The Role of Correlation in the Doubly Dirty Fading MAC with Side Information at the Transmitters
Farshad Rostami Ghadi, Ghosheh Abed Hodtani, F. Javier Lopez-Martinez
11 The Role of Correlation in the Doubly Dirty FadingMAC with Side Information at the Transmitters
Farshad Rostami Ghadi, Ghosheh Abed Hodtani, and F. Javier L´opez-Mart´ınez
Abstract —We investigate the impact of fading correlation onthe performance of the doubly dirty fading multiple accesschannel (MAC) with non-causally known side information attransmitters. Using Copula theory, we derive closed-form ex-pressions for the outage probability and the coverage regionunder arbitrary dependence conditions. We show that a positivedependence structure between the fading channel coefficients isbeneficial for the system performance, as it improves the outageprobability and extends the coverage region compared to thecase of independent fading. Conversely, a negative dependencestructure has a detrimental effect on both performance metrics.
Index Terms —Doubly dirty multiple access channel, correlatedRayleigh fading, side information, outage probability, coverageregion, Copula theory.
I. I
NTRODUCTION
Achieving reliability constraints in applications like con-nected robotics and autonomous systems [1] is a key openchallenge in the roadmap to sixth-generation (6G) technology.In this regard, multi-user wireless communications techniquesthat take advantage of side information (SI) at the transmitterscan be of great interest, since such knowledge – either channelstate information (CSI), or interference awareness – can beleveraged to intelligently encode their information. By doingso, the destructive effects of the interference can be reduced,and reliable communication with higher data rates can beachieved.The use of SI at the transmitter was first studied by Shannonin the context of single-user communication systems [2].For a multi-user setting, Jafar provided a general capacityregion for a discrete and memoryless multiple-access channel(MAC) with causal and non-causal independent SI in [3].By exploiting a random binning technique, Philosof − Zamirextended Jafar’s work and presented achievable rate regionsfor the discrete and memoryless MAC with correlated SIknown non-causally at the encoders [4]. The case of a two-userGaussian MAC with SI at both transmitters (i.e, doubly dirtyMAC) for the high-SNR and strong interference regimes wasstudied in [5], on which the achievable rate regions suffer from
Manuscript received March xx, 2021; revised XXX. This work has beenfunded in part by the Spanish Government and the European Fund forRegional Development FEDER (project TEC2017-87913-R) and by Junta deAndalucia (project P18-RT-3175). The review of this paper was coordinatedby XXXX.F.R. Ghadi and G.A. Hodtani are with Department of ElectricalEngineering, Ferdowsi University of Mashhad, Mashhad, Iran. (e-mail: { f . rostami . gh , ghodtani } @gmail . com ).F.J. Lopez-Martinez is with Departmento de Ingenieria de Comunicaciones,Universidad de Malaga - Campus de Excelencia Internacional AndaluciaTech., Malaga 29071, Spain (e-mail: fjlopezm@ic . uma . es ).Digital Object Identifier 10.1109/XXX.2021.XXXXXXX a bottleneck effect dominated by the weaker user compared tothe case of a clean MAC (i.e., without interference).In wireless communication theory, dependence structuresassociated to random phenomena in temporal, frequency orspatial scales are often neglected for the sake of tractability[6]. This is the case, for instance, of multi-user channels,where due to physical proximity of the transmitters the chan-nel coefficients observed by each user are in general notindependent. One plausible approach to incorporate arbitrarydependence structures that is recently gaining momentum inthe wireless communication arena is the use of Copula theory[7], [8]. Copulas are widely used in statistics, survival anal-ysis, image processing, machine learning, and have becomepopular in the context of performance analysis of wirelesscommunication systems; specifically: general bounds on theoutage performance for dependent slow-fading channels wasanalyzed in [8]. The authors in [9] studied the performanceof physical layer security under a correlated Rayleigh fadingwiretap channel and derived closed-form expressions for somesecrecy performance metrics by exploiting