Analog and digital filters with α-splines

Analytical expressions of analog and digital α-spline filters with continuous transition bands are derived. Previous works based on β-spline functions consider the order of the spline function, which must be a natural number, as a parameter to sharpen the transition band. However, the inclusion of an α-spline model provides greater flexibility since its order can be a real number, it depends on three parameters, and it can be used to model the transition band in the frequency domain. This paper presents the formulation of analog α-spline filters and the corresponding discretization to obtain the frequency and the impulse responses of their digital counterparts. There is also shown the design of α-spline digital filters minimizing two different least integral squared errors. As these methods do not provide direct control over the stopband attenuation, a third method which leads to principally flat digital filters is introduced. Several numerical examples demonstrate the efficiency and flexibility of the proposed techniques.