. For maximum ratio combining (MRC), we characterize the maximum ZOC within afinite number of bits. Finally, the results are evaluated explicitly for the special cases of Rayleigh fadingand Nakagami- m fading in order to quantify the ZOCs for common fading models. Index Terms
Copula, Joint distributions, Fading channels, Delay-limited Capacity, Outage probability.
I. I
NTRODUCTION
Current visions for emerging technologies like 6G include new applications like autonomoussystems which have extremely high reliability constraints [2], [3]. It is therefore of great interest,how these strict reliability constraints can be achieved.
Parts of this work have been presented at the 54th Asilomar Conference on Signals, Systems, and Computers, November2020 [1].The authors are with the Institute of Communications Technology, Technische Universität Braunschweig, 38106 Braunschweig,Germany (email: {k.besser, p.lin, e.jorswieck}@tu-bs.de).This work is supported in part by the German Research Foundation (DFG) under grant JO 801/23-1.
For slow-fading channels, a common performance metric is the ε -outage capacity. It is definedas the maximum transmission rate for which the outage probability is not greater than ε . Ofparticular interest is the zero-outage capacity (ZOC), i.e., the maximum transmission rate atwhich we can transmit without any outages. This will be especially relevant in the context ofultra-reliable low latency communication (URLLC) [3], [4].It is well-known that the ZOC is zero in most of the common scenarios considered in literature,e.g., single link single-input single-output (SISO) with independent Rayleigh fading [5]. How-ever, with diversity and average power constraint, a positive ZOC is possible [6]. It was shown in[7], that the ZOC can be positive for two Rayleigh fading links without channel-state information at the transmitter (CSI-T),if the two channels are negatively correlated. In [6], the authors derive bounds on the ε -outagecapacity for communication systems with n dependent fading links. It is shown that the upper-bound on the ε -outage capacity, i.e., for the best-case joint distribution, is strictly positive.This implies that there exists at least one joint distribution for which the ZOC is positivefor given arbitrary marginals. However, it remains an open question, if there exist multiplejoint distributions that achieve a positive ZOC. Additionally, it is unclear, if there exist jointdistributions for which the ZOC is positive but strictly less than the upper bound derived in [6].A positive ZOC implies that a transmission without any outages is possible which is relevant,e.g., in the context of URLLC, since no re-transmissions are necessary which allows a verylow latency. Besides, if there exists only a single joint distribution with a positive ZOC, thiswould be an unstable operation point. We are therefore also interested in other joint distributionsachieving positive ZOCs.In this work, we will answer both of these open questions. We show that there exists an infinitenumber of joint distributions, for which the ZOC is strictly between zero and the best-case upperbound. The proof is constructive in the way that we explicitly give a parameterized family ofjoint distributions that achieve positive ZOCs. Our results hold for general fading distributions,and are evaluated for the special case of Rayleigh fading and Nakagami- m fading.The basis of our derivations is copula theory [8], which allows a flexible modeling of depen-dency structures between random variables. Copulas have been used in communications beforeto model dependency between channels. One of the first works on applying this technique in thecontext of wireless communications is [9]. There, a Clayton copula is used to introduce a newfading model for Nakagami- m fading with tail dependence. In [10], [11], copulas have beenused to model dependency in multiple-input multiple-output (MIMO) channels. In [12], it was shown on measurements that real channels can show a tail dependency which can be described bycopulas. The outage probability of the two-user Rayleigh fading multiple access channel (MAC),where the two links follow a specific copula, is evaluated in [13]. Copulas have also beenused to model interference in internet of things (IoT) networks [14] and in the area of physicallayer security [15]–[17]. General bounds on the outage performance for dependent slow-fadingchannels can be found in [18].The tools from copula theory are not only helpful for analyzing the outage capacity for slowfading channels, but also for the ergodic capacities for fast fading multi-user channels [19]. Inparticular, when the multi-user channel has the same marginal property, copulas have been usedto derive capacities regions for Gaussian interference channels, Gaussian broadcast channels andthe secrecy capacity for Gaussian wiretap channels [20]–[22].However, in this work, we focus on the ZOC for n dependent slow-fading links. Our contri-butions are summarized as follows. • We show that the ZOC in a multi-connectivity setting can be strictly positive for dependentchannels with arbitrary marginal fading distributions, even when only statistical CSI-T isavailable. We consider both maximum ratio combining (MRC) and selection combining (SC)at the receiver. • We show that the ε -outage capacity in general, and the zero-outage capacity in particular,are maximized by countermonotonic channel gains in the case of two dimensions. • We show that there exists an infinite number of joint distributions, for which the ZOC isstrictly between zero and the best-case (upper) bound. The proof is constructive in the waythat we explicitly state a parameterized family of joint distributions that achieve positiveZOCs. • We provide explicit expressions for the maximum ZOC for SC at the receiver with n homogeneous links. • We derive inner and outer bounds for the maximum ZOC in the general case of n > channels with homogeneous channel gains X and MRC at the receiver. The bounds arewithin a finite gap for all n equal to log ( E [ X ]) − log (cid:0) F − X (1 / e) (cid:1) . • All results are evaluated for the special cases of Rayleigh fading and Nakagami- m fading.In addition, we provide all plots presented in this work as interactive versions in [23].The practical implications of the results are the following. If one is able to tune the dependencystructure between different channels, e.g., by using smart relays [24] or reconfigurable meta- surfaces [25], it is possible to transmit data over fading channels without any outages. The thirdresult implies a certain robustness. Assume that we are able to parameterize the dependencystructure within a desired range. Even if the designed parameters are not set perfectly, the ZOCcan still be positive, if one hits one of the infinitely many joint distributions with positive ZOC.The rest of the paper is organized as follows. In Section II, we state the system model andproblem formulation of this work. We also introduce some necessary mathematical backgroundfrom copula theory. Some general observation and results are given in Section III. The twodiversity combining techniques MRC and SC are investigated in detail in Sections IV and V,respectively. Finally, Section VI concludes the paper. Notation:
Throughout this work, we use the following notation. Random variables are denotedin capital boldface letters, e.g., X , and their realizations in small letters, e.g., x . We will use F and f for a probability distribution and its density, respectively. The expectation is denotedby E and the probability of an event by Pr . It is assumed that all considered distributions arecontinuous. The uniform distribution on the interval [ a, b ] is denoted as U [ a, b ] . The normaldistribution with mean µ and variance σ is denoted as N ( µ, σ ) . The derivative of a function f is denoted by f ′ . As a shorthand, we use [ x ] + = max [ x, . The real numbers, non-negative realnumbers, and extended real numbers are denoted by R , R + , and ¯ R , respectively. Logarithms, ifnot stated otherwise, are assumed to be with respect to the natural base.II. P RELIMINARIES AND S YSTEM M ODEL
We consider a slow fading channel with n receive antennas. The symbol M is transmitted witha rate R over the slow fading links H i , i = 1 , . . . , n . The received signal Y = ( Y , . . . , Y n ) ∈ C n is given as Y i = H i M + N i , (1)where N i are independent complex Gaussian noise terms with zero mean and variance σ i . Thetransmit signal-to-noise ratios (SNRs) of the individual channels are given as ρ i = P/σ i , where P is the average transmit power constraint.We assume that the receiver has perfect channel-state information (CSI) while the transmitteronly has statistical CSI. The receiver applies the diversity combining strategy L : R n + → R .In this work, we consider strategies in the form L ( X , . . . , X n ) with X i = ρ i | H i | . In thiscase, an outage occurs, if the instantaneous channel capacity is less than the rate R used for the transmission, i.e., log (1 + L ( X , . . . , X n )) < R . The maximum rate R ε for which theprobability of such an outage is less than ε is called ε -outage capacity and defined as [5] R ε = sup R ≥ n R | Pr (cid:16) log (1 + L ( X , . . . , X n )) < R (cid:17) ≤ ε o . (2)This can be reformulated as the (receive) SNR optimization s ⋆ ( ε ) = sup s ≥ { s | Pr ( L ( X , . . . , X n ) < s ) ≤ ε } (3)with R ε = log (1 + s ⋆ ( ε )) . A. Problem Formulation
We consider a communication scenario with n slow-fading channels which can be dependent.The receiver has perfect CSI, while we only assume statistical CSI-T. The quantity of interestis the zero-outage capacity R , which is given as the maximum rate fulfilling Pr (cid:16) L ( X , . . . , X n ) < R − (cid:17) = 0 . (4)The main question that we will answer in this work is: Given marginal distributions of the indi-vidual wireless fading channels, do there exist joint distributions (and how many) which achievepositive ZOCs and what ZOCs can be achieved by different diversity combining techniques?
