Cellular automata are considered to be a nice model of complex systems in which an infinite one-dimensional array of finite state machines (cells) updates itself in synchronous manner according to a uniform local rule. We study a generalized synchronization problem that gives a finite-state protocol for synchronizing a large scale of cellular automata with an initial initiator at any generalized position. We focus on the fundamental generalized firing squad synchronization algorithms operating in optimum- and non-optimum-steps on one-dimensional cellular arrays. A new eight-state algorithm is proposed in this paper. The eight-state optimum-step algorithm is the smallest one known at present in the class of generalized optimum-step firing squad synchronization protocols. Some new viewpoints and results are also given.