Abstract: For a class of discrete-time systems, the design and analysis of direct model reference adaptive control (MRAC) with normalized adaptive law are investigated. We reprove the discrete-time conclusions on the Lp and L2δ relationship properties between the input and the output, and the discrete-time swapping Lemmas 1 and 2. We also establish the properties of discrete-time adaptive law, define the normalizing signal, and relate the signal with all the signals in the closed-loop system. Thus, the stability and convergence properties of the discrete-time MRAC scheme are analyzed rigorously in a systematic fashion as in the continuous-time case.