Synchronization problems are inevitable in in-band full-duplex (IBFD) systems due to arrival time misalignment between self-interference (SI) and the desired signal. We begin by showing that the timing misalignment significantly affects the performance of digital SI cancellation (SIC) or signal detection when conventional synchronization is used. The conventional synchronization method introduces inter-symbol interference between SI signals, thus reducing the performance of the digital SIC. In order to guarantee the performance of digital SIC and signal detection against the timing misalignment, we propose a two-step synchronization method. Lastly, using numerical results, we compare the performance of digital SIC and signal detection according to the addressed synchronization methods and show that the superiority of the proposed methods under asynchronous environments.