TERSE-KF: Event-Trigger Diffusion Kalman Filter with Application to Localization and Time Synchronization

by   Amr Alanwar, et al.

The performance of a distributed network state estimation problem depends strongly on collaborative signal processing, which often involves excessive communication and computation overheads on a resource-constrained sensor node. In this work, we approach the distributed estimation problem from the viewpoint of sensor networks to design a more efficient algorithm with reduced overheads, while still achieving the required performance bounds on the results. We propose an event-trigger diffusion Kalman filter, specifying when to communicate relative measurements between nodes based on a local signal indicative of the network error performance. This holistic approach leads to an energy-aware state estimation algorithm, which we then apply to the distributed simultaneous localization and time synchronization problem. We analytically prove that this algorithm leads to bounded error performance. Our algorithm is then evaluated on a physical testbed of a mobile quadrotor node moving through a network of stationary custom ultra-wideband wireless devices. We observe the trade-off between communication cost and error performance. For instance, we are able to save 86 degradation in the performance.



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I Introduction

Wireless sensor network consists of spatially sensor devices which play a vital role to monitor the state of network [1]. State estimation algorithms across wireless network offer many advantages and services in emergency rescue, homeland security, military operations, habitat monitoring, and home automation services. Such critical services would require maintaining some guarantees on the accuracy of the state estimation during operating hours. However, there are many power constraints [2], limitations in terms of bandwidth [3], and limitations in computation and communication [4, 5].

Thus, many design challenges are imposed in the required state estimation algorithms which get many attentions in the recent years among the researchers [6]. Also, due to the importance of resource awareness in a broad range of emerging wireless network algorithms running on resource-constrained commodity platforms, it is imperative to rethink how such resources are handled in the state estimation algorithms across the sensor networks. And what is the trade-off between system performance and the consumed resources in the wireless sensor network?

Figure 1: Distributed event-trigger state estimation system. Every sensor node is running a distributed state estimator to get the network state . Besides distributed estimator, there is a trigger logic running on each node. One of the trigger logic processes is elected to be the leader based on the network state on interest. Trigger logic is based mainly on monitoring , thus linking the transmit decision to the estimation performance. The gray blocks constitute the event-based state estimator to be designed herein. The depicted idea can be applied in different scenarios where resources are limited.

Many algorithms are introduced to estimate an unobserved state based on sensors’ measurements. One of them is the distributed Kalman filtering algorithm which proves to have a good performance; however, it requires a tremendous amount of message exchanges and computation overhead. To address this knotty problem, event-triggered transmission is raised. The event-triggered transmission strategies consist of monitoring a trigger condition and updating the communication and computation rate based on the application requirements. They are used to regulate the resource consumption in wireless networks and ensure the effectiveness of every consumed resource. On the one hand, the performance of the distributed diffusion Kalman filtering [7] depends heavily on the collaboration strategy employed between the contributed nodes. So, the more message exchange and collaboration, the more accurate estimation we get. On the other hand, capabilities of individual nodes are very limited, and they are battery powered, as stated before. So, decreasing the communication overhead is a great concern from power a perspective. Combining both hands introduces conflicting objectives which require unorthodox solutions for many situations. Therefore, we are encouraged to propose TERSE-KF and establish its associated study.

TERSE-KF proposes a novel approach to event trigger the distributed diffusion Kalman filter protocols. We are able to significantly reduce the number of transmission messages compared to the nominal distributed diffusion Kalman filter protocols, illustrating the usefulness of the event-trigger strategy to wireless networked systems. In particular, we characterize the trade-off between the spent resources and the corresponding network performance.

An entirely representative application in distributed state estimation is localization and time synchronization. With the growing prevalence of wireless devices, it is important to coordinate timing among IoT and to provide contextual information, such as location. Also, position estimation is necessary for different fields such as military [8], indoor and outdoor localization [9], security surveillance, and wildlife habitat monitoring. Also, maintaining a shared notion of time is critical to the performance of many cyber-physical systems (CPS), Big Science [10], swarm robotics [11], high-frequency trading [12], and global-scale databases [13]. Furthermore, localization and time synchronization algorithms involve a significant amount of collaboration between individual sensors to perform complex signal processing algorithms. This introduces a considerable communication and energy overhead for networked devices that are oftentimes resource-constrained. This collaborative signal processing nature of sensor networks requires significant research efforts for energy management. For example, just the decision of whether to do the collaborative message exchanges or some local processing has a significant implication on the overall energy and lifetime. Therefore, we picked distributed localization and time synchronization as the application driver to illustrate our proposed event-trigger algorithm. More specifically, we aim to apply our event trigger diffusion Kalman filer on D-SLATS [14], which is distributed simultaneous localization and time synchronization framework.

