Cell-Free UAV Networks: Asymptotic Analysis and Deployment Optimization

Recently, cell-free (CF) architectures, in which every user can potentially communicate with every base station, have received a lot of attention. This paper considers the uplink of fully and partially centralized CF networks where unmanned aerial vehicles serve as flying base stations (FBSs). A subset of FBSs participates in the reception of each user and a subset of users is received by each FBS. Deterministic equivalent expressions, exact asymptotically in the subset sizes and approximate for finite dimensions thereof, are derived for the spectral efficiency under Rician fading. Capitalizing on these expressions, the FBS deployment problem is investigated for different receiver architectures. The nonconvex deployment problem, tackled through a combination of gradient-based and Gibbs sampling algorithms, results in a superior performance with respect to a square grid deployment; this superiority extends to the minimum and aggregate spectral efficiency for both fully and partially centralized cell-free networks.

for the sake of scalability, subsets of users communicate with subsets of base stations, and the central processing goes one step beyond cooperation. By leaving behind the concept of a cell, it becomes possible to (i) provide a much more uniform degree of service, (ii) turn interference into useful signal, and (iii) further densify the network. The foregoing works, and references therein, tackle different subproblems within CF structures, e.g., power allocation, subset creation, or pilot assignment, but a major opportunity remains: optimizing the base station deployment.
Flying base stations (FBSs) embodied by unmanned aerial vehicles (UAVs) are attractive in that respect. Also, FBSs provide superior coverage and can be deployed quickly and on-demand; this latter aspect is especially relevant in situations such as natural or man-caused disasters, when the fixed infrastructure is damaged or outright non-existent. The challenge of deploying FBSs under various optimality criteria has received considerable attention [25], [26], [27], [28], [29], [30], [31], [32], [33], [34], [35], [36], [37], [38], [39]. However, with the exception of some recent works such as [21] and [24], the UAV literature is restricted to the cellular paradigm and, to the best of our knowledge, the deployment of UAVs as FBSs in CF networks has not been thoroughly studied. This paper aims at filling that gap.
Compared with terrestrial CF networks, their aerial counterparts present major differences and new challenges. First, air-to-ground links are dominated by the line-of-sight (LoS) component [40], [40], [41] and thus a Rician channel model is needed for these scenarios. This results in larger channel coherence bandwidths relative to those of ground networks [42], [43], [44]. The phase of the LoS component, often modelled deterministically to reflect the tracking and correction effected by the receiver's phase lock loop, may have to be modelled stochastically to reflect drifting [21], [22], [23]. Second, large arrays are not feasible, which precludes the deployment of massive MIMO; however, smaller directional antennas can be carried onboard [25], [27], [45]. Finally, the computational load at the cloud radio access network (C-RAN) gives rise to a tension between fully and partially centralized CF structures. While the performance of fully centralized systems is superior, certain functionalities such as large matrix inversions or fast channel estimation may represent an excessive burden. Partially centralized systems where some of these tasks are distributed may therefore be preferable whenever the C-RAN is subject to certain constraints. This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ Focusing on the FBS deployment problem, ingredients that are important, namely imperfect channel state information (CSI) and directional antennas, are explicitly accounted for in this paper. The analysis entails considering a large number of FBSs and ground users (GUs). However, as a network-wide receiver is not scalable, only a subset of FBSs can participate in the decoding of each user and each FBS can only be involved in the reception of a subset of users. Elaborating on certain random matrix theory results [46], [47], [48], [49], [50], deterministic equivalents (finite-dimensional approximations that become exact asymptotically in the subset sizes) for the spectral efficiency under either a deterministic or a random model for the LoS phases are provided. Various receiver architectures are entertained, namely fully centralized minimum mean square error (MMSE) [15], [16], [17] and fully/partially centralized maximum ratio combiner (MRC) [12], [13], [14], [15]. Capitalizing on these deterministic equivalents, the FBS deployment optimization is studied. Specifically, and with the aim of increasing fairness, the stated objective is to maximize the minimum spectral efficiency across the network. Although the problem is nonconvex, the combination of gradient based (GB) and Gibbs sampling (GS) techniques [51], [52] results in remarkable gains in terms of minimum and sum spectral efficiency. The main contributions of the paper can be summarized as follows: • The uplink of CF UAV networks is analyzed, including imperfect CSI, MMSE/MRC combining, and a realistic antenna radiation pattern for UAVs. • Linear MMSE and MRC receivers are formulated and novel asymptotic expressions are provided for the spectral efficiency over Rician channels with deterministic and random phases in the LoS components. • The CF UAV deployment problem is studied. In particular, the maximization of the minimum GU spectral efficiency is formulated. Capitalizing on the obtained expressions for the spectral efficiency, analytical forms are provided for the gradients thereof. Given the nonconvexity of the problem, the gradient updates are combined with the GS technique to avoid low-quality local solutions. • The relationship between deployment gain and geometry is established. Extensive simulations show the dependence of the former on the latter, analyzing the impact of the number of FBSs and GUs, and of the directivity of the receiver antennas, among others elements.
The remainder of the paper is organized as follows. Section II presents the complete system model of the CF UAV network and the communication process. Fully and partially centralized CF UAV architectures are studied in Sections III and IV, respectively. Section V focuses on the deployment optimization problem while numerical results are presented and discussed in Section VI. Concluding remarks are set forth in Section VII.
Notation: lowercase letters, lowercase bold letters, and capital bold letters denote scalars, vectors, and matrices, respectively. The Hadamard product is indicated by •. A circularly symmetric complex Gaussian random variable (r.v.) is denoted by N (a, b) where β 0 and κ are the path loss at a reference distance of 1 m and the path loss exponent, respectively, while d k,m denotes the distance. The Rician factor is for environment-dependent parameters A 1 and A 2 [41] whereas a k,m ∼ N (0, 1) accounts for the small-scale fading. The antenna gain at the mth FBS is where α m controls the trade-off between coverage and directivity [45]. The generalization to multi-antenna FBS will be straightforward if the fading is IID within each FBS, as then a multi-antenna FBS can be regarded as multiple collocated single-antenna FBSs. If the fading is not IID within each FBS, then the problem formulation will be the same, with the final results affected by the correlation statistics. If the LoS phase is perfectly tracked by the FBS, g k,m ∼ N (g k,m , r k,m ) with and Alternatively, if the LoS phase component is more appropriately modelled as random, i.e. ψ k,m ∼ U[0, 2π], the channel reduces to a zero-mean r.v. with