Asa consequence, this also includes the maximum ZOC with respect to all joint distributions withthe given marginals.By the construction in [7], we know that there exists at least one joint distribution with positivezero-outage capacity. However, a singleton is an unstable operating point and it is therefore ofinterest, how robust a positive ZOC is with respect to the joint distribution. In addition, we areinterested in the maximum ZOC since it shows what the best case performance can be.
B. Mathematical Background
To describe and analyze the structure of joint distributions, we will use tools from copulatheory [8], which we introduce in the following.
Definition 1 (Copula) . A copula is an n -dimensional distribution function with standard uniformmarginals.The practical relevance of copulas stems from Sklar’s theorem, which we restate in thefollowing Theorem 1. Theorem 1 (Sklar’s Theorem [8, Thm. 2.10.9]) . Let H be an n -dimensional distribution functionwith margins F , . . . , F n . Then there exists a copula C such that for all x ∈ ¯ R n , H ( x , . . . , x n ) = C ( F ( x ) , . . . , F n ( x n )) . (5) If F , . . . , F n are all continuous, then C is unique. Conversely, if C is a copula and F , . . . , F n are distribution functions, then H defined by (5) is an n -dimensional distribution function withmargins F , . . . , F n . This theorem implies that copulas can be used to describe dependency structures betweenrandom variables, regardless of their marginal distributions. This allows us to separate thedependency structure (described by the copula C ) from the marginal distributions F , . . . , F n .We will see in the following section that the ZOC depends on the underlying copula between thechannel gains. The results in this work are based on the Fréchet-Hoeffding bounds for copulas,which we state in the following theorem. Theorem 2 (Fréchet-Hoeffding Bounds [8, Thm. 2.10.12]) . Let C be a copula. Then for every u ∈ [0 , n W ( u ) ≤ C ( u ) ≤ M ( u ) (6) with W ( u ) = max { u + · · · + u n − n + 1 , } , (7) M ( u ) = min { u , . . . , u n } . (8)In the case that n = 2 , W is a copula and two random variables whose joint distributionfollows the copula W are called countermonotonic. The upper bound M is a copula for all n and random variables that follow M are called comonotonic [8].III. G ENERAL C ONSIDERATIONS AND R ESULTS
In this section, we will make some general observations and derivations about the consideredproblem. Throughout this work, we will assume that the diversity combining function L onlydepends on the channel gains X i = ρ i | H i | and is non-decreasing in each variable.The underlying observation for our derivations in this work is the following. As stated inSection II, we know that the outage probability corresponds to the probability of the event BS MRC S SC s ⋆ SC s ⋆ MRC s ⋆ SC s ⋆ MRC x x BS MRC S SC Figure 1. Areas corresponding to the ZOC. Area B shows the area where F X , X = 0 . Area S MRC shows the integration areafrom (10) to calculate the outage probability when MRC is used at the receiver. The value s ⋆ MRC denotes the maximum value of s such that S MRC is still a subset of B . The analogue for SC at the receiver is denoted by the index “SC”. L ( X , . . . , X n ) < s . An equivalent way of expressing this is via the integral of the jointdistribution F X ,..., X n over the area S = (cid:8) ( x , . . . , x n ) ∈ R n + | L ( x , . . . , x n ) < s (cid:9) . (9)Then the outage probability, given R , can be written as ε = Z S d C ( F X ( x ) , . . . , F X n ( x n )) , (10)where we use the copula representation of the joint distribution based on Theorem 1.Recall that our goal is to have zero outages, i.e., ε = 0 . With reference to (10), this is achievedif the probability of the joint distribution is zero in S . The joint distribution is zero if and onlyif its copula C is zero, and the corresponding area of ( X , . . . , X n ) can be written as B = { ( x , . . . , x n ) | C ( F X ( x ) , . . . , F X n ( x n )) = 0 } . (11)In other words, S is not inside the support of F X ,..., X n . This idea is exemplarily shown in Fig. 1for the two diversity schemes MRC and SC. Detailed explanations and results for both diversityschemes will be given in the following sections. A. Two-Dimensional Case
First, we show that the general ε -outage capacity R ε is upper bounded by countermonotonicchannels for n = 2 . Lemma 3 (Maximum ε -outage capacity) . The maximum ε -outage capacity for two links withchannel gains X and X is attained by countermonotonic random variables, i.e., for F X , X ( x, y ) = W ( F X ( x ) , F X ( x )) . Proof.
The proof can be found in Appendix A.From this, it immediately follows, that also the ZOC is upper bounded by the case ofcountermonotonic channels.
Corollary 4 (Maximum ZOC) . The maximum ZOC for two links with channel gains X and X is attained by countermonotonic random variables.Remark . It is well-known that the ZOC can be zero, e.g., for channels withindependent links [5]. Since the capacity is a non-negative quantity, we can conclude that zerois the lower bound on the ZOC.
B. General Case
The extension to the general n -dimensional case with n > is not straightforward, since W is only a valid copula for n = 2 . However, we can reformulate the problem in the generalcase using the discussions from above. Our goal is to find the maximum value of R in (2) (orequivalently, s ∗ in (3)), such that S is still a subset of B . Note that the boundary of B is notnecessarily convex. We therefore rewrite the optimization problem (3) as s ⋆ = max ( x ,...,x n ) ∈B L ( x , . . . , x n ) . Since we know from the monotonicity of L that the maximum will be on the boundary of B defined by B , s ⋆ can also be written as minimizing the function L over the boundary B as s ⋆ = min ( x ,...,x n ) L ( x , . . . , x n ) s. t. B ( x , . . . , x n ) = 0 . (12)The structure of the boundary function B is determined by the underlying joint distribution andwe will give particular examples in the following sections.IV. M AXIMUM R ATIO C OMBINING
First, we will investigate the case that the receiver applies MRC as the diversity combiningtechnique. In this case, the combination function L is the sum of the channel gains X i [5] L MRC ( X , . . . , X n ) = n X i =1 X i . In order to introduce the basic concepts, we will start with the two-dimensional case and thenextend the results to the general n -dimensional case. The results will be illustrated with twoexamples, namely Rayleigh fading and Nakagami- m fading. A. Two-Dimensional Case
We start with the two-dimensional case n = 2 . From Corollary 4, we know that the maximumZOC is attained for countermonotonic ( X , X ) . In the following theorem, we show that foreach value c between zero and the maximum ZOC, there exists a joint distribution for whichthe ZOC is equal to c , i.e., the ZOC is continuous with respect to the joint distribution. Theorem 5 (Zero-Outage Capacities for Two Links with MRC) . Let X and X be non-negativecontinuous random variables representing the channel gains of two communication links. Thereceiver applies MRC as diversity combining technique. Then there exist joint distributions of X and X with the following zero-outage capacities for t ∈ [0 , R ( t ) = log (cid:0) (cid:8) F − X ( t ) , F − X ( t ) , x ⋆ + B t ( x ⋆ ) (cid:9)(cid:1) (13) with B t ( x ) = F − X ( t − F X ( x )) (14) and x ⋆ = arg min x ≥ (cid:8) x + B t ( x ) | f X ( x ) = f X (cid:0) F − X ( t − F X ( x )) (cid:1)(cid:9) . (15) Proof.
