Efficient Computation of Probability Density Function in Interfering 𝜅-𝜇 Communication Channels and Application to Fitting Measurement Data

De Gruyter Journal YYYY; aop This is the authors’ accepted manuscript of the article that has been accepted for publication in Frequenz. This is the pre-review version. The final version will be available at https://www.degruyterbrill.com/journal/key/freq/html after publication. Research Article Jelena R. Djordjević-Kozarov, Danijela A. Aleksić, Jelena A. Anastasov*, Aleksandar M. Kovačević, and Dejan N. Milić Efficient Computation of Probability Density Function in Interfering 𝜅-𝜇 Communication Channels and Application to Fitting Measurement Data https://doi.org/10.1515/sample-YYYY-XXXX Received Month DD, YYYY; revised Month DD, YYYY; accepted Month DD, YYYY Abstract: In this paper, we propose an efficient algorithm for computing the probability density function (PDF) of signal-to-interference ratio over 𝜅 – 𝜇 fading environment in wireless communications. At first, we examine the convergence of the sum involved in the PDF expression. After that, we introduce the reminder estimate in truncated summation by updating it with each additional term until the requested accuracy is met. Then, we present the pseudocode of the PDF summation algorithm. In order to hint the importance of this algorithm, we show its comparison with integration PDF approach, in terms of execution time and precision performance. Also, due to the wide applicability of 𝜅 – 𝜇 fading model, we show one example of using the proposed algorithm in overlapping measured data from indoor communication scenario with numerical curve fit. Keywords: Probability density function, Summation algorithm, Radio channel, Signal-to-interference ratio. 1 Introduction Theoretical analysis of wireless communication links is of huge importance as the first step in the design of numerous telecommunication systems. Preliminary, statistical characterization of fading channel, as essential part in signal transmission, is needed [1, 2]. In large number of published papers, effect of multipath propagation throughout open indoor/outdoor media i.e. fading phenomena has been discussed in details and experimentally fitted with existed theoretical fading models [ 3 – 5 ], in order to provide accommodative baseline for crucial system performance evaluations. Also, numerous distributions that describe multipath homogeneous/non-homogeneous propagation exist in the literature. Among all of them, 𝜅 – 𝜇 fading model is proposed as generic one with physical phenomena aspects [ 6 , 7 ]. The model is parametrized by two shaping factors 𝜅 ( 𝜅 > 0 . 5) and 𝜇 ( 𝜇 > 0 . 