1. Introduction
The adaptive networks work with several agents distributed in an area gathering data in order to estimate a desired entity [1-7]. All of these agents have access to some measurements and with the help of their processing unit, they perform local estimations. These estimations are usually performed by simple adaptive algorithms such as least mean square (LMS) in order to reduce processing power. But despite these simple algorithms, the use of complicated cooperation schemes makes adaptive networks a powerful tool for distributed estimation. Several cooperation techniques are presented for the collaboration of agents, but the most useful ones are incremental and diffusion strategies [2].
Diffusion strategy is a more practical mode of cooperation because finding a Hamiltonian cycle for the network topology of the incremental mode is sometimes not possible [5]. Recently, the diffusion strategy has been used in several applications including sparse system identification [6] and multitask networks [8] where the adaptive network is assigned to perform estimations for more than one desired variable.
Although the performances of adaptive networks have been considered in several different environmental conditions like: noisy links ([5], [15] and [16]), imperfect links ([3], [5]) and fading channels ([1], [2] and [4]), we can still find unchecked scenarios in this problem. Also, some new challenges are continually being presented for wireless channels in future wireless communication systems [11-13]. The performance of the incremental strategy in Rayleigh fading condition has been considered in [1], while the authors of [2] have considered the performance in the same Rayleigh channel for diffusion strategy. However, in [2] the exact impact of fading channel on the adaptive network has not been considered directly. Instead, the authors have presumed that the channel estimation is available in sensor nodes and equalization is possible in order to delete the effect of fading. In [14] the use of adaptive networks in channel estimation has been considered. However, in [14] the effect and estimation of Rayleigh fading have only been analyzed with Incremental strategy.
Based on previous studies, in this paper, we examine the direct effect of fading channels on the performance of diffusion adaptation strategy. Besides the Rayleigh fading model, we also consider the scenario of Rician channel. The assumption of Rician model for links among nodes is a more practical one than the Rayleigh fading model because, in this model, we can usually consider a powerful link between adjacent nodes. In this case, we can expect that the performance of networks becomes much better than the previously presented results in [1] and [4]. Our analysis showed that unlike other cases, the fading effect can be more reduced by CTA algorithm and in this case, the performance of the diffusion CTA algorithm is superior to the performance of ATC diffusion algorithm. Also, we showed that the performance of both diffusion schemes in Rician fading is better than Rayleigh fading because in Rician case, as we mentioned, there is a powerful line of sight between the transmitter and the receiver nodes and this phenomenon results in a more accurate data communication between them. The rest of this paper is designed as follows: In part II, we introduce the adaptive model and problem statement. Furthermore, the details of the impact of the fading channels on the diffusion algorithms are presented in this part. In section III. We reviewed the fading models that are used in this paper. In part IV, we presented the simulation results to investigate the channel effects. And in part V, we concluded our paper and suggested further insights.
Notation: We utilized small bold-faced letters to show vectors and capital bold-faced letters for matrices. The operator [⋅]∗ shows complex conjugate for scalar values and Hermitian transposition for matrices. Furthermore, \(\mathbb{E}[\cdot]\) is used for presenting the mathematical expectation and the operator ‖⋅‖ is utilized for showing the Euclidian norm of a vector.
2. Problem statement
Usually, for adaptive networks we assume a linear relation linking the desired values and the input data vectors[1]:
\(d_{k}(i)=\boldsymbol{u}_{k, i} \boldsymbol{w}^{o}+v_{k}(i)\) (1)
where dk(i) represents the desired values, uk,i represents the regressor vectors, and vk(i) shows the measurement noise that follows Gaussian distribution. Our purpose is to estimate the optimal weight vector wo by minimizing the global cost:
\(J^{g l o b}(\boldsymbol{w})=\sum_{k=1}^{N} \mathbb{E}\left|d_{k}(i)-\boldsymbol{u}_{k, i} \boldsymbol{w}\right|^{2}\) (2)
By minimizing this cost function with respect to w we seek to achieve the optimum weight vector as [7]:
\(\boldsymbol{w}^{o}=\left(\sum_{k=1}^{N} \boldsymbol{R}_{u, k}\right)^{-1}\left(\sum_{k=1}^{N} \boldsymbol{R}_{d u, k}\right)\) (3)
where \(\boldsymbol{R}_{u, k}=\mathbb{E}\left[\boldsymbol{u}_{k, i}^{*} \boldsymbol{u}_{k, i}\right] \text { and } \boldsymbol{R}_{d u, k}=\mathbb{E}\left[d_{k}(i) \boldsymbol{u}_{k, i}^{*}\right]\)represent the correlation of inputs (or regressors) and the cross-correlation of inputs and desired values. Due to the fact that we do not have these correlation quantities, we must use the iterative or adaptive schemes to arrive at the results. Up until now, many centralized and distributed adaptive algorithms have been presented to solve this problem [7, 9]. In distributed algorithms, each sensor node collects data and applies local estimations of the unknown vector namely \(\boldsymbol{\psi}_{k}^{(i)}\), then sends this estimation to its neighboring nodes. However, through this sharing of information, the neighboring nodes receive these local estimations with some deficiencies. These deficiencies may occur in the shape of channel noise or fading. In Fig. 1, the additive noise signals and channel coefficients are present in all the links between nodes.