A. Channel Estimation
The channel coherence T c depends on the maximum UAV velocity, v max , the carrier frequency f c , and the speed of light, c, as T c ≈ c/fc 2vmax Other reasonable figures for these parameters will result in similarly large coherence values. Therefore, a large number of orthogonal pilot dimensions are available and pilot contamination can be kept to a minimum [20]. Thereby neglecting contamination, upon observation at the mth FBS of the pilot transmitted by GU k, the linear MMSE channel estimateĝ k,m satisfies g k,m =ĝ k,m +g k,m , where [54], [55] with for given τ and p, denoting the pilot length and transmit power, respectively, while σ 2 is the noise power at the receiver. The uncorrelated error followsg k,m ∼ N (0, c k,m ) with c k,m = r k,m − γ k,m . Ultimately, if the LoS component experiences a random phase, the channel estimate is zero-mean and γ k,m is as in (10), with

B. UL Data Transmission
On a given time-frequency resource, the uplink channel matrix is where g k ∈ M×1 is the channel between GU k and all FBSs, satisfying G =Ĝ +G, withĜ andG being the channel estimation and error matrices, respectively. The subset of FBSs participating in the reception of each GU is determined by the binary matrix M (s) = (m (s) 1 , . . . , m (s) where U m is the set of GUs regarded as signal by the mth FBS. Its complementary matrix is M (i) = 1 − M (s) , with nonzero entries indicating the GUs that each FBS treats as noise. Pooling the observations from the M FBSs, where x = ( √ ps 1 , . . . , √ ps K ) T with unit power symbols s k , p denotes the transmit power, and noise n ∼ N (0, σ 2 I).
The effective noise v is zero-mean with covariance Σ = {vv * } = D 1 + D 2 + σ 2 I given and When the LoS components' phases are random, the channel estimates are zero-mean and the above holds with only minor modifications in the statistics of the involved terms.
Based on the C-RAN capabilities, two classes of architectures emerge, namely fully centralized (FC) and partially centralized (PC), as depicted in Fig. 1. While the former is superior in performance, the latter is suitable for settings in which some operations are best distributed, e.g., when the C-RAN is itself a flying/orbiting device or when the network of FBSs operates in an ad hoc fashion. In the sequel, we study FC networks where the FBSs convey high-resolution complex observations and PC networks where the FBSs convey the linear combination of data observations with locally gathered channel estimates.

