Broadcasting with fidelity criteria

Consider the problem of broadcasting an i.i.d. source sequence X = {X_i} _i=1^N (possibly N → ∞) to n listeners over a discrete broadcast channel, consisting of n channels with capacities C₁ = C_max ≥ C₂ ≥...≥ C_n = C_min. Let the tuple D = (D₁, D₂,...,D_n) represent the average distortion in reconstructing sources at the n listeners. The problem of characterizing all achievable tuples D is still open for a general case. For a fairly general class of discrete channels, we prove the achievability of the tuple n(ρ₁,ρ₂,...,ρ_n) = (D_X(ρ₁C₁ ζ), D_X(ρ₂C₂ - ζ),...,D_X(ρ_nC_n - ζ)), provided that λ_i = (ρ_iC_i - ρ_i+₁C_i+1)/C_i > 0, for 1 ≤ i ≤ n $1, λ_n = ρ_n and Σ_i=1^n-1 λ_i ≤ 1, where D_X (R) is the distortion rate function of X. The penalty term ζ = 1/2 for a general source with real alphabets and is ζ = 0 if X is progressively refinable. The factor 0 < ρ_i ≤ 1 is called the utilization of the i^th channel. As an example, for n = 2, we show that ²(1/(2 - C₂/C₁), 1/(2 $C₁/C₂)) is achievable for any C₁, C₂. Furthermore, the common utilization of ρ = (1 + ln(C_max/C_min))^-1 is shown to be achievable for all channels. Conversely, we find examples of channels, namely erasure switch-to-talk channels, for which the proposed achievable utilizations are tight. In particular, while ³ (2/3, 2/3, 2/3) is achievable for any compatible broadcast channel with capacities C₁ = 2C₂ = 2C₃, for any δ > 0, we find examples of channels for which ³(2/3+δ,2/3+δ,2/3+ δ) is not achievable.