 Research
 Open Access
An analytical model for the intercell interference power in the downlink of wireless cellular networks
 Benoit Pijcke^{1, 2, 3},
 Marie ZwingelsteinColin^{1, 2, 3}Email author,
 Marc Gazalet^{1, 2, 3},
 Mohamed Gharbi^{1, 2, 3} and
 Patrick Corlay^{1, 2, 3}
https://doi.org/10.1186/16871499201195
© Pijcke et al; licensee Springer. 2011
 Received: 18 February 2011
 Accepted: 12 September 2011
 Published: 12 September 2011
Abstract
In this paper, we propose a methodology for estimating the statistics of the intercell interference power in the downlink of a multicellular network. We first establish an analytical expression for the probability law of the interference power when only Rayleigh multipath fading is considered. Next, focusing on a propagation environment where smallscale Rayleigh fading as well as largescale effects, including attenuation with distance and lognormal shadowing, are taken into consideration, we elaborate a semianalytical method to build up the histogram of the interference power distribution. From the results obtained for this combined small and largescale fading context, we then develop a statistical model for the interference power distribution. The interest of this model lies in the fact that it can be applied to a large range of values of the shadowing parameter. The proposed methods can also be easily extended to other types of networks.
Keywords
 Intercell interference power
 Statistical modeling
 Wireless networks
 Rayleigh fading
 lognormal shadowing
I Introduction
In the emerging wireless communication standards LTEAdvanced and Mobile WiMAX, aggressive spectrum reuse is mandatory in order to achieve the increased spectral efficiency required by IMTAdvanced for the 4th generation of standard telephony. However, since spectrum reuse comes at the expense of increased intercell interference, these standards explicitly require interference management as a basic system functionality [1–3]. The research area related to the development and analysis of interference management techniques, mostly in relation with the more general subject of radio resource management, is very dynamic, as witnessed by the high number of relevant recent contributions in this area [4–10]. All these new standards use OFDMA as the modulation and the multiple access scheme. In an OFDMA system, there is no intracell interference as the users remain orthogonal, even through multipath channels. However, when users from different cells are present at the same time on the same subchannel, which is the case under aggressive frequency reuse, signals superpose, leading to some form of intercell interference.
Providing statistical models of the interference power is essential to allow for an accurate evaluation of network performances without the need for lengthy and costly Monte Carlo simulations. The statistical characterization of the interferences has been investigated for a long time, under lots of different scenarios, and following several approaches. The distribution of cumulated instantaneous interference power in a Rayleigh fading channel was investigated in [11], where an infinite number of interfering stations was considered. In [12], the interference power statistics is obtained analytically for the uplink and downlink of a cellular system, but in the presence of largescale fading only. Interference modeling when considering only largescale fading effects has also been investigated in [13–15], where the emphasis is on finding a good approximation of the lognormal sum distribution. In [16], an analytical derivation of the probability density function (pdf) of the adjacent channel interference is derived for the uplink. More recently, in [17], the pdf of the downlink SINR was derived in the context of randomly located femtocells via a semianalytical method. Other contributions have focused directly on the analysis of a particular performance measure that is influenced by intercell interference, like the probability of outage and the radio spectrum efficiency [18–20]. The analysis of interference in dense asynchronous networks, such as ad hoc networks, is also an active research area, for which a deep review of the recent developments can be found in [21, 22].
In this paper, we derive a semianalytical methodology to estimate the statistics of the intercell interference power in a wireless cellular network, when the combined effects of largescale and smallscale multipath fading are taken into consideration. Largescale effects include attenuation with distance (path loss) as well as lognormal shadowing, and the smallscale fading is Rayleigh distributed. We consider a distributed wireless multicellular network, in both cases where power control and no power control are applied. The proposed methodology is semianalytical, in that the statistical estimate of the interference power resulting from N > 1 interferers is obtained by numerical techniques from an analytically derived interference model for one interferer. The methodology is valid in a quite general framework; we have chosen to present it using a hexagonal network layout, although it can handle any other topology. We validate the proposed methods by comparing the moments of the estimates to the exact moments of the distribution which can be derived analytically. Using this methodology, we are able to provide a very good estimate of the pdf of the interference power, for different values of the shadowing standard deviation, σ_{dB}. Based on these estimates, we then propose an analytical statistical model of the interference power, based on a modified Burr distribution, which includes five parameters. This analytical, parameterized by σ_{dB}, model will hopefully serve as a practical tool for the assessment and simulation of wireless cellular networks when the effect of shadowing is to be considered.
The main contributions of this paper are as follows:

