This paper is concerned with discrete-time negative imaginary (DT-NI) functions. First, a new definition of DT-NI functions is introduced. Then, by means of the relations between discrete-time positive real and DT-NI functions, two different versions of DT-NI lemmas are established to characterize the DT-NI properties based on state-space realizations. Also, a necessary and sufficient condition is presented to guarantee the internal stability of positive feedback interconnected DT-NI systems. Meanwhile, some other properties of DT-NI functions are studied. Several numerical examples are presented to illustrate the main results of this paper. Compared to the previous results, our results remove the symmetric assumption in rational case. (C) 2017 Elsevier Ltd. All rights reserved.