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 D-T, and (ii) the remaining degrees of freedom D-R. Due to the symmetries or antisymmetries in the impulse responses of the spectral factors, D-T 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 D-R parameters can then be used to improve the stopband characteristics of the overall Lth-band filter. We derive general formulas for D-R 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 D-R despite increasing the arithmetic complexity, order, and the delay.

Nyckelord

Pulse shaping; Lth-band filters; Spectral factors; FIR filters; Linear-phase

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