Alex S. Leong

On Kalman filtering over fading wireless channels with controlled transmission powers

We study stochastic stability of centralized Kalman filtering for linear time-varying systems equipped with wireless sensors. Transmission is over fading channels where variable channel gains are counteracted by power control to alleviate the effects of packet drops. We establish sufficient conditions for the expected value of the Kalman filter covariance matrix to be exponentially bounded in norm. The conditions obtained are then used to formulate stabilizing power control policies which minimize the total sensor power budget. In deriving the optimal power control laws, both statistical channel information and full channel information are considered. The effect of system instability on the power budget is also investigated for both these cases.

Kalman filtering with faded measurements

This paper considers a sensor network where single or multiple sensors amplify and forward their measurements of a common linear dynamical system (analog uncoded transmission) to a remote fusion centre via noisy fading wireless channels. We show that the expected error covariance (with respect to the fading process) of the time-varying Kalman filter is bounded and converges to a steady state value, based on some general earlier results on asymptotic stability of Kalman filters with random parameters. More importantly, we provide explicit expressions for sequences which can be used as upper bounds on the expected error covariance, for specific instances of fading distributions and scalar measurements (per sensor). Numerical results illustrate the effectiveness of these bounds.

Optimal Energy Allocation for Kalman Filtering Over Packet Dropping Links With Imperfect Acknowledgments and Energy Harvesting Constraints

This paper presents a design methodology for optimal transmission energy allocation at a sensor equipped with energy harvesting technology for remote state estimation of linear stochastic dynamical systems. In this framework, the sensor measurements as noisy versions of the system states are sent to the receiver over a packet dropping communication channel. The packet dropout probabilities of the channel depend on both the sensors transmission energies and time varying wireless fading channel gains. The sensor has access to an energy harvesting source which is an everlasting but unreliable energy source compared to conventional batteries with fixed energy storages. The receiver performs optimal state estimation with random packet dropouts to minimize the estimation error covariances based on received measurements. The receiver also sends packet receipt acknowledgments to the sensor via an erroneous feedback communication channel which is itself packet dropping.

Power Allocation for Outage Minimization in State Estimation Over Fading Channels

This paper studies the outage probability minimization problem for state estimation of linear dynamical systems using multiple sensors, where an estimation outage is defined as an event when the state estimation error exceeds a pre-determined threshold. The sensors amplify-and-forward their measurements (using uncoded analog transmission) to a remote fusion center over wireless fading channels. For stable systems, the resulting infinite horizon problem can be formulated as a constrained average cost Markov decision process (MDP) control problem. A suboptimal power allocation that is less computationally intensive is proposed and numerical results demonstrate very close performance to the power allocation obtained from the solution of the MDP based average cost optimality equation. Motivated by practical considerations, assuming that sensors can transmit with only a finite number of power levels, optimization of the values of these levels is also considered using a stochastic approximation technique. In the case of unstable systems, a finite horizon formulation of the estimation outage minization problem is presented and solved. An extension to the problem of minimization of the expected error covariance is also studied.

Asymptotics and Power Allocation for State Estimation Over Fading Channels

State estimation of linear systems using analog amplify and forwarding with multiple sensors, for both multiple access and orthogonal access schemes is considered. Optimal state estimation can be achieved at the fusion center using a time-varying Kalman filter. We show that in many situations, the estimation error covariance decays at a rate of 1/M when the number of sensors M is large. We consider optimal allocation of transmission powers that 1) minimizes the sum power usage subject to an error covariance constraint, and 2) minimizes the error covariance subject to a sum power constraint. In the case of fading channels with channel-state information, the optimization problems are solved using a greedy approach, while for fading channels without channel state information (CSI) but with channel statistics available, a suboptimal linear estimator is derived.

On Kalman Smoothing With Random Packet Loss

This correspondence studies the performance of Kalman fixed lag smoothers with random packet losses and its comparison with the Kalman filter with packet loss. In terms of estimator stability via boundedness of the expectation of the error covariance, we show that smoothing does not provide any benefit over filtering. On the other hand, it is demonstrated that using a probabilistic notion of performance, smoothing can provide significant gains when compared to Kalman filtering. An analysis of Kalman filtering using two simple retransmission schemes and its comparison with Kalman smoothing is also made.