The difference between incremental and diffusion distributed strategies is in this data sharing process. In the incremental strategy, each node only needs one local estimation of its immediate neighboring node to perform its own estimation. But, in the diffusion strategy, each node collects several local estimations from neighboring nodes and by combining them, performs estimation. This shows that the diffusion strategy is more prone to the channel deficiencies. The performance of these strategies must be evaluated and compared in different scenarios.
For assessing the diffusion network estimation performance, two main error criteria are proposed in the literature namely: the Excess Mean Square Error (EMSE) and the Mean Square Deviation (MSD) [1, 7]. These values can be obtained by the definition of the following error vectors:
\(\widetilde{\boldsymbol{\psi}}_{k}^{(i)} \triangleq \boldsymbol{w}^{o}-\boldsymbol{\psi}_{k}^{(i)}\) (4)
\(e_{a, k}(i) \triangleq \boldsymbol{u}_{k, i} \widetilde{\boldsymbol{\psi}}_{k-1}^{(i)}\) (5)
where \(\widetilde{\boldsymbol{\psi}}_{k}^{(i)} \) represents the weight error vector. Also, \(e_{a, k}(i) \) denotes the apriori local error at node k and iteration i. The values of MSD and EMSE in the steady state condition are then:
\(M S D \triangleq \mathbb{E}\left\|\widetilde{\boldsymbol{\psi}}_{k}^{(\infty)}\right\|^{2}\) (6)
\(E M S E \triangleq \mathbb{E}\left|\boldsymbol{e}_{a, k}(\infty)\right|^{2}\) (7)
We only need the MSD values for presenting our simulation results. As mentioned, in the diffusion strategy each node needs to combine several local estimations of its neighboring nodes and then perform its own estimation. This process entails two different methods described as below:
2.1 The Combine Then Adapt (CTA) diffusion strategy with noisy and fading links
For the CTA strategy first of all the network nodes combine their estimated and sent data and then update their local estimation. We must take heed to the point that the combined estimates that are received estimations from neighboring nodes are affected by channel noise and fading coefficients, through the links. Therefore, we have:
\(\boldsymbol{t}_{k, l}^{(i)}=h_{k, l}^{(i)} \boldsymbol{\psi}_{k}^{(i)}+\boldsymbol{q}_{k, l}^{(i)}, \quad l \in \mathcal{N}_{k}\) (8)
where \(\boldsymbol{t}_{k, l}^{(i)}\) represents the received estimations from node to node k, \(h_{k, l}^{(i)} \) is the channel coefficient among these nodes at each iteration i and \(\boldsymbol{q}_{k, l}^{(i)}\) is the channel noise that follows Gaussian distribution with zero mean and the covariance matrix of \(\boldsymbol{Q}_{k, l}=\mathbb{E}\left[\boldsymbol{q}_{k, l}^{(i)} \boldsymbol{q}_{k, l}^{(i) *}\right]\). As we can see in Fig. 1, the channel coefficients (\(h_{k, l}^{(i)} \)) are multiplied to the exchanged data and channel noises (\(\boldsymbol{q}_{k, l}^{(i)}\)) are added to them.
Fig. 1. The diffusion adaptive network with fading and noisy channels.