Inspired by the discussions above, we made the following key contributions in this paper:

  • Introducing the event-triggered distributed diffusion Kalman filter to reduce the communication and computation overhead.

  • Applying the proposed strategy in the problem of localizing and time synchronizing distributed nodes in an ad-hoc network.

  • Proving the stability of the event trigger D-SLATS [14].

  • Evaluating the proposed strategy on a real testbed using custom ultra-wideband wireless devices and a quadrotor, representing a network of both static and mobile nodes.

The rest of the paper is organized as follows: Section II gives an overview of the relevant work in our domain. Section III provides an overall overview of the system model under our study. We then go through our proposed algorithm in Section IV. Section VI illustrates the experimental setup and evaluates the proposed algorithm on static and mobile network of nodes. Finally, Section VII lists some concluding and discussion remarks.

Ii Related Work

The related work can be categorized into the following categories:

Ii-a State Estimation algorithms

Under this umbrella, we have centralized and distributed estimation algorithms. We are more interested in the distributed algorithms due to their advantages over the centralized ones. Also, they are widely used in the wireless network to reconstruct the system state of the plant from measurements with external disturbances. Therefore, algorithms for diffusion LMS [15], diffusion RLS [16] and diffusion Kalman filtering [17] have been proposed. Also, estimation algorithms based on average consensus have been analyzed in [18, 19, 20]. The proposed distributed estimation algorithm in [21] deals with extremely large scale systems. The core idea is to approximate the inverse of the large covariance matrix by make use of L-Banded Inverse and DICI-OR method [22]. These algorithms lack of the resource aware aspect which is an essential feature in the wireless network. Thus, due to the limited the resources in the wireless sensor network, many investigations have been done to decrease the communication and computation overhead while having a relevant good estimation while leads to the next category work.

Ii-B Centralized Event-Triggered Estimation Algorithms

The event-triggered scheme is first proposed in centralized estimation problems. The Send-on-delta (SOD) is proposed on Kalman filter in [23], where sensor data values are transmitted only at encountering a change more than the specified value. In [24]

, an event-based sensor data scheduler has been proposed based on the minimum mean-squared error (MMSE). The variance-based triggering scheme has been developed in

[25], where each node runs a copy of the Kalman filter and transmits its measurement only if the associated measurement prediction variance exceeds a chosen threshold. The properties of set-valued Kalman filters with multiple sensor measurements have been analyzed in [26]. The event trigger scheme has been applied in network estimation problem.

In general, when the sensor and the estimator are not on the same node, the communication is needed for the information exchange between nodes, and the event-trigger scheme can reduce the necessary number of communications, as in [27, 28]. Also, a discrete time approach is proposed in [29] to address the same concern. The importance of including the effects of external disturbances and measurement noise in the analysis of the event-triggered control systems is shown in [30]. Event-Based centralized state estimation for linear Gaussian systems is proposed in [31]. Finally, covariance intersection algorithm is investigated in order to get a centralized event-trigger estimator [32].

Ii-C Distributed Event-Triggered Estimation Algorithms

As one of the main goal of sensor networks is to perform estimation distributively, event-triggered scheme is also applied in such scenario, including Kalman filter with covariance intersection [33], consensus Kalman filter [34, 35]. Interestingly, send-on-delta data transmission mechanism is proposed in the Event-triggered Kalman consensus filter [36]. Moreover, event triggering on the sensor-to-estimator channel and estimator-to-estimator channel are investigated in distributed Kalman consensus [37].

Transmission delays and data drops in distributed event-triggered networked control system was considered in [38, 39]. More analysis and proving the stability of the system given abound delays by a given deadlines are presented before as well [40]. Also, a multiple distributed sensor nodes scenario is considered in [41], where the sensors observe a dynamic process and sporadically exchange their measurements in order to estimate the full state of the dynamic system. Significant deviation from the information predicted from the last transmitted one is monitored in order to get a data-driven distributed Kalman filter [42]. For more related work in the domain, the reader is advised to check out [43].