In the special situation where only path loss and Rayleigh fading are considered (no shadowing), we derive a very accurate approximated analytical expression for the pdf and the cumulative distribution function (cdf) of the intercell interference power;

We propose a semianalytical method for the estimation of the pdf of the intercell interference power in a multicellular network when the combined propagation effects of path loss, Rayleigh fading and lognormal shadowing are considered;

Based on this method, we derive an analytical model for the pdf of the intercell interference power by slightly modifying a Burr probability distribution. This model is parameterized by the lognormal standard deviation σ_{dB}, and its interest resides in the fact that it is valid on the whole [0, 12]dB range of values.
The remainder of this paper is organized as follows. In Section II, we describe the multicell downlink transmission environment, and we provide the expression of the interference power for which we want to find a statistical model. In Section III, the original methodology for estimating the statistics of the interference power is presented. For this purpose, we examine in Section IIIA the particular case where path loss and Rayleigh fastfading are the only fading phenomena considered. In Section IIIB, we include the shadowing effect and we consider in the first instance the contribution of one interfering cell. We then generalize to N > 1 interferers. In Section IV, we apply the proposed method to estimate the pdf of the interference power in a typical multicellular network, under two frequency reuse scenarios. Section V is dedicated to the parametric analytical modeling of the interference power. Section VI concludes the paper by summarizing the proposed methods and by presenting some perspectives.
We will use the following notation for the rest of the paper. Nonbold letters such as x are used to denote scalar variables, and x is the magnitude of x. Bold letters like x denote vectors. We use $E\left\{X\right\}$ to denote the expectation of X. The pdf and cdf of the random variable (r.v.) X will be denoted as p_{ X } (x) and F_{ X } (x), respectively.
II Multicell downlink transmission model
Here, x_{0} (m, ℓ) represents the information symbol intended to UT_{0} and x_{ n } (m, ℓ), n ≠ 0, the n th interfering symbol (this symbol is sent from AP n to its respective user). The coefficient h_{ n } (m, ℓ) denotes the instantaneous gain of the ℓ th (interfering) subchannel from AP n to UT_{0}. Each subchannel ℓ is subject to additive white Gaussian noise w (m, ℓ). In the following, we will focus without loss of generality on a single OFDMA subchannel, thereby omitting subchannel index ℓ in all subsequent notations.
Two frequency reuse scenarios will be considered (see Figure 1):

the full frequency reuse pattern, denoted FR1, where all APs in the network transmit at the same time using the same frequency range (N = 18 intercell interferers);

