Our investigation focuses on the performance of fullduplex systems where both nodes simultaneously exchange their signals using a single spectrum for two-way communication.To maximize the achievable sum rate for full-duplex systems in multi-spectrum environments, we address a new selection strategy that considers the channel state of both nodes. The achievable sum rate for a multi-spectral bi-directional full-duplex (M-BFD) system is obtained using the proposed selection strategy and compared with conventional multi-spectral bi-directional half-duplex (M-BHD) systems. We derive the achievable sum rate as closed-form expressions and quantify the effect of selection diversity on the achievable sum rates for M-BFD and M-BHD systems. In addition, we present practical-case Monte-Carlo simulation results by considering the effects of self-interference (SI) and channel correlation. The numerical results show that the achievable sum rate (ASR) gain for M-BFD is larger than that for M-BHD.