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 di▲erent 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 e▲ectiveness 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 coe▼cient that maximizes the sum rate. Finally, we provide an in-depth discussion of the structure of the Pareto boundary, characterized by the channel coe▼cient, the degree of impropriety measured by the covariance, and the pseudo-covariance of signals transmitted by two users.