5), assuming line of sight (LOS) or non LoS (NLoS) scenarios. It encompasses Ricean and Nakagami𝑚 distributions and thought One-sided Gaussian and Rayleigh, too. Jelena R. Djordjević-Kozarov, Jelena A. Anastasov, Dejan N. Milić, University of Niš, Faculty of Electronic Engineering, Niš, Serbia, e-mails: k[email protected], [email protected], [email protected] Danijela A. Aleksić, ATVSS, Niš, Serbia, e-mail: [email protected] Aleksandar M. Kovačević, University of Kragujevac, Faculty of Technical Sciences, Čačak, Serbia, e-mail: aleksanda[email protected] 2J. Djordjević-Kozarov et al., Efficient Computation of PDF in Interfering Communication Channels This fading model have shown generality and widespread application in realistic scenarios of wireless communication links. On the other hand, 𝜅 – 𝜇 envelope’s mathematical tractability of its probability density function (PDF) showed to be challenging in numerous system’s performance evaluations. In [ 8 ], simulations are exceptionally given, because the PDF or cumulative distribution function (CDF) are in a form of infinite number of terms in summation. Also, in [9] the PDF of 𝜅–𝜇variable was used in terms of sum. In nowadays emerging telecommunication systems such as cellular networks, device-to-device (D2D), body area networks (BAN), another important metric is interfering signals that can seriously corrupt performance. The level of noise is often negligible compared to the level of interference in these so-called interference-limited systems. Therein, the signal-to-interference ratio (SIR) is the criteria that merits [ 10 , 11 ]. From the designer’s point of view, it is substantially to determine the proper received power signal in order to diminish influence of different links operating on the same frequency. In [ 4 ] authors have emphasized the importance of inter-BAN co-channel interference. Importance of the interference-limited communication have been also highlighted in [ 12 ]. In [ 13 ], analytical framework for evaluating the PDF and the cumulative distribution function (CDF) of the ratio of two 𝛼 - 𝜇 variates, two 𝜂 - 𝜇 and two 𝜅 – 𝜇 variates, is proposed. In this work the expressions of SIR’s PDF is in a form of one infinite sums and the CDF is given in a form of two infinite sums. The authors haven’t deal with the convergence of those expressions. Furthermore, in [ 14 ] a detailed SIR characterization is presented based on collected received signal strength indicators, in laboratory controlled conditions, when diversity reception at two base stations is employed. The analysis has shown that besides open outdoor space, also indoor field measurements render 𝜅 – 𝜇 fading model suitable for wireless channel propagation description. Numerous statistical approaches can be found in literature for defining the propagation path between two communication nodes [15]. In [16] an extensive analysis is presented on the product of two 𝜅–𝜇variables in a way of defining the fundamental statistics of interest i.e. the PDF, the CDF and moment-generating function. In the work it in also noted that from derived expressions, the distributions of the ratio of 𝜅 – 𝜇 / 𝜅 – 𝜇 can be deduced. However, analytical formulations there are in series representations, as in many other papers. In