Distortion bounds for broadcasting with bandwidth expansion

We consider the problem of broadcasting a single Gaussian source to two listeners over a Gaussian broadcast channel, with ρ channel uses per source sample, where ρ > 1. A distortion pair (D₁, D₂) is said to be achievable if one can simultaneously achieve a mean-squared error (MSE) D₁ at receiver 1 and D₂ at receiver 2. The main result of this correspondence is an outer bound for the set of all achievable distortion pairs. That is, we find necessary conditions under which (D₁, D₂) is achievable. We then apply this result to the problem of point-to-point transmission over a Gaussian channel with unknown signal-to-noise ratio (SNR) and ρ > 1. We show that if a system must be optimal at a certain SNR_min then, asymptotically, the system distortion cannot decay faster than O(1/SNR). As for achievability, we show that a previously reported scheme, due to Mittal and Phamdo (2002), is optimal at high SNR. We introduce two new schemes for broadcasting with bandwidth expansion, combining digital and analog transmissions. We finally show how a system with a partial feedback, returning from the bad receiver to the transmitter and to the good receiver, achieves a distortion pair that lies on the outer bound derived here.