## Sum-rate maximization for active channels

##### Abstract

In conventional wireless channel models, there is no control on the gains of different
subchannels. In such channels, the transmitted signal undergoes attenuation and
phase shift and is subject to multi-path propagation effects. We herein refer to such
channels as passive channels. In this dissertation, we study the problem of joint power
allocation and channel design for a parallel channel which conveys information from a
source to a destination through multiple orthogonal subchannels. In such a link, the
power over each subchannel can be adjusted not only at the source but also at each
subchannel. We refer to this link as an active parallel channel. For such a channel, we
study the problem of sum-rate maximization under the assumption that the source
power as well as the energy of the active channel are constrained. This problem is
investigated for equal and unequal noise power at different subchannels.
For equal noise power over different subchannels, although the sum-rate maximization
problem is not convex, we propose a closed-form solution to this maximization
problem. An interesting aspect of this solution is that it requires only a subset of
the subchannels to be active and the remaining subchannels should be switched off.
This is in contrast with passive parallel channels with equal subchannel signal-tonoise-
ratios (SNRs), where water-filling solution to the sum-rate maximization under
a total source power constraint leads to an equal power allocation among all subchannels.
Furthermore, we prove that the number of active channels depends on the
product of the source and channel powers. We also prove that if the total power
available to the source and to the channel is limited, then in order to maximize the
sum-rate via optimal power allocation to the source and to the active channel, half
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of the total available power should be allocated to the source and the remaining half
should be allocated to the active channel.
We extend our analysis to the case where the noise powers are unequal over different
subchannels. we show that the sum-rate maximization problem is not convex.
Nevertheless, with the aid of Karush-Kuhn-Tucker (KKT) conditions, we propose a
computationally efficient algorithm for optimal source and channel power allocation.
To this end, first, we obtain the feasible number of active subchannels. Then, we show
that the optimal solution can be obtained by comparing a finite number of points
in the feasible set and by choosing the best point which yields the best sum-rate
performance. The worst-case computational complexity of this solution is linear in
terms of number of subchannels.