Channel capacity and state estimation for state-dependent Gaussian channels

We formulate a problem of state information transmission over a state-dependent channel with states known at the transmitter. In particular, we solve a problem of minimizing the mean-squared channel state estimation error E∥Sⁿ - Ŝⁿ∥ for a state-dependent additive Gaussian channel Yⁿ = Xⁿ + Sⁿ + Zⁿ with an independent and identically distributed (i.i.d.) Gaussian state sequence Sⁿ = (S₁,..., S_n) known at the transmitter and an unknown i.i.d. additive Gaussian noise Zⁿ. We show that a simple technique of direct state amplification (i.e., Xⁿ = αSⁿ), where the transmitter uses its entire power budget to amplify the channel state, yields the minimum mean-squared state estimation error. This same channel can also be used to send additional independent information at the expense of a higher channel state estimation error. We characterize the optimal tradeoff between the rate R of the independent information that can be reliably transmitted and the mean-squared state estimation error D. We show that any optimal (R, D) tradeoff pair can be achieved via a simple power-sharing technique, whereby the transmitter power is appropriately allocated between pure information transmission and state amplification.