Ii-D Event-Triggered Diffusion Kalman Filters

When it comes to event-trigger diffusion Kalman filter which is the main concern of TERSE-KF, we found two previous work only to the best of our knowledge at the time of writing. First one is partial diffusion Kalman filter [44]

, where the concern of the authors is just the diffusion step. Every wireless node shares only a subset of its intermediate estimate vectors at each iteration among its neighbors. However, there is no saving at the measurement update step which already includes a high communication head. Also, it is not totally shutting down the communication at the diffusion step. On the other hand, the concern of the second work

[45] is the measurement update step, and they neglect the diffusion step which already also has a significant overhead as proved in the previous work [45]. Our concern is the diffusion step and the measurement step. So, we totally shut-down the communication between the nodes. Also, we do not depend on monitoring the change between the expected state and the calculated one. To the best of our knowledge, TERSE-KF is the first work to propose event trigger on the diffusion Kalman filter on both steps based on an internal signal. Also, we are evaluating the mechanism on a real testbed with the application to localization and time synchronization which is not the trend in all the previous work.

Iii System Model

Consider the following linear time-varying system


where is the state of node at time . is the measurement sent to node from the neighborhood node . The process and measurement noise are assumed to be uncorrelated, and zero mean white Gaussian noise. They are denoted by and , respectively. and are the process covariance and the measurement noise matrix at time , respectively. The state update and measurement functions are denoted by and , respectively. In this work, we consider distributed state estimation problem over a network of nodes indexed by . Nodes are spatially distributed over some region in space. Our node represents some sensors and estimator. Also, we say that two nodes are connected, if they can communicate directly with each other.

One of the illustrative application to the event trigger body of work is the problem of distributed localization and time synchronization due to it power hungry feature, excessive communication and computation overhead. Moreover, these types of algorithms are quite useful among sensors network device where the finite resources is a great concern. Therefore, we picked this application to show the practicality of our proposed algorithm in real life. Thus, our state vector consists of three dimensional position vector , clock time offset , and clock frequency bias . We adopt a convention where both and are described with respect to the global time clock which is usually the clock of a master node, which can be any node. So, in summary, our state vector is as following:


The clock parameters evolve according to the first-order affine approximation of the following dynamics and , where given that is the root node time which is the global time. Therefore, we can write the update function as follows:


Our framework supports three types of measurements which are distinguished by the number of messages exchanged between a pair of nodes. The measurement vector sent from node to node has the following form


where, , represents the counter difference at time which is the measurement of the difference between the offset of the two nodes clocks and at time . , on the other hand, represents a noisy measurement due to frequency bias discrepancies between and which is formally represented by single-sided two-way range. Finally, is another distance measurement between nodes and based on a trio of messages between the nodes at time . This is a more accurate estimate than due to mitigation of frequency bias errors from the additional message. It is formally called double-sided two-way range. For more details about the three types of measurements, we encourage the reader to check [14].

We would like to note that subset of these measurements may be used rather than the full set, i.e., we can have experiments involving just , , or . and define, respectively, the response and the round-trip durations between the appropriate pair of these timestamps. The propagation velocity of radio is taken to be the speed of light in a vacuum, denoted by . Thus, our measurement function is in the following form:


Iv Event-Trigger Diffusion EKF Algorithm

Exposing diffusion distributed Kalman filter capabilities towards satisfying the system constraints using efficient duty cycling is our ultimate goal. Our proposed algorithm works well with the diffusion distributed Kalman filter, and the proof of convergence should be a straight forward step after Sayed work [7]. We should mention that the convergence guarantees would work only under the two assumptions in [7], where the model parameter are time-invariant. However, our application, namely, localization and time synchronization and many others require nonlinear time variant system model. Thus, we propose our algorithm to deal with the nonlinear time variant case which is the more general case as shown in Algorithm 1, where we have the nonlinearly solved by Equation 6.

We denote by the estimate of given the observations up to time where every node seeks to minimize the mean-square error . The elements represent the weights that are used by the diffusion algorithm to combine neighborhood estimates. The algorithm starts with the measurement update where every node obtains at time step . Next, in the diffusion step, information from the neighbors of node are combined in a convex manner to produce a new state estimate for the node. Then, every node performs the time update step. Starting from the second round, step 1 and step 2, namely Measurement update and Diffusion update, are only executed if the trace of intended part of the matrix . Explicitly, the event is defined as .

