Link Gain Matrix Estimation in Distributed Large-Scale Wireless Networks
© Jing Lei et al. 2010
Received: 9 June 2009
Accepted: 25 November 2009
Published: 4 January 2010
In planning and using large-scale distributed wireless networks, knowledge of the link gain matrix can be highly valuable. If the number of radio nodes is large, measuring node-to-node link gains can be prohibitive. This motivates us to devise a methodology that measures a fraction of the links and accurately estimates the rest. Our method partitions the set of transmit-receive links into mutually exclusive categories, based on the number of obstructions or walls on the path; then it derives a separate link gain model for each category. The model is derived using gain measurements on only a small fraction of the links, selected on the basis of a maximum entropy. To evaluate the new method, we use ray-tracing to compute the "true" path gains for all links in the network. We use knowledge of a subset of those gains to derive the models and then use those models to predict the remaining path gains. We do this for three different environments of distributed nodes, including an office building with many obstructing walls. We find in all cases that the partitioning method yields acceptably low path gain estimation errors with a significantly reduced number of measurements.
The powerful technology and market trends towards portable computing and communications imply an increasingly important role for wireless access in the next-generation Internet. Moreover, distributed and pervasive computing applications are proliferating and expected to drive large-scale deployments of embedded computing devices interconnected via wireless links. Large-scale distributed wireless networks arise in a variety of forms. Examples include sensor networks, wherein data processing is distributed among the nodes ; ad hoc mesh networks, wherein nodes act as relays for each other ; and the laboratory testbeds used to evaluate sensor and mesh network protocols . In all these cases, the operation for the network can benefit from knowing the link gain matrix, which describes the transmitter-receiver power ratio among all the nodes in the network taken pairwise.
In , we considered the use of spatial interpolation for pathloss estimation. In this approach, a subset of link gains is measured, and then we invoke the assumption of smooth spatial variations to infer all other path gains via interpolation. For the 400-node ORBIT testbed, we found that the use of spatial interpolation methods permitted reasonably accurate estimates to be obtained using only a few thousand measurements instead of 80,000. However, we report here on an alternative approach that provides even better accuracies with even fewer measurements. The key to this new approach, and what makes it novel, is that the set of all node-to-node paths is partitioned into 3 or more categories, and a separate stochastic model is derived for each. By using a suitable means of categorization, we find that each model of pathloss versus log-distance fits a simple mathematical function with a low-standard deviation of values about the fit. Specifically, we show that using 1,000 measurements, and a heuristic method for choosing which links to measure, the RMS value of the errors in estimating link gains can be kept below 3 dB. As noted, the testbed environment we study is characterized by multipath and also by obstructions on many of the paths. To test our approach, we emulate the measurements of link gains by using WiSE, a ray-tracing tool developed by Bell Labs . In addition to the perfect square grid on the ORBIT testbed, we also apply the proposed method to a similar lab with larger obstructions and an irregular node layout in a building with many obstructing walls. We find that, even for a difficult scenario with numerous obstructions, the partitioning method yields acceptably low-path gain estimation errors with a much-reduced number of measurements.
The rest of this paper is organized as follows. Models of both the environment and the node-to-node link gain (or pathloss) are given in Section 2. The new method for estimating link gains from limited measurements is described and exemplified in Section 3, and a method based on entropy is described for choosing the subset of transmitting nodes. In Section 4, two alternative, more complicated distributed network scenarios are postulated. For each, the entropy method for choosing transmitters and the new method for estimating link gains are applied, and numerical results are presented. Section 5 concludes the paper.
2. System Model
2.1. Classification of Link Gains
With the help of WiSE , we can obtain the set of all link gains for a specific environment as a function of its geometry. We have observed that in an indoor environment, the link gains deviate from the law of free-space propagation, due to the impacts of reflection, diffraction, and scattering. Furthermore, we have found that obstructed links, that is, those without a clear LOS path, usually undergo more severe attenuation than those with an LOS path, and the added attenuation caused by the obstructions is almost unrelated to the T-R separation distance .
Accordingly, the link between a given transmitter and receiver can be classified into one of several different categories according to the number of obstructing objects lying between them. A partition-based path loss analysis for in-home and residential areas at 5.85 GHz was conducted by Durgin et al. in . In this paper, we generalize their framework to distributed wireless networks and propose to estimate both the path loss exponent and attenuation factors using selectively sampled measurements.
More generally, let us assume that, in a given network of distributed wireless nodes, there are some paths between node pairs with as many as different types of obstructions. Then, according to our approach, there will be distinct categories (with one LOS category and NLOS categories) for the pathloss formula, where pathloss is the negative dB value of link gain (received power divided by transmitted power). Assume that is a conveniently chosen reference distance, which is typically meter in indoor environments; that is the pathloss at for a single direct ray free-space pathloss; that is the pathloss exponent for the th category; that denotes the pathloss of type at T-R separation distance . A generalized expression for the LOS (type 0) and NLOS (type 1 to type ) pathloss estimate can be given by
3. Link Gain Matrix Estimation Based on Classified Links
3.1. MMSE Estimation for the Model Parameters
denotes the difference between the actual path loss (measured by equipment or emulated using ray tracing tools) and the estimate based on our model in (1).