literature, there are also algorithms for generating 𝜅 – 𝜇 variables for purposes of simulations. In [ 17 ], the authors propose efficient random sequence generator of generating 𝜅 – 𝜇 variates. The generator is adapted for arbitrary model shaping factors. Novel unified importance sampling estimators are proposed in [8] for efficient estimation of 𝜅–𝜇variates. In this paper, we propose a novel algorithm that efficiently evaluates the PDF of SIR over 𝜅 – 𝜇 fading channels. In more details, we introduce an analytical framework throughout the investigation of the convergence of sum in the PDF of SIR, the estimation of reminder in the partial sum and the required number of terms in summation. We present the algorithm that computes PDF by increasing the number of terms when the reminder estimate is below the predefined threshold. To illustrate the efficiency of this algorithm and also its highly required application, we compare it with the integration procedure of SIR’s PDF for experimentally gained 𝜅–𝜇fading shaping factors in indoors. In brief, the paper is structured as follows. The system model that is later on used for measurement fitting is described in Section 2. Also, in this Section 𝜅 – 𝜇 fading model as well as the ratio of the desired signal variable and interfering signal variable is analytically defined. In Section 3, we deal with the convergence of the summation which appears in the PDF of SIR and we determined the required number of terms in order to reach predefined accuracy. In Section 4, we propose a novel algorithm for evaluating the PDF of SIR of 𝜅 – 𝜇 fading model. The section contains comparison of the algorithm and integration in terms of computation facilities and accuracy. Also, in this section the algorithm is used to overlap the acquired measurement data. Finally, the main concluding remarks are stated in Section 5. 2 System model and problem formulation There are two devices in the network of base station, which work in the same frequency range - cellphone and interferer device (Fig. 1). The first device, the cellphone, which represents the transceiver, moves along J. Djordjević-Kozarov et al., Efficient Computation of PDF in Interfering Communication Channels 3 Base Sta on Device - cellphone Base sta on coverage area Guide rail 2000 mm Interferer device Fig. 1: System model of a physical communication scenario. a metal rail of 2 m, with resolution of 1 mm, so that in total we have 2000 discrete samples. While moving along the rail, the device emits a continuous unmodulated signal. The second device that is an interferer, which can be located inside or outside the coverage area of the observed base station, creates interference. Direct measurements for two different movement lines are labelled as “Signal” and “Interference”. Clearly, two signals in the same frequency range mutually interfere and the unwanted signal is regarded as an interference. The desired signal level to the interference level ratio is labelled as SIR. In this paper, we use the 𝜅 – 𝜇 fading model in fitting with actual measurement data, according to the fact that it is a general model corresponding to LOS/NLOS environments. 