III. FULLY CENTRALIZED CF NETWORKS
In FC networks, FBSs share the pilot observations, z m , and the collected data signals, y m with the C-RAN.
. . , M be the subset of FBSs involved in the reception of GU k at the C-RAN. From the rows of y whose indices are in F k , we obtain the |F k | × 1 vector where whileĜ k ∈ |F k |×K and v k ∈ |F k |×1 . The combiner that maximizes the SINR is the |F k | × 1 MMSE filter [16] whereĝ k ∈ |F k |×1 and Σ k are the downsized versions of the original M -dimensional channel estimate of GU k and the corresponding covariance of the effective noise. From (23), the SINR at GU k is An alternative combiner is the MRC, w k =ĝ k . For given channel realizations, GU k then attains From whichever form applies for the SINR, the FC ergodic spectral efficiency of GU k is where τ τc accounts for the pilot overhead.

A. Large-Dimensional Analysis
To evaluate (26), we consider the large-dimensional regime, M, K → ∞ with finite M/K, and investigate the convergence of (24) and (25) to nonrandom limits. This requires that the subsets themselves grow with the network, i.e., |F k |, |U m | → ∞ ∀ k, m, accounting for the non-zero entries in the random matrices. The premises for this convergence are different depending on whether the LoS phases are deterministic [46], [47] or random [49], [50], but they all need the covariance matrix of the channel estimate between User k and all FBSs, defined by and G k to satisfy some technical conditions. Specifically, we require that (a) the inverse of the resolvent matrix in (24) exists, which is ensured by the presence of Σ k , and that (b) Γ k and G have uniformly bounded spectral norms. The latter condition prevents the received energy from concentrating on a fixed subset of dimensions as |F k |, |U m | → ∞ and thus the analysis in the sequel applies under this premise. Let us first discuss the MMSE-based receiver with deterministic LoS phases, corresponding to (24). Recent works have shown the asymptotic behavior of bilinear forms based on the resolvent of large random matrices with nonzero mean, as is the case in (24), when the variance profiles are separable [46], [47]. In cellular MIMO, correlations among the BS antennas are common to all users, based on which asymptotic expressions for M, K → ∞ over Rician channels have been given [48]. In CF networks, however, the correlations among antennas at distinct BSs are different for every user, i.e., [Γ k ] m,m = [Γ k ] n,n ∀m = n. Therefore, the variance profile is non-separable and none of the previous analyses is valid for CF networks when using centralized MMSE with deterministic LoS phases.
On the other hand, when the LoS phase is random and uniformly distributed on [0, 2π], the channel estimates are zero-mean. Therefore, application of the derivations in [49] and [50] to a CF setup yields the following result.
The coefficients e j,k are obtained iteratively with e j,k = lim n→∞ e Proof: Details on how (29) emanates from [49] and [50] can be found in Appendix B.
Next, we turn our attention to the MRC case in (25), considering both deterministic and random phases in the LoS components.
Proposition 2: For |F k |, |U m | → ∞ ∀ k, m with MRC subset combining and deterministic phases in the LoS components, → 0 with SINR k given in (31), shown at the bottom of the next page.
Proof: See Appendix C.