Farlie-Gumbel-Morgenstern (FGM) Copula. Besides, bounds on the secrecyoutage probability for secure communications under dependentfading channels were obtained in [10]. Copulas have also beenused for analyzing the impact of interference correlation in thecontext of ad hoc networks [11]. Finally, the authors in [12]derived closed-form expressions for the outage probability andthe coverage region in the correlated Rayleigh fading cleanMAC, bringing out the negative effect of a positive dependencebetween fading channels in the system performance.In this work, we study the impact of fading correlationon the performance of doubly dirty MAC with non-causallyknown SI at transmitters. Differently from the case on whichinterferences are not present, our theoretical results show thatpositive dependence between the fading channel coefficients isbeneficial, since it allows for reducing the outage probabilityand extending the coverage region compared to the baselinecase of independent fading.II. S YSTEM MODEL AND DEFINITIONS
A. The wireless doubly dirty MAC
We consider a two-user wireless doubly dirty MAC with twoknown interferences S and S (see Fig. 1), where, transmitters(users) t and t send the inputs X and X , respectively.Therefore, the received signal Y at receiver (base station) r can be defined as: Y = h X + h X + S + S + Z (1) a r X i v : . [ c s . I T ] F e b Fig. 1. System model depicting a wireless doubly dirty MAC where Z represents the Additive White Gaussian Noise(AWGN) with zero mean and variance N (i.e., Z ∼ N (0 , N ) )at the receiver r , and h and h are the corresponding fadingchannel Rayleigh coefficients, meaning that the channel powergains (i.e., g = | h | and g = | h | ) are exponentiallydistributed. We consider the general case on which the fadingprocesses h and h are correlated. We assume that the inter-ference signals S and S with variances Q ( S ∼ N (0 , Q ) )and Q ( S ∼ N (0 , Q ) ) are known non-causally at thetransmitters t and t , respectively; and the inputs X and X sent by transmitters t and t over the channels aresubjected to the average power constraint as E [ | X | ] ≤ P and E [ | X | ] ≤ P , respectively. Besides, we define the signal-to-noise ratio (SNR) at transmitters t and t as γ = P | h | N and γ = P | h | N , so that the corresponding average SNRsare given by ¯ γ = P E [ | h | ] N and ¯ γ = P E [ | h | ] N , respectively.Therefore, the marginal distributions for the SNR γ i , i = 1 , are given by f ( γ i ) = e − γi ¯ γi ¯ γ i , F ( γ i ) = 1 − e − γi ¯ γi . B. Preliminary definitions
Theorem 1.
In a block fading doubly dirty MAC with thecoherent receiver (fading coefficients h and h are knownat the receiver) and two independent interferences S and S non-causally known at transmitters t and t , the instanta-neous capacity region is determined as follows as long as theinterferences S and S are strong (i.e., Q , Q → ∞ ) [5] R + R ≤
12 log (cid:16) { P | h | d α N , P | h | d α N } (cid:17) (2) where R and R are the desired transmission rates for trans-mitters t and t located at distances d and d , respectively,and α > is the path loss exponent. We now briefly review some basic definitions and propertiesof the two-dimensional Copulas [7].
Definition 1 (Copula) . Let S = ( S , S ) be a vector oftwo random variables with marginal cumulative distributionfunctions (CDFs) F ( s j ) = Pr( S j ≤ s j ) for j = 1 , ,respectively. The relevant bivariate CDF is defined as: F ( s , s ) = Pr( S ≤ s , S ≤ s ) (3) Then, the Copula function C ( u , u ) of the random vector S =( S , S ) defined on the unit hypercube [0 , with uniformly distributed random variables U j := F ( s j ) for j = 1 , over [0 , is given by C ( u , u ) = Pr( U ≤ u , U ≤ u ) (4) Theorem 2 (Sklar’s theorem) . Let F ( s , s ) be a joint CDFof random variables with margins F ( s j ) for j = 1 , . Then,there exists one Copula function C such that for all s j in theextended real line domain ¯ R , F ( s , s ) = C (cid:0) F ( s ) , F ( s )) (cid:1) . (5) Corollary 1.
By applying the chain rule to (5) , the jointprobability density function (PDF) f ( s , s ) is derived as: f ( s , s ) = f ( s ) f ( s ) c (cid:0) F ( s ) , F ( s ) (cid:1) (6) where c (cid:0) F ( s ) , F ( s ) (cid:1) = ∂ C ( F ( s ) ,F ( s )) ∂s ∂s is the Copuladensity function and f ( s j ) for j = 1 , are the marginal PDFs,respectively. Definition 2.
