In communications theory, the calculation of channel capacity for M parallel Gaussian channels typically includes a “water-filling” methodology to allocate the received power P_m in each channel, m = 1,2,…M, subject to a total average power constraint. (See equations (36) and (37) in the link provided for instance).
Suppose, however, that we know that our transmitter cannot change the transmit power in each channel, and that the received power P_m in each channel therefore cannot be adjusted away from noisy channels toward quieter channels as the “water-filling” methodology undertakes to do. Then the received power P_m in a channel will be independent of the noise power in channel m, and independent of the noise power in all of the other channels.
Admittedly the communication will generally be sub-optimal in such cases. Nevertheless, does it make sense to define a channel capacity for this operational constraint, simply by adding the independent channel capacity CC_m of all of the channels m = 1,2…M together?
This straight-forward sum would be a total channel capacity subject to a practical power-management constraint (noise-indifferent power allocation) that happens to differ from the average power constraint implemented in the water-filling methodology.
Do you know of a relevant reference on that point?
Ronald
The missing link is here
http://www-mobile.ecs.soton.ac.uk/lly/slides/Channel-Capacity-4up.pdf
Yes, you can define the capacity under a particular type of power constraint, such as the one that you describe. The capacity is generally defined as the mutual information, maximized with respect to the input distribution (over a given set of feasible input distributions). In your case, the input distribution is limited in such a way that there is a maximum power in each parallel channel and one cannot move any power between the channels. This will lead to equal power allocation.
The channel capacity is set by the Shanon formula as:
C= B Log2(1+S/(N+I)),
with B is the channel bandwidth, S is the received signal at the receiver and I is the interference. For parallel channels we we sum the above expression a number of times equals to the number of parallel channels.
Assuming that the parallel channels have the same bandwidth B,
and Assuming certain noise power distribution among the channels which means different values of the noise power with different sensitivities of the receiver, then it remains to allocate the power P for the different channels. One needs to maximize the summation of the above expression above over all the parallel channels. I think the factor which which determines the interference signal I is the power allocation among the parallel channels. one can assume that I is proportional to P so, one can allocate for the less noisy channel neighbored by a more noisy channel a larger power, this would lead to increased average channel capacity.
Best wishes