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The both Riccati equation approach and state space approach are presented using separation principle for the observer based discrete-time left sub-static-output and weighted right sub-static-output feedback (LWRSSOF) H2 optimal control problem. Necessary and sufficient conditions for the existence of all controllers solving the problem are given in terms of two Riccati equations. Based on these conditions, all controllers solving the problem are parameterized. Using the same two Riccati equations, necessary and sufficient conditions for the existence of all observer based state feedback H2 optimal controllers can be obtained. However, unlike the observer based state feedback H2 optimal control problem, the feedback of the problem contains all information of measured output.