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The receiving signal strength (RSS) is crucial for the Internet of Things (IoT), as it is the key foundation for communication resource allocation, localization, interference management, sensing, and so on. Aside from its significance, the measurement process could be tedious, time consuming, inaccurate, and involving human operations. The state-of-the-art works usually applied the fashion of “measure a few, predict many,” which use measurement calibrated models to generate the RSS for the whole networks. However, this kind of methods still cannot provide accurate results in a short duration with low measurement cost. In addition, they also require careful scheduling of the measurement which is vulnerable to measurement conflict. In this paper, we propose a compressive sensing- (CS-) based RSS measurement solution, which is conflict-tolerant, time-efficient, and accuracy-guaranteed without any model-calibrate operation. The CS-based solution takes advantage of compressive sensing theory to enable simultaneous measurement in the same channel, which reduces the time cost to the level of

With the ubiquitous wireless networking devices, we envision realizing the Internet of Things, which requires the wireless networks to develop with higher spectrum utilization, less transmission delay, and lower energy consumption. This trend gives birth to many emerging technologies (e.g., OFDMA, network coding, and cognitive radio [

Towards the efficient and accurate RSS measurement and its application in resource allocation and localization, most existing works [

To obtain accurate RSS values, exhausting measurement on all wireless links will incur unacceptable time cost. First, the RSS collection process, for example, the SINR-optimization process, requires the RSS values between every pair of nodes in the network, so the number of RSS values to be measured grows quadratically with the network size. As is illustrated in Figure

Traditional exhaustive RSS measurement.

Nonconflict RSS measurement.

Both the measurement cost and the time consumption are unacceptable for a network with large size of nodes. To tackle this, we propose a compressive sensing-based solution, whose basic idea could be summarized as two aspects:

The idea of partial measurement is illustrated in Figure

Compressive sensing-based RSS measurement.

The idea of simultaneous measurement is illustrated in Figure

Simultaneous RSS measurement.

The problems associated with the partial measurement and simultaneous measurement could be jointly solved in our proposed compressive sensing-based solution, which mainly introduce the idea of compressive sensing and lead to the measurement cost and time cost to the level of

The contributions of this paper are summarized as follows:

We reveal the important problem of accurate and efficient RSS measurement for communication and sensing in IoT.

We modeled the RSS measurement process as a linear system and propose a basic framework to perform simultaneous measurement of RSS instead of the traditional nonconflict measurement.

We further present a compressive sensing- (CS-) based solution to achieve partial measurement. It can achieve the time efficiency of

We introduce a LDPC-based measurement matrix, which only generates a small number of measurements. It greatly reduces the energy consumption for IoT.

We conduct extensive experiments using real communication traces collected from a wireless mesh network testbed, which show the efficiency of the proposed solutions.

The rest of this paper is organized as follows. Section

In this section, we explore the first step to achieve partial measurement and simultaneous measurement, which is to model the efficient RSS measurement problem as a linear system. Before proposing the modeling, we first propose the network model and some important metrics to evaluate our solution.

We consider a synchronized, time-slotted wireless network consisting of

Our main task is to

A measurement scheme could be evaluated via the following metrics:

Time cost: it is the total time slots to accomplish the measurement process.

Overhead: it is the total number of measurements in all nodes and channels.

Accuracy: we defined two levels of accuracy, which are link-wise accuracy and network-wide accuracy, receptively. Regarding link-wise accuracy, the result of measurement should be within a certain level of confidence

It is worth mentioning that the relation between link-wise accuracy and network-wide accuracy is not exclusive. One can achieve both of them when restrictions are put in the frequency in single link measurement and also optimization accuracy is put in global optimization results. However, such dual restriction will lead to unacceptable overhead. In fact, one can achieve the link-wise accuracy through a measurement method [

The accuracy is assured by an adequate number of measurements, while the overhead and time cost metric require as few measurements as possible. Thus, our target is to design the solution that achieves good tradeoffs among these metrics.

Basically, the measurement process can be modeled as a linear system. By applying the prevalent compressive sensing [

Before introducing the solution with partial measurement and simultaneous measurement, we first formulate our problem in the form of a linear system.

According to the SINR model, the RSS is approximately linear additive. This property implies that when several nodes in the network send signals in the same slot, the RSS of a certain node is the sum of the RSSs from all of the sending nodes. Formally, in one time slot, we have

From this linear system perspective, our problem could be stated as follows.