The proof can be found in Appendix B.The parameter t characterizes the joint distribution of the two fading links. For t = 0 and t = 1 , X and X are comonotonic and countermonotonic, respectively. The maximum ZOC istherefore achieved for t = 1 . Remark . In the proof of Theorem 5, we used thecopula C t from [26, Eq. (3.8)]. However, the only property we actually need for the proof isthe fact that C t ( a, b ) = 0 for a + b ≤ t . Thus, any copula, that has this property, achieves thesame result. Another example having this property is a generalization of the circular copula [8,Eq. (3.1.5)] C t, circ ( a, b ) = M ( a, b ) if | a − b | > tW ( a, b ) if | a + b − | < − t a + b − t otherwise , which is illustrated in Fig. 8b in Appendix B. There are also families of absolutely continuouscopulas whose support is not the full unit square. An example is the Clayton copula given by [8,Eq. (4.2.1)] C θ, clay ( a, b ) = (cid:0) max (cid:2) a − θ + b − θ − , (cid:3)(cid:1) − θ , θ ∈ [ − , ∞ ) \ { } , for negative values of the parameter θ . B. General Case
From Theorem 5, it can be seen that there exists an infinite number of joint fading distributionsthat achieve a positive ZOC. This answers our question from Section II-A and shows that the setof joint distributions with a positive ZOC is not a singleton. In the following, we will thereforeonly focus on the maximum
ZOC with respect to the set of joint distributions.As shown in Corollary 4, for the two-dimensional case, the maximum is achieved for counter-monotonic channel gains, i.e., X and X follow the W -copula, cf. Theorem 2. Unfortunately,the extension to the general n -dimensional case is not straightforward since the Fréchet-Hoeffdinglower bound W is not a copula anymore for n > . The most general case of arbitrary marginaldistributions F X , . . . , F X n with n > , therefore, remains an open problem. However, we willderive new results for the n -dimensional case under the following assumptions. We only considerthe homogeneous case, i.e., all marginal distributions are the same, F X = · · · = F X n = F X . Inaddition, the distribution function F X fulfills the following definition. Definition 2 ( B -SYM Distribution) . Let X , . . . , X n be non-negative continuous random vari-ables with distribution function F X = . . . = F X n = F X with finite first moment. Thedistribution function F X is a B -SYM distribution, if the solution to the optimization problem s ⋆ = min n X i =1 x i s. t. B ( x , . . . , x n ) = 0 (16)lies on the identity line.A characterization of some B -SYM distribution functions for two relevant B is given in thefollowing lemmas. Lemma 6.
For B ( x , . . . , x n ) = P ni =1 F X ( x i ) − n + 1 , a continuous distributions F X with astrictly quasi-concave density f X is a B -SYM distribution, if the following sufficient conditionholds f ′ X (cid:18) F − X (cid:18) − n (cid:19)(cid:19) < . (17) Proof.
The proof can be found in Appendix C.
Lemma 7.
For B ( x , . . . , x n ) = P ni =1 ( F X ( x i )) n − − n + 1 , a continuous distribution F X witha strictly quasi-concave density f X is a B -SYM distribution, if the following two sufficientconditions hold. The function g ( x ) = 1 n − F X ( x ) − nn − f X ( x ) (18) is strictly quasi-concave, and f ′ X ( x ⋆ )( f X ( x ⋆ )) < n − n − (cid:18) − n (cid:19) − n (19) with x ⋆ = F − X (cid:18) − n (cid:19) n − ! . (20) Proof.
The proof can be found in Appendix D.
Remark . A stricter condition than (19) is f ′ X (cid:0) F − X (e − ) (cid:1) < . Based on the unimodality of f X , this can also be written as F X ( mode ) < e − ≈ . , where “mode” refers to the mode ofdistribution F X . Similarly, we can rewrite (17) as F X ( mode ) < − n . Since these conditionsare stricter than those from the above lemmas, they are also sufficient to characterize a B -SYMdistribution. Remark . Most of the common fading distributions fulfill thesufficient conditions from Lemmas 6 and 7. In the case of Rayleigh fading, the channel gains X i are exponentially distributed and therefore have a monotone density, i.e., f ′ ( x ) < for all x . Other common distributions like log-normal, Gamma, and χ also fulfill the conditions. Anexample of a distribution that is not B -SYM for the considered B , is a Weibull distribution withscale parameter λ = 1 and shape parameter k = 6 for n = 2 . For this example, f ′ ( F − (0 . ≈ . > . The mentioned examples are also illustrated in [23].
1) Outer Bounds on the Maximum ZOC:
First, we will give two outer bounds on the maximumZOC. The first one is based on the Fréchet-Hoeffding lower bound W . Even though it is not avalid copula for n > , the bound from Theorem 2 still holds and can therefore be used as aloose upper bound on the actual maximum ZOC. Theorem 8 (Outer Bound on the Maximum MRC-ZOC for n Homogeneous Links based on W ) . Let X , . . . , X n be n non-negative continuous random variables representing the channelgains of n communication channels. All X i follow the same distribution, i.e., F X i = F X for i = 1 , . . . , n , which is a B -SYM distribution for B ( x , . . . , x n ) = P ni =1 F X ( x i ) − n + 1 . Themaximum zero-outage capacity R for MRC at the receiver is then upper bounded by R n = log (cid:18) nF − X (cid:18) − n (cid:19)(cid:19) . (21) Proof.
The proof can be found in Appendix E.As already mentioned, W is not a copula in the case of n > . The outer bound fromTheorem 8 will therefore be loose. In the following examples in Section IV-C and IV-D, we willsee that the gap to the actual maximum ZOC can grow arbitrarily large for large n . A betterouter bound can be obtained by using joint mixability [27]. Based on [28, Thm. 2.6], a generalbound on the best-case ε -outage capacity is derived in [6, Thm. 7]. We will use this to derivean outer bound on the maximum ZOC in the following corollary. Corollary 9 (Outer Bound on the Maximum MRC-ZOC for n Homogeneous Links based on[6, Thm. 7]) . Let X , . . . , X n be n non-negative continuous random variables representing thechannel gains of n communication channels. All X i follow the same distribution, i.e., F X i = F X for i = 1 , . . . , n . The maximum zero-outage capacity R for MRC at the receiver is then upperbounded by R n = log (1 + n E [ X ]) . (22) Proof.
The proof directly follows from [6, Thm. 7]. We only need to set ε = 0 and obtain(22).As stated in [28], the bound is tight, if the distribution F X is n -completely mixable. Unfor-tunately, as shown in [27, Rem. 2.2], distributions with a one-sided unbounded support can notbe completely mixable. For details on this matter, we refer the readers to [27], [29]. However,as shown in [6], the bound from Corollary 9 might come arbitrarily close to the exact value in some special cases when n → ∞ . We will also observe this behavior for the Rayleigh fadingexample in Section IV-C.