On Scaling Laws of Diversity Schemes in Decentralized Estimation

This paper is concerned with decentralized estimation of a Gaussian source using multiple sensors. We consider a diversity scheme where only the sensor with the best channel sends their measurements over a fading channel to a fusion center, using the analog amplify and forwarding technique. The fusion centre reconstructs a minimum mean squared error (MMSE) estimate of the source based on the received measurements. A distributed version of the diversity scheme where sensors decide whether to transmit based only on their local channel information is also considered. We derive asymptotic expressions for the expected distortion (of the MMSE estimate at the fusion centre) of these schemes as the number of sensors becomes large. For comparison, asymptotic expressions for the expected distortion for a coherent multiaccess scheme and an orthogonal access scheme are derived. It is seen that as opposed to the coherent multiaccess scheme and the orthogonal scheme (where the expected distortion decays as 1/M, M being the number of sensors), the expected distortion decays only as 1/ln(M) for the diversity schemes. This reduction of the decay rate can be seen as a tradeoff between the simplicity of the diversity schemes and the strict synchronization and large bandwidth requirements for the coherent multiaccess and the orthogonal schemes, respectively. We study for the diversity schemes, the optimal power allocation for minimizing the expected distortion subject to average power constraints. The effect of optimizing the probability of transmission on the expected distortion in the distributed scenario is also studied. It is proved that for Rayleigh fading optimal sensor transmit power allocation achieves the same asymptotic scaling law as the constant power allocation scheme, whereas it is observed that optimizing the sensor transmission probability (with or without optimal power allocation) in the distributed case makes very little difference to the asymptotic scaling laws.

An optimal transmission strategy for kalman filtering over packet dropping links with imperfect acknowledgements

This paper presents a novel design methodology for optimal transmission policies at a smart sensor to remotely estimate the state of a stable linear stochastic dynamical system. The sensor makes measurements of the process and forms estimates of the state using a local Kalman filter. The sensor transmits quantized information over a packet dropping link to the remote receiver. The receiver sends packet receipt acknowledgments back to the sensor via an erroneous feedback communication channel which is itself packet dropping. The key novelty of this formulation is that the smart sensor decides, at each discrete time instant, whether to transmit a quantized version of either its local state estimate or its local innovation. The objective is to design optimal transmission policies in order to minimize a long-term average cost function as a convex combination of the receivers expected estimation error covariance and the energy needed to transmit the packets. Under high-resolution quantization assumptions, the optimal transmission policy is obtained by the use of dynamic programming techniques. Using the concept of submodularity, the optimality of a threshold policy in the case of scalar systems with perfect packet receipt acknowledgments is proved. Suboptimal solutions and their structural results are also discussed. Numerical results are presented, illustrating the performance of the optimal and suboptimal transmission policies.

Quantized Filtering Schemes for Multi-Sensor Linear State Estimation: Stability and Performance Under High Rate Quantization

In this paper we consider state estimation of a discrete time linear system using multiple sensors, where the sensors quantize their individual innovations, which are then combined at the fusion center to form a global state estimate. We prove the stability of the estimation scheme under sufficiently high bit rates. We obtain asymptotic approximations for the error covariance matrix that relates the system parameters and quantization levels used by the different sensors. Numerical results show close agreement with the true error covariance for quantization at high rates. An optimal rate allocation problem amongst the different sensors is also considered.

On Kalman filtering with faded measurements

This paper considers a sensor network where single or multiple sensors amplify and forward their measurements of a common linear dynamical system (analog uncoded transmission) to a remote fusion centre via noisy fading wireless channels. We show that the expected error covariance (with respect to the fading process) of the time-varying Kalman filter is bounded and converges to a steady state value, based on some general earlier results on asymptotic stability of Kalman filters with random parameters. More importantly, we provide explicit expressions for sequences which can be used as upper bounds on the expected error covariance, for specific instances of fading distributions and scalar measurements (per sensor). Numerical results illustrate the effectiveness of these bounds.