On Capacity-Achieving Distributions Over Complex AWGN Channels Under Nonlinear Power Constraints and their Applications to SWIPT

The capacity of deterministic, complex and discrete time memoryless Additive White Gaussian Noise (AWGN) channel under three constraints, namely, channel input average power, channel input amplitude and delivered power at the channel output is considered. The delivered power constraint is modelled as a linear combination of even-moment statistics of the channel input being larger than a threshold. It is shown that the capacity of an AWGN channel under transmit average power and receiver delivered power constraints is the same as the capacity of an AWGN channel under an average power constraint, however, depending on the two constraints, it can be either achieved or arbitrarily approached. It is also shown that under average power, amplitude and delivered power constraints, the optimal capacity achieving distributions are discrete with a finite number of mass points. To establish the results, the confluent hypergeometric functions as well as the output rate of decay of complex Gaussian channels are utilized extensively. As an application, a simultaneous information and power transfer (SWIPT) problem is studied, where an experimentally-validated nonlinear model of the harvester is used. Relying on small signal analysis approximation, a general form of the delivered Direct-Current (DC) power in terms of system baseband parameters is derived for independent and identically distributed (iid) inputs. It is shown that the delivered power depends on higher order statistics of the channel input. By defining the rate-power (RP) region, two inner bounds, one based on complex Gaussian inputs and the other based on convexifying the optimization probability space, are obtained. For Gaussian inputs, it is shown that the optimal inputs are zero mean and a tradeoff between information and power is recognized by considering asymmetric power allocations between Inphase and Quadrature subchannels. To obtain the second inner bound, numerical programming is utilized. Through numerical algorithms, it is observed that the numerically obtained input (NOI) distributions attain larger RP region compared to Gaussian input counterparts.