A practical communication channel often suffers from constraints on input other than the average power, such as the peak power constraint. In order to compare achievable rates with different constellations as well as the channel capacity under such constraints, it is crucial to take these constraints into consideration properly. In this paper, we propose a direct approach to compare the achievable rates of practical input constellations and the capacity under such constraints. As an example, we study the discrete-time complex-valued additive white Gaussian noise (AWGN) channel and compare the capacity under the peak power constraint with the achievable rates of phase shift keying (PSK) and quadrature amplitude modulation (QAM) input constellations.
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