For a vector of two random variables S =( S , S ) with joint CDF F ( s , s ) and marginal survivalfunctions ¯ F ( s j ) = Pr( S j > s j ) = 1 − F ( s j ) for j = 1 , , thejoint survival function ¯ F ( s , s ) is given by ¯ F ( s , s ) = Pr( S > s , S > s ) (7) = ¯ F ( s ) + ¯ F ( s ) − C (1 − ¯ F ( s ) , − ¯ F ( s )) (8) = ˆ C ( ¯ F ( s ) , ¯ F ( s )) (9) where ˆ C ( u , u ) = u + u − C (1 − u , − v ) is thesurvival Copula of S = ( S , S ) . Definition 3. [FGM Copula] The bivariate FGM Copula withdependence parameter θ F ∈ [ − , is defined as: C F ( u , u ) = u u (1 + θ F (1 − u )(1 − u )) (10) where θ F ∈ [ − , and θ F ∈ (0 , denote the negative andpositive dependence structures respectively, while θ F = 0 in-dicates the independence structure. Besides, it can be derivedthat the FGM survival Copula is the same as FGM Copula,meaning that ˆ C F ( u , u ) = C F ( u , u ) . III. O
UTAGE PROBABILITY
The outage probability is a key metric to evaluate theperformance of communication systems operating over fadingchannels, and is defined as the probability that the channelcapacity is less than a certain information rate R > . Thus,we have: P out = Pr( R + R ≤ R ) (11) = Pr (cid:16)
12 log (cid:0) { γ d α , γ d α } (cid:1) ≤ R (cid:17) (12) = Pr (cid:0) min { γ d α , γ d α } ≤ R − (cid:1) (13) = 1 − Pr (cid:0) γ > β , γ > β (cid:1) (14) = 1 − ˆ C ( ¯ F γ ( β ) , ¯ F γ ( β )) (15)where β = d α (2 R − and β = d α (2 R − . Theorem 3.
The outage probability over correlated Rayleighfading doubly dirty MAC with defined parameters ¯ γ , ¯ γ , θ F , β , and β is given by P out = 1 − e − ( β γ + β γ ) (cid:16) θ F (1 − e − β γ )(1 − e − β γ ) (cid:17) (16) Proof.
By utilizing the FGM Copula and the relevant survivalCopula from Definition 3, the outage probability is obtainedas (16). IV. C
OVERAGE REGION
In this section, by exploiting the concept of coverage regionprovided in [13], we determine the expression for the coverageregion of the system model in Fig. 1. For simplicity andwithout loss of generality, we assume that receiver r is locatedat the origin (0 , . Then, we define the coverage region as thegeographic zone for which the sum rate R + R is guaranteed,with R , R > , i.e. G ( d , d ) def = { d , d , C ( d , d ) > R + R } (17)where C ( d , d ) = log (cid:0) { P | h | Nd α , P | h | Nd α } (cid:1) denotesthe channel capacity when transmitters t and t are locatedat d and d , respectively. Theorem 4.
The coverage region for the concerned correlatedRayleigh fading doubly dirty MAC with defined parameters ¯ γ , ¯ γ , θ F , α , R , and R is given by (22) .Proof. In order to achieve certain rates, the expectation ofrandom SNRs γ and γ should be computed. Thus, thecoverage region can mathematically be expressed as: R + R ≤ E γ ,γ (cid:34)
12 log (cid:16) { γ d α , γ d α } (cid:17)(cid:35) (18) = (cid:90) ∞ (cid:90) ∞
12 log (cid:16) { γ d α , γ d α } (cid:17) f ( γ , γ ) dγ dγ (19) = (cid:90) ∞ (cid:32) (cid:90) γ
12 log (cid:0) γ d α (cid:1) f ( γ , γ ) dγ + (cid:90) ∞ γ
12 log (cid:0) γ d α (cid:1) f ( γ , γ ) dγ (cid:33) dγ (20)where f ( γ , γ ) is the joint PDF of SNRs and is obtained asfollows by exploiting FGM Copula: f ( γ , γ ) = e − γ γ − γ γ ¯ γ ¯ γ (cid:104) θ F (cid:0) − e − γ γ (cid:1)(cid:0) − e − γ γ (cid:1)(cid:105) (21)By substituting the joint PDF from (21) into (20), and calcu-lating the above integrals, the coverage region is obtained as(22). The details of the proof are in Appendix A. V. S IMULATION R ESULTS
