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We introduce two new methods for the demodulation of acoustic signals by posing the problem in a convex optimization framework. This allows the parameters of the modulator and carrier to be explicitly defined as constraints in an optimization problem. We first show the theory used to define the demodulation relationship within the rules of convex programming. Then, for the two approaches introduced, we derive specific cost functions and constraints to solve for modulators specifically motivated by perceptual rules. The methods described here perform well with simple, harmonic, and stochastic carriers, and also in the presence of noise.