a partial frequency reuse pattern, denoted FR3, with reuse factor 3 (N = 6 interferers).
where ξ = 10/ln(10) [23]. Note that the importance of the shadowing phenomenon is directly related to the standard deviation σ_{dB}. For a given σ_{dB}, the parameter μ_{dB} is determined to ensure a unit mean shadowing gain: $E\left\{{G}_{s,n}\right\}=1$, which leads to ${\mu}_{dB}={\sigma}_{dB}^{2}\u2215\left(2\xi \right)$. As r.v.'s G_{f,n}and G_{s,n}are independent from each other, and as $E\left\{{G}_{f,n}\right\}=E\left\{{G}_{s,n}\right\}=1$, we have, from (1), $E\left\{{G}_{n}\left({r}_{n}\right)\right\}={G}_{pl,n}\left({r}_{n}\right)$, which reflects the fact that the n th interfering channel's Rayleigh fading and shadowing components cause the actual gain G_{ n } (r_{ n } ) to fluctuate about its mean value G_{pl,n}(r_{ n } ).
The total interference power undergone by UT_{0} can then be written as $I={\sum}_{n=1}^{N}{P}_{n}{G}_{n}\left({r}_{n}\right)$, where ${P}_{n}=E\left\{{\left{x}_{n}\right}^{2}\right\}$ is the power emitted by AP n. In what follows, we consider that all APs transmit at the same power, i.e., P_{ n } = P for all n. This corresponds to, e.g., a fastfading environment where no channel state information feeds back from mobile users to APs, which results in a no power control scheme where all APs transmit at the maximum power; although crude, this scheme can be seen as a lower bound on performance for real systems. Considering that each AP transmits at the same power P also applies to a more practical scenario where APs have access to channel state information, and power control is associated with the opportunistic scheduling policy proposed (and proved to be sumrate optimal) in [10], when the number of users per cell is high (since in this case, it can be expected that the channels between users scheduled at the same time and their serving APs have about the same power gains). Thus, the interference simplifies to $I=P{\sum}_{n=1}^{N}{G}_{n}\left({r}_{n}\right)$.
(Note that G is a function of UT_{0}'s location through the distances r_{ n } .) So, as I = PG, characterizing the interference power I is equivalent to studying the interference gain G. We will concentrate on the latter in the subsequent sections.
III Methodology
We are now interested in finding an estimate of the pdf of the random interference gain (2). Since direct calculation of the pdf does not seem possible, we aim at producing an accurate histogram for the interference gain G that will then be modeled using a specified statistical distribution. Such a histogram is constructed from a set of samples called a typical set, i.e., a discrete ensemble of values that accurately represents a random phenomenon. Traditionally (and especially in the telecommunications area), this typical set is issued from Monte Carlo simulations, which might, at first sight, produce satisfying results. However, in a propagation environment that is subject to intense shadowing (i.e., for large values of the [0, 12]dB range under consideration), the classical Monte Carlo method fails at producing a representative set of sampled gains [24, 25]. This can be explained by examining the particular distribution involved, for one single as well as for multiple interfering cells. A typical cdf of the interference gain (single or multiple interferers) for a high value of σ_{dB} belongs to the class of heavytailed distributions [26], for which the leastfrequently occurring valuesalso called rare eventsare the most important ones, as a proportion of the total population, in terms of moments. A finitetime random drawing process performed on this cdf never produces these rare events because of their very low probabilities, which causes the resulting set to be not typical. Hence, the need for a new approach.
 1.
Produce a typical set of gains for one interferer using the generalized inverse method. This method consists in generating a typical set of samples corresponding to an arbitrary continuous cdf F and is based upon the following property: if U is a uniform [0, 1] r.v., then F ^{1} (U) has cdf F;
 2.
Produce a typical set for multiple interferers by adequately combining typical sets from single interferers and the Monte Carlo computational technique.
A Special case: no shadowing
using the notation r_{ n } = f_{ n } (r_{0}, θ), n = 1, 2,..., N, where (r_{0}, θ) are UT_{0}'s polar coordinates, as depicted in Figure 2. By examining (4), we see that, under this approximation, G_{ n } does not depend on UT_{0}'s varying position anymore.
In Section IVA, it is first shown that approximation (4) is valid in the case of one single interfering cell. This consequently validates the proposed model (7) in the case of multiple interfering cells, which we show for both frequency reuse patterns FR1 and FR3.
