Probabilistic shaping (PS) is a well-known method to achieve improved performance upon a regular quadrature amplitude modulation (QAM) by taking the target constellation and making the distribution of underlying points non-uniform. It has been extensively studied over the years for the additive white Gaussian noise (AWGN) and Rayleigh fading channels. However, the potential of probabilistic shaping in the multiple-input and multiple-output (MIMO) setting needs further investigations. In this paper, we prove that if shaped symbols follow Maxwell-Boltzmann distribution, the optimal maximum a posteriori (MAP) detection is equivalent to the case of uniform constellation with a simple preprocessing step. Our approach has multiple benefits. It enables to utilize the same processing chain for both shaped and unshaped system, which simplifies the receiver architecture. This technique can be applied for both linear MMSE and nonlinear (near-)MAP demapper types. In addition, the complexity of adaptive methods such as sphere decoding can be reduced.

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