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It is well known that over the parallel Gaussian point-to-point channel, coding separately over the individual subchannels (carriers) achieves capacity subject to optimal power allocation. Thus, the capacity of the parallel Gaussian point-to-point channel is equal to the sum of the capacities of the point-to-point Gaussian subchannels with corresponding powers chosen through the waterfilling algorithm. Similarly, it has also been shown that separate coding over each carrier is optimal for parallel Gaussian Multiple Access (MAC) and Broadast (BC) Channels.
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