B General case: attenuation with distance, shadowing and multipath fading
Let us now focus on characterizing the distribution of the intercell interference gain G in a propagation environment where Rayleigh fading as well as shadowing (due to obstacles between the transmitter and receiver that attenuate signal power) are taken into account. To the best of our knowledge, no closed form expression for the interference gain G exists in the literature. But, as will be seen in Section IIIB.2, we determine an analytical formula (under integral form) of the distribution of the interference gain for one interferer. Using this result, we are able to obtain a histogram for G's distribution in the presence of multiple interferers.
For this purpose, we proceed in two steps: first, we compute a typical set for the interference gain produced by one single interferer. As described in Section IIIB.2, this is done by numerical computation (from the integralform cdf), followed by nonuniform partitioning, and then inversion, of the cdf. Then, we generate a typical set for N interferers using an appropriate combination of the (weighted by λ_{ n } ) typical sets of each single interferer (Section IIIB.3). The accuracy of the proposed method will be evaluated in both single and multipleinterferer cases by comparing the actual moments computed from the typical sets with the exact moments of the interference gain distribution (which can be formulated analytically, as will be seen in Section IIIB.1).
1) Preliminaries: We begin this section by examining two important points.
is a quantity of particular interest because it is proportional to the average power of the interference signal.
As closed form expressions of moments have been determined, they may be used in evaluating the accuracy of typical sets for both single and multipleinterferer statistical laws.
We are now interested in generating a typical set of the interference gain G_{ n } ; we denote this typical set by ${\mathcal{S}}_{n}^{\ell}$, where ℓ is the number of elements in the set. It was mentioned in Section III that, though widely used in telecommunications, the Monte Carlo computational technique proves inefficient for large values of σ_{dB}. An interesting alternative method is the generalized inverse method, for which an ℓelement typical set for a given distribution is obtained by an ℓlevel uniform partitioning, followed by inversion, of the cdf. Now, we know that, for large values of σ_{dB}, the distribution of G_{ n } exhibits the heavytailed property, which means, as described before, that the least frequently occurring values (i.e., the highest gains) are the most important ones in terms of moments. Therefore, taking these highest amplitudes into consideration using the 'classical' generalized inverse method would require a finer partitioning of the cdf, which would produce a typical set made up of a huge amount of elements.
3) Multiple interferers: We now focus on finding an $\mathcal{L}$element typical setdenoted ${\mathcal{S}}^{\mathcal{L}}$for the interference gain G that must be computed from N typical sets ${\mathcal{S}}_{n}^{\ell}$, n = 1,2,...,N.
We first note that interferer n's typical set can be directly obtained by weighting each element of ${\mathcal{S}}_{n}^{\ell}$ by its average path loss λ_{ n } ; we will denote interferer n's typical set by ${\lambda}_{n}{\mathcal{S}}_{n}^{\ell}$. Let us now find a way to produce the ensemble ${\mathcal{S}}^{\mathcal{L}}$ from the typical sets ${\lambda}_{n}{\mathcal{S}}_{n}^{\ell}$.
Ideally, ${\mathcal{S}}^{\mathcal{L}}$ should be constructed by considering all combinations of the elements of the typical sets ${\lambda}_{n}{\mathcal{S}}_{n}^{\ell}$, but the cardinality of the resulting set, $\mathcal{L}={\ell}^{N}={\left(JP\right)}^{N}$, would rapidly become prohibitive as the number N of interferers increases.
To get rid of this complexity, we point out that the abovementioned ideal (exhaustive) solution can also be viewed as an exhaustive combination of intervals (J^{ N } combinations) associated with an exhaustive combination of elements within each interval combination (P^{ N } combinations). And, we observe that the most important part of this exhaustive solution pertains to the combination of intervals, i.e., the combination of elements belonging to interval j of typical set ${\lambda}_{n}{\mathcal{S}}_{n}^{\ell}$ with elements belonging to interval k, k ≠ j typical set ${\lambda}_{m}{\mathcal{S}}_{m}^{\ell},m\ne n$. So, a way to construct a (nearoptimal) typical set for G could be to perform exhaustive combinations of the intervals (as in the exhaustive solution) and to approximate the exhaustive combination of the elements within each interval combination by the following procedure: for each of the J^{ N } combinations of NPpoint intervals,

