The Optimum Linear Modulator for a Gaussian Source Used with a Gaussian Channel

A Decision Theoretic Approach to Gaussian Sensor Networks

“…Transmission noises could anyway be included straightforwardly in our treatment. It is worth noting that, since the variance of the source components is uniform, in the presence of a Gaussian vector channel with uniform variance of the noise components, linear (i.e., uncoded) strategies would be optimal, as noted in [8]; however, the optimum gain matrix would be non-diagonal, and would therefore require a centralized solution, whereas we are seeking a decentralized one, where each sensor encodes its observed variable. In [11], uncoded transmission is anyway adopted for the sensors; then, by exploiting the fact that the source observations are correlated, the minimum number of sensors that need to be activated to achieve nearly optimal distortion is sought, out of the total number of deployed sensors.…”

“…This is a natural choice, due to the distributed solution of Problem 1 that is assumed in [11]. In situations where centralized strategies are applicable, optimality would be guaranteed by a centralized exploitation of the covariance matrix [8].…”

“…We will refer to (8) and (9) here as linear strategies. We must note that, even if (8) and (9) result in non-exactly optimal strategies, they represent a reasonable choice for a distributed environment with possibly unknown sensor locations and a useful tool to derive insights into the performance sensitivity of Problem 1.…”

“…Indeed, besides the well-known case of a scalar Gaussian random variable to be transmitted over a Gaussian channel, where uncoded and "instantaneous" (also known as "singleletter" or "real-time") transmission is optimal (in the sense of achieving minimum quadratic distortion under a power constraint) [4], there are recent interesting results that prove the optimality of such uncoded schemes for a class of Gaussian sensor networks in general [5] (after a scaling-law optimality of uncoded transmission had been shown previously to hold for the same class of networks [6,7]). Notwithstanding these cases, however, the separation theorem would still apply in other more general situations as, among others, the Gaussian vector channel considered in [8] (where the best linear encoder-decoder pairs are sought). Interestingly enough, the best linear solution here (i.e., the uncoded one, in the sense that the input to the channel is only obtained by multiplying the observed values by a matrix, subject to the overall power constraint) turns out to be also instantaneous (i.e., with block length of 1), and may prevent some components with low signal-to-noise ratio to be transmitted, in favor of others.…”

Nash and Stackelberg equilibria for dynamic cheap talk and signaling games

“…Matching essentially requires that the capacity achieving source probabilities and the rate-distortion achieving channel probabilistic characteristics are simultaneously realized for a given system; this is precisely the case for a scalar Gaussian source transmitted over a scalar additive Gaussian channel. One special case where such a matching holds is the case when the noise and signal power levels are identical in every channel and the distortion criterion is identical for all scalar components [19]. For further discussions on multidimensional Gaussian source and channel pairs, we refer the reader to [17]- [24].…”