Conditions for Lth-band filters of order 2N as cascades of identical linear-phase FIR spectral factors of order N

This paper presents formulas for the number of optimization parameters (degrees of freedom) when designing Type I linear-phase finite-length impulse response (FIR) Lth-band filters of order 2N as cascades of identical linear-phase FIR spectral factors of order N. We deal with two types of degrees of freedom referred to as (i) the total degrees of freedom DT, and (ii) the remaining degrees of freedom DR. Due to the symmetries or antisymmetries in the impulse responses of the spectral factors, DT roughly equals N/2. Some of these parameters are specifically needed to meet the Lth-band conditions because, in an Lth-band filter, every Lth coefficient is zero and the center tap equals 1/L. The remaining DR parameters can then be used to improve the stopband characteristics of the overall Lth-band filter. We derive general formulas for DR with given pairs of L and N. It is shown that for a fixed L, the choices of N, in a close neighborhood, may even decrease DR despite increasing the arithmetic complexity, order, and the delay.