In this section, the analytical and Monte-Carlo simulationresults for the outage probability and coverage region arepresented, with special focus on comparing the performancesin the presence/absence of fading correlation.Fig. 2 shows the behavior of the outage probability basedon the variation of ¯ γ for selected values of θ F . For simplicity,we set d = d = 1 in this scenario. We see that theoutage probability continuously decreases by increasing ¯ γ for a given value of ¯ γ , which is reasonable because thechannel condition between transmitter t and receiver r isimproved. From the correlation viewpoint, we see that underthe positive dependence structure ( θ F ∈ (0 , ), the correlatedfading (CF) case has achieved a better performance, i.e., alower outage probability, as compared with the uncorrelatedfading (UF) case. We now illustrate in Fig. 3 the evolution ofthe outage probability as the average SNRs ¯ γ and ¯ γ varyunder perfect positive correlation ( θ F = 1 ). We see that theoutage probability tends to zero for in the high SNR regime.The effect of the threshold rate R o on the outage probabilityfor selected values of θ F and three scenarios ¯ γ > ¯ γ , ¯ γ = ¯ γ ,and ¯ γ < ¯ γ is evaluated in Fig. 4. In all three scenarios, itis shown that as R o increases, the outage probability tendsto 1, which is coherent with the fact that the communicationbecomes impossible at very high rates. We also notice thatunder the positive dependence structure, the outage probabilityachieves lower values for the CF case compared to the UFcase, which suggests that positive dependence has a beneficialrole on system performance. The coverage region for selectedvalues of θ F and ¯ γ is illustrated in Fig. 5. We see that as ¯ γ increases, a wider coverage region is achieved. We observethat when ¯ γ reaches ¯ γ , the distance d also approaches d under the positive dependence structure. We see that thebottleneck effect in the capacity region (2), which is limitedby the minimum SNR of the users, is relaxed in the presenceof a positive dependence. This implies that the coverage regionis improved compared to the case of independent fading, asobserved in the figure. It is interesting to highlight that thisis in stark contrast with the observations made in [12] in theabsence of interference. Hence, we see that considering thenon-causally known SI at transmitters in MAC can improvethe performance of outage probability and coverage regionunder the positive dependence structure.VI. C ONCLUSION
In this letter, we evaluated the performance analysis of dou-bly dirty multiple access channels with non-causally knownside information at transmitters, where the corresponding fad-ing channel coefficients are assumed correlated. Specifically,we derived the closed-form expressions for outage probabilityand coverage region using FGM Copula, analyzing the ef-fect of correlated fading case in both negative and positivedependence structures. We showed that in the latter case, thesystem performance is improved in terms of outage probabilityreduction and coverage region extension. Results confirm thebeneficial impact of positive fading correlation in the doublydirty MAC channel due to strong interference, compared tothe case of an interference-free clean MAC. R + R ≤ √ π (cid:32) ¯ γ e dα γ γ γ γ (1 − π ) + ¯ γ e dα γ γ γ γ (1 − π ) γ + ¯ γ )+ θ F (cid:34) ¯ γ e dα γ γ γ γ (1 − π ) (cid:16) e dα γ γ γ γ (1 − π ) (cid:17) + ¯ γ e dα γ γ γ γ (1 − π ) (cid:16) e dα γ γ γ γ (1 − π ) (cid:17) γ + ¯ γ ) − ¯ γ e dα γ γ γ γ (1 − π ) + ¯ γ e dα γ γ γ γ (1 − π ) (2¯ γ + ¯ γ ) − γ e dα γ γ γ γ (1 − π ) + ¯ γ e dα γ γ γ γ (1 − π ) γ + 2¯ γ ) (cid:35)(cid:33) (22) Fig. 2. Outage probability versus ¯ γ for selected values of dependenceparameter θ F Fig. 3. Outage probability versus ¯ γ and ¯ γ for θ F = 1 Fig. 4. Outage probability versus threshold rate R o for selected values ofdependence parameter θ F -4 -3 -2 -1 0 1 2 3 4-4-3-2-101234 Fig. 5. Coverage region for selected values of dependence parameter θ F A PPENDIX AP ROOF OF T HEOREM f ( γ , γ ) in (20) and exploit-ing the linearity rules of integration, (20) can be decomposedas R + R ≤ (cid:90) ∞ (cid:90) γ e − γ γ − γ γ γ ¯ γ log (cid:0) γ d α (cid:1) × (cid:104) θ F (cid:0) − e − γ γ (cid:1)(cid:0) − e − γ γ (cid:1)(cid:105) dγ dγ + (cid:90) ∞ (cid:90) ∞ γ e − γ γ − γ γ γ ¯ γ log (cid:0) γ d α (cid:1) × (cid:104) θ F (cid:0) − e − γ γ (cid:1)(cid:0) − e − γ γ (cid:1)(cid:105) dγ dγ (23) = A + θ F ( A − A − A + 4 A )+ B + θ F ( B − B − B + 4 B ) (24)where the integrals in (24) follow the following formats: (cid:90) e − ζx log (1 + ηx ) dx = 1 ζ ln 2 (cid:104) e ζη Ei (cid:0) − ( ζη + ζx ) (cid:1) − e − ζx ln(1 + ηx ) (cid:105) (25) (cid:90) ∞ e − ζx log (1 + ηx ) dx = − e ζη ζ ln 2 Ei (cid:16) − ζη (cid:17) (26) (cid:90) ∞ e − ζx Ei (cid:0) − ( κ + ηx ) (cid:1) dx = 1 ζ (cid:104) Ei( − κ ) − e ζκη Ei (cid:0) − ( ζ + η ) κη (cid:1)(cid:105) (27) Now, by exploiting (25), (26), and (27), we have: A = (cid:90) ∞ (cid:90) γ e − γ γ − γ γ γ ¯ γ log (cid:0) γ d α (cid:1) dγ dγ = (cid:90) ∞ e − γ γ γ ln 2 (cid:34) e dα γ Ei (cid:16) − γ + d α ¯ γ (cid:17) − e − γ γ ln(1 + γ d α ) − e dα γ Ei (cid:16) − d α ¯ γ (cid:17)(cid:35) dγ = − ¯ γ e d α ( ¯ γ γ γ γ ) γ + ¯ γ ) ln 2 Ei (cid:16) − d α (cid:16) ¯ γ + ¯ γ ¯ γ ¯ γ (cid:17)(cid:17) (28) A = (cid:90) ∞ (cid:90) γ e − γ γ − γ γ γ ¯ γ log (cid:0) γ d α (cid:1) dγ dγ = − ¯ γ e d α ( γ γ γ γ ) γ + ¯ γ ) ln 2 Ei (cid:16) − d α (cid:16) γ + ¯ γ ¯ γ ¯ γ (cid:17)(cid:17) (29) A = (cid:90) ∞ (cid:90) γ e − γ γ − γ γ γ ¯ γ log (cid:0) γ d α (cid:1) dγ dγ = − ¯ γ e d α ( ¯ γ γ γ γ ) γ + 2¯ γ ) ln 2 Ei (cid:16) − d α (cid:16) ¯ γ + 2¯ γ ¯ γ ¯ γ (cid:17)(cid:17) (30) A = (cid:90) ∞ (cid:90) γ e − γ γ − γ γ γ ¯ γ log (cid:0) γ d α (cid:1) dγ dγ = − ¯ γ e d α ( ¯ γ γ γ γ ) γ + ¯ γ ) ln 2 Ei (cid:16) − d α (cid:16) ¯ γ + ¯ γ ¯ γ ¯ γ (cid:17)(cid:17) (31)Similarly, by utilizing (26), we have: B = 12¯ γ ¯ γ (cid:90) ∞ (cid:90) ∞ γ e − γ γ − γ γ log (cid:0) γ d α (cid:1) dγ dγ = (cid:90) ∞ e − γ ( γ + γ ) γ log (cid:0) γ d α (cid:1) dγ = − ¯ γ e dα γ γ γ γ γ + ¯ γ ) ln 2 Ei (cid:16) − d α (¯ γ + ¯ γ ) γ γ (cid:17) (32) B = 12¯ γ ¯ γ (cid:90) ∞ (cid:90) ∞ γ e − γ γ − γ γ log (cid:0) γ d α (cid:1) dγ dγ = − ¯ γ e dα γ γ γ γ γ + ¯ γ ) ln 2 Ei (cid:16) − d α (2¯ γ + ¯ γ ) γ γ (cid:17) (33) B = 12¯ γ ¯ γ (cid:90) ∞ (cid:90) ∞ γ e − γ γ − γ γ log (cid:0) γ d α (cid:1) dγ dγ = − ¯ γ e dα γ γ γ γ γ + 2¯ γ ) ln 2 Ei (cid:16) − d α (¯ γ + ¯2 γ ) γ γ (cid:17) (34) B = 12¯ γ ¯ γ (cid:90) ∞ (cid:90) ∞ γ e − γ γ − γ γ log (cid:0) γ d α (cid:1) dγ dγ = − ¯ γ e dα γ γ γ γ γ + ¯ γ ) ln 2 Ei (cid:16) − d α (¯ γ + ¯ γ ) γ γ (cid:17) (35)Now, by inserting (28)-(35) into (24) and applying the approx-imation Ei( − x ) ∼ − √ π e − ( π ) x [14], the proof is completed.R EFERENCES[1] W. Saad, M. Bennis, and M. Chen, “A vision of 6g wireless systems:Applications, trends, technologies, and open research problems,”
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