m and MRC subset combiner with random phases in the LoS components,
Proof: The proof follows the procedure in Appendix C, by setting the mean components to zero, i.e., g k,m = 0 ∀k, m, and thus with r k,m = 2(α m + 1) β 0 . Note that γ k,m depends on r k,m as derived in (10).
From the continuous mapping theorem [56], the ergodic spectral efficiency in (26) satisfies given the applicable form for SINR k . The deterministic equivalents SINR k and SE fc k depend only on the channel statistics.
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B. Fully Centralized Problem Formulation
Armed with the deterministic equivalent expressions, we can turn to optimizing the FBS deployment. Defining the set of FBS locations by Q = {q m , m = 1, . . . , M}, and with the aim of increasing fairness in the network, we can formulate the maximization of the minimum spectral efficiency as which is nonconvex. We combine GB and GS updates to iteratively relocate the FBSs. Given that the optimization of the spectral efficiency is equivalent to that of the SINR, we obtain the gradients w.r.t. the latter from Props. 1, 2 and Cor. 1.
As the combination of GB and GS is common to the partially centralized case (see Sec. V), in the remainder of this section we only derive the gradient expressions for later use. 1) FC MMSE Deployment: For fully centralized MMSE combining at the C-RAN and random phases in the LoS component, the optimization boils down to For fixed e i,k , the gradient w.r.t. the mth FBS location is where and ∇ q m Den m is the gradient of (37). The derivatives of γ k,m and r i,m w.r.t. x m , can be found in [1] and the derivatives w.r.t. y m can be obtained similarly, altogether completing (36).
2) FC MRC Deployment: For fully centralized MRC combining at the C-RAN and deterministic phases in the LoS component, the optimization problem reduces to with SINR k given in (31) on top of this page. The gradient w.r.t. the mth FBS is where Num m and Den m are the numerator and denominator of (31), respectively, and ∇ q m Num m and ∇ q m Den m are the corresponding gradients. The computation of (39) w.r.t. x m is sketched in Appendix D, and a similar process yields the derivatives w.r.t. y m and thus the overall gradient in (39). All the expressions are simplified further if the LoS components have random phases as in Cor. 1, since then g k,m = 0 ∀k, m.

IV. PARTIALLY CENTRALIZED CF NETWORKS
In PC networks, some combining operations are performed locally at every FBS such that the C-RAN need not be privy to the channel estimates. Each FBS requires channel estimates for the GUs served by that FBS and does not forward the estimates to the C-RAN.
Once the mth FBS observes y m , it is locally combined with w k,m to yield a signal estimatex k,m = w * k,m y m . Next,x k,m is sent through the fronthaul. Once {x k,m : m ∈ F k } are at the C-RAN, they are combined with v k,m to provide the final signal estimatê The equivalent noise n k, where n m is the vectorized form of n k,m . We also define the nonzero-mean vector where v k is the vectorized form of v k,m . Proposition 3: In a PC network, the highest spectral efficiency achievable by GU k is where SINR k is provided in (44), shown at the bottom of the next page. Proof: See Appendix E. The combiner required for (44) is MMSE-based, yet simpler large-scale-based solutions could be applied, specifically local MRC (w k,m =ĝ k,m ) with equal gain combining (EGC) at the C-RAN (v k = 1).

A. Large-Dimensional Analysis
Expressions can be derived for |F k |, |U m | → ∞ with local MRC and ECG at the C-RAN under both deterministic and random phases in the LoS components.
→ 0 with SINR k defined in (45), shown at the bottom of the next page, where Σ k = (M Proof: See Appendix F. Proof: The proof follows the procedure in Appendix F, only with g k,m = 0 ∀k, m and thus with r k,m = 2(α m + 1) β 0 . Note that γ k,m depends on r k,m as derived in (10).
From the continuous mapping theorem [56], the spectral efficiency in Prop. 3 satisfies given the applicable form for SINR k .

B. Problem Formulation
The maximization of the minimum spectral efficiency, hence the minimum SINR, entails with SINR k defined in (45) where Num m and Den m are the numerator and denominator of (45), respectively, with ∇ q m Num m and ∇ q m Den m the corresponding gradients. The computation of (49) is akin to that of (39), which is detailed in Appendix D. If random phases are considered in the LoS components, a similar optimization problem can be formulated based on Cor. 2.