1 Start with and for all , and at every time instant , compute at every node : Step 1: Time update:
if there exist such that  then
2       Step 2: Measurement update:
Step 3: Diffusion update:
Algorithm 1 Event-Trigger Diffusion EKF

V Theoretical Analysis

Despite the fact that EKF has proven to work well in many practical applications, its general convergence guarantees, even in the centralized version, cannot be proved [46]. Thus, we limit our convergence analysis to our application. Therefore, we are going to prove that our algorithm is bounded for the D-SLATS application. While the algorithm applies EKF, where the coefficients depend on the state, we need to find an upper bound of the covariance update that is time-invariant to proceed the analysis. With the time-invariant bound, we can apply the conventional conclusion in diffusion Kalman filter to obtain the boundedness properties.

To begin with, we define the augmented state-error vector for the whole system as

In the following, we prove that the convergence of local estimators leads to that of the global estimation in this event-triggered scheme.

Theorem 1.

If every local Kalman filter in each nodes converges, i.e. and , for , then the covariance of global estimates converges in Algorithm 1. Explicitly, converges.


Based on (5), the measurement matrix is given by


with . Since the sensor noise is finite in any practical setting, it is not difficult to find such that . We then can find the upper bound in the measurement update with

The finite property of depends on the fact that the estimation error of with some finite , which is practical in D-SLATS application. By choosing such that

Therefore, we can replace the measurement update by


which has a time-invariant coefficient. Furthermore, with the same initial condition, the covariance update given by (9) is always larger than the one updated by the EKF update in Algorithm 1.

With the time-invariant upper bound on the local estimator, we can now turn to the exact estimation performance in the global scenario. Let denoted the number of time updates in the consecutive measurement/diffusion update. Since is nonzero, we have

In other words, is finite. Combining the error transition in measurement update

and time update,

we have the total error transition between consecutive measurement/diffusion updates as

We can now consider the global error propagation with the block-diagonal matrices defined as follow:

With the definition, we have the error update for the whole system as


where the identity matrix

has the dimension of .

Consequently, the error update for the whole system can be written as

With the assumption that every local Kalman filter converges, the convergence of Algorithm 1 follows by the similar conclusion in original diffusion Kalman filter in [17]. ∎

The above analysis shows that under certain reasonable conditions, the event-trigger diffusion Kalman filter with application to D-SLATS algorithm has bounded covariance, which asserts its effectiveness.

Vi Evaluation

We are considering mainly the communication overhead and its associated accuracy with different network topologies to show the effectiveness of our proposed algorithm. We start by describing our experimental setup, where we conducted our experiments. Then, we are going to do case studies to have a satisfying evaluation.

Figure 2: Experimental setup, including, UWB Anchor nodes, motion capture cameras, and mobile quadrotor UWB nodes

Vi-a Experimental Setup

We evaluated the performance of the proposed event-triggered algorithm on a custom ultra-wideband RF test bed based on the DecaWave DW1000 IR-UWB radio [47]. The overall setup is shown in Figure 2. The main components of our test bed can be summarized in the following points.

Figure 3: Custom anchor node with ARM Cortex M4 processor and UWB expansion.
Figure 4: Ceiling-mounted anchor with DW1000 UWB radio in 3D-printed enclosure.
Figure 5: CrazyFlie 2.0 quadrotor helicopter with DW1000 UWB expansion.
  • A Motion capture system capable of 3D rigid body position measurement with less than 0.5 mm accuracy. The system consists of an eight-camera which are deployed in order to provide accurate ground truth position measurements. The results presented in this work treat the motion capture estimates as true position, though we qualify here that all results are accurate to within the motion capture accuracy. The ground truth position estimates from the motion capture cameras are sent to a centralized server which uses the Robot Operating System (ROS) [48] with a custom package. We adopt a right-handed coordinate system where is the vertical axis, and and make up the horizontal plane.

  • The Fixed nodes used in the following experiments consist of custom-built circuit boards equipped with ARM Cortex M4 processors t 196 MHz powered over Ethernet and communicating to Decawave DW1000 ultra-wideband radios as shown in Figure 5 and Figure 5. Each anchor performs a single and double-sided two-way range with its neighbors. The used Decawave radio is equipped with a temperature-compensated crystal oscillator with frequency equals 38.4 MHz and a stated frequency stability of ppm. We installed eight UWB anchor nodes in different positions in a lab. More specifically, six anchors are placed on the ceiling at about high, and two were placed at waist height at about in order to better disambiguate positions on the vertical axis. Each anchor node is fully controllable over a TCP/IP command structure from the central server. These nodes are placed to remain mostly free from obstructions, maximizing line-of-sight barring pedestrian interference.