2) Inner Bound on the Maximum ZOC:
Next, we derive an inner bound on the maximumZOC. It is based on the Archimedean copula stated in [30, Prop. 4.6]. Archimedean copulasare a popular class of single-parameter copulas for an arbitrary number of dimensions n . Thesimple construction for an arbitrary dimension n > is one of the reasons for their popularity [8,Chap. 4]. In the following, we need this extension to n > and since it is a valid copula, thederived value is achievable and we obtain an inner bound. We use this particular copula sinceit is a lower bound on all Archimedean copulas [30, Prop. 4.6].Again, to obtain closed-form solutions, we assume that F X is a B -SYM-distribution. However,with the mentioned copula, an inner bound can also be obtained for arbitrary fading distributions. Theorem 10 (Inner Bound on the Maximum MRC-ZOC for n Homogeneous Links) . Let X , . . . , X n be n non-negative continuous random variables representing the channel gains of n communica-tion channels. All X i follow the same distribution, i.e., F X i = F X for i = 1 , . . . , n , which is a B -SYM distribution for B ( x , . . . , x n ) = P ni =1 ( F X ( x i )) n − − n + 1 . The maximum zero-outagecapacity R for MRC at the receiver is then lower bounded by R n = log nF − X (cid:18) − n (cid:19) n − !! . (23) Proof.
The proof can be found in Appendix F.
3) Gap Between Inner and Outer Bound:
For distributions that are B -SYM distributions forthe boundary function in Theorem 10, we know that the exact value of the maximum ZOCis between the outer bound given in Corollary 9 and the inner bound from Theorem 10. It istherefore of interest to analyze the gap between the bounds. This will be summarized in thefollowing corollary. Corollary 11 (Maximum Gap between Inner and Outer Bound on the Maximum MRC-ZOC) . Let X , . . . , X n be n non-negative continuous random variables representing the channel gains of n communication channels. All X i follow the same distribution, i.e., F X i = F X for i = 1 , . . . , n ,which is B -SYM for the function B from Theorem 10. The gap between the inner bound on themaximum ZOC from Theorem 10 and the outer bound from Corollary 9 is at most R n − R n ≤ log (cid:18) E [ X ] F − X (e − ) (cid:19) . (24) Proof.
The proof can be found in Appendix G.Corollary 11, therefore, allows us to calculate the maximum ZOC for n homogeneous fad-ing links and MRC at the receiver within a finite amount of bits equal to log ( E [ X ]) − log (cid:0) F − X (1 / e) (cid:1) . C. Example: Rayleigh Fading
In the following, we will illustrate the general results with the example of Rayleigh fading. Inthis case, all channel gains | H i | are exponentially distributed with mean . This gives ρ i | H i | = X i ∼ exp(1 /ρ i ) . The cumulative distribution function (CDF) and inverse CDF of X i ∼ exp( λ i ) are given by F X i ( x ) = if x < − exp ( − λ i x ) if x ≥ and F − X i ( u ) = − log(1 − u ) λ i if ≤ u < ∞ if u = 1 , respectively. Note that the expected value of X i is /λ i = ρ i .
1) Two Links:
First, we will take a look at the two-dimensional case. In order to do this, wewill evaluate (13) from Theorem 5 in the following.We start with determining x ⋆ according to (15). This gives the following expression x ⋆ ( t ) = − λ log (cid:18) λ λ + λ (2 − t ) (cid:19) . Note that the range of x ⋆ is bounded by and F − X ( t ) . The boundary of B is computed accordingto (14) as B t ( x ) = − log (2 − t − exp( − λ x )) λ , and therefore, we get B t ( x ⋆ ) = − log (cid:16) (2 − t ) λ λ + λ (cid:17) λ , (25)when < x ⋆ < F − X ( t ) . For the extreme cases, we find that x ⋆ + B ( x ⋆ ) = F − X ( t ) if x ⋆ = 0 F − X ( t ) if x ⋆ = F − X ( t ) . Thus, we can combine the above results according to (13) to get the expression of the ZOC fortwo Rayleigh fading links as R ( t ) = log (cid:18) x ⋆ ( t ) − log (2 − t − exp( − λ x ⋆ ( t ))) λ (cid:19) (26) . . . . . . . . . . . . . . . . . . x x B t ( x ) for t = 0 . x + x = F − X (0 . x + x = F − X (0 . B t ( x ) for t = 0 . x + x = F − X (0 . x + x = x ⋆ + B ( x ⋆ ) Figure 2. Boundary B t ( x ) for Rayleigh fading channels with SNRs ρ = 0 dB and ρ = − for the values t = 0 . and t = 0 . . with x ⋆ ( t ) = (cid:20) min (cid:26) − λ log (cid:18) λ λ + λ (2 − t ) (cid:19) , F − X ( t ) (cid:27)(cid:21) + . (27)Figure 2 shows examples for the function B t and the possible candidates for s = 2 R − .Recall that the idea is to find the line x + x = s with the maximum s such that the line isstill below B t . The different curves are shown for two values of t , namely t = 0 . and t = 0 . .It can be seen that the values F − X ( t ) and F − X ( t ) increase with increasing t . These are the firstcandidates for the optimal R from (13). They are represented by the lines x + x = F − X ( t ) and x + x = F − X ( t ) , respectively. In the case of t = 0 . , the optimal s is given by F − X (0 . since there is no other line of the form x + x = s with a larger s which is still below B t ( x ) .In contrast, the line x + x = F − X ( t ) is not below B t . For the larger value of t = 0 . , thereexists a tangent point at around x ⋆ = 0 . which gives the maximum s = 2 R − of around . .This can then be used to determine the ZOC R .The ZOC R from (26) is shown for different values of ρ and ρ in Fig. 3a. As expected, theZOC increases for increasing SNR values ρ i , since the channel quality increases. An interestingphenomenon can be seen from the asymmetric constellations of ρ and ρ . Especially, if thereis a big difference between them, e.g., ρ = − and ρ = 10 dB , the ZOC is low and growsonly slowly for small t . However, for larger t , the rate of growth increases. The reason for thisis that R is only determined by the weaker channel for small t . This can easily be seen whencomparing the case of ρ = − and ρ = 10 dB with the case of ρ = − and ρ = 5 dB .For t up to around . , both cases achieve the same ZOC R , since their weaker channel hasthe same SNR of − . However, for t > . , the constellation with the better second channel, . . . . Copula Parameter t R ( t ) ρ = ρ = 0 dB ρ = ρ = 5 dB ρ = 0 dB , ρ = 5 dB ρ = − , ρ = 5 dB ρ = − , ρ = 10 dB (a) ZOC for different copula parameters t . . . . . . . . . . . . . . . . . − − − − SNR ρ [dB] S N R ρ [ d B ] (b) The copula parameter is set to t = 0 . . The marked pointscorrespond to the SNR combinations shown in Fig. 3a.Figure 3. Achievable zero-outage capacities R for two dependent Rayleigh fading channels with different SNR values ρ and ρ . The two channels follow the copula C t . i.e., higher ρ , is able to achieve a larger R . The same behavior can be seen for the cases of ρ = ρ = 0 dB and ρ = 0 dB , ρ = 5 dB . The only difference is that the value of t at whichthe curves start to differ is lower, at around . . This can also be seen from Fig. 3b, where theZOCs of combinations of ρ and ρ are shown for t = 0 . . All of the presented figures arealso available as interactive versions at [23]. We encourage the interested readers to change theparameters on their own and explore the behavior of the presented results.The maximum ZOC is achieved for countermonotonic X and X . This corresponds to avalue of t = 1 . The joint distribution F X , Y in this case is supported on the line x = F − X (1 − F X ( x )) = − log (1 − exp( − λ x )) λ . From (27), we get x ⋆ (1) = − (cid:16) log λ λ + λ (cid:17) /λ and together with (26) this yields the maximumZOC R (1) = log λ + λ λ λ + log λ + λ λ λ ! . (28)In the case that λ = λ = 1 , this evaluates to R (1) = log (1 + 2 log 2) . The result for thisspecial case was also derived in [7, Thm. 1] and [6, Ex. 3] by different approaches.
