Capacity of the Product and Sum of Two Unmatched Broadcast Channels

Assume that $p(y_1,z_1|x_1)=p(y_1|x_1)p(z_1|y_1)$ and $p(y_2,z_2|x_2)=p(z_2|x_2)p(y_2|z_2)$ are two degraded broadcast channels. A broadcast channel with transmitter $(X_1,X_2)$ and receivers $(Y_1,Y_2)$ and $(Z_1,Z_2)$ is called a product of two unmatched degraded broadcast channels. Similarly, the sum of two unmatched degraded broadcast channels is defined as a broadcast channel with transmitter $(X_1\cup X_2)$ and receivers $(Y_1\cup Y_2)$ and $(Z_1\cup Z_2)$. Unlike the original broadcast channels, neither the sum nor the product is a degraded broadcast channel. The region of the capacity is established for the following: i) the product of two unmatched degraded discrete memoryless broadcast channels; ii) a spectral Gaussian broadcast channel; iii) the sum of two unmatched degraded discrete memoryless broadcast channels. These theorems concerning the capacity contain particular results regarding the product of discrete memoryless channels that were obtained by Poltyrev, as well as results concerning spectral Gaussian broadcast channels obtained by Hughes-Hartogs. They also demonstrate that the region of rates obtained by Hughes-Hartogs is optimal for zero overall rate.