On Molecular Flow Velocity Meters
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The concentration of molecules in the medium can provide us very useful information about the medium. In this paper, we use this information and design a molecular flow velocity meter using a molecule releasing node and a receiver that counts these molecules. We first assume M hypotheses according to M possible medium flow velocity values and an L-sample decoder at the receiver and obtain the flow velocity detector using maximum-a-posteriori (MAP) method. To analyze the performance of the proposed flow velocity detector, we obtain the error probability, and its Gaussian approximation and Chernoff information (CI) upper bound. We obtain the optimum sampling times which minimize the error probability and the sub-optimum sampling times which minimize the Gaussian approximation and the CI upper bound. When we have binary hypothesis, we show that the sub-optimum sampling times which minimize the CI upper bound are equal. When we have M hypotheses and L →∞, we show that the sub-optimum sampling times that minimize the CI upper bound yield to M 2 sampling times with M 2 weights. Then, we assume a randomly chosen constant flow velocity and obtain the MAP and minimum mean square error (MMSE) estimators for the L-sample receiver. We consider the mean square error (MSE) to investigate the error performance of the flow velocity estimators and obtain the Bayesian Cramer-Rao (BCR) and expected Cramer-Rao (ECR) lower bounds on the MSE of the estimators. Further, we obtain the sampling times which minimize the MSE. We show that when the flow velocity is in the direction of the connecting line between the releasing node and the receiver with uniform distribution for the magnitude of the flow velocity, and L →∞, two different sampling times are enough for the MAP estimator.
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