  • The Mobile Node used in the experiments are battery-powered mobile nodes also with ARM Cortex M4 processors based on the CrazyFlie 2.0 helicopter [49] and equipped with the very same DW1000 radio as shown in Figure 5. This allows for compatibility in the single and double sided ranging technique used.

Vi-B Experiments

We demonstrate TERSE-KF performance on distributed simultaneous localization and time synchronization problem. To begin, static nodes are placed in distinct locations around the area, as described in our experimental setup. We are concerned in applying the event trigger algorithm based on the location of the mobile node. In other words, we use as following:


At a high level, the wireless network has a relevant state that varies over time. The network needs to efficiently utilize its computation and communication resources while achieving the required performance. To give the reader an intuition of how we are going to evaluate TERSE-KF, we show the results of running a portion of the experiments in Figure 6. The mobile node is flying freely with different speeds in our lab, while trying to save its computation and communication resources. The threshold is set to be . The second sub-figure in Figure 6 shows the behavior of . While the is less than the , all the nodes are only executing the Time update Step. So, we are decreasing the spent sensing power, message exchange, and computation overheads during that period. The effect on the estimated localization error of the mobile node can be seen clearly in the first sub-figure. Once, the hits the threshold , all the nodes are triggered to doing measurements and exchange messages in order to decrease the back to the allowed range. The third sub-figure in Figure 6 demonstrates the time where Measurement update Step is executed at all nodes. For instance, we can see that the Measurement update Step happens at the following time instances:

Similarly, Diffusion update step is happening just after the measurement update step, as shown in the forth sub-figure. Finally, the last sub-figure shows that the Time update step is happening all the time.

Figure 6: A snapshot of 20 seconds of our experiments. The threshold is set to 4. The 3D localization error and trace value are shown in the first and second sub-figures. The measurement, diffusion, and time update steps flags are shown in the remaining sub-figures where, a value of indicates of executing the step, while means skipping the step at the corresponding time instance.

Vi-B1 Fully Connected Network Case Study

We are going to illustrate the effectiveness of TERSE-KF triggering algorithm by showing the amount of communication-saving and the associated localization error by applying the algorithm over a fully connected network. The crazyflie is driven through our lab over four different sessions. Each session was conducted on a different day with a different number of students working around for each day. Also, the path of the crazyflie was a random walk for each day. The total duration of the session was about 10 minutes. We repeated the experiments while setting a different value of the threshold, then calculate the localization error reported by the motion capture system. We should note that our proposed algorithm could be used also with the general distributed Kalman where the fully connected case is more applicable.

Communication Analysis

Figure 6(a) shows the effect of changing the threshold value on the percentage of the saved message for a fully connected network. Zero Threshold refers to the case of sending all the messages and running the three steps of the algorithm in a normal fashion. In other words, it corresponds to sending messages. Remember that every message already corresponds to single-sided or double-sided two-way ranges. Thus, the reported number of messages should be multiplied by two or three depending on the message ranging type. Setting the threshold to lead to saving about of the overall number messages. Namely, saving messages. Also, a threshold of ends up with saving of the total number of messages. Finally, saving of the total number of messages can be achieved by threshold.

Accuracy Analysis

Event Trigger algorithms affect the performance. However, there is no meaning of spending more resources while the application need is much less. So the question should not be how much the algorithm could achieve; however, it should be is the algorithm capable to the satisfy the application needs, while saving the resources?. Therefore, we conduct the following study to see what the trade-off between 3D-localization error with each value of the Threshold .

The error plot in Figure 6(c)

summarizes our results to this case study. The red rectangles correspond to the mean value of the localization error, while the vertical lines represent the standard deviation around that mean value. At threshold

, we are not saving any resources, and we could achieve mean localization error with a standard deviation of m. While results in a m mean error with a standard deviation of m. Finally, achieves a m mean error with a standard deviation of m. It is up to the application need to set the appropriate threshold based on its need.

