Theorem (Lossy Source-Channel Coding Theorem)

Suppose the DMS $\{ X_{i} \}_{i=1}^\infty$ has rate distortion function $R(D)$ and the DMC has capacity $C$ then

1. Forward Part: For any $D>0$ such that $$R(D)<C$$ $\exists \{ (f_{n}^{sc},g_{n}^{sc}) \}$ of lossy source-channel codes such that $$\limsup_{ n \to \infty } E[d(X^{n},\hat{X}^{n})]\le D$$
2. Converse: For any blocklength $n$, if the lossy source-channel code $(f_{n}^{sc},g_{n}^{sc})$ has distortion $$E[d(X^{n},\hat{X}^n)]\le D$$then we must have that $$R(D)\le C$$