This paper analyzes the performance of a single-input multiple-output (SIMO) wireless communication system employing one- and two-sided amplitude shift keying (ASK) modulation schemes for data transmission and operating under correlated Rician fading channels. The receiver deploys an optimal noncoherent maximum likelihood detector, which exploits statistical knowledge of the channel state information for signal decoding. An optimal receiver structure is derived, from which series-form and closed-form expressions for the union bound on the symbol error probability (SEP) are obtained for general and massive SIMO systems, respectively. Furthermore, an optimization framework to derive the optimal one- and two-sided ASK modulation schemes is proposed, which focuses on minimizing SEP performance under an average transmit energy constraint. The conducted numerical investigations for various system parameters demonstrate that the proposed noncoherent SIMO system with the designed optimal ASK modulation schemes achieves superior error performance compared to traditional equispaced ASK modulation. It is also shown that, when the proposed system employs traditional two-sided ASK modulation, superior error performance from the case of using one-sided ASK is obtained.