We extend a fast technique for solving semidefinite programs involving nonnnegative trigonometric polynomials to problems derived from the discrete-time Kalman-Yakubovich- Popov (KYP) lemma and some of its generalizations. The frequency-domain inequality associated with the generalized KYP lemma is first expressed as a weighted sum of squares of rational functions. By taking a sufficient number of samples of the sum-of-squares expression, an equivalent standard-form semidefinite program with low-rank structure is obtained. This low-rank structure is easily exploited in implementations of primal-dual interior-point algorithms. A complexity analysis and numerical examples are provided to support the performance improvement over standard semidefinite programming solvers.