For those links which are not measured, we can learn their T-R separations as well as their path type (i.e., value of ) through a simple geometric analysis. Then, by plugging the MMSE estimates into (1), the unknown link gains can be predicted.
3.2. A Heuristic Approach and Some Results
In order to achieve a good tradeoff between estimation accuracy and the complexity of measurement, an appropriate choice for the sampling set of link gains is important. Unfortunately, this is beyond the scope of classical sampling theory. Therefore, we need to resort to some heuristics.
To begin, let us consider the ORBIT testbed in Figure 2, which shows the 2-D top view of the 400-nodes. The three rectangles in the middle represent the obstructing pillars, and the uniformly-spaced dots denote the possible node locations. Through a simple geometrical calculation, we learned that to have a full diversity of links, that is, types , , and all included, the transmitters have to be placed on one of the 21 locations (within the ellipses highlighted), while the remaining locations do not have type links. Therefore, these 21 transmitter locations have more uncertainty than the remaining ones in terms of link types.
Our design is to measure a total of 1,000 link gains, and to do so by using transmitters at the 21 locations within the highlighted ellipses. Then we randomly choose 350, 600 and 50 samples from links of types , , and , respectively. The numbers of samples are chosen to be proportional to the total number of link gains in each category. This set of choices constitutes a trial. For each trial, we estimate the link gain model parameters via (2), and then substitute them into (1) to obtain the set of link gain estimates, say . For each transmitter-receiver pair on the grid, we employed the ray tracing result from WiSE as our benchmark set of type link gain "measurements", say . The estimation error for a type link is then given by
We repeat the above experiments for 100 trials.
3.3. Maximum Entropy Sampling
Despite the success of the heuristic strategy for the ORBIT testbed, for a more general setup a quantitative or semianalytic approach is desired. The first problem we need to solve is the selection of measurements. Given the size of samples, our objective is to select a most informative subset of link gains. As is traditional, we use entropy as our measure of information since it is a robust measure of the information available from a set of random variables . To this end, let us assume that through site-specific analysis, the relative frequencies of type link gains over the node ensemble of size are known a priori, and are given by , . For links in each category, we characterize their "importance" or entropy by a constant 
As a test for the proposed maximum-entropy sampling strategy, we calculated the empirical entropy for all the transmitter locations in Figure 2. It is not surprising that the 21 transmitter locations highlighted in Figure 2 stand out as the ones having the largest entropy.
In light of (8), we can identify the locations for the transmitter-receiver pairs whose link gains are going to be measured. Specifically, assume that is the total number of link gain measurements for a size network, and that we will measure all the link gains between a transmitter, say , and its receivers. We can choose transmitter locations for sampling, which correspond to the first indices in (8). Considering the reciprocity of link gains, is a lower bound for candidate transmitter locations. It is worth noting that spatial correlation is not taken into account by (8). In other words, provided some of the nodes are close enough in space, the adjacent neighbors may exhibit similar entropy values because they are subject to very similar obstruction situations. To remove the redundancy incurred by spatial correlation, we can employ a spatial mask over a sufficiently small area to "filter out" the node with representative entropy value and have it serve as the centroid of a clustered neighborhood.
4. Results for More Complex Scenarios
The results so far are for a fairly benign scenario: A square grid in an open lab with three small obstructions. Here, we postulate two scenarios that are more difficult and evaluate the new method for each one, using the maximum-entropy strategy for selecting the links that are measured. The first scenario assumes an ORBIT-like testbed, except that the three obstructing pillars are irregularly placed and of various sizes. The second scenario assumes an irregular layout of nodes distributed throughout an office building with separation walls. Specifically, we consider the first floor of the Alcatel-Lucent building at Crawford Hill in Holmdel, New Jersey. This building has been the focus of numerous studies using the WiSE ray-tracing tool [13–15].
4.1. Modified Testbed Environment
In this exercise, we will specify measurements, considerably more than the 1,000 measurements for the simpler, more regular ORBIT Lab. This means that there will be transmitting nodes, each sending signals to be measured by the other 399 nodes.
4.2. Mesh Network in an Office Building
Before proceeding, we note that there is a small number of weak links where the pathloss falls below dB. As a rule-of-thumb, we can ignore any link in this category, since the pathloss is so large that the two nodes can be regarded as "disconnected". As a result, there are link gains to model, of which are LOS (type 0) and are NLOS.
Most mesh network scenarios will probably have a complexity lying between the two extremes of the ORBIT Lab and the Crawford Hill example, above. In that case, the RMS gain estimation errors for most cases are likely to lie between and dB. The latter value might be reduced further, not by increasing but by alternative, novel arrangements for choosing the links to be measured. This is a topic for further research.
We have developed a link gain matrix estimation methodology for distributed nodes in wireless networks. In contrast to stochastic pathloss models with but one set of parameters, the proposed approach distinguishes among links with different numbers of path obstructions (or walls) and partitions them into separate models. We also developed a maximum-entropy method for selecting, in a structured way, the links to be measured. The results show that all gain matrix elements can be predicted with reasonable accuracy by measuring only a small fraction of all network links. Finally, the proposed method could be extended to outdoor networks, assuming the availability of site-specific data. This is due to the generality of the pathloss modeling, link partitioning, and transmitter selection approaches described here.
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