Symbolic MAPLE-based design of allpole filters

To compute the filter coefficients of a digital filter, it is often necessary the solution of a set of linear equations using numerical methods. Unfortunately, if the number of unknowns is high, then the solution is likely to be error prone. To avoid this problem, this paper presents the use of the symbolic tool MAPLE to compute the filter coefficients. Consequently, the proposed solution is given in symbolic form. The corresponding numerical values are obtained from the proposed closed form relations. Particularly, we consider the design flat group delay allpole filters with real and complex coefficients due to promising applications like allpass filter design, IIR filters design, Hilbert transformer design, and filter bank design. The resulting allpole filter design has the following design constraints: 1) complex filter coefficients are obtained using even values of the allpole filter order, otherwise they are real, 2) in real and complex cases, the degree of flatness at ω = 0 and ω = π is N - 2, where N is the order of the allpole filter, 3) the phase value at the frequency point ω_p is φ _D(ω_p), in the real case ω_p ≠ O and arbitrary, and in the opposite case we have ω_p = 0. Simulation examples show the effectiveness of the proposed approach. Finally, the Appendix shows the MAPLE code used for the filter coefficients computation for two specific cases; N = 4 and N = 5.