Design of recursive 1-D variable filters with guaranteed stability

The digital filters with adjustable frequency-domain characteristics are called variable filters. Variable filters are used in many signal processing fields, but the recursive variable filters are extremely difficult to design due to the stability problem. This paper proposes a new method for designing recursive one-dimensional (1-D) variable filters whose stability is guaranteed. The method finds the coefficients of the transfer function of a variable digital filter as the multidimensional (M-D) polynomials of a few variables. The variables specify different frequency-domain characteristics, thus, we call the variables the spectral parameters. In applying the resulting variable filters, substituting different values of the spectral parameters into the M-D polynomials will obtain different filter coefficients and, thus, obtain different frequency-domain characteristics. To guarantee the stability, we first perform coefficient substitutions on the denominator coefficients such that they satisfy the stability conditions. Then both denominator and numerator coefficients are determined as M-D polynomials. In determining the M-D polynomials, we also propose an efficient least-squares approximation method that requires only solving simultaneous linear equations. Two examples are given to show the effectiveness of the proposed variable filter design technique.