This paper revisits an alternative formulation of the Kalman-Yakubovich-Popov (KYP) Lemma, relating an infinite dimensional Frequency Domain Inequality (FDI) to a pair of finite dimensional Linear Matrix Inequalities (LMI). It is shown that this alternative formulation allows a certain class of the coefficient matrix of the FDI to be frequency dependent without introducing conservatism. The construction provided in the present paper is helpful in other problems where system augmentation is commonly used, such as those involving rational or polynomial multipliers for stability and performance analysis. (C) 2010 Elsevier Ltd. All rights reserved.