V. GB-GS DEPLOYMENT
Equipped with the gradients for each of the scenarios, at iteration number j, the GB updates are where ρ (j) is a decreasing function of j for convergence reasons. Due to the lack of convexity, the updates in (50) may quickly converge to low-quality solutions. Furthermore, an exhaustive search on the 2D plane at altitude H would be computationally prohibitive. Therefore, affordable techniques should be investigated. A potential alternative is simulated annealing [1]. However, the exploration in simulated annealing is limited to only one possible direction. Although the results are promising, exploring other locations may result in a better cost function. We thus resort to stochastic optimization, and more particularly to the GS technique [51], to improve the results. Given a discrete search space Θ, GS aims at solving The FBS locations are iteratively updated according to a certain probability distribution [51]. Precisely, at Iteration t, the coordinates of the FBSs serving the GU with the lowest SINR (t) are updated; the index of this GU is denoted by k   can be calculated as [51] Pr where γ is a fixed parameter and Θ t+1 represents the possible locations that FBS m can explore at iteration t + 1. To reduce the search space, we limit the number of possible neighbors to six, i.e., |Θ t+1 | = 6 (see Fig. 2). The six alternatives are to stay, move north, move south, move east, move west, and move in the direction of the gradient in (50). Note that the gradient can point at any location in the 2D plane at altitude H. Hence, the search space is continuous. Moreover, at each iteration, matrix M (s) is updated. A summary of the iterative process is included in Alg. 1 where and I max control when to stop.
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Find the user with lowest cost function, k (t) min and the FBS connected to it, The reduced search space with six possible locations is created and denoted by Θ t+1 .
The cost function at the six possible new locations is computed:

VI. SIMULATION RESULTS
For the purpose of performance evaluation, we consider a 1 km 2 universe, wrapped around to avoid boundary effects. Unless otherwise specified, Table I lists the typical parameters used in the simulations, selected based on the CF and UAV literature [13], [57], [58], [59], [60]. The channel coherence is larger than that of ground networks because aerial settings exhibit higher coherence bandwidths; this, in turn, enables channel estimation with low pilot contamination. Henceforth, contamination is thereby neglected, and readers interested in results that include it are referred to [1]. Unless otherwise specified, and, consistently with the neglect of contamination, we consider a higher value of τ ; precisely, τ = 200 for a 3.2% pilot overhead. The [m, k] entry of M (s) is 1 if d k,m ≤ R max for R max = 400 m, which ensures connectivity to multiple FBSs per GU. Regarding the GB-GS implementation, ρ (t) must be a decreasing function of t for convergence reasons, in this case ρ (t) = 10 · 1.005 −t though other choices would work as well. The value of D GS is set to D GS = 0.6 m (see Fig. 2) while γ = 30, = 0.01 and the maximum number of iterations is I max = 500. Varying these parameters would shift the tradeoff between speed of convergence and final cost function; for example, diminishing D GS increases the resolution of the search space and thus the number of iterations, while increasing γ improves the chances of the GS algorithm choosing the direction that yields the highest cost function, lowering the stochastic component of the method. Additionally, the entries of M (s) are updated at every iteration of the GS-GB algorithm following the aforementioned distance-based rule. The GU locations abide by a Poisson Point Process (PPP) and we test our optimization algorithm over 100 deployments, sufficiently many to gauge the performance. The optimized deployment is denoted by a-opt (after optimization) while a square grid FBS deployment, denoted by b-opt (before optimization), serves as a benchmark.

A. FC Networks
Before proceeding to the deployment optimization in FC networks, we measure the gap in terms of average SINR between the deterministic phase (DP) and the random phase  (RP) models for the LoS components. This is presented in Fig. 3a for different values of H, given M = 100, K = 75, and FC MMSE reception. The gap is only 0.1-0.5 dB, as there are enough pilot symbols to accurately estimate the channels. The high coherence of aerial channels enables large values for τ , in contrast with [22] and [23], where extremely low values are assumed.
Then, Fig. 3b shows that relatively small networks, with their correspondingly small subsets, suffice for the deterministic equivalent in Prop. 1 to be accurate.
The counterpart of Fig. 3b for MRC reception is presented in Fig. 4 for DP and RP models.   increase in sum spectral efficiency. Given the space limitations, only the DP model is presented, although similar results are achieved for the RP model. For MMSE, the increase in sum spectral efficiency is between 30% and 45% while for MRC the range of improvement varies between 15% and 40%. Fig. 11 examines whether augmenting the optimization to include also the altitude can further improve the deployments. Then, twelve additional locations, same six 2D spots with a higher and lower heights, are explored at each iteration, with the altitude being constrained to 20 ≤ H m ≤ 80 m and the same D GS applied vertically. Given the space limitations, only results for the RP MMSE case are presented, over different values of κ. The improvement in terms of minimum SINR remains in the range of 15 to 25 dB, hence adjusting the altitude does not seem to provide substantial additional gains once the horizontal position is already being optimized.
Finally, in Sec. V it is shown that the number of complex multiplications at each iteration is O |F k (t) min |K . In Fig. 12, the number of complex multiplications is plotted after averaging over the number of iterations varying K and for different M under MMSE reception. Note that M , and therefore |F k |, controls the slope while the growth is linear in K.

