The lattice DSP concept map

How the classical ideas fit together

lattice-dsp uses terminology from several communities: speech linear prediction, digital filters, orthogonal polynomials, operator theory, rational approximation, multirate filter banks, and MIMO signal processing. This page is a map of that vocabulary. It is not meant to be a complete mathematical treatment. Its purpose is to help readers recognize why the examples in the package belong together. For the interpolation and numerical conditioning motivation behind Schur/reflection coordinates, see Interpolation, Schur stability, and why lattice coordinates help.

The short version is:

Lattice filters are not just implementation tricks. They are computational forms of Schur/Szegő recursions. Reflection coefficients encode stability; prediction errors expose orthogonality; all-pass and paraunitary completions expose energy preservation; Hankel and Padé viewpoints explain rational approximation and model reduction; matrix/MIMO versions replace scalar reflection coefficients by contraction matrices.

1. The core object: a stable recursive filter

The scalar IIR object is a rational transfer function

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

For a causal discrete-time filter, stability is controlled by the roots of A. Directly updating the denominator coefficients a_i is fragile: a small-looking coefficient update can move a pole outside the unit disk. The lattice viewpoint replaces the polynomial coefficients by reflection/PARCOR coefficients

\[k_1, k_2, \ldots, k_p.\]

For the scalar all-pole recursion, the simple condition

\[|k_i| < 1 \quad \text{for all } i\]

is the practical stability handle used throughout the package.

Where to look: Scalar lattice and lattice-ladder IIR filters, Reflection conversion, and Stability vs direct IIR.

2. Schur recursion: stability by reflection coefficients

The Schur recursion is the classical way to peel a stable analytic function or stable polynomial into a sequence of reflection coefficients. In signal processing language, the same idea appears as step-up/step-down conversion between denominator coefficients and PARCOR coefficients.

A typical step-up form is

\[A_m(z) = A_{m-1}(z) + k_m z^{-m} A_{m-1}^{\sharp}(z),\]

where sharp denotes the reversed/conjugate polynomial. In real scalar code, this becomes the familiar denominator update implemented by reflection_to_denominator.

Practical interpretation:

• step-up recursion builds a stable denominator from reflection coefficients;

• step-down recursion tests or extracts reflection coefficients from a denominator;

• a reflection coefficient with magnitude at least one signals loss of stability.

Where to look: Reflection conversion.

3. Szegő polynomials: the orthogonal-polynomial view

Szegő recursions are the orthogonal-polynomial-on-the-unit-circle version of the same structure. Instead of viewing the recursion as a filter realization, one views it as building orthogonal polynomials with respect to a spectral measure. The reflection coefficients are then the unit-circle analogues of recursion parameters; in the OPUC literature they are often called Verblunsky coefficients.

For AR modeling, this is the conceptual bridge:

\[\text{autocorrelation moments} \quad \longrightarrow \quad \text{orthogonal prediction polynomials} \quad \longrightarrow \quad \text{reflection coefficients}.\]

That is why Levinson-Durbin, Burg estimation, prediction-error filters, and lattice filters are so tightly connected.

Where to look: AR estimation: autocorrelation, Levinson-Durbin, and Burg, Burg and Levinson AR tools, and AR spectral estimation.

4. Christoffel-Darboux: why prediction errors and kernels appear

The Christoffel-Darboux identity is a compact identity for sums of orthogonal polynomials. In this package it is not exposed as a public function, but it is part of the background explaining why AR spectra, prediction-error energies, and reproducing kernels are related.

A schematic version of the message is:

\[K_n(z,w) = \sum_{j=0}^n \phi_j(z)\overline{\phi_j(w)}\]

can be rewritten using only the latest orthogonal polynomials and their reversed partners. In filtering language, many cumulative prediction quantities can be represented through the current forward/backward prediction-error polynomials instead of through the whole history.

Practical interpretation:

• prediction-error filters are not arbitrary residual generators;

• they are orthogonalization objects;

• spectral peaks and prediction error are two views of the same AR model.

Where to look: Spectral diagnostics: periodogram, AR, Burg, and Capon, Periodogram vs AR spectrum, and Spectral diagnostics comparison.

5. Orthogonal filters: lattice stages as rotations/reflections

A scalar lattice stage can be read as a local transformation of forward and backward prediction errors. In stable/lossless normalizations, lattice stages behave like elementary rotations or hyperbolic rotations. This is why lattice filters appear naturally in speech coding, AR modeling, wave digital filters, and filter-bank realizations.

For MIMO and matrix-valued systems, the scalar coefficient k is replaced by a matrix K. The scalar bound becomes a contraction condition:

\[|k| < 1 \qquad \leadsto \qquad \|K\|_2 < 1.\]

This is one of the most important analogies in the package.

Where to look: Matrix lattice all-pass filters, Matrix lattice all-pass, and ML unitary convolution demo.

6. All-pass completion: from a stable denominator to a lossless system

An all-pass system has unit magnitude response in the scalar case:

\[|H(e^{i\omega})| = 1.\]

A matrix all-pass or paraunitary system satisfies the stronger matrix identity

\[H(e^{i\omega})^* H(e^{i\omega}) = I.\]

This means the system redistributes energy across time, frequency, or channels without changing total energy. Lattice and matrix-lattice structures are useful because they can parameterize such energy-preserving systems through stable local factors.