𝜅 – 𝜇 distribution is defined by the following PDF of envelope [6, 7] for the desired signal 𝑓𝑅1(𝑟1) = 2𝜇1(1 + 𝑘1)𝜇1+1 2 𝑘 𝜇1−1 2 1exp (𝜇1𝑘1) 𝑟1𝜇1  Ω(𝜇1+1) 1 exp (︃−𝜇1(1 + 𝑘1)(︂𝑟1  Ω1)︂2)︃𝐼𝜇1−1(︂2𝜇1√︀𝑘1(1 + 𝑘1)𝑟1  Ω1)︂, (1) as well as for interfering signal 𝑓𝑅2(𝑟2) = 2𝜇2(1 + 𝑘2)𝜇2+1 2 𝑘 𝜇2−1 2 2exp (𝜇2𝑘2) 𝑟2𝜇2  Ω(𝜇2+1) 2 exp (︃−𝜇2(1 + 𝑘2)(︂𝑟2  Ω2)︂2)︃𝐼𝜇2−1(︂2𝜇2√︀𝑘2(1 + 𝑘2)𝑟2  Ω2)︂,(2) where  Ω2 = 𝐸 ( 𝑅 )is expectation of envelope, and 𝜅 and 𝜇 are fading shaping factors. In more details, 𝜅 determines ratio of the total power of the dominant components to the total power of the scattered waves, while 𝜇defines the number of multipath clusters. In order to investigate SIR statistics, in mathematical manner, Bessel function should be presented in a form of infinite summation relying on [ 18 , (03.02.02.0001.01)], 𝐼𝜈 ( 𝑥 ) = +∞ ∑︀ 𝑖=0 1 Γ(𝑖+𝜈+ 1)𝑖!(︂𝑥 2)︂2𝑖+𝜈 . Thus, 4J. Djordjević-Kozarov et al., Efficient Computation of PDF in Interfering Communication Channels the previous two expressions should be rewritten as 𝑓𝑅*(𝑟*) =2𝜇*𝜇*(1 + 𝜅*)𝜇*  Ω2𝜇* * 𝑟* 2𝜇*−1exp (︃−𝜇*𝜅*−𝜇*(1 + 𝜅*)(︂𝑟*  Ω*)︂2)︃ +∞ ∑︁ 𝑖=0 (︂√︀𝜅*(1 + 𝜅*)𝜇*  Ω* 𝑟*)︂2𝑖 𝑖! Γ(𝑖+𝜇*),(3) with subscript * being 1 or 2 to denote the PDF of desired signal or interfering signal PDF, respectively. When considering the SIR characteristics, the PDF of quotient of two 𝜅 – 𝜇 variables i.e. 𝑍 = |𝑅1|/|𝑅2| , can be found as 𝑓𝑍(𝑥) = +∞ ∫︁0 𝑟2𝑓𝑅1(𝑟2𝑥)𝑓𝑅2(𝑟2) d𝑟2, 𝑥 ≥0.(4) Using series representation of the 𝜅–𝜇PDF given in (3), and using [19], the integral can be solved as 𝑓𝑍(𝑥) = 2𝜉𝜇2 e(𝜅1𝜇1+𝜅2𝜇2) 𝑥2𝜇1−1 (𝜉+𝑥2)𝜇1+𝜇2 +∞ ∑︁ 𝑖=0 (︀𝜅1𝜇1𝑥2/(𝜉+𝑥2))︀𝑖 𝑖! Γ(𝑖+𝜇1) +∞ ∑︁ 𝑗=0 Γ(𝑖+𝑗+𝜇1+𝜇2)(︀𝜅2𝜇2𝜉/(𝜉+𝑥2))︀𝑗 𝑗! Γ(𝑗+𝜇2),(5) where 𝜉=1 + 𝜅2 1 + 𝜅1 𝜇2 𝜇1 Ω2 2 Ω2 1 . Parameters 𝜅1 and 𝜇1 correspond to random variable 𝑅1 , while 𝜅2 and 𝜇2 correspond to 𝑅2 . We assume that random variables 𝑅1 and 𝑅2 are distributed according to 𝜅 – 𝜇 distribution, and consequently PDF of 𝑍is given by (5). Furthermore, by utilizing 1𝐹1(︂𝑎 𝑏𝑧)︂ = +∞ ∑︀ 𝑘=0 (𝑎)𝑘𝑧𝑘 (𝑏)𝑘𝑘! [ 18 , (07.20.06.0002.01)] considering summation over 𝑗, we rewrite (5), in the following form 𝑓𝑍(𝑥) =2𝜇2𝜉𝜇2𝑥2𝜇1−1 (𝑥2+𝜉)𝜇1+𝜇2exp(−𝜅1𝜇1−𝜅2𝜇2) +∞ ∑︁ 𝑖=0 (︂𝜅1𝜇1 𝑥2 𝑥2+𝜉)︂𝑖1 𝑖!(︂𝑖+𝜇1+𝜇2−1 𝜇2)︂1𝐹1(︂𝑖+𝜇1+𝜇2 𝜇2 𝜅2𝜇2𝜉 𝑥2+𝜉)︂,(6) where 1𝐹1(︂𝑎 𝑏𝑐)︂ represents Kummer confluent hypergeometric function. On the other hand, utilizing [ 18 , (07.20.16.0001.01)] and [18, (07.20.27.0001.01)], we get the diverse form of the SIR’s PDF, as 𝑓𝑍(𝑥) = 2𝜉𝜇2𝑥2𝜇1−1 (𝑥2+𝜉)𝜇1+𝜇2exp (︂−𝜅1𝜇1−𝜅2𝜇2𝑥2 𝑥2+𝜉)︂ +∞ ∑︁ 𝑖=0 𝑖+𝜇1 𝑖!(︂𝜅1𝜇1𝑥2 𝑥2+𝜉)︂𝑖 L𝜇2−1 𝑖+𝜇1(︂−𝜅2𝜇2𝜉 𝑥2+𝜉)︂,(7) where L𝜆 𝜈(𝑥)is generalized Laguerre function. With the same procedure, considering the summation over 𝑖in (6), we get the alternative form as 𝑓𝑍(𝑥) = 2𝜉𝜇2𝑥2𝜇1−1 (𝑥2+𝜉)𝜇1+𝜇2exp (︂−𝜅2𝜇2−𝜅1𝜇1𝜉 𝑥2+𝜉)︂ +∞ ∑︁ 𝑖=0 𝑖+𝜇2 𝑖!(︂𝜅2𝜇2𝜉 𝑥2+𝜉)︂𝑖 L𝜇1−1 𝑖+𝜇2(︂−𝜅1𝜇1𝑥2 𝑥2+𝜉)︂.