Bisection algorithm for computing the frequency response gain of sampled-data systems

This paper presents a bisection algorithm for computing the frequency response gain of sampled-data systems composed of a continuous-time and a discrete-time system together with a sampler and a hold. Although a bisection algorithm is known in the literature, it is applicable only to those frequencies at which the frequency response gain is greater than //D// where D denotes the direct feed-through term in the lifted transfer function of the sampled-data system. This paper gives a new bisection algorithm that can be applied to any frequencies. The derivation given in this paper is based on the lifting technique, and the properties of the congruent transformation (the Schur complement and the Sylvester law of inertia) play a key role. To facilitate the algorithm, a bisection algorithm for computing the singular values of D is also presented with the combined use of the lifting and FR-operator techniques.