Abstract: We propose a thoroughly optimal signal design strategy to achieve the Pareto boundary (boundary of the achievable rate region) with improper Gaussian signaling (IGS) on the Z-interference channel (Z-IC) under the assumption that the interference is treated as additive Gaussian noise. Specifically, we show that the Pareto boundary has two different schemes determined by the two paths manifesting the characteristic of improperly transmitted signals. In each scheme, we derive several concise closed-form expressions to calculate each user’s optimally transmitted power, covariance, and pseudo-covariance of improperly transmitted signals. The effectiveness of the proposed optimal signal design strategy is supported by simulations, and the results clearly show the superiority of IGS. The proposed optimal signal design strategy also provides a simple way to achieve the required rate region, with which we also derive a closed-form solution to quickly find the circularity coefficient that maximizes the sum rate. Finally, we provide an in-depth discussion of the structure of the Pareto boundary, characterized by the channel coefficient, the degree of impropriety measured by the covariance, and the pseudo-covariance of signals transmitted by two users.

Key words: Z-interference channel, Improper Gaussian signaling, Sum-rate, Pareto boundary, Covariance, Pseudo-covariance

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