On stability across a Gaussian product channel

We present sufficient conditions for stabilizing a scalar discrete-time LTI plant in the mean squared sense when a sensor transmits the plant state information to a remotely placed controller across a Gaussian product channel. The Gaussian product channel models a continuous-time waveform Gaussian channel, where the encoder transmits information to the receiver across multiple noisy paths. It is known that linear coding schemes may lead to overly restrictive stabilizability conditions in such scenarios. We present a non-linear coding scheme and present the resulting stablizability conditions. When these conditions are satisfied with equality, the proposed coding scheme transmits data across the product channel at a rate equal to the capacity of the channel; thus, the conditions are conjectured to be necessary as well.