The Capacity Region of Gaussian Broadcast Channels with Intersymbol Interference

We derive the two-user capacity region with and without common information for a Gaussian broadcast channel with intersymbol interference, assuming colored Gaussian noise and an input power constraint. The proof is achieved by rst deening a similar channel model, the circular broadcast channel, which can be decomposed into a set of parallel degraded broadcast channels. The capacity region for parallel degraded broadcast channels is known. We show that the capacity region of the original broadcast channel equals that of the circular broadcast channel in the limit of innnite block length, and we obtain an explicit formula for the resulting capacity region. The coding strategy used to achieve each point on the convex hull of the capacity region uses superposition coding on some or all of the parallel channels and dedicated transmission on the others. For the case without common information the optimal power allocation for any point in the capacity region is obtained via a multilevel water-lling. We derive this optimal power allocation and the resulting capacity region for several broadcast channel models.