The CTA algorithm formulation with the presence of the channel fading and noise is then given as:
\(\boldsymbol{\phi}_{k}^{(i-1)}=\sum_{l \in \mathcal{N}_{k}} a_{k, l} \boldsymbol{t}_{k, l}^{(i-1)}\) (9)
\(\boldsymbol{\psi}_{k}^{(i)}=\boldsymbol{\phi}_{k}^{(i-1)}+\mu_{k} \boldsymbol{u}_{k, i}^{*}\left(d_{k}(i)-\boldsymbol{u}_{k, i} \boldsymbol{\phi}_{k}^{(i-1)}\right)\) (10)
where \(a_{k, l}\), represent the combination coefficients that play an important role in the performance of the diffusion network. There are several techniques to choose these values that are described in [7]. In this paper, we used the Uniform combination method for determining these coefficients. In this method we have:
\(a_{l, k}=\left\{\begin{array}{ll} \frac{1}{n_{k}}, & l \in \mathcal{N}_{k} \\ 0, & \text { otherwise } \end{array}\right.\) (11)
Where \(n_{k} \triangleq\left|\mathcal{N}_{k}\right|\) represents the size of the neighborhood of node that can be 2 to 7. All of the neighbors of node are assigned uniformly with the same weights, \(\frac{1}{n_k}\) .
2.2 Adapt Then Combine (ATC) diffusion strategy with noisy and fading links
The next diffusion algorithm is the ATC where the network nodes first update their local estimations and then combine their estimations with the neighboring nodes:
\(\boldsymbol{\phi}_{k}^{(i)}=\boldsymbol{\psi}_{k}^{(i-1)}+\mu_{k} \boldsymbol{u}_{k, i}^{*}\left(d_{k}(i)-\boldsymbol{u}_{k, i} \boldsymbol{\psi}_{k}^{(i-1)}\right)\) (12)
For the CTA algorithm, we used the \(\boldsymbol{r}_{k, l}^{(i)}\) notation to represent the received information from node l to node k and can write
\(\boldsymbol{r}_{k, l}^{(i)}=h_{k, l}^{(i)} \boldsymbol{\phi}_{k}^{(i)}+\boldsymbol{q}_{k, l}^{(i)}, \quad l \in \mathcal{N}_{k}\) (13)
\(\boldsymbol{\psi}_{k}^{(i)}=\sum_{l \in \mathcal{N}_{k}} a_{k, l} \boldsymbol{r}_{k, l}^{(i)}\) (14)
Again, we used the Uniform combination method for determining the ak,l values in this algorithm.
3. Overview of fading channel models
To the best of our knowledge, the effects of the no wireless radio fading models other than the Rayleigh distribution have been considered in the performance of the adaptive networks [1]-[4], while it is a very important issue. In this paper, we studied the exact impact of fading channel models on the performance of diffusion adaptation strategies, here we consider two main fading channel models. In order to simulate the fading condition, we can produce the channel coefficients between each pair of nodes and separately according to the presumed model.
3.1 Rayleigh fading channel model
The fading channel coefficients can be produced by Rician or Rayleigh distributions, based on the existence or the absence of the specular signal component or the line of sight [10-11]. Fading can be modeled as Rayleigh if the multiple reflective paths are many and there is no main line-of-sight (LOS) path between the transmitter and the receiver. The fading coefficient hi at the ith time instant can be given as the combination of two Gaussian variables:
\(h_{k, l}^{(i)}=\sqrt{\left(x_{k, l}^{(i)}\right)^{2}+\left(y_{k, l}^{(i)}\right)^{2}}\) (15)
where \(x_{k, l}^{(i)}\) and \(y_{k, l}^{(i)}\) represent two independent zero mean Gaussian variables with variance \(\sigma_0^2\) and in this case the Rayleigh distribution of channel envelop is given as:
\(f_{\text {Rayleigh}}(h)=\frac{h}{\sigma_{0}^{2}} \exp \left[-\frac{\left(h^{2}\right)}{2 \sigma_{0}^{2}}\right] \quad, r \geq 0\) (16)
3.2 Rician fading channel model
In [10-11], it is mentioned that if there is a main LOS path in the communication link between transmitter and receiver, then the fading can be modeled as Rician. The fading coefficient \(h_{k, l}^{(i)}\)at the i th time instant can be given as:
\(h_{k, l}^{(i)}=\sqrt{\left(x_{k, l}^{(i)}+\beta\right)^{2}+\left(y_{k, l}^{(i)}\right)^{2}}\) (17)
where \(\beta\) is the amplitude of the specular component and we define the Rician K factor (not to be mistaken with k the node indices) as:
\(K=\frac{\beta^{2}}{2 \sigma_{0}^{2}}\) (18)
that shows the ratio of the specular to the diffuse energy, and changing its values has a significant impact on the performance of the network. In this case, the Rician distribution of channel envelop is given by:
\(f_{\text {Rice}}(h)=\frac{h}{\sigma_{0}^{2}} \exp \left[-\frac{\left(h^{2}+\beta^{2}\right)}{2 \sigma_{0}^{2}}\right] I_{0}\left[\frac{h \beta}{\sigma_{0}^{2}}\right], r \geq 0\) (19)
where \(I_{0}[.]\) is the zeroth-order modified Bessel function [11] which is shown in Fig. 2.