(8) J. Djordjević-Kozarov et al., Efficient Computation of PDF in Interfering Communication Channels 5 3 Proposed algorithm In order to determine the algorithm that will efficiently evaluate (7), we analyse the convergence of the infinite sum. By the limit comparison test, we compare 𝑎𝑖=𝑖+𝜇1 𝑖!(︂𝜅1𝜇1𝑥2 𝑥2+𝜉)︂𝑖 L𝜇2−1 𝑖+𝜇1(︂−𝜅2𝜇2𝜉 𝑥2+𝜉)︂,(9) to 𝑏𝑖=𝑖+𝜇1 𝑖!(︂𝜅1𝜇1𝑥2 𝑥2+𝜉)︂𝑖(︂𝜅2𝜇2𝜉 𝑥2+𝜉)︂𝑖+𝜇1𝑖 (𝑖+𝜇1)! (10) Obviously, 𝑏𝑖< 𝑎𝑖. Taking 𝛼=𝜅2𝜇2𝜉/(𝑥2+𝜉), we get the limit lim 𝑖→+∞ 𝑎𝑖 𝑏𝑖 = lim 𝑖→+∞ (𝑖+𝜇1)!L𝜇2−1 𝑖+𝜇1(−𝛼) 𝑖 𝛼𝑖+𝜇1= 1.(11) By the limit comparison test, we prove that the PDF series converges if and only if 𝑏𝑖 converges. Now we proceed to prove convergence of the following series +∞ ∑︁ 𝑖=1 𝑏𝑖= +∞ ∑︁ 𝑖=1 𝛾𝑖 (𝑖−1)!(𝑖+𝜇1−1)!,(12) where 𝛾=𝛼𝜇1𝜅1𝜇1 𝑥2+𝜉. We know that +∞ ∑︁ 𝑖=1 𝛾𝑖 (𝑖−1)! =𝛾 +∞ ∑︁ 𝑖=0 𝛾𝑖 𝑖!=𝛾e𝛾.(13) Since each series member is 𝛾𝑖 (𝑖−1)!(𝑖+𝜇1−1)! <𝛾𝑖 (𝑖−1)! , the series converges, which in turn proves convergence of (7). 3.1 Remainder estimate Since the series converges, the PDF can be approximated by a partial sum that has a pre-defined numerical error threshold 𝑓𝑍(𝑥) = 2𝜉𝜇2𝑥2𝜇1−1 (𝑥2+𝜉)𝜇1+𝜇2exp (︂−𝜅1𝜇1−𝜅2𝜇2𝑥2 𝑥2+𝜉)︂ [︁𝑁 ∑︁ 𝑖=0 𝑖+𝜇1 𝑖!(︂𝜅1𝜇1𝑥2 𝑥2+𝜉)︂𝑖 L𝜇2−1 𝑖+𝜇1(︂−𝜅2𝜇2𝜉 𝑥2+𝜉)︂ +𝑅𝑁+1]︁,(14) where 𝑅𝑁+1 is the remainder after taking only 𝑁 terms in the summation. Observing the remainder series 𝑅𝑁= +∞ ∑︁ 𝑖=𝑁 𝑖+𝜇1 𝑖!𝛽𝑖L𝜇2−1 𝑖+𝜇1(−𝛼),(15) where 𝛽 = 𝜅1𝜇1𝑥2/ ( 𝑥2 + 𝜉 ), and 𝛼 = 𝜅2𝜇2𝜉/ ( 𝑥2 + 𝜉 ), we note that a useful property of Laguerre polynomials by using recurrence relation from [21, (5.1.13)] L𝜆 𝑛(−𝛼)<L𝜆 𝑛+1(−𝛼), 𝛼, 𝜆, 𝑛 ≥0.(16) 6J. Djordjević-Kozarov et al., Efficient Computation of PDF in Interfering Communication Channels This could be proofed as 𝑛L𝜆 𝑛(−𝛼)=(𝛼+ 2𝑛+𝜆−1)L𝜆 𝑛−1(−𝛼)−(𝑛+𝜆−1)L𝜆 𝑛−2(−𝛼),(17) and it can be used to bound the remainder from above. Further, it stands that 𝑅𝑁≤L𝜇2−1 𝑁+𝜇1(−𝑎) +∞ ∑︁ 𝑖=𝑁 𝑖+𝜇1 𝑖!𝛽𝑖 = L𝜇2−1 𝑁+𝜇1(−𝛼)[︁𝛽𝑁 (𝑁−1)! + e𝛽(𝛽+𝜇1)(︂1−Γ(𝑁, 𝛽) (𝑁−1)!)︂]︁ =L𝜇2−1 𝑁+𝜇1(−𝛼) (𝑁−1)! [︁𝛽𝑁+ e𝛽(𝛽+𝜇1)𝛾(𝑁, 𝛽)]︁, 𝑁, 𝛽 > 0,(18) with Γ( , )denoting upper incomplete Gamma function [ 18 , (06.06.02.0001.01)], [ 22 ] and 𝛾 ( , )being lower incomplete Gamma function [22, (8.8.2)]. Asymptotic behavior of Laguerre functions is known to be [20] L𝜆 𝑛(−𝑥)∼(𝑛+ 1)𝜆/2−1/4e2√𝑥(𝑛+1) 2√𝜋𝑥𝜆/2+1/4e𝑥/2.(19) Using the asymptotic expression, it is straightforward to prove that lim 𝑁→+∞ 𝑅𝑁= 0, as it should. However, this form is not amenable for finding 𝑁 in a relatively simple form, given the value of 𝑅𝑁 . However, this form is not amenable for finding 𝑁 in a relatively simple form, given the value of 𝑅𝑁 . It would be convenient to know in advance the number of terms 𝑁 required for specified precision. However, the remainder estimate form is not amenable for finding 𝑁 in a relatively simple form. Therefore, the next best option is to have continuously updatable estimate with each summation step, and it is also imperative that this estimate update does not take a lot of computing resources in order not to overwhelm the usability of the summation process. It turns out that this is an achievable task, and further results are incorporated in the following subsection. 