Perform a random permutation of the P elements within each of the NPpoint intervals^{c};

Add up these N permuted Ppoint intervals to obtain one resulting Pelement interval.
This last Pelement interval approximates the P^{ N } element interval that would have resulted from an exhaustive combination of elements within the considered interval combination. Now, as there are J^{ N } interval combinations, the resulting typical set would contain J^{ N }P elements, which can still be prohibitive, so this second solutionwhich we will refer to as the nearoptimal solutioncan not be applied as such.
We eventually propose a novel approach which makes use of this nearoptimal solution and is based on the following twostep algorithm:
Step 1 Apply exhaustive combinations of intervals to a subset of M interfering links;
Step 2 Perform Monte Carlo simulations for the N  M remaining links.
We now detail the principle of the proposed method. In Step 1, we apply the nearoptimal solution described previously, but to a subset of M < N interfering links which we will call compelled links. The compelled links are chosen to have the highest average path losses (λ_{1} ≥ ⋯ λ_{ M } ≥ ⋯ λ_{ N } ) so as to minimize errors in other (noncompelled) interfering links. The exhaustive combination of the J intervals for M compelled links obtained from the nearoptimal solution thus results in one set of J^{ M }P elements. In Step 2, we build up a J^{ M }Pelement set for each of the N  M remaining, noncompelled, links by performing J^{ M } random drawings of intervals according to the probability set {δ_{ j } }, j = 1, 2,..., J. As in the nearoptimal solution, a random permutation of the elements is applied at each drawing. The ensemble of amplitudes of the intercell interference gain Gthe socalled typical set ${\mathcal{S}}^{\mathcal{L}}$is then constructed by adding up these N  M + 1 sets; it is of cardinality $\mathcal{L}={J}^{M}P$. Associated to ${\mathcal{S}}^{\mathcal{L}}$ is a probability set determined as follows: to each interval is associated a weight which is the product of probabilities δ_{ k } of intervals issued from compelled links (for noncompelled links, probabilities are accounted for by means of the random selection process); these weights are then normalized to obtain probabilities. Finally, the histogram of the interference gain G can be constructed from these resulting amplitude and probability sets. It is important to note, however, that, as a random drawing process is involved, a number of iterations might be needed in order for this process to converge (elements of S^{ L } and associated probabilities are averaged at each iteration). We will call this semianalytical technique the Monte Carlopanel method (MCP, in short)^{d}.
is a normalizing constant such that ${\sum}_{j=1}^{\mathcal{J}}{\delta}_{j}^{\prime}=1$ (using the particular nonuniform partitioning described previously, we have: $\alpha =1\u2215\left(10.{1}^{\mathcal{J}}\right)\gtrsim 1$.
Now, as was mentioned before, high amplitudes play an important role in terms of moments. Although the impact of neglecting them in noncompelled links is globally limited because these links are weighted by smaller average path losses λ_{ n } (n = M +1,..., N), it has to be compensated in order to satisfy the 1storder moment constraint (i.e., the sampled mean has to converge to the exact value^{f}). For this purpose, small (resp. large) amplitudes need to be underweighted (resp. overweighted). Thus, an underweighting multiplicative factor, denoted f^{}, is applied to amplitudes of the $\mathcal{J}$ first intervals of compelled links; similarly, an overweighting multiplicative factor f^{+} is applied to amplitudes of the last $N\mathcal{J}$ intervals. (Computation details of factors f^{} and f^{+} are given in Appendix D.)
Let us last notice that the choice for values of M and $\mathcal{J}$ is a tradeoff between different aspects: cardinality of the resulting typical set (i.e., tractable number of points), number of simulation runs and accuracy of the histogram. We have determined that M = 2 and $\mathcal{J}=3$ meet all these requirements.
IV Numerical results
In this section, we present numerical results related to the different methods introduced in the preceding section. In Section IVA, we first examine the validity of the original approximation introduced in Section III, stating that the interference gain G_{ n } (and, consequently, G) does not depend on the user's position within its cell. For this purpose, we compare the approximation of G given by (6) with the 'exact' formula (3). Then, in Section IVB, we obtain the histogram of the interference gain G_{ n } (one single interferer) by applying the nonuniform partitioning generalized inverse method described in IIIB.2. Finally, the MCP method (see IIIB.3) is used to build up the histogram of the interference gain for multiple interferers in Section IVC.
Average path losses λ_{ n }, n = 1, 2,..., N, defined by (6), in decreasing order of importance
FR1 (N= 18)  FR3 (N= 6)  

