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In this paper, the design of reduced-order and full-order interval observers for discrete-time system is investigated. First, the original system is transformed into a descriptor system with the ability to estimate the unknown noise of sensors. Next, a reduced-order interval observer design method for descriptor system is presented. The state matrix of the system is obtained by linear transformation, and a Sylvester equation is constructed to find the appropriate observer gain matrix. Then, an existing design method of full-order interval observer is introduced in this paper. Finally, a simulation example is given to demonstrate that the reduced-order interval observer has more accurate interval estimation results than the full-order interval observer.