3.2 Algorithm We can compute PDF by increasing 𝑁 until the remainder estimate is bellow the predefined threshold 𝜖 . Moreover, one can use recurrence relations for Laguerre polynomials [ 21 , (5.1.10)] and incomplete Gamma functions [ 22 , (8.8.2)] to simplify the computation. This kind of customization provides a way to compute PDF values much faster than using generic summation algorithms that many of the software packages provide. Using the previous derivations, it turns out that computing the value of (6) can be much simpler than at first glance. Also, the following form stands 𝑓𝑍(𝑥) =2𝜇1𝜉𝜇2𝑥2𝜇1−1 (𝑥2+𝜉)𝜇1+𝜇2exp(−𝜅1𝜇1−𝜅2𝜇2) +∞ ∑︁ 𝑖=0 (︂𝜅2𝜇2 𝜉 𝑥2+𝜉)︂𝑖1 𝑖!(︂𝑖+𝜇1+𝜇2−1 𝜇1)︂1𝐹1(︂𝑖+𝜇1+𝜇2 𝜇1 𝜅1𝜇1𝑥2 𝑥2+𝜉)︂,(20) leading to alternative form of the PDF, expressed as: 𝑓𝑍(𝑥) = 2𝜉𝜇2𝑥2𝜇1−1 (𝑥2+𝜉)𝜇1+𝜇2exp (︂−𝜅2𝜇2−𝜅1𝜇1𝜉 𝑥2+𝜉)︂ +∞ ∑︁ 𝑖=0 𝑖+𝜇2 𝑖!(︂𝜅2𝜇2𝜉 𝑥2+𝜉)︂𝑖 L𝜇1−1 𝑖+𝜇2(︂−𝜅1𝜇1𝑥2 𝑥2+𝜉)︂,(21) J. Djordjević-Kozarov et al., Efficient Computation of PDF in Interfering Communication Channels 7 1: function pdfSum(𝜇1, 𝜇2, 𝛼, 𝛽) 2: 𝑠←0◁Initial value of the sum 3: 𝜖←10−6◁Setting requested precision 4: 𝐸←𝜖 ◁ Initial value of relative error threshold 5: 𝑟←1◁Initial estimate of remainder 6: 𝑖←0◁Summation counter 7: 𝑓←1◁Holder variable for 1/𝑖! 8: 𝐵←1◁Holder variable for 𝛽𝑖 9: 𝐿←L𝜇2−1 𝜇1(−𝛼)◁ Initial Laguerre function values 10: 𝐿*←L𝜇2−1 𝜇1−1(−𝛼) 11: 𝑐←e𝛽(𝛽+𝜇1)◁A constant value, needed later 12: 𝐺←Γ(0, 𝛽)◁ Initial value for upper incomplete Gamma function, 𝛽 > 0 13: while 𝑟≥𝐸do ◁While remainder estimate is above the threshold 14: 𝑠←𝑠+𝑓(𝑖+𝜇1)·𝐵·𝐿 ◁ Add a term 15: 𝑡←𝐿 ◁ Temporary variable 16: 𝐿←2𝑖+ 2𝜇1+𝜇2+𝛼 𝑖+𝜇1+ 1 𝐿 ◁ Three-term recurrence relation 17: −𝑖+𝜇1+𝜇2−1 𝑖+𝜇1+ 1 𝐿* 18: 𝐿*←𝑡 ◁ Recover temporary value 19: 𝐺←𝑖𝐺 +𝐵e−𝛽◁Recurrence relation for incomplete Gamma 20: 𝐵←𝛽𝐵 ◁ Update 𝛽𝑖 21: 𝑟←𝐿[𝑓𝐵 +𝑐(1 −𝑓𝐺)] ◁Estimate remainder 22: 𝑖←𝑖+ 1 ◁Increase counter 23: 𝑓←𝑓/𝑖 ◁ Update 1/𝑖!variable 24: 𝐸←𝑠·𝜖 ◁ Update relative error threshold 25: end while ◁Until prescribed precision is met 26: return 𝑠 ◁ Return the partial sum value 27: end function Fig. 2: Pseudocode of the PDF summation algorithm For the alternative form of (21), we need to examine 𝑅𝑁= +∞ ∑︁ 𝑖=𝑁 𝑖+𝜇2 𝑖!𝛼𝑖L𝜇1−1 𝑖+𝜇2(−𝛽),(22) and by analogy the alternative remainder estimate 𝑅𝑁≤L𝜇1−1 𝑁+𝜇2(−𝛼)[︁𝛼𝑁 (𝑁−1)! + e𝛼(𝛼+𝜇2)(︂1−Γ(𝑁, 𝛼) (𝑁−1)!)︂]︁ =L𝜇1−1 𝑁+𝜇2(−𝛽) (𝑁−1)! [︁𝛼𝑁+ e𝛼(𝛼+𝜇2)𝛾(𝑁, 𝛼)]︁, 𝑁, 𝛼 > 0.(23) Relying on the analysis so far, we propose the pseudocode of the PDF summation algorithm (Fig. 2). 