2) General Case:
For the extension to the n -dimensional case, we now assume homogeneouslinks, i.e., F X = · · · = F X n = F X . We can then apply the results from Section IV-B to find . . . . . . . . Number of Links n M a x i m u m Z O C R n Exact Values from [6]Outer Bound from Theorem 8Outer Bound from Corollary 9Inner Bound from Theorem 10
Figure 4. Exact values and bounds on the maximum ZOC for n Rayleigh fading links with ρ = 0 dB . bounds on the maximum ZOC. In addition, if F X has a strictly monotone density, the exactvalues for the maximum ZOC are derived in [6]. An example with this property is Rayleighfading, since the channel gains X i are exponentially distributed. In the following example, we cantherefore compare the exact values from [6] to the bounds from Theorems 8, 10, and Corollary 9for Rayleigh fading. The results are shown in Fig. 4.The first upper bound R n , based on the Fréchet-Hoeffding lower bound W , is evaluatedaccording to (21). The second upper bound from Corollary 9 is calculated to R n = log (1 + nρ ) . (29)As shown in [6], the exact value approaches R n for n → ∞ in the case of Rayleigh fading. Theinner bound from Theorem 10, is evaluated to R n = log − ρn log − (cid:18) − n (cid:19) n − !! . (30)The exact value of the ZOC R n is between the inner and outer bound, i.e., R n ≤ R n ≤ R n . Asdescribed in Corollary 11, the gap between the bounds is upper bounded by lim n →∞ R n − R n = − log (1 − log(e − ≈ .
12 bit . (31)We want to emphasize that the exact values derived in [6] only hold for monotonic densitiesof the channel gains X i . It can therefore not be used for all types of fading, e.g., it can not beused for log-normal fading. Therefore the upper and lower bounds derived in the present workare useful and more general. . . . . . . . . . Copula Parameter t R ( t ) ρ = ρ = 0 dB ρ = ρ = 5 dB ρ = 0 dB , ρ = 5 dB ρ = − , ρ = 5 dB ρ = − , ρ = 10 dB Figure 5. Achievable zero-outage capacities R for two dependent Nakagami- m fading channels with different SNR values ρ and ρ and m = 5 . The two channels follow the copula C t defined in Remark 2. D. Example: Nakagami- m Fading
We will now give an example of Nakagami- m fading. In this case, | H i | is distributed accordingto a Nakagami- m distribution, i.e., | H i | ∼ Nakagami ( m, [31]. Therefore, the channel gains X i = ρ i | H i | are distributed according to a Gamma-distribution, i.e., X i ∼ Γ( m, ρ i /m ) .The ZOCs are calculated numerically according to Theorem 5. The source code of thecalculations together with interactive versions of the presented figures can be found in [23].Figure 5 shows R ( t ) for two Nakagami- m fading channels with m = 5 for different SNRcombinations. As expected, the achievable ZOC increases with increasing SNRs. For asymmetricSNR values ρ and ρ , we can observe a similar behavior compared to the case of Rayleighfading discussed in the previous section. Up to a certain value of t , the achievable ZOC is onlydetermined by the worse channel. In Fig. 5 this can be seen, e.g., for ρ = 0 dB and ρ = 0 dB or . Up to t around . , both constellations achieve the same ZOC. Above this value, thescenario with the higher ρ achieves higher R . We encourage the interested readers to explorethis behavior in the interactive version in [23] with further parameter constellations.For the homogeneous case with n > , we select the parameters m = 5 and ρ = 0 dB . TheGamma distribution with these parameters fulfills the inequality that the mode is less than themedian, mode = m − m ρ = 0 . < median = 0 . , and we can therefore apply Theorem 8. It is also straightforward to confirm that the distributionfulfills the condition from Lemma 7, which allows us to use Theorem 10. The results for this . . . . . . . . n M a x i m u m R n Outer Bound from Theorem 8Outer Bound from Corollary 9Inner Bound from Theorem 10
Figure 6. Inner and outer bounds on the maximum ZOC for n Nakagami- m fading links with m = 5 and ρ = 0 dB . example are shown in Fig. 6. Similar to the case of Rayleigh fading, the gap between the innerbound R n and the outer bound R n from Theorem 8 grows indefinitely with increasing n . Onthe other hand, the gap between between the inner bound R n and the outer bound R n fromCorollary 9 is, according to Corollary 11, always less than around .
328 bit . E. Summary
In this section, we showed that there exists an infinite number of joint fading distributions forwhich the ZOC is positive, if the receiver employs MRC. For the two-dimensional scenario, wederived a general expression for the maximum ZOC, which can also be applied to heterogeneousmarginals. For homogeneous marginals, we extended the results to the case of n > under someconstraints and provide upper and lower bounds on the maximum ZOC which admit a finite gap.An overview of the results can be found in Table I. The most general scenario of n heterogeneousmarginals remains an open problem at this point. Table IM
AXIMUM
ZOC
FOR M AXIMUM R ATIO C OMBINING L MRC ( X , . . . , X n ) = P ni =1 X i Homogeneous Links Heterogeneous Links B -SYM Distribution General n = log (1 + 2 median( X )) See Heterogeneous Links log (cid:0) x ⋆ + F − X (1 − F X ( x ⋆ )) (cid:1) n > R n − R n ≤ log E [ X ] F − X (cid:0) (cid:1) ! Open Problem Open Problem V. S
ELECTION C OMBINING
In this section, we will consider SC as another diversity combining scheme. In this case, onlythe strongest link is selected at the receiver [32]. The combining function L is therefore givenas L SC ( X , . . . , X n ) = max { X , . . . , X n } . Analogue to Section IV, we start with the two-dimensional case and extend it to the general n -dimensional case. We also evaluate the results for the examples of Rayleigh fading and Nakagami- m fading. A. Two-Dimensional Case
Similar to MRC, the maximum ZOC for SC with two dimensions is achieved for counter-monotonic X and X [26]. The optimization problem (12) can therefore be written as min F X ( x )+ F X ( x )=1 max { x , x } . (32)With the substitutions x = F − X ( p ) and x = F − X (1 − p ) , we rewrite the problem as min p ∈ [0 , max (cid:8) F − X ( p ) , F − X (1 − p ) (cid:9) . (33)Recall that the quantile function F − X is an increasing function. Combined with the assumptionthat all X i are supported on the non-negative real numbers, we can derive that the minimum isattained at p ⋆ for which F − X ( p ⋆ ) = F − X (1 − p ⋆ ) (34)holds. The maximum ZOC is then given as R = log (cid:0) F − X ( p ⋆ ) (cid:1) . (35)In the case of homogeneous marginals, i.e., F X = F X = F X , we get p ⋆ = 0 . . The ZOCfrom (35) can then be simplified to R = log (1 + median( X )) . B. General Case
For the extension to the n -dimensional case, first recall the general problem formulationfrom (3). We are interested in the case that the outage probability is zero, i.e., Pr( L ( X , . . . , X n )