Given a network of

Note that, if we choose

Before presenting our solutions, we briefly introduce the compressive sensing theory. Compressive sensing (or sampling) (CS) [

In other words, signals can be accurately rebuilt based on the following conditions:

The a priori knowledge of sparsity or compressibility of signals is known.

A small number of global linear measurements are provided.

Let

Compressive sensing theory states that an

The

It has been established that Gaussian matrix

As aforementioned, efficient RSS measurement relies on the process of partial measurement and simultaneous measurement. The former could be achieved with solving the linear system modeling, while the latter relies on how we recover the RSS matrix with only a few time slot measurements. Our basic idea is applying the compressive sensing theory to the linear system.

The process of partial measurement and simultaneous measurement is mainly enabled by the careful design of measurement matrix in the compressive sensing, which owns the ability to recover the full signal from partially measured signal information and also has the ability to distinguish the overlapped signal when they are linearly combined. We will discuss it specifically when we propose the design of measurement matrix.

The success of the solution depends on two crucial components. The first one is the generation of measurement matrix with good RIP. According to the theorem in [

Assume that the RSS matrix

According to (

In our measurement, the measurement matrix is a binary matrix, with each row as the sending plan of all nodes in one time slot. The number of rows represents the number of time slots used to perform measurements. In addition, because the representation basis is usually orthonormal matrix, the RIP of matrix

To design the measurement matrix satisfying the partial measurement requirement, we only have to make sure the measurement matrix is in form of

To design the measurement matrix satisfying the simultaneous measurement requirement, we only have to make sure the measurement matrix is in form of

To begin with, we derive the size of rows for the measurement matrix. This will inherently control the network-wide accuracy. According to [

Then, we derive the elements of measurement matrix. As aforementioned, the measurement matrix in CS is usually drawing from a random matrix whose entries are i.i.d. Gaussian variables complying to

A binary matrix

According to [

The measurement overhead could be further reduced by a matrix manipulating trick. We split the measurement matrix

If matrix

When

According to this theorem, we find that if we choose an orthonormal matrix as

In this section, we describe how to control the sparsity of RSS matrix. Note that, in the RSS matrix, most of the elements are in fact close to but not equal to zero. This situation requires us to carefully drop some elements to make the matrix sparse. With all these considerations, we apply singular value decomposition here.

Simply stated, a

To determine the representation basis

As we learned from several signal propagation models, the RSSs increase linearly with the sending power. The sending power, however, impacts the interference range of the node. In a CS-based solution, we prefer a low-rank RSS matrix. A higher sending power will result in a larger interference and, in turn, make more entries in RSS matrix not close to zero. Meanwhile, if the sending power is tuned too small, the RSSs (which are close to zero) tend to be too vulnerable when encountering noises and recovery errors. Thus, when applying the result derived from too small sending power to the ordinary scenarios, the error ratio will be amplified. A proper sending power is needed for a better performance of the CS-based solutions. According to the experiment result in Figure

Different from traditional work with CS, where the recovery target is a vector, our work aims to recover a matrix with CS. According to [

In summary, the CS-based solution achieves

The accuracy control consists of two parts, namely, controlling the row number of measurement matrix and controlling the sparsity of representation basis.

Regarding the row number of the measurement matrix, it inherently controls the network-wide accuracy. According to [

Regarding the sparsity of the representation basis, the accuracy of the recovery will decrease as we drop some of the small singular values of the original RSS matrix. Thus, to increase the recovery accuracy, the representation basis should comply with the sparsity

Apart from using the CS solution as a single solution, it could also be used as an extension to mitigate the harm due to background noises. In practice, the RSS is not all from the nodes inside network; the background noise is usually sporadic and affects almost all nodes.

The measured result could be divided into two parts: the noise matrix and the RSS matrix. Formally put,

Note that the background noise usually affects a large area of network. The nodes in network could measure the average background noise for each part and derive a low-rank matrix

The CS solution could be easily transformed into a distributed solution. As each node could easily collect the RSS from the nodes in the networks, assuming the measurement matrix is known, the measurement process in the node

Thus, this is a classical compressive sensing form, as long as

For the measurement matrix, we can still use the LDPC matrix. As the measurement matrix is a global measurement schedule, it must be synchronized. The random generation could be synchronized for all nodes if they use the same random seed. This seed could be the global clock or the others that have already been synchronized.

