nx_lpc_levinson.nx

nx_lpc_levinson.nx -- Levinson-Durbin recursion in Q30 fixed-point. Arc 1 Phase A.2 (after nx_lpc_autocorr). Given autocorrelation R[0..p] (i64 from nx_lpc_autocorr), computes: - reflection coefficients (PARCOR) k[1..p] in Q30 (|k_i| < 1 for stable) - LPC predictor coefficients a[1..p] in Q30 - prediction error E[p] in i64 (= R[0] * Π(1 - k_i²)) Re-derived from Levinson-Durbin / Yule-Walker. Patent-clean math. Per [[feedback-bits-up-exceed-never-match]] — no Opus, no MLow, no proprietary codec dependency, integer-only. Algorithm (per any DSP textbook, e.g. Rabiner & Schafer 1978 §8.3): E = R[0] for i = 1 to p: num = -R[i] - Σ_{j=1..i-1} a[j] * R[i-j] (with proper scaling) k_i = num / E new_a[i] = k_i for j = 1 to i-1: new_a[j] = a[j] + k_i * a[i-j] E = E * (1 - k_i²) a = new_a Fixed-point convention: - R[*] in raw i64 (autocorrelation sums) - E in raw i64 (always positive, decreasing each iteration) - k[*] in Q30: real_k = stored_k / 2^30, |real_k| < 1 ⇒ |stored_k| < 2^30 - a[*] in Q30: same convention; final coefs bounded by O(2^p) typically Returns 0 on success, -1 if R[0] <= 0 (degenerate / silence) or p > 30 (would overflow Q30 in a coefs).