We will now evaluate the results for SC at the receiver for Rayleigh fading and Nakagami- m fading. The first example is for two heterogeneous links. The first link is Rayleigh fading,i.e., X ∼ exp(1 /ρ ) , while the second is Nakagami- m fading, i.e., X ∼ Γ( m, ρ /m ) . Theparameters are set to m = 5 and ρ = ρ = 10 dB . The maximum ZOC is evaluated accordingto (35), which yields p ⋆ = 0 . , s ⋆ SC = 8 . , and R = 3 . .Next, we consider the homogeneous case with n ≥ . In this case, the maximum ZOC isgiven by (37). For Rayleigh fading, we get R n = log (1 + ρ log n ) , (38)and for Nakagami- m fading, we get R n = log (cid:18) ρm P − (cid:18) m, − n (cid:19)(cid:19) , (39)where P − ( a, z ) is the inverse of the regularized lower incomplete Gamma function P ( a, z ) [34,Eq. 6.5.1]. The maximum ZOCs for these fading types are shown in Fig. 7. First, recall thatRayleigh fading is a special case of Nakagami- m fading. It is achieved by setting m = 1 . FromFig. 7, it can be seen that the maximum ZOC increases for all fading distributions with increasing n . However, the maximum ZOC increases slower for higher m . This is due to the shape of theGamma-distribution. For increasing m , the probability mass in the upper tail gets smaller. The Number of Links n M a x i m u m Z O C R n Rayleigh FadingNakagami- m Fading – m = 2 Nakagami- m Fading – m = 5 Nakagami- m Fading – m = 10 Figure 7. Maximum ZOC for n links with Rayleigh and Nakagami- m fading with SNR ρ = 10 dB and SC at the receiver. changes in the quantile function F − X are therefore very small, if we look at probabilities closeto one, which is the case for − /n with increasing n . Therefore the changes in the maximumZOC are also small when m is high. However, it should be emphasized that the maximum ZOCstill goes to infinity for n → ∞ . On the other hand, the median increases with increasing m andthe maximum ZOC therefore also increases with m for n = 2 . D. Summary
In this section, we derived expressions for the maximum ZOC, if the receiver employs SC.An overview of the results can be found in Table II. For homogeneous marginals, we derivedthe maximum ZOC for all n ≥ . For heterogeneous marginal distributions, we give a generalexpression for the two-dimensional case. The most general scenario of n heterogeneous marginalsremains an open problem at this moment. Table IIM
AXIMUM
ZOC
FOR S ELECTION C OMBINING L SC ( X , . . . , X n ) = max( X , . . . , X n ) Homogeneous Links Heterogeneous Links n = log (cid:18) F − X (cid:18) − n (cid:19)(cid:19) log (cid:0) F − X ( p ⋆ ) (cid:1) n > Open Problem VI. C
ONCLUSION
In this work, we investigated the ZOC for fading channels with a dependency structure. Inter-estingly, there exist joint distributions for which the ZOC is strictly positive, without requiringperfect CSI-T, i.e., without power control and only with an average power constraint. First, wegive a parameterized description of a dependency structure in form of a copula that achievespositive ZOCs. This shows that the set of joint distribution, for which the ZOC is positive, isnot a singleton. Next, we investigated the maximum ZOC over all joint distributions with givenmarginals. In the homogeneous case, i.e., all marginal distributions are the same, we provide anexplicit expression for the maximum ZOC when SC is used at the receiver as diversity combiningtechnique. For MRC at the receiver, we derive upper and lower bounds on the maximum ZOC.The gap between these bounds is finite for marginal distributions that fulfill certain requirements.For heterogeneous marginals, we describe the solution in the case of n = 2 . The general n -dimensional case with heterogeneous marginal distributions remains an open problem at thismoment.This work gives a theoretical analysis and shows how dependency control among randomvariables can enable ultra-reliable communications, which is of particular interest for URLLC.In future work, it will be necessary to address ways of implementing active dependency controlin real communication systems. A key technology enabling this might be smart relaying [24] orreconfigurable intelligent surfaces (RISs) [25].A PPENDIX AP ROOF OF L EMMA s , we can lower bound (10) as follows: ε = Z S d C ( F X ( x ) , F X ( x )) ( a ) = lim max i | I i |→ n X i =1 | C ( F X ( I i,x ) , F X ( I i,y )) | ( b ) ≥ lim max i | I i |→ n X i =1 | W ( F X ( I i,x ) , F X ( I i,y )) | ( c ) = Z S d W ( F X ( x ) , F X ( x )) , a + b − t a b t t a b (a) Copula from [26, Eq. (3.8)] a + b − a ba + b − t t − t t − t a b (b) Generalized Circular CopulaFigure 8. Two different copulas with parameter t that allow to achieve a positive ZOC. where (a) is from the fact that Riemann-Stieltjes integral can be obtained from a limit of aRiemann-Stieltjes sum, where I i = [ X i , X i ] × [ Y i , Y i ] ∈ R with S = S ni =1 I i and the area | I i | = ( y i − y i )( x i − x i ) ; (b) is from the Fréchet-Hoeffding bounds in Theorem 2, i.e., W is thelower bound of all copulas; and (c) follows from reversely using the Riemann-Stieltjes integralform in (a). Note that Pr ( L ( X , X ) < s ) is a monotonic increasing function with respect to s . If we set R S d W ( F X ( x ) , F X ( x )) = ε , the corresponding s will be the maximum amongthose calculated from all copulas, which completes the proof.A PPENDIX BP ROOF OF T HEOREM X and X be determined by the copula C t ( a, b ) = max [ a + b − t, if ( a, b ) ∈ [0 , t ] M ( a, b ) otherwise (40)with t ∈ [0 , . The copula is also shown in Fig. 8a. For t = 0 and t = 1 , C t is equal to theFréchet-Hoeffding upper and lower bounds, respectively. Observe that C t ( a, b ) = 0 if a + b ≤ t .Since we are interested in the area S , cf. (10), we need to transform the area from the copulaspace into the space of x and y . From Sklar’s theorem, we know that F X , X ( x, y ) = 0 , if F X ( x ) + F X ( x ) ≤ t . We will refer to this area as B as defined in (11) in the following.We denote the boundary of B as B t ( x ) = F − Y ( t − F X ( x )) . Since we are looking for a tangentwith slope − , we have the following condition ∂B t ∂x = − f X ( x ) f Y (cid:0) F − Y ( t − F X ( x )) (cid:1) ! = − , (41) B B t ( x ⋆ ) + x ⋆ F − X ( t ) F − X ( t ) x x F X ( x ) + F X ( x ) ≤ tx + x = F − X ( t ) x + x = F − X ( t ) x + x = B t ( x (1) ) + x (1) x + x = B t ( x ⋆ ) + x ⋆ Figure 9. Area with zero probability mass of the joint distribution F X , X and different candidates for corresponding ZOC. where we set u = t − F X ( x ) and apply the well-known rules for derivatives. The solutions to(41) are denoted as x ( i ) . Recall that we are looking for lines in the form of x + x = s and thatthe above condition gives these tangents on B at x ( i ) . The next candidate solutions are thereforein the form x + x = x ( i ) + B t ( x ( i ) ) . An exemplary B can be seen in Fig. 9. It is easy to seethat there exist multiple solutions x ( i ) to (41). However, only one of the lines x ( i ) + B ( x ( i ) ) liescompletely in B . For the optimal solution x ⋆ , we therefore need to take the minimum over all x ( i ) , i.e., x ⋆ = arg min x ( i ) (cid:8) x ( i ) + B ( x ( i ) ) (cid:9) .If the boundary B is not convex or it has no slope of − in the first quadrant of the x - x plane, we need to consider the limit points, which are given at x = 0 and x = 0 and we have x ≤ F − X ( t ) and x ≤ F − X ( t ) , respectively. Recall that the original problem x + x ≤ s canbe viewed as finding the line x = s − x with the maximum s such that the line is in B . Withthe first bounds on x and x , we have the simple lines x + x = F − X ( t ) and x + x = F − X ( t ) as possible candidates. This is illustrated in Fig. 9. From this exemplary plot, it is also goodto see that both lines do not fully lie in B . Thus, there is a positive probability mass in theareas x + x ≤ F − X ( t ) and x + x ≤ F − X ( t ) . In this case, we need to find the tangent on theboundary of B . We now have the following possible candidates for ss = x ⋆ + B t ( x ⋆ ) , s = F − X ( t ) , s = F − Y ( t ) . Recall that s = 2 R − with the ZOC R when the joint distribution of X and X followcopula C t . With reference to Fig. 9, it becomes clear that we need to take the minimum of all s i in order to guarantee that x + x ≤ s is a subset of B . Combining all of the above finallystates the theorem. A PPENDIX CP ROOF OF L EMMA P ni =1 x i , we areinterested in the tangent planes P ni =1 x i = s on B ( x ) = 0 . The condition for the critical pointsis therefore given by ∂B∂x i = ∂B∂x j , ∀ i, j . (42)For the considered B , this means that f ( x i ) = f ( x j ) , ∀ i, j . (43)First, it is easy to see from this, that there will always be a critical point on the identity line x = · · · = x n that solves (43). If the density f is strictly decreasing, e.g., the exponentialdistribution, it has an inverse function and the point on the identity line will be the only solution(43). For such distributions, the proof is concluded.For general distributions with quasiconcave densities, we next check the curvature of B atthe point on the identity line that we are interested in. If it is negative, this corresponds to amaximum of the curve and is not a solution of (16). If we have a minimum on the identity line,it is a valid candidate to be the solution to (16).With the implicit function theorem [35, Sec. 8.3], we will express the implicit function B ( x , . . . , x n ) = 0 by an explicit function b ( x , . . . , x n − ) as B ( x , . . . , x n ) = 0 ⇔ b ( x , . . . , x n − ) = x n . (44)From the implicit function theorem, we can determine the gradient of b as ∂b∂x i = − (cid:18) ∂B∂x n (cid:19) − ∂B∂x i , i = 1 , . . . , n − , (45)and the second derivatives, i.e., the Hessian matrix, whose entries are then given by ∂ b∂x i ∂x j = − (cid:18) ∂B∂x n (cid:19) − (cid:18) ∂ B∂x i ∂x j ∂B∂x n − ∂B∂x i ∂ B∂x n ∂x j (cid:19) . (46)For the considered B , this gives ∂ b∂x i = − f ′ ( x i ) f ( x n ) (47)and ∂ b∂x i ∂x j = 0 , i = j, (48) where f ′ is the derivative of the density. From this, it can be seen that the Hessian matrix of B is a diagonal matrix and its eigenvalues are therefore the entries on the main diagonal which aregiven by (47). In order to have a minimum on the identity line, the Hessian needs to be positivedefinite at this point, i.e., all of its eigenvalues need to be positive. From (47), we see that thisis the case if f ′ ( x ⋆ ) = f ′ (cid:18) F − (cid:18) − n (cid:19)(cid:19) < , (49)which is exactly (17).We have shown that there is a local minimum on the identity line for distributions that fulfillthe above condition, and we now show that this is then also a global minimum. For this, recallthat all other critical points are given by (43). For the general case of a strictly quasiconcavedensity, we know that there are two solutions to f ( x ) = α for each level set α , due to theunimodality. One of the solutions is greater than the mode while the other is less than the mode.If all x i are on the same side from the mode, they are all equal and this is, therefore, the point onthe identity line that we already considered. We therefore have to check the remaining solutions,i.e., when at least two x i are on the different sides of the mode. In this case, the x i that is lessthan the mode is on the increasing part of the density, i.e., f ′ ( x i ) > , while f ′ ( x j ) < , when x j is greater than the mode. With (47), we can therefore conclude that the Hessian matrix isindefinite at all remaining critical points, i.e., they are saddle points.Since we assume unbounded support of F X , the range of each x i is [0 , ∞ ) . Therefore, if x i → ∞ for any i , the sum x + · · · + x n also tends to ∞ . In other words, the function iscoercive [36, Def. 2.31] with respect to the sum.We can summarize the above discussion as follows. A distribution that fulfills condition (17)from the lemma has a unique minimum of the sum on the identity line. All other stationary pointsare saddle points. Combining this with the coerciveness, we can conclude that the minimum onthe identity line is global. A PPENDIX DP ROOF OF L EMMA B ( x , . . . , x n ) = P ni =1 ( F X ( x i )) n − − n + 1 . The first derivatives of B are given as ∂B∂x i = g ( x i ) = 1 n − F ( x i )) − nn − f ( x i ) . (50) Thus, the critical points for the tangent plane are now given by the condition ( F ( x i )) − nn − f ( x i ) = ( F ( x j )) − nn − f ( x j ) , ∀ i, j . (51)The second partial derivatives of B are given as ∂ B∂x i = g ′ ( x i ) = ( F ( x i )) − nn − n − f ′ ( x i ) + 2 − nn − f ( x i )) F ( x i ) ! (52)and ∂ B∂x i ∂x j = 0 , i = j . (53)Combining this with (46), we can see that the Hessian matrix again is a diagonal matrix, wherethe entries on the main diagonal are given by (52). Similarly to the proof of Lemma 6, we havea minimum on the identity line, if the Hessian matrix is positive definite at this point. From thecondition B ( x , . . . , x n ) = 0 , we get the point on the identity line as x = . . . = x n = x ⋆ = F − (cid:18) − n (cid:19) n − ! . All of the eigenvalues of the Hessian are positive if f ′ ( x ⋆ ) < − − nn − (cid:18) − n (cid:19) − n f ( x ⋆ ) . (54)This is condition the (19) from the lemma to prove.By the assumption in the lemma, g ( x ) is a unimodal function. We can therefore use the sameargument as in the proof of Lemma 6 in App. C. With the above observations and reference toApp. C, this concludes the proof. A PPENDIX EP ROOF OF T HEOREM n X i =1 x i = s, (55)which touches B given by P ni =1 F X i ( x i ) + 1 − n = 0 . Since we assume that the distribution is B -SYM, we know this tangent point x ⋆ will be on the identity line, i.e., x ⋆ = x = . . . = x n .We therefore get x ⋆ = F − X (cid:0) n − n (cid:1) , and applying this to (55) gives the corresponding s ⋆ as s ⋆ = nx ⋆ = nF − X (cid:0) n − n (cid:1) . With the definition of s , we get the statement (21) from the theorem. A PPENDIX FP ROOF OF T HEOREM C n ( u , . . . , u n ) = " n X i =1 u n − i − n + 1 + ! n − . Since this is a valid copula, the derived bound in (23) is achievable and therefore an inner boundon the maximum ZOC. A