N  λ _{ n }  AP _{ m }  n  λ _{ n }  AP _{ m } 
1  6.467  1  1  0.568  8 
2  3.588  2  2  0.426  18 
3  1.708  6  3  0.307  10 
4  1.069  3  4  0.219  16 
5  0.767  5  5  0.178  12 
6  0.663  4  6  0.158  14 
7  0.568  8  
8  0.426  18  
9  0.316  7  
10  0.307  10  
11  0.260  9  
12  0.219  16  
13  0.188  17  
14  0.178  12  
15  0.158  14  
16  0.145  11  
17  0.118  15  
18  0.107  13 
A No shadowing
B Shadowing, one interferer
Exact and approximated moments for one single interferer and for multiple interferers
No shadowing (σ_{dB} = 0 dB)  Intense shadowing (σ_{dB} = 12 dB)  

Exact  Approximated  Exact  Approximated  
$E\left\{\left({G}_{n}\right)\right\}$  1  1  1  0.990 
$E\left\{{\left({G}_{n}\right)}^{2}\right\}$  2  2  4.138 · 10^{3}  1.119 · 10^{3} 
$E\left\{{\left({G}_{n}\right)}^{3}\right\}$  6  6  53.127 · 10^{9}  13.246 · 10^{6} 
$E\left\{\left(G\right)\right\}\left(FR1\right)$  17.25  17.10  17.25  17.08 
$E\left\{\left(G\right)\right\}\left(FR3\right)$  1.857  1.857  1.857  1.855 
C Shadowing, multiple interferers
We now evaluate the MCP method developed in Section IIIB.3. We have determined that 20,000 iterations of the base MCP algorithm guarantee that the 1storder moment computed from any typical set (whatever σ_{dB} value is considered) converges to its exact value (13). Table 2 presents the values of the 1storder moment of G, both exact (analytical) and approximated (computed from the typical set). We can see that the proposed method performs very well for the whole range of σ_{dB} values.
V Statistical model
In Section III, we developed analytical and numerical methods to build up a good approximation of the histogram of the interference gain G. In this section, we aim at using this result to elaborate a statistical model for G, i.e., a closed form expression of the probability law, characterized by the shadowing parameter σ_{dB}. This task is challenging in that one single parametric law is required, that is valid for propagation environments which considerably vary depending upon the shadowing phenomenon (parameter σ_{dB}), and that is applicable to various frequency reuse scenarios (FR1 and FR3).
We initialize the modeling process by extracting useful information from a careful analysis of the histograms of the interference gain G (see Figures 10, 11). We first note that G is a positive continuous r.v. We then observe that all curves are asymmetric, and this property is even more pronounced for large values of σ_{dB}. In this case, G's pdf's also have a sharper peak and a longer, fatter tail, the last of which being a characteristic of heavytailed distributions (a.k.a. power distributions), as already mentioned.
Due to the strongly skewed nature of the interference gain distribution for large σ_{dB}'s, a powertype statistical model turns out to be suitable here.
Coefficients a_{ i }, i = 1, 2,..., 6, of the empirical laws of parameters η, α, k, and β (FR1 and FR3 scenarios)
FR1  FR3  

a _{ 1 }  a _{ 2 }  a _{ 3 }  a _{ 4 }  a _{ 5 }  a _{ 6 }  a _{ 1 }  a _{ 2 }  a _{ 3 }  a _{ 4 }  a _{ 5 }  a _{ 6 }  
η  4  0  1  1  1  1  0  1  1  1  1  1 
α  0.93  0.87  65  1  7.2  3.2  0.38  0.94  39. 90  2.00  8.30  3.00 
k  0.65  2.18  3.3  0.39  4.75  2.06  0  12.70  2.35  2.07  11.00  6.47 
β  0.04  16.44  13.45  9  6.35  2.56  1.81  24.35  3.60  2.77  1.77  1.31 
Coefficients a_{ i }, i = 1, 2,..., 6, of the empirical laws of parameter x_{ t }(FR1 and FR3 scenarios)
a _{1}  a _{2}  a _{3}  a _{4}  a _{5}  

