Frequency Transformation for Linear State-Space Systems and Its Application to High-Performance Analog/Digital Filters

Frequency transformation is one of the well-known techniques for design of analog and digital filters [1, 2]. This technique is based on variable substitution in a transfer function and allows us to easily convert a given prototype low-pass filter into any kind of frequency selective filter such as low-pass filters of different cutoff frequencies, high-pass filters, band-pass filters, and band-stop filters. It is also well-known that the transformed filters retain some properties of the prototype filter such as the stability and the shape of the magnitude response. For example, if a prototype filter is stable and has the Butterworth magnitude response, any filter given by the frequency transformation is also stable and of the Butterworth characteristic. Due to this useful fact, the frequency transformation is suitable not only to the filter design but also to the real-time tuning of cutoff frequencies, which can be applied to design of variable filters [3] and to adaptive notch filtering [4, 5]. Hence the frequency transformation plays important roles in many modern applications of signal processing from both the theoretical and practical points of view.