Generalizations and new proof of the discrete-time positive real lemma and bounded real lemma

There are three different restatements claimed to be equivalent to the definition of discrete-time positive realness (DTPR) in the literature. These restatements were obtained by assuming that they are similar to the results of continuous-time positive realness when the transfer function has poles on the stability boundary. In this paper it is shown that only one of them is equivalent to the DTPR lemma and others are disproved by counter-examples. Furthermore, the DTPR lemma is specialized for minimal systems which have all poles on the unit cycle, the DTPR lemma is also generalized for nonminimal systems, the discrete-time bounded real (DTBR) lemma is proven by a simple method, and then the DTBR lemma is extended to the nonminimal case. Continuous-time results are also briefly considered in the Appendix.