Interpolation, Schur stability, and why lattice coordinates help

This page explains the motivation behind several design choices in lattice-dsp. The short version is:

Stable IIR design and reduction are often hard not because the formulas are unknown, but because the direct coefficient coordinates are numerically poor. Lattice, Schur, reflection, and Hankel coordinates expose the contractive and finite-rank structure more directly.

The discussion is intentionally finite-dimensional and numerical. It gives the mathematical map behind the tutorials; it is not a claim that the package solves all Nevanlinna–Pick, AAK, or matrix-valued interpolation problems.

Why interpolation problems become numerically hard

A classical scalar Nevanlinna–Pick problem asks for an analytic bounded function f on the unit disk that matches prescribed values,

\[f(z_i) = w_i, \qquad |f(z)| \le 1 \quad (|z| < 1).\]

In the Schur-class normalization, feasibility is characterized by the Pick matrix

\[P_{ij} = \frac{1 - w_i \overline{w_j}}{1 - z_i \overline{z_j}}.\]

The exact mathematical condition is elegant: a solution exists when P is positive semidefinite. Numerically, however, this matrix can be very difficult:

  • interpolation nodes close to each other produce nearly dependent kernel columns;

  • nodes close to the unit circle amplify denominator sensitivity;

  • values w_i close to the unit circle make the problem nearly extremal;

  • high-order data can produce very large condition numbers even when the mathematical problem is feasible;

  • tiny perturbations can change a nearly singular feasibility certificate into an apparently indefinite one.

This is the first reason direct coefficient methods are fragile. Polynomial, state-space, or denominator-coefficient solves can hide the contractive structure that the problem is really about. A numerically small coefficient perturbation can correspond to a large movement of poles, zeros, or interpolation residuals.

Why IIR and lattice coordinates are natural

IIR filters are a natural representation for rational stable systems: a modest number of poles can represent long memory. The difficulty is that the denominator controls stability. In direct form,

\[H(z) = \frac{B(z)}{A(z)}, \qquad A(z) = 1 + a_1 z^{-1} + \cdots + a_p z^{-p},\]

stability requires all roots of A to lie strictly inside the unit disk. The stable set in the direct coefficient vector a is not a simple box. Adaptive updates, least-squares fits, or model-reduction post-processing can easily cross the stability boundary.

Reflection/PARCOR coefficients give a different scalar coordinate system. In Schur or Levinson recursion, a stable denominator is built stage by stage from coefficients

\[k_1, k_2, \ldots, k_p, \qquad |k_m| < 1.\]

For scalar all-pole denominators, this boundedness condition is the key practical benefit: the hard root-location condition becomes a per-stage contractivity condition. Implementations can enforce a margin, for example

\[|k_m| \le 1 - \varepsilon,\]

by clipping or by optimizing an unconstrained parameter through tanh. The result is not a guarantee that every adaptive objective is solved well, but it keeps the denominator inside a stable parameterization while the objective is being optimized.

Inner and outer factors

Hardy-space language also explains why lattice and all-pass examples appear next to Hankel and reduction examples.

A scalar Hardy-space transfer function can often be viewed through an inner–outer factorization,

\[G(z) = G_{\mathrm{inner}}(z)\,G_{\mathrm{outer}}(z).\]

Informally:

  • the inner factor is energy-preserving on the unit circle. In SISO rational examples this is closely related to all-pass/Blaschke behavior;

  • the outer factor carries the minimum-phase amplitude information and is tied to spectral factorization and prediction-error filters;

  • Schur parameters and lattice stages are a constructive way to expose contractive/all-pass structure one stage at a time.

For MIMO systems, inner/outer factorization becomes matrix-valued. The same words still identify the conceptual roles, but the numerical problem is much harder: factors may be rectangular, singular directions interact, and scalar root tests are replaced by matrix contractivity and state-space conditions. This is why the package treats MIMO lattice/all-pass material as specialized scaffolding and diagnostics rather than as a complete matrix-valued interpolation solver.

Advanced context: tangential Schur and Potapov–Blaschke products

Matrix-valued lattice filters are related to tangential Schur recursions and Potapov–Blaschke products. A scalar Blaschke factor is the simplest unit-disk inner function; a Potapov–Blaschke factor is its matrix/signature analogue and is \(J\)-inner on the unit circle. These factors are the building blocks used in linear-fractional Schur transformations and in lossless/all-pass system parametrizations.

The public API exposes a finite, definite baseline: right tangential Pick matrices, RKHS/Pick positivity checks, interpolation residual checks, constant-solution sanity checks, and elementary J-inner Potapov factors. It still does not claim full generalized indefinite matrix Schur synthesis or the complete recursive Hanzon–Olivi–Peeters/Marmorat manifold parametrization. See Tangential Schur, Pick matrices, and J-inner factors and Tangential Schur Pick and J-inner diagnostics.

Kronecker theorem and finite Hankel rank

The Hankel point of view gives a second motivation for the model-reduction tools. For a sequence of Markov parameters g_0, g_1, ..., the Hankel matrix

\[\begin{split}H = \begin{bmatrix} g_1 & g_2 & g_3 & \cdots \\ g_2 & g_3 & g_4 & \cdots \\ g_3 & g_4 & g_5 & \cdots \\ \vdots & \vdots & \vdots & \ddots \end{bmatrix}\end{split}\]

summarizes how past inputs influence future outputs. A classical Kronecker-type statement says, in practical DSP terms:

A Hankel operator has finite rank exactly when the corresponding sequence is generated by a rational transfer function; the rank corresponds to the realization order or McMillan degree under the usual minimality assumptions.

This is the reason singular values of finite Hankel blocks are useful. Exact finite rank suggests an exact low-order rational model. Fast decay suggests that a low-order model may be a good numerical approximation. In SISO, this can lead back to an IIR denominator and reflection coordinates. In MIMO, the same idea appears as block-Hankel/ERA-style state-space reduction.

How this motivates the package workflow

The package documentation uses the following chain:

interpolation and bounded analytic functions
   -> Schur contractivity and reflection coefficients
   -> stable scalar IIR/lattice coordinates
   -> Hankel finite-rank and model-reduction diagnostics
   -> MIMO block-Hankel/state-space examples
   -> matrix-lattice/all-pass scaffolds

The package is most useful when a user has data that can already be represented as arrays: impulse responses, Markov parameters, time series, or synthetic room responses. The numerical routines then expose stability, order, and reduction information in coordinates that are less fragile than direct polynomial updates.

Scope boundary

The page motivates the implementation but does not expand the public claims. In this release:

  • scalar reflection coefficients are used as a practical stability parameterization for SISO IIR examples;

  • finite Hankel and Ho–Kalman-style routines are finite numerical diagnostics and reducers;

  • finite Nehari/AAK helpers expose finite-section candidate checks and certificates;

  • MIMO routines operate on finite Markov tensors and reduced state-space models;

  • definite right-tangential Pick/J-inner utilities are included as finite diagnostics; generalized indefinite matrix Schur synthesis, exact infinite-dimensional Nehari optimization, and complete MIMO inner–outer synthesis are outside the package scope.