Given a discrete stationary source $\{V_{i}\}^\infty_{i=1}$ with alphabet $\mathcal{V}$ and entropy rate $H(\mathcal{V})$, and a DMC $(\mathcal{X},\mathcal{Y},P_{Y|X})$ with capacity $C$, where both $H(\mathcal{V})$ and $C$ are measured in the same units, then we have: 1. Forward Part (Achievability): For any $0<\epsilon<1$ and given that the stationary source is also ergodic (i.e. it is stationary ergodic), if $$H(\mathcal{V})<C$$then $\exists$ sequence of rate-one source-channel codes $\{\mathcal{C}_{m,m}\}^\infty_{m=1}$ with error probability $P_{e}(\mathcal{C}_{m,m})$ satisfying $$P_{e}(\mathcal{C}_{m,m})<\epsilon$$for $m$ sufficiently large. 2. Converse Part: If $$H(\mathcal{V})>C$$then any sequence of rate-one source-channel codes $\{\mathcal{C}_{m,m}\}^\infty_{m=1}$ for the source and the channel has error probability $P_{e}(\mathcal{C}_{m,m})$ satisfying $$\liminf_{m\to\infty}P_{e}(\mathcal{C}_{m,m})>0$$

Given a discrete stationary source $\{V_{i}\}^\infty_{i=1}$ with alphabet $\mathcal{V}$ and entropy rate $H(\mathcal{V})$, and a DMC $(\mathcal{X},\mathcal{Y},P_{Y|X})$ with capacity $C$, where both $H(\mathcal{V})$ and $C$ are measured in the same units, then we have: 1. Forward Part (Achievability): For any $0<\epsilon<1$ and given that the stationary source is also ergodic (i.e. it is stationary ergodic), if $$\limsup_{m\to\infty} \frac{m}{n_{m}} H(\mathcal{V})<C$$then $\exists$ sequence of $m$-to-$n_{m}$ source-channel codes $\{\mathcal{C}_{m,n_{m}}\}^\infty_{m=1}$ with error probability $P_{e}(\mathcal{C}_{m,n_{m}})$ satisfying $$P_{e}(\mathcal{C}_{m,n_{m}})<\epsilon$$for $m$ sufficiently large. 2. Converse Part: If $$\limsup_{m\to\infty} \frac{m}{n_{m}} H(\mathcal{V})>C$$then any sequence of $m$-to-$n_{m}$ source-channel codes $\{\mathcal{C}_{m,n_{m}}\}^\infty_{m=1}$ for the source and the channel has error probability $P_{e}(\mathcal{C}_{m,n_{m}})$ satisfying $$\liminf_{m\to\infty}P_{e}(\mathcal{C}_{m,n_{m}})>0$$