Scaling Laws for Dense Gaussian Sensor Networks and the Order Optimality of Separation

We investigate the optimal performance of dense sensor networks by studying the joint source–channel coding problem. There are $N$ uniformly spaced sensor nodes sampling noiselessly a one-dimensional spatial random process over an interval $[0,U_{0}]$ . The overall goal of the sensor network is for the sensor nodes to code and transmit the measurement samples to a collector node over a cooperative multiple-access channel with noisy feedback, and for the collector node to reconstruct the entire random process with minimum expected distortion. We provide separation-based lower and upper bounds for the minimum achievable expected distortion when the underlying random process is Gaussian. When the Gaussian random process satisfies some general conditions, such as the eigenvalues of its Karhunen–Loeve expansion decrease roughly inverse polynomially in order $x$, i.e., the $k$th eigenvalue is roughly $k^{-x}$ , we evaluate the lower and upper bounds explicitly, and show that they are of the same order for a wide range of power constraints. Thus, for these random processes, under these power constraints, we show that the minimum achievable expected distortion decreases as $\left (\log NP(N) \right )^{1-x}$, where $P(N)$ is the sum power constraint on the sensor nodes. Further, we show that the achievability scheme that achieves the lower bound on the distortion is a separation-based scheme that is composed of multiterminal rate-distortion coding and amplify-