Practical interpretation:

• all-pass completions turn stability constraints into lossless systems;

• paraunitary filter banks are multichannel all-pass/orthogonal systems in polyphase form;

• MIMO lattice stages are compact parameterizations of structured unitary or contractive responses.

Where to look: Matrix lattice all-pass filters, Matrix unitary response compression, Paraunitary filter-bank demo.

7. Hankel forms and Padé: rational approximation and model reduction

A recursive filter is a rational approximation to an input-output map. Two classical ways to understand reduced-order rational models are:

• Padé approximation: match a finite number of moments or Taylor/Markov coefficients;

• Hankel/operator approximation: approximate the map from past inputs to future outputs.

Given an impulse response

\[h_0, h_1, h_2, \ldots,\]

one finite Hankel matrix is

\[\begin{split}\mathcal H = \begin{bmatrix} h_1 & h_2 & h_3 & \cdots \\ h_2 & h_3 & h_4 & \cdots \\ h_3 & h_4 & h_5 & \cdots \\ \vdots & \vdots & \vdots & \ddots \end{bmatrix}.\end{split}\]

Its singular values indicate how much input-output memory is carried by successive directions. That is why Hankel singular values are natural order selection diagnostics for reduced IIR models.

The current package includes finite model-reduction benchmarks and finite-section Nehari/AAK-style diagnostics. Exact infinite-dimensional Nehari/AAK reduction is outside the public scope. The theory pages explain this scope map.

Where to look: Hardy, Hankel, reachability, and observability, Interpolation, Schur stability, and why lattice coordinates help, Model reduction, Hankel operators, Nehari, and AAK, and Model-reduction benchmark.

8. Matrix/MIMO generalization: contractions replace scalar coefficients

The matrix/MIMO path is the rare part of the package. The scalar lattice story has the following matrix analogue:

Scalar lattice idea	Matrix/MIMO analogue
reflection coefficient k	reflection/contraction matrix K
stability condition |k| < 1	contraction condition ||K||_2 < 1
scalar all-pass response	matrix all-pass/unitary response
scalar AR prediction error	vector AR prediction-error covariance
denominator polynomial	matrix polynomial / block Toeplitz system
Levinson recursion	block Levinson / LWR-style recursion

This is why scalar lattice filters, multichannel AR, paraunitary filter banks, and ML-inspired unitary convolutions are grouped together in the docs. They are all examples of structured stable or energy-preserving transformations.

Where to look: Multichannel AR and block Levinson, Matrix lattice all-pass filters, Multichannel Levinson AR, and MIMO lattice vs block Levinson, Diagonal MIMO equals independent SISO, and Finite block-Hankel MIMO reduction.

9. Where each concept appears in the package

Concept	What it means	Where it appears
Schur recursion	Converts stable denominators to and from reflection coefficients.	Reflection conversion; Stability vs direct IIR
Szegő recursion	Orthogonal-polynomial version of the same reflection-coefficient recursion.	AR estimation: autocorrelation, Levinson-Durbin, and Burg; Burg/Levinson AR tools
Schur/Jury stability test	Tests whether polynomial roots lie inside the unit disk; failed step-down coefficients reveal instability.	Scalar lattice and lattice-ladder IIR filters; Stability vs direct IIR
Christoffel-Darboux identity	Reproducing-kernel identity behind compact prediction-error and spectral formulas.	Spectral diagnostics: periodogram, AR, Burg, and Capon; AR spectral estimation
Orthogonal lattice filters	Lattice stages as local rotations/reflections of forward/backward errors.	Matrix lattice all-pass filters; ML unitary convolution
All-pass completion	Completes a stable scalar or matrix factor into a lossless/energy-preserving response.	Matrix lattice all-pass; Paraunitary filter bank
Hankel forms	Encode input-output memory and moment/impulse-response matching.	Model reduction, Hankel operators, Nehari, and AAK; Finite-Hankel SISO reduction; Finite block-Hankel MIMO reduction
Padé approximation	Builds rational approximants from moment or series matching.	Model reduction, Hankel operators, Nehari, and AAK; rational-approximation diagnostics
Walsh and rational approximation viewpoints	Organize tables/families of rational approximants and explain degeneracies or near-cancellations.	Rational-approximation diagnostics
Schmidt pairs / Schmidt form	Singular-vector structure of Hankel operators used in AAK theory.	Model reduction, Hankel operators, Nehari, and AAK; finite-section AAK diagnostics
Matrix Schur / matrix lattice forms	MIMO generalization where reflection coefficients become contraction matrices.	Matrix lattice all-pass filters; MIMO lattice vs block Levinson; Diagonal MIMO equals independent SISO

Suggested reading order

For a new reader, the most useful path is:

1. Scalar lattice and lattice-ladder IIR filters for scalar stability and reflection coefficients.

2. AR estimation: autocorrelation, Levinson-Durbin, and Burg and Spectral diagnostics: periodogram, AR, Burg, and Capon for prediction and spectra.

3. Adaptive lattice filters: NLMS and RLS and LMS as a minimax robust filter for adaptive filtering.

4. Model reduction, Hankel operators, Nehari, and AAK for the model-reduction scope map.

5. Multichannel AR and block Levinson and Matrix lattice all-pass filters for the matrix/MIMO direction.

References

For classical background, see References and further reading, especially the sections on scalar lattice filters, OPUC/Schur/Szegő theory, multirate/paraunitary systems, and Hankel/Nehari/AAK model reduction.