B. PC Networks
Prop. 4 Fig. 16 shows the increase in sum spectral efficiency for the DP model, which ranges between 15% and 40%.

VII. CONCLUSION
This paper has considered fully and partially centralized architectures for the uplink of aerial cell-free networks. Two  Capitalizing on these deterministic equivalents, a deployment optimization problem has been proposed for each of the architectures. The analytical gradients have been obtained and, given the lack of convexity, a combined GB-GS approach has been followed. The resulting improvements in minimum  Avenues for follow-up work include incorporating power control [61], accounting for the residual effects of pilot contamination, considering multiantenna transceivers, or studying the impact of the user location distributions and how such distributions could be estimated dynamically.  where with coefficients e k = lim n− →∞ e with initial values e   and Ω k = |F k |, Ω k . Then, (24) can be written as In our case, the role of HH * + S + zI M ) −1 in Theorem 1 is played by Ω k . There is a direct mapping between the terms in the aforementioned theorem and our problem, namely (i) D = Γ k p, (ii) R j = Γ j p, and (iii) S + zI M = 1 |F k | Σ k with matrix T k following the structure of T in Theorem 1, i.e., where the contribution of GU k is removed as in Ω k . The necessary coefficients can be calculated as e j,k = lim n→∞ e where m (s) k,j,m = ½{j ∈ U m for m ∈ F k }. The fixed-point algorithm can be used to compute e (n) j,k and has been proved to converge [49]. Finally, since matrices Γ k and T k are diagonal, (60) can be written as Authorized licensed use limited to the terms of the applicable license agreement with IEEE. Restrictions apply.
which yields the expression in Prop. 1 after some straightforward algebra.
APPENDIX C PROOF OF PROP. 2 From (25), we can compute the deterministic equivalence of each term in the numerator and denominator separately, letting |F k |, |U m | → ∞. We divide each by 1/|F k | 2 and recall that g k,m = g k,m + ζ k,m with the distribution of ζ k,m derived in (10). Then, the signal satisfies as a consequence of [49,Lemma 4]. Therefore, from the continuous mapping theorem, For the interfering terms, a similar procedure is used. Applying [49,Lemma 4] twice, the interfering terms follow Finally, following a similar approach for the noise, Plugging the derived deterministic equivalents into (25), Prop. 2 is obtained.

APPENDIX D GRADIENT COMPUTATIONS
We first calculate some partial derivatives needed for (38). We do so with respect to x m , and a similar procedure can be used for y m . First, note that (3) can be written as with and The calculation of the derivatives for each term is straightforward, involving only polynomial terms that depend on g i,m and γ i,m , whose derivatives are derived in (69) and [1], respectively. After some algebra, the derivative of each interfering term is obtained. Finally, the noise can be decomposed as As for the interference terms, obtaining the derivative w.r.t. the FBS locations from (74) is tedious but straightforward. Combining the derivatives obtained from (73) and (74), the partial derivative w.r.t. x m of Den m arises. A similar procedure can be followed to obtain the derivative w.r.t. y m and, with that, the overall gradient.

APPENDIX E PROOF OF PROP. 3
In PC networks, only the FBSs have access to the channel estimates. Thus, the C-RAN regards v * k {µ k,k } as the true channel and the signal model therein iŝ  The second, third, and fourth terms, which are uncorrelated, are pooled as effective noise. As uncorrelated Gaussian noise represents the worst case in terms of the achievable spectral efficiency, we obtain the lower bound in Prop. 3.

APPENDIX F PROOF OF PROP. 4
As for the centralized MRC, we compute the equivalents for each of the terms in (44). Again, we first divide both numerator and denominator by 1/|F k | 2 . Then, the signal term satisfies 1 For the interference terms, where R i = diag{m Authorized licensed use limited to the terms of the applicable license agreement with IEEE. Restrictions apply.
as a consequence of [49,Lemma 3]. A similar procedure can be used to derive the contribution of the noise, After plugging the various terms into (44), Prop. 4 is obtained.