Broadcasting with fidelity criteria
Consider the problem of broadcasting an i.i.d. source sequence X = {X/sub i/} /sub i=1//sup N/ (possibly N /spl rarr/ /spl infin/) to n listeners over a discrete broadcast channel, consisting of n channels with capacities C/sub 1/ = C/sub max/ /spl ges/ C/sub 2/ /spl ges/.../spl ges/ C/sub n/ = C/sub min/. Let the tuple D = (D/sub 1/, D/sub 2/,...,D/sub 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(/spl rho//sub 1/,/spl rho//sub 2/,...,/spl rho//sub n/) = (D/sub X/(/spl rho//sub 1/C/sub 1/ /spl zeta/), D/sub X/(/spl rho//sub 2/C/sub 2/ - /spl zeta/),...,D/sub X/(/spl rho//sub n/C/sub n/ - /spl zeta/)), provided that /spl lambda//sub i/ = (/spl rho//sub i/C/sub i/ - /spl rho//sub i/+/sub 1/C/sub i+1/)/C/sub i/ > 0, for 1 /spl les/ i /spl les/ n $1, /spl lambda//sub n/ = /spl rho//sub n/ and /spl Sigma//sub i=1//sup n-1/ /spl lambda//sub i/ /spl les/ 1, where D/sub X/ (R) is the distortion rate function of X. The penalty term /spl zeta/ = 1/2 for a general source with real alphabets and is /spl zeta/ = 0 if X is progressively refinable. The factor 0 < /spl rho//sub i/ /spl les/ 1 is called the utilization of the i/sup th/ channel. As an example, for n = 2, we show that /sup 2/(1/(2 - C/sub 2//C/sub 1/), 1/(2 $C/sub 1//C/sub 2/)) is achievable for any C/sub 1/, C/sub 2/. Furthermore, the common utilization of /spl rho/ = (1 + ln(C/sub max//C/sub min/))/sup -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 /sup 3/ (2/3, 2/3, 2/3) is achievable for any compatible broadcast channel with capacities C/sub 1/ = 2C/sub 2/ = 2C/sub 3/, for any /spl delta/ > 0, we find examples of channels for which /sup 3/(2/3+/spl delta/,2/3+/spl delta/,2/3+ /spl delta/) is not achievable.