Generalized Laguerre filter and generalized Jacobi filter

Bandpass analog and digital filters are proposed which have maximally flat attenuation and group delay characteristics. Since the proposed analog filter corresponds to or is an extension of the low-pass Laguerre filter, it is referred to as the generalized Laguerre filter, and for the same reason the proposed digital filter is called the generalized Jacobi filter. The numerator and denominator polynomials of the generalized Laguerre filter are obtained from the solutions of two differential equations having common eigenvalues. In order for these two differential equations to have always identical eigenvalues and polynomial solutions with real coefficients, it is sufficient that the orders of the numerator and the denominator of the transfer function be even and that the smallest eigenvalues of each equation be taken. The transfer function of the generalized Jacobi filter is similarly obtained. The relations between these two types of filters and other known filters are demonstrated. Examples of the frequency responses of these two types of filters are illustrated.