PPENDIX GP ROOF OF C OROLLARY R n in (22) and R n in (23) can be written as R n − R n = log n E [ X ]1 + nF − X (cid:16)(cid:0) − n (cid:1) n − (cid:17) . The gap increases with increasing n and is therefore upper bounded by lim n →∞ R n − R n = log (cid:18) E [ X ] F − X ( ) (cid:19) , which is (24). R EFERENCES [1] K.-L. Besser, P.-H. Lin, and E. A. Jorswieck, “On the set of joint Rayleigh fading distributions achieving positivezero-outage capacities,” in , IEEE, Nov. 2020.[2] W. Saad, M. Bennis, and M. Chen, “A vision of 6G wireless systems: Applications, trends, technologies, and openresearch problems,”
IEEE Network , pp. 1–9, 2019.[3] J. Park, S. Samarakoon, H. Shiri, M. K. Abdel-Aziz, T. Nishio, A. Elgabli, and M. Bennis,
Extreme URLLC: Vision,challenges, and key enablers , Jan. 2020.[4] M. Bennis, M. Debbah, and H. V. Poor, “Ultrareliable and low-latency wireless communication: Tail, risk, and scale,”
Proceedings of the IEEE , vol. 106, no. 10, pp. 1834–1853, Oct. 2018.[5] D. Tse and P. Viswanath,
Fundamentals of Wireless Communications . Cambridge University Press, 2005.[6] K.-L. Besser and E. A. Jorswieck, “Reliability bounds for dependent fading wireless channels,”
IEEE Transactions onWireless Communications , vol. 19, no. 9, pp. 5833–5845, Sep. 2020.[7] E. Jorswieck and P.-H. Lin, “Ultra-reliable multi-connectivity with negatively dependent fading channels,” in , IEEE, Aug. 2019, pp. 373–378.[8] R. B. Nelsen,
An Introduction to Copulas , 2nd ed., ser. Springer Series in Statistics. Springer New York, 2006. [9] J. A. Ritcey, “Copula models for wireless fading and their impact on wireless diversity combining,” in , IEEE, Nov. 2007, pp. 1564–1567.[10] J. Kitchen and W. Moran, “Copula techniques in wireless communications,” ANZIAM Journal , vol. 51, pp. 526–540,Aug. 2010.[11] M. H. Gholizadeh, H. Amindavar, and J. A. Ritcey, “On the capacity of MIMO correlated Nakagami- m fading channelsusing copula,” EURASIP Journal on Wireless Communications and Networking , vol. 2015, no. 1, p. 138, Dec. 2015.[12] G. W. Peters, T. A. Myrvoll, T. Matsui, I. Nevat, and F. Septier, “Communications meets copula modeling: Non-standarddependence features in wireless fading channels,” in , IEEE, Dec. 2014, pp. 1224–1228.[13] F. R. Ghadi and G. A. Hodtani, “Copula function-based analysis of outage probability and coverage region for wirelessmultiple access communications with correlated fading channels,”
IET Communications , vol. 14, no. 11, pp. 1804–1810,Jul. 2020.[14] C. Zheng, M. Egan, L. Clavier, G. W. Peters, and J.-M. Gorce, “Copula-based interference models for IoT wirelessnetworks,” in
ICC 2019 - 2019 IEEE International Conference on Communications (ICC) , IEEE, May 2019.[15] K.-L. Besser and E. A. Jorswieck, “Bounds on the secrecy outage probability for dependent fading channels,”
IEEETransactions on Communications , vol. 69, no. 1, pp. 443–456, Jan. 2021.[16] ——, “Bounds on the ergodic secret-key capacity for dependent fading channels,” in , VDE, Feb. 2020.[17] F. R. Ghadi and G. A. Hodtani, “Copula-based analysis of physical layer security performances over correlated Rayleighfading channels,”
IEEE Transactions on Information Forensics and Security , vol. 16, pp. 431–440, 2021.[18] K.-L. Besser and E. A. Jorswieck, “Copula-based bounds for multi-user communications – Part II: Outage Performance,”
IEEE Communications Letters , vol. 25, no. 1, pp. 8–12, 2021.[19] E. A. Jorswieck and K.-L. Besser, “Copula-based bounds for multi-user communications – Part I: Average Performance,”
IEEE Communications Letters , vol. 25, no. 1, pp. 3–7, 2021.[20] P.-H. Lin, E. A. Jorswieck, R. F. Schaefer, C. R. Janda, and M. Mittelbach, “Copulas and multi-user channel orders,” in
ICC 2019 - 2019 IEEE International Conference on Communications (ICC) , IEEE, May 2019.[21] P.-H. Lin, E. A. Jorswieck, C. R. Janda, M. Mittelbach, and R. F. Schaefer, “On stochastic orders and fading Gaussianmulti-user channels with statistical CSIT,” in , IEEE,Jul. 2019.[22] P.-H. Lin, E. A. Jorswieck, R. F. Schaefer, M. Mittelbach, and C. R. Janda, “New capacity results for fading Gaussianmultiuser channels with statistical CSIT,”
IEEE Transactions on Communications , vol. 68, no. 11, pp. 6761–6774, 2020.[23] K.-L. Besser. (2020). “Set of joint distributions achieving positive zero-outage capacities, Supplementary material,”[Online]. Available: https://gitlab.com/klb2/zero-outage-joint-distributions.[24] T. Wang, G. Giannakis, and R. Wang, “Smart regenerative relays for link-adaptive cooperative communications,”
IEEETransactions on Communications , vol. 56, no. 11, pp. 1950–1960, Nov. 2008.[25] M. Di Renzo, M. Debbah, D.-T. Phan-Huy, A. Zappone, M.-S. Alouini, C. Yuen, V. Sciancalepore, G. C. Alexandropoulos,J. Hoydis, H. Gacanin, J. de Rosny, A. Bounceur, G. Lerosey, and M. Fink, “Smart radio environments empowered byreconfigurable AI meta-surfaces: An idea whose time has come,”
EURASIP Journal on Wireless Communications andNetworking , vol. 2019, no. 1, p. 129, Dec. 2019.[26] M. J. Frank, R. B. Nelsen, and B. Schweizer, “Best-possible bounds for the distribution of a sum – a problem ofKolmogorov,”
Probability Theory and Related Fields , vol. 74, no. 2, pp. 199–211, 1987.[27] B. Wang and R. Wang, “Joint mixability,”
Mathematics of Operations Research , vol. 41, no. 3, pp. 808–826, Aug. 2016. [28] R. Wang, L. Peng, and J. Yang, “Bounds for the sum of dependent risks and worst value-at-risk with monotone marginaldensities,” Finance and Stochastics , vol. 17, no. 2, pp. 395–417, Apr. 2013.[29] G. Puccetti, B. Wang, and R. Wang, “Advances in complete mixability,”
Journal of Applied Probability , vol. 49, no. 2,pp. 430–440, Jun. 2012.[30] A. J. McNeil and J. Nešlehová, “Multivariate Archimedean copulas, d -monotone functions and ℓ -norm symmetricdistributions,” Annals of Statistics , vol. 37, no. 5B, pp. 3059–3097, Oct. 2009.[31] M. D. Yacoub, “The α - µ distribution: A general fading distribution,” in The 13th IEEE International Symposium onPersonal, Indoor and Mobile Radio Communications , IEEE, 2002, pp. 629–633.[32] D. Brennan, “Linear diversity combining techniques,”
Proceedings of the IRE , vol. 47, no. 6, pp. 1075–1102, Jun. 1959.[33] T. L. Lai and H. Robbins, “A class of dependent random variables and their maxima,”
Zeitschrift für Wahrscheinlichkeit-stheorie und Verwandte Gebiete , vol. 42, no. 2, pp. 89–111, 1978.[34] M. Abramowitz and I. A. Stegun,
Handbook of Mathematical Functions: With Formulas, Graphs, and MathematicalTables , 10th Ed. 1972.[35] K. Binmore and J. Davies,
Calculus: Concepts and Methods . Cambridge University Press, 2001.[36] A. Beck,
Introduction to Nonlinear Optimization, Theory, Algorithms, and Applications with MATLAB . Society forIndustrial and Applied Mathematics (SIAM), Oct. 2014.[37] W. Lee and J. Y. Ahn, “On the multidimensional extension of countermonotonicity and its applications,”