Location estimation in wireless sensor networks is the problem of estimating the location of an object from a set of noisy measurements. These measurements are acquired in a distributed manner by a set of sensors.
Use Many civilian and military applications require monitoring that can identify objects in a specific area, such as monitoring the front entrance of a private house by a single camera. Monitored areas that are large relative to objects of interest often require multiple sensors (e.g., infra-red detectors) at multiple locations. A centralized observer or computer application monitors the sensors. The communication to power and bandwidth requirements call for efficient design of the sensor, transmission, and processing. The CodeBlue system of Harvard University is an example where a vast number of sensors distributed among hospital facilities allow staff to locate a patient in distress. In addition, the sensor array enables online recording of medical information while allowing the patient to move around. Military applications (e.g. locating an intruder into a secured area) are also good candidates for setting a wireless sensor network.
Setting
Let θ {\displaystyle \theta } denote the position of interest. A set of N {\displaystyle N} sensors acquire measurements x n = θ + w n {\displaystyle x_{n}=\theta +w_{n}} contaminated by an additive noise w n {\displaystyle w_{n}} owing some known or unknown probability density function (PDF). The sensors transmit measurements to a central processor. The n {\displaystyle n} th sensor encodes
x n {\displaystyle x_{n}} by a function m n ( x n ) {\displaystyle m_{n}(x_{n})} . The application processing the data applies a pre-defined estimation rule
θ ^ = f ( m 1 ( x 1 ) , ⋅ , m N ( x N ) ) {\displaystyle {\hat {\theta }}=f(m_{1}(x_{1}),\cdot ,m_{N}(x_{N}))} . The set of message functions
m n , 1 ≤ n ≤ N {\displaystyle m_{n},\,1\leq n\leq N} and the fusion rule f ( m 1 ( x 1 ) , ⋅ , m N ( x N ) ) {\displaystyle f(m_{1}(x_{1}),\cdot ,m_{N}(x_{N}))} are designed to minimize estimation error. For example: minimizing the mean squared error (MSE),
E ‖ θ − θ ^ ‖ 2 {\displaystyle \mathbb {E} \|\theta -{\hat {\theta }}\|^{2}} . Ideally, sensors transmit their measurements x n {\displaystyle x_{n}}
right to the processing center, that is m n ( x n ) = x n {\displaystyle m_{n}(x_{n})=x_{n}} . In this settings, the maximum likelihood estimator (MLE) θ ^ = 1 N ∑ n = 1 N x n {\displaystyle {\hat {\theta }}={\frac {1}{N}}\sum _{n=1}^{N}x_{n}} is an unbiased estimator whose MSE is
E ‖ θ − θ ^ ‖ 2 = var ( θ ^ ) = σ 2 N {\displaystyle \mathbb {E} \|\theta -{\hat {\theta }}\|^{2}={\text{var}}({\hat {\theta }})={\frac {\sigma ^{2}}{N}}} assuming a white Gaussian noise
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