Guaranteeing Variable-Filter Stability Utilizing Inverse Trigonometric Functions

Digital filtering is a fundamental signal processing technique, which is applied in almost every area nowadays. Some of the filtering operations require the filter to update filter’s frequency responses as quickly as possible in the process of filtering operations. This goal can be accomplished by utilizing a variable digital filter (VDF) because a VDF has considerably high flexibility to update its frequency responses instantly. When designing a recursive VDF that contains feedback circuits, one must deal with the instability problem seriously. A recursive VDF is useless if it is unstable. This paper first examines the methodology for guaranteeing the VDF’s stability based on coefficient transformations, and then presents two novel functions that can be exploited in carrying out the transformations. The two functions are based on two inverse trigonometric functions, and any of them satisfies a special requirement. Incorporating any of them into the process of designing a recursive VDF results in a recursive VDF whose stability of is surely guaranteed.