Power allocation for robust distributed best-linear-unbiased estimation against sensing noise variance uncertainty

Motivated by the fact that system parameter mismatch occurs in real-world sensing environments, this paper proposes power allocation schemes for robust distributed best-linear-unbiased estimation (BLUE) that take account of the uncertainty in the local sensing noise levels. Assuming that (i) the sensing noise variance follows a statistical distribution widely used in the literature and (ii) the link channel gains between sensor nodes and the fusion center (FC) are i.i.d. Rayleigh fading, we propose to use the average reciprocal mean square error (ARMSE), averaged with respect to the distributions of sensing noise variance and fading channels, as the distortion measure. A fundamental inequality characterizing the relation between ARMSE and the average mean square error (AMSE) is established to justify the proposed design metric. While the exact formula for ARMSE is difficult to find, we derive an associated closed-form lower bound which involves the incomplete gamma function. To further ease analysis, we further derive a key inequality that specifies the range of the ARMSE lower bound. Particularly, it is shown that the boundary points of this inequality are characterized by a common function, which involves the Gaussian-tail Q(·) and is thus more analytically appealing. By conducting optimization on the basis of such a function, we obtain closed-form robust solutions for two power allocation problems: (i) optimizing distortion metric under a total power constraint, and (ii) minimizing total power under a target distortion requirement. In case that instantaneous channel state information (CSI) is available to the FC, the proposed approach can be easily modified to derive analytic robust power allocation factors best matched to the CSI realizations. Computer simulations evidence the effectiveness of the proposed schemes.