8J. Djordjević-Kozarov et al., Efficient Computation of PDF in Interfering Communication Channels Tab. 1: Execution times comparison of the proposed algorithm and a general integration approach using commercial software Curve Parameters Integration Algorithm execution time [s] execution time [s] 1𝜇1= 10.8 𝜅1= 2.3 𝜇2= 3.1 𝜅2= 1.8 1.172 0.042 2𝜇1=𝜇2= 1.057 𝜅1=𝜅2= 1.837 0.932 0.015 3𝜇1= 1 𝜅1= 1.4 𝜇2= 4.8 𝜅2= 8.1 1.035 0.011 4 Numerical results and discussion First, we start with comparing the results of our summation algorithm to numerical results obtained by integration algorithms available for general purpose integration in Mathematica. Numerical integration follows direct application of (4), and the exact code segment used is shown in Fig. 3. The results are show in Fig. 4, where different values of parameters are used to show different PDF curves. There is visual overlap between the two approaches indicating that the summation yields precise results visually indistinguishable from the results of summation algorithm. Parameters of the shown curves are chosen to represent typical values expected in a communication environment corresponding to a small office or a laboratory [14]. To evaluate the proposed numerical approach against the general summation approach we compare the obtained results in terms of execution time on a typical desktop computer. We have computed values of PDF curves for different parameters, and each curve is calculated in 60 equidistant points of the interval [0 , 3]. The parameter values are the same as in Fig 4, and curve designations follow those of the Figure. The computations are performed on an Intel ® Core ™ i7-2600 CPU @ 3.40GHz. Each curve has been computed 1000 times in sequence, and the total time is obtained by dividing the total time taken by 1000. This comparison approach effectively uses a range of 𝑥 values, which avoids potentially strong influence of specific “good” or “bad” points that may affect the integration or the summation algorithm. The precision level is set to 10 −6 in both approaches, for a fair comparison. The results indicate that the proposed summation algorithm performs much better in terms of execution time than the general integration algorithm available in commercial software (Table 1). Rough estimate is that the proposed summation is 30 to a 100 times faster, which is not surprising since it is tailored for the specific purpose. Fig. 3: Code segment for direct PDF computation using numerical integration in Mathematica Demonstrating the speed of the algorithm, we continue to show its precision performance, and the balance between the two. We choose a single 𝑥 value as 𝑥 = 1 and then proceed to progressively increase the precision requirements, while observing the performance of algorithm. In Table 2, we summarize the results for requested relative precision levels of 10 0 to 10 −12 . As the requested precision level becomes higher ( 𝜖 effectively becomes lower), the number of terms in summation increases, and corresponding execution time also increases. On the other hand, the number of observed accurate digits in the obtained result increases. Single PDF point results summarized in Table 2 clearly show that the number of terms in the summation is higher when the requested precision level is higher. This confirms that the remainder estimate is properly