FR1  61.56  6.06  1.84  5.27  2.51 
FR3  1.71  5.10  1.89  6.40  2.30 
VI Conclusion and future work
In this paper, we have proposed a methodology to estimate the statistics of the intercell interference power in the downlink of a multicellular network. In a propagation environment subject only to path loss and multipath Rayleigh fading, we have established an accurate approximated analytical expression for the interference power distribution. Then, considering the combined effects of path loss, lognormal shadowing and Rayleigh fading, we have proposed a semianalytical method for the estimation of the pdf of the interference power. Finally, we have developed a statistical model parameterized by the shadowing parameter σ_{dB} and valid on a large range of values ([0, 12] dB). It is our hope that the methods described in this paper are sufficiently detailed to enable the reader to apply them to other types of environments.
A future work will pertain to improving the statistical interference power model by more closely linking the proposed model developed for a combined Rayleigh fadinglognormal shadowing environment to the 'exact' analytical formula obtained in the case where only Rayleigh fading was considered. Another perspective is to apply the proposed methods to other wireless network topologies (e.g., ad hoc networks,...).
Appendix
A Normalized channel power gain
where r_{ n } represents the distance between UT_{0} and AP n (distances r_{ n } are functions of UT_{0}'s position within its cell), and H_{pl,n}(rn), G_{f,n}and G_{s,n}represent the path loss, multipath Rayleigh fading and shadowing components, respectively. We now further describe these last three components.
where (29) derives from (24) and (28).
B Computation of moments for one interferer
Replacing (31) and (32) in (30) leads to (11).
C Computation of moments for multiple interferers
where the following notation is used:

a = (α_{1}, α_{2},..., α_{ N }), α_{ n }∈ ℕ, n = 1, 2, dots, N, is an Ndimensional vector whose sum of components is$\lefta\right\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}=\sum _{n=1}^{N}{\alpha}_{n};$

the multifactorial a! is such that$a!=\prod _{n=1}^{N}\left({\alpha}_{n}!\right);$

the variable Z^{ a }is defined as follows:${Z}^{a}={\left({\lambda}_{1}{G}_{f,1}{G}_{s,1}\right)}^{{\alpha}_{1}}{\left({\lambda}_{2}{G}_{f,2}{G}_{s,2}\right)}^{{\alpha}_{2}}\cdot \cdot \cdot {\left({\lambda}_{N}{G}_{f,N}{G}_{s,N}\right)}^{{\alpha}_{N}}.$
Using (30), we can further develop (33), which gives (12).
D Computation of correction factors
We determine the correction factors used in the MCP method described in Section B. Recall that the technique consists, for noncompelled links, in randomly selecting intervals from a subset containing only the $\mathcal{J}$ highestprobability (i.e., smallestamplitude) intervals. But, as highamplitude intervals never appear in this random process, small amplitudes get overweighted in noncompelled links, which must be compensated in compelled links, where small (resp. large) amplitudes need to be underweighted (resp. overweighted), in such a way that the 1storder sampled moment converges to its exact value. Thus, in order to satisfy the mean constraint, an underweighting multiplicative factor, denoted f^{}, is applied to amplitudes of the $\mathcal{J}$ first intervals of compelled links; similarly, an overweighting multiplicative factor f^{+} is applied to amplitudes of the last $N\mathcal{J}$ intervals. We now compute these two correction factors.
is the exact mean (13).
Note that we have f^{+}> 1 and, as α ≿ 1, f^{} ≾1.
Endnotes
^{a}As this paper will focus on power gains only, the term power will then be omitted in subsequent paragraphs. ^{b}To produce moments of the same accuracy, the traditional uniform partitioning approach would require about ℓ = 900 × 10^{25} points. ^{c}Two interval combinations of the same rank j are supposed to be orthogonal because of the high number of points in each interval (P = 900), which guarantees the independence of permutations. ^{d}The term 'panel' refers to survey panels used by polling organizations. ^{e}The probability set is obtained by normalizing the set of weights. ^{f}We recall that the mean $E\left\{G\right\}$ is of particular importance because it is proportional to the average interference power. ^{g}Note that, for the sake of simplification, each Pelement interval is reduced to its center of massdenoted g_{ j } .
Declarations
Authors’ Affiliations
References
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