Module dsp::ztransform
General N-th order Laplace-to-Z (bilinear) transform + Z-domain frequency response evaluator
biquad.h's Biquad::bilinear and Biquad::response_mag/phase cover the 2nd-order case (the common one — cascaded biquads build any even order in practice). This file generalizes both to arbitrary order M, for the less common case of designing/analyzing a single higher-order IIR section directly instead of as a cascade of biquads: an analog prototype H(s) = (sum_{k=0}^{M} B[k] s^k) / (sum_{k=0}^{M} A[k] s^k), via the same substitution s = 2fs(z-1)/(z+1) Biquad::bilinear uses, producing bOut/aOut directly in the negative-power-of-z form filter.h's IirFilter/FIirFilter already expect (H(z) = (sum b[i] z^-i) / (sum a[i] z^-i)) — bilinear_nth_order's output can be handed straight to IirFilter::new.
No polynomial root-finder is included (finding a filter's poles/zeros individually needs an iterative eigenvalue or Durand-Kerner-type algorithm — a separate, substantial piece of numerical code that isn't needed for either designing a filter via the bilinear transform or evaluating its frequency response, both of which work directly on the coefficient arrays).
Functions
| bilinear_nth_order | B, A: order+1 analog (s-domain) coefficients each, ascending powers of s (B[0]/A[0] is the constant term, B[order]/A[order] is the s^order term). bOut, aOut: order+1 digital (z-domain) coefficients each, written in the negative-power-of-z convention (bOut[0]/aOut[0] is the z^0 term, bOut[order]/aOut[order] is the z^-order term), normalized so aOut[0] = 1 — the same normalization Biquad::bilinear's a0-divide-out already uses, and what IirFilter expects. |
| ztransform_response_mag | Z-domain frequency response of an order-'order' digital filter (b[]/a[] in the negative-power-of-z convention above) at 'freqHz' — the general-order counterpart of Biquad::response_mag/phase. |
| ztransform_response_phase |