Duality and Sum Capacity of Gaussian MIMO Broadcast Channels

In this report, we survey the landmark papers on the sum capacity of Gaussian multiple-input multiple-output (MIMO) broadcast channels (BC). Three different groups have reached the same conclusion that the sum rate capacity of Gaussian MIMO BC is achieved by the dirty paper coding (DPC). In particular, we thoroughly investigate the approach which utilized the duality relation between the DPC achievable region of Gaussian MIMO BC and the capacity region of Gaussian MIMO multiple access channels (MAC). This report starts with the explanation of the DPC in Gaussian MIMO BC and then establishes the duality between Gaussian scalar MAC and BC. The duality result is extended to the duality between the capacity region of Gaussian MIMO MAC and the DPC achievable region of Gaussian MIMO BC. Using this extended duality as well as the notion of Sato’s upper bound, we finally show that the DPC region achieves the sum capacity of Gaussian MIMO BC.