(a) Effect of changing the threshold value on the percentage of the saved message for a fully connected network.
(b) Effect of changing the threshold value on the percentage of the saved message of the network for a partially connected network where every node is connected to only neighbors.
(c) Effect of changing the threshold value on the 3D localization error of the crazyflie for a fully connected network.
(d) Effect of changing the threshold value on the 3D localization error of the crazyflie for a partially connected network where every node is connected to only neighbors.
Figure 7: Effect of changing the threshold value on a fully connected network and partially connected one.
Figure 8: The trade-off between the communication overhead saving and the mean 3D localization error of the crazyflie for a fully connected network.

We investigated TERSE-KF which restricts the amount of processing, sensing, and communication. Such a chosen restriction dramatically reduces the amount of communication overhead in the network, but potentially results in a reduced network performance. Thus, there is a trade-off between the number of messages sent in the wireless network and the estimation algorithms performance. A better understanding of this trade-off results in a useful insight on how much resources one needs to spend in order to obtain the desired performance. Figure 8 shows the trade-off between the communication overhead and the mean 3D localization error. Interestingly, saving of communication overhead leads to increase in the localization error. This has been calculated by considering the mean localization error plus the standard deviation at threshold and .

Vi-B2 Partially connected Network case study

We considered another case study where every node of the nine nodes is connected to only four neighbors instead of eight neighbors in the previous case study. Again, we are going to analyze the communication saving and the associated localization error.

Communication Analysis

Figure 6(b) summarizes the results. Interestingly, setting the threshold to m lead to saving about of the overall number messages. Also, a threshold of ends up with saving of the total number of messages.

The error plot in Figure 6(d) summarizes our results to this case study. Again, the red rectangles correspond to the mean value of the localization error, while the vertical lines represent the standard deviation around that mean value. At threshold , we are not saving any resources, and we could achieve mean localization error with a standard deviation of m, where the network is partially connected as described before. While results in a m mean error with a standard deviation of m. Finally, achieves a m mean error with a standard deviation of m.

Vii Conclusion

TERSE-KF investigated the energy aware-aspect of the distributed estimation problem for a multi-sensor system with event-triggered processing schedules. More specifically, we proposed event-trigger diffusion distributed Kalman filter for the wireless network. Then, we picked distributed localization and time synchronization as a representative application for our estimation algorithm. The stability of the proposed algorithm is ensured. Several experiments using real, custom ultra-wideband wireless anchor nodes and mobile quadrotor nodes were conducted and they indicate that the proposed algorithm is reliable in terms of performance, and efficient in the use of computational and communication resources. Future directions will deal with testing over a large-scale system.


This research is funded in part by the National Science Foundation under awards # CNS-1329755 and CNS-1329644. The U.S. Government is authorized to reproduce and distribute reprints for Governmental purposes notwithstanding any copyright notation thereon. The views and conclusions contained herein are those of the authors and should not be interpreted as necessarily representing the official policies or endorsements, either expressed or implied, of NSF, or the U.S. Government.


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Step 1 Step 2 Step 3

Increase Decrease No Change
Increase Decrease [50, 21] Governed by (14)
Table I: Summary of the effect of each step of Diffusion Kalman Filter on and

Inherent in the original centralized Kalman filter a powerful tool which error covariance matrix. It is well maintained and time updated by the underlying core of Kalman filter. Such a covariance is a perfect measure of the estimated accuracy of the state. However, when it comes to the distributed diffusion Kalman filter, it is important to note it lost such nice feature. Thus the covariance of the estimation error is not available locally anymore, since the diffusion update does not take into account the recursions for the covariance, and it messes up the notion of the state error.

Therefore, one of the main problems in diffusion Kalman filter is that the matrix used in every local nodes is not necessarily the real covariance. Even though one collects all ’s, one still can not reconstruct the overall covariance of the system . While the triggering event is defined on local , we need to address the relation between and for clarity. We discuss three steps of diffusion Kalman filter in the following.

Vii-a Step 1: Time update

Every local updates by . In terms of local Kalman filter, one has


with defined in (11) and in positive-semidefinite sense. While the overall covariance is updated by , we have


Vii-B Step 2: Measurement update

updates according to

as in standard Kalman filter and one arrives . However, the overall covariance updates by


The covariance should decrease as in [50, 21]. However, we do not prove this in this work.

Vii-C Step 3: Diffusion update

In diffusion Kalman filter, local ’s are not varied in this step, even though the estimations are changed. Correspondingly, the overall covariance changes by


However, the order of and can not be guaranteed in general.

The tentative result can be summarized in Table I. While diffusion update is widely applied, the interplay between local and global estimations needs to be explored furthermore.