Fig. 2. The zeroth-order modified Bessel function.
Following the relations of (16) and (19), the difference between the Rayleigh and the Rician PDFs are shown in Fig. 3.
Fig. 3. The PDFs of Rayleigh and Rician distributions based on the Rician factor.
4. Simulation studies
After the consideration of the proposed channel fading models, here are presented the impacts of the Rayleigh and Rician channel models on the performance diffusion ATC and CTA algorithms. We designed a network with 20 nodes with the structure of Fig. 4.
Fig. 4. The structure of a diffusion adaptive network with 20 nodes.
For the topology of Fig. 4, the amount of neighboring nodes for each agent is different and it can be between 2 and 7 nodes. For the combination coefficients ak,l, we specified them due to these number of neighboring nodes. Our goal is to estimate the unknown optimal vector with the size of \(M=4 \text { like } \boldsymbol{w}^{o}=[1111]^{T} / \sqrt{4}\). The detailed features of the utilized variables in our simulations are given in Fig. 5.
Fig. 5. The statistical values of the utilized parameters.
In Fig. 6, we presumed that the variance of the measurement noise is equal to \(\sigma^2_{v,k}\), 2 for each node k and the covariance matrix of channel noise between the node k, l is equivalent to \(\boldsymbol{Q}_{k, l}=\sigma_{c, k}^{2} \boldsymbol{I}\), where I is the M × M identity matrix, and therefore we only need to know the variance values. For generating our fading channel coefficients as in (15) and (17) we used zero mean Gaussian variables with variance \(\sigma_0^2 = 1\). In this case, if we consider K for three values of 0, 5 and 7, the \(\beta\) becomes 0, 3.16 and 3.74 respectively.
First, we compare the performance of the network in a noisy environment and a noisy plus fading condition using MSD criteria. Also, in Fig. 6, we presented that the performance of the network in a noisy environment is much better than its performance in a fading environment. In all of the simulations, the performances of Both ATC and CTA algorithms are presented. The important fact is that the performance of CTA diffusion strategy is more acceptable than ATC diffusion when we have fading. We assume that this is because of the combination method of the CTA diffusion algorithm that performs adaptation process after the combination, and this allows the adaptive algorithm in each sensor to reduce some of the effects of fading in the adaptation step.
Fig. 6. Algorithms comparisons in noisy and noisy plus fading environments.
In Fig. 7, MSD curves are depicted for several cases of the Rician factor. We expected that the performance of the diffusion network relies on the value of and as the value of it increases, the power of the prominent path between nodes increase and consequently the performance of network gets better. We ran our simulations for 3 values of and presented the results for MSD of the network. It is important to mention when K = 0, the fading is Rayleigh. In this figure, we can see that the performance of the network in Rician fading is much better than Rayleigh fading and as we mentioned earlier, the Rician fading model is a better assumption for adaptive networks than Rayleigh fading.
Fig. 7. Algorithms comparisons in noisy and noisy plus fading environments.
In order to show the MSD comparisons under different Rician factor , we depict the MSD curves, where the performance of the diffusion network is presented for all rational values of factor at the same time.
In Fig. 8, we can see that as the value of factor increases, the performance diffusion network gets better. Also, in all the conditions the performance of CTA diffusion strategy is more acceptable than that of ATC diffusion scheme.
Fig. 8. Algorithm comparisons in Rician fading with different values.
5. Conclusion
In this paper, we investigated the performance of diffusion adaptive networks in the presence of Rayleigh and Rician fadings, respectively. First of all, we showed that the Rician fading for wireless adaptive networks is a more practical model than Rayleigh fading model. Secondly, we presented that the effect of fading is much more degrading for networks in comparison with noisy links. Finally, the computer simulation results demonstrated that in the presence of fading, the performance of the CTA diffusion strategy is more acceptable than ATC diffusion strategy. Other more advanced fading models can also be applied to the adaptive networks. In future works, we will work on the combining weights of diffusion strategy in order to find a suitable combination policy for reducing the effects of fading.
Acknowledgements
National Natural Science Foundation of China grants (No. 61401069), Jiangsu Specially Appointed Professor Grant (RK002STP16001), High-level talent startup grant of Nanjing University of Posts and Telecommunications (XK0010915026) and “1311 Talent Plan” of Nanjing University of Posts and Telecommunications.
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