The problem of decentralized sequential detection with conditionally independent observations is studied. The sensors form a star topology with a central node called fusion center as the hub. The sensors make noisy observations of a parameter that changes from an initial state to a final state at a random time where the random change time has a geometric distribution. The sensors amplify and forward the observations over a wireless Gaussian multiple access channel and operate under either a power constraint or an energy constraint. The optimal transmission strategy at each stage is shown to be the one that maximizes a certain Ali-Silvey distance between the distributions for the hypotheses before and after the change. Simulations demonstrate that the proposed analog technique has lower detection delays when compared with existing schemes. Simulations further demonstrate that the energy-constrained formulation enables better use of the total available energy than the power-constrained formulation in the change detection problem.
Deep Dive into Decentralized sequential change detection using physical layer fusion.
The problem of decentralized sequential detection with conditionally independent observations is studied. The sensors form a star topology with a central node called fusion center as the hub. The sensors make noisy observations of a parameter that changes from an initial state to a final state at a random time where the random change time has a geometric distribution. The sensors amplify and forward the observations over a wireless Gaussian multiple access channel and operate under either a power constraint or an energy constraint. The optimal transmission strategy at each stage is shown to be the one that maximizes a certain Ali-Silvey distance between the distributions for the hypotheses before and after the change. Simulations demonstrate that the proposed analog technique has lower detection delays when compared with existing schemes. Simulations further demonstrate that the energy-constrained formulation enables better use of the total available energy than the power-constrained f
Consider the use of a wireless sensor network for detection of a disruption or a change in environment. The change is required to be detected with minimum delay subject to a false alarm constraint. The standard medium access control and physical layer design for such a network (e.g., IEEE 802.15.4 standard) is one where sensors quantize their observations and send them to a fusion center via random access over a wireless Gaussian multiple-access channel (GMAC). The transmitted data are typically quantized individual log-likelihood ratios (LLR) of the hypotheses representing the environment before and after the change. The fusion center collects each sensor's LLR and adds them to get a fused statistic, if observations at sensors are independent conditioned on the state of the environment; this would be the case when the observation noises are additive and independent from sensor to sensor 1 . Such a design has a few drawbacks.
- It does not exploit the spatial correlation in observations across sensors. 2) It does not exploit the superposition available on the GMAC.
Leena Zacharias is with Beceem Communications Pvt. Ltd., Bangalore, India, and Rajesh Sundaresan is with the Department of Electrical Communication Engineering, Indian Institute of Science, Bangalore, India.
This work was supported by the Defence Research & Development Organisation (DRDO), Ministry of Defence, Government of India under a research grant on wireless sensor networks (DRDO 571, IISc). 1 As we will see later, conditional independence notwithstanding, sensor observations are correlated.
- It employs an ad hoc separation between quantization or compression on one hand, and transmission across the channel on the other; the latter requires adequate coding for noiseless reception and correct further processing at the fusion center. 4) It requires sufficient time slots for sensors to resolve all channel contentions2 .
Our goal in this paper is to detect change in environment in a manner that addresses the aforementioned drawbacks. Specifically, we consider a “star” topology of sensors. Sensors make an affine transformation of the observed data and transmit the output in an analog fashion over the GMAC. Given that observations at sensors at any instant are spatially correlated, only the sum of the LLRs is relevant to the decision maker, i.e., it is a sufficient statistic to decide on the change. By making the sensors simultaneously transmit an affine function of their LLRs in an analog fashion, and via distributed transmit beamforming, we exploit the spatial correlation in sensor data and the superposition available on the GMAC -the channel computes the required sum. Moreover, the analog data is in loose terms matched to the channel and does not require explicit channel coding. Finally, the sum is available at the fusion center in a single transmit duration unlike the situation in the random access case.
The biggest challenge in our proposed technique is the practicality of distributed transmit beamforming. The transmitters’ clocks should be synchronized to some extent, so that carrier, phase, and symbol ticks align. A technique similar to the master-slave architecture proposed by Mudumbai, Barriac & Madhow [1] can be used to achieve this synchronization. The scheme exploits channel reciprocity in a time-division duplex (TDD) system.
Organization and preview of main results: In Section II, we formulate and solve a change detection problem under a power-constrained setting 3 . We arrive at a Markov decision problem framework and show that parameters of the affine transformation should minimize the variance of the combined observation and GMAC noises, which turns out to be a nonconvex optimization problem. We then provide an explicit algorithm to compute the optimal control parameters. Section III considers an energy-constrained setting. Section IV compares the simulation performance of our scheme with a previously known scheme. It also compares the energy-constrained formulation of Section III with the power-constrained formulation of Section II. Appendix I contains a new characterization of optimal control: maximize a certain Ali-Silvey distance [2] between the distributions of the fusion center’s observation before and after the change. This is used to arrive at the minimum variance criterion of Section II.
Prior work: Change detection problems were solved in a centralized setting by Page [3], Lorden [4], and Shiryayev [5]. Shiryayev considered a Bayesian setting which is of relevance to our work. Veeravalli [6] solved the decentralized version of this problem with parallel error-free bit pipes of limited capacity from the sensors to the fusion center and identified the optimal stopping policy and quantizer structure. These results are analogous to those for hypothesis testing and sequential hypothesis testing (Tsitsiklis [7], Veeravalli et al. [8]). Prasanthi [9] considered access and decision delays in sequential detection
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