For the representation basis, each node could have their own basis without affecting the recovery results. We apply the standard representation basis here. Specifically, we use Discrete Cosine Transform Basis (DCT) [

With the synchronized measurement matrix

To implement our algorithm to the real deployed wireless networks, we have to consider some implementation issues, especially those with the distributed algorithms. Consequently, in this subsection, we discuss two major issues, time synchronization and measurement matrix generation.

Time synchronization has been constantly drawing research attentions ever since the distributed systems to wireless sensor networks and the mac protocol in wireless networks. Our algorithm mainly relies on the network synchronization in two folds. The first one is that our measurements are performed in a slotted fashion, such that the clock must be synchronized among all the nodes. The second one is that the measurement matrices are generated by the same random function with the same seed. This seed is usually the time clock. Thus, the synchronization is critical for our algorithm. The synchronization frequency and the time cost is the major overhead in real implementation. These costs could be alleviated by selecting suitable synchronization scheme. One suitable solution for our situation is the diffusion algorithm proposed in literature [

Another critical concern in the synchronization is how to discover the inconsistency of the measurement matrix between each node. This situation is mainly caused by the clock shift. In our algorithm the inconsistency could be easily discovered by neighbor exchange of the measurement matrix or certain form of checksum, for example, the sum of all elements in measurement matrix.

As aforementioned, the measurement matrix is generated complying with LDPC matrix. Traditional scheme is to generate a binary matrix with entries complying to Bernoulli distribution with success probability

In this section, we analyze the performance of the proposed solutions with experiments. We first present the experimental methodology and simulation settings; then, we discuss the numerical results.

Our simulations are based on the data collected from the SWIM platform [

We generated several experimental scenarios from this data set. The experimental scenarios consist of

The benchmarks used to quantify our solutions are how the RSS metrics obtained via the solutions impact the performance of the throughput optimization. The accuracy is quantified using the MPE (Mean Percentage Error), which is formally defined as

Regarding the CS-based solution, we mainly examine how this method improves the performance of SINR-based optimization. We also provide a comparison of the performance between the CS solution and the solution proposed in [

First, we compare the fundamental performance of CS-based solution and model-based solution [

The CDF of throughput under model-based and CS-based solutions.

We also examine how the recovery is affected by the introduction of LDPC-based measurement matrix. As aforementioned, the LDPC code has better RIP comparing to the Gaussian matrix. It also has the advantage of less measurement cost, as there are limited nonzero elements per row. The latter advantage could be easily examined with mathematical computation. Thus, we do not provide the numerical result here. The experiment examines the recovery advantage of LDPC-based measurement matrix as shown in Figure

The recovery advantage of LDPC-based measurement matrix.

The comparison of model-based and CS-based solutions in different network densities.

We further compare these two solutions in different network densities. As mentioned above, dense networks give rise to a RSS matrix with higher rank, which in turn will compromise the accuracy. This is illustrated using a MSE versus nodes density graph. In Figure

Finally, we examine the time cost of CS-based solution in different network size. The result is shown in Figure

The total time slots used by model-based and CS-based solution versus different network sizes.

In summary, CS-based solution performed superiorly in the scenario of low density and large network size in terms of time efficiency.

In [

These works endeavour to make more accurate signal attenuation models in a single transmission. But, none of them focus on making a group measurement on all potential links and all channels, which are crucial for the throughput optimization in wireless networks.

Compressive sensing theory was firstly introduced to recover sparse signal with less sampling [

Our method is different from the above method in that our method is used in a distributed fashion and we proposed a new measurement matrix based on LDPC and its variant, which could reduce the measurement cost and time cost greater than the traditional way. Comparing to our previous work, we further consider the feature of the IoT systems and propose distributed algorithm with improved energy efficient measurement matrix.

The efficiency and accuracy of the RSS measurement in the wireless networks are of great importance for throughput optimization, localization, and wireless sensing in the Internet of Things. Traditional efficient RSS measurement adopt a “measure a few, predict many” fashion with calibrating the parameters in the propagation models. However, we claim that these kinds of methods are not good enough as they miss the chance of simultaneous measurement and controllable partial measurement, which are all achieved with the compressive sensing-based solution we proposed. With CS-based solution, the whole measurement process could be finished in time of

The authors declare that they have no conflicts of interest.

This work is supported in part by the National NSF of China (under Grant nos. 61602238, 61672278, 61672283, and 61373128), the Key Project of Jiangsu Research Program Grant (BK20160805), and the China Postdoctoral Science Foundation (no. 2016M590451).