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This paper investigates the leader-following consensus problem of discrete-time multi-agent systems. The dynamics of the leader and all following agents adopt the same general form of a linear model that can be of any order. The interconnection topology among the agents is assumed to be switching and undirected. To track the active leader, two kinds of distributed observer-based consensus protocols are proposed for each following agent, whose distributed observers are used to estimate the leader's state and the tracking error based on the relative outputs of neighboring agents, respectively. In light of the modified discrete-time algebraic Riccati equality and Lyapunov method, we prove that the discrete-time leader-following consensus problem can be solved by proposing the distributed observer-based consensus protocol under switching topologies. Finally, a numerical example is given to illustrate the obtained result.