Generalised water-filling: costly power optimally allocated to sub-carriers under a general concave performance function

A data terminal seeks the optimal power allocation for M sub-carriers under a general concave performance function (¿capacity¿), and a per-Watt price. The value to the terminal of one transferred information bit, as well as the noise-normalised channel gains for each sub-carrier are the critical parameters. We provide the structure of the general solution, and give closed-form expressions for the logarithmic form. The analysis reveals that if a sub-channel gain is less than or equal to the suitably normalised power price said sub-channel is not usable. Our results are directly applicable for power allocation in a time-division OFDM system. They can also be useful in an OFDMA system, if complemented with some sensible sub-channel allocation scheme, as we propose separately.