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The positive real analysis problem for discrete singular systems is considered. By giving an augmented system for general discrete singular system, and using the equivalent admissibility between augmented system and original system and the character of linear matrix inequalities, a new necessary and sufficient condition for that discrete singular systems are admissible is derived. And the new admissible condition can deduce the theorem of the existing literature. Then the extended strict positive real problem for discrete singular systems is discussed, a new necessary and sufficient condition for that discrete singular systems are admissible and extended strictly positive real (ESPR) is given. Finally, through numerical examples illustrate the criterion of extended strictly positive real discrete singular systems less conservative than previous results.