handled in the algorithm, and no unnecessary computation takes place. This in turn ensures the speed of computation is kept as fast as it could be, provided the specific remainder estimate derived in this paper. On the other hand, we need to make sure this estimate remains within the requested precision level for a J. Djordjević-Kozarov et al., Efficient Computation of PDF in Interfering Communication Channels 9 0.0 0.5 1.0 1.5 2.0 2.5 3.0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 Algorithm fZ(x) x 1=2.3 1=10.8 2=1.8 2=3.1 1= 2=1.837 1= 2=1.057 1=1.4 1=1 2=8.1 2=4.8 Integration 12 3 1 2 3 Fig. 4: PDF comparison-Integration vs. proposed algorithm. range of PDF abscissa 𝑥 values. Therefore, we examine how the obtained precision behaves for a range of PDF abscissas versus the requested precision level. This is shown in Fig. 5. The observed error clearly shows saw-like behavior consistent with changing number of terms 𝑁 that depends on each specific 𝑥 value. The observed precisions are shown as percentage, and it is evident that these values always stay bellow the requested 𝜖, as absolutely required. 4.1 Connection to measurement results In Fig. 6 we show a set of measurement data from Laboratory of Telecommunication Systems, Faculty of Electronic Engineering, University of Nis. The data represent signal levels versus the user transceiver position for 2000 specific positions on a specific line inside the laboratory. The SIR represents the target random process whose PDF is the object of consideration in this paper. In order to characterize the PDF of this ratio, it is necessary to obtain the distribution parameter values that provide the best fit of the proposed model to the measured data. The model has four independent parameters 𝜇1 , 𝜅1 , 𝜇2 , 𝜅2 that need to be determined in such a way that the target function of difference between the model and acquired data is minimized. Therefore, the speed of calculation of model PDF values for different parameters is an imperative, if the fitting results are to be obtained in a reasonable time period. After running a global minimization strategy based on simulated annealing, which is available in Mathematica software package, we were able to obtain the parameter values as 𝜇1 = 𝜇2 = 1 . 057, 𝜅1 = 𝜅2 = 1 . 837.Ωis set to value of one, as this is convenient in order to reduce the number of parameters. In general, Ωcan be varied by the specific technology by changing the output power, and is not of interest in our specific case, although it can be accommodated by proper data normalization. The Fig. 7 shows a comparison in logarithmic scale between the histogram of acquired data and the model with the specified parameters. The logarithmic scale is important in this sense because the distribution tails affects overall error probability during data transmission. Also, the linear scale comparison is shown in Fig. 8 where the agreement is clearly observed for the main distribution mass.