This paper considers the robust observer design problem for linear time-invariant dynamic systems subject to external disturbances. In this paper, robustness against disturbances is considered within a finite frequency range, instead of the standard infinite frequency range solution; this is achieved via the well established Generalized Kalman-Yakubovich-Popov (GKYP) results. In order to obtain a convex optimization problem, Finsler's Lemma is used to pose the GKYP Lemma as minimization problem in the form of a Linear Matrix Inequality which represents a finite range version of the Bounded Real Lemma. In contrast to other GKYP LMI formulations, the results presented in this work may be considered more conservative, but may be argued to be more simple and tractable than existing formulations. The effectiveness of the proposed procedure is shown through a simulation example.