References and further reading¶
This page lists books and papers that are useful for understanding the algorithms
and examples in lattice-dsp. The package documentation does not attempt to
replace these sources; it explains how the package uses the ideas.
Scalar lattice filters, PARCOR, and AR modeling¶
J. Makhoul, “Linear Prediction: A Tutorial Review,” Proceedings of the IEEE, 1975. A classic tutorial on linear prediction, AR models, prediction-error filters, and reflection/PARCOR ideas.
J. D. Markel and A. H. Gray, Jr., Linear Prediction of Speech. Springer, 1976. A classic source for speech linear prediction and PARCOR/lattice structures.
J. Durbin, “The Fitting of Time-Series Models,” Revue de l’Institut International de Statistique, 1960. Classic Toeplitz/Yule-Walker recursion background.
J. P. Burg, “Maximum Entropy Spectral Analysis,” presented at the 37th Annual International SEG Meeting, 1967. Original maximum-entropy/Burg spectral estimation work.
T. J. Ulrych and T. N. Bishop, “Maximum entropy spectral analysis and autoregressive decomposition,” Reviews of Geophysics, 1975. A widely cited review connecting maximum entropy spectral analysis and AR modeling.
Schur, Szegő, OPUC, and rational approximation background¶
G. Pick, “Über die Beschränkungen analytischer Funktionen, welche durch vorgegebene Funktionswerte bewirkt werden,” Mathematische Annalen, 1916. Classical source for the interpolation condition now known as the Pick or Nevanlinna–Pick problem.
H. Dym, J Contractive Matrix Functions, Reproducing Kernel Hilbert Spaces and Interpolation, American Mathematical Society, 1989. Reference for Schur-class interpolation, reproducing kernels, and contractive matrix-function viewpoints.
I. Schur, “Über Potenzreihen, die im Innern des Einheitskreises beschränkt sind,” Journal für die reine und angewandte Mathematik, 1917. Classical source for the Schur algorithm and Schur functions.
G. Szegő, Orthogonal Polynomials, 4th ed., American Mathematical Society, 1975. Classical reference for orthogonal polynomial recurrences and Christoffel-Darboux identities.
B. Simon, Orthogonal Polynomials on the Unit Circle, Parts 1 and 2, American Mathematical Society, 2005. Modern reference for OPUC, Verblunsky coefficients, Szegő recurrences, and unit-circle spectral theory.
A. Bultheel and M. Van Barel, “Rational approximation in linear systems and control,” Journal of Computational and Applied Mathematics, 2000. Useful bridge between Schur algorithms, rational approximation, and control-oriented system theory.
V. P. Potapov, “The multiplicative structure of J-contractive matrix functions,” Trudy Moskov. Mat. Obshch., 4, 1955. Classical source for Potapov factorization and J-contractive matrix functions.
A. Bultheel and K. Müller, “On several aspects of J-inner functions in Schur analysis,” Bulletin of the Belgian Mathematical Society - Simon Stevin, 1998. Survey-style discussion of J-inner functions, Potapov factorization, and generalized Schur interpolation context.
B. Hanzon, M. Olivi, and R. L. M. Peeters, “Balanced realizations of discrete-time stable all-pass systems and the tangential Schur algorithm,” arXiv:1012.3272, 2010. Explains the connection between tangential Schur algorithms, RKHS ideas, Schur parameters, and balanced realizations of multivariable lossless systems.
M. Olivi, J.-P. Marmorat, B. Hanzon, and R. L. M. Peeters, “Schur parametrizations and balanced realizations of real discrete-time stable all-pass systems,” CDC, 2003. Real tangential Schur parametrization and balanced-realization context for stable all-pass systems.
J.-P. Marmorat, B. Hanzon, M. Olivi, and R. L. M. Peeters, “Matrix rational H2 approximation: a state-space approach using Schur parameters,” CDC, 2002. Uses Schur-parameter coordinates on the manifold of stable all-pass/lossless systems for matrix rational approximation.
P. Gonnet, S. Güttel, and L. N. Trefethen, “Robust Padé Approximation via SVD,” SIAM Review, 2013. Practical numerical-linear-algebra view of Padé approximation and near-degenerate rational approximants.
L. N. Trefethen, “Square blocks and equioscillation in the Padé, Walsh, and Carathéodory-Fejér tables,” 2023. Modern discussion of Padé/Walsh rational-approximation tables and degeneracy structure.
Adaptive filtering¶
This section includes the Widrow–Hoff LMS origin, Odile Macchi’s transmission-oriented LMS treatment, standard adaptive-filter texts, and the H∞/minimax reinterpretation of LMS.
B. Widrow and M. E. Hoff, Jr., “Adaptive Switching Circuits,” 1960 IRE WESCON Convention Record, 1960. Classical origin point for the LMS/delta-rule adaptive-update idea.
Odile Macchi, Adaptive Processing: The Least Mean Squares Approach with Applications in Transmission, Wiley, 1995. Focused treatment of LMS-based adaptive processing, transmission/equalization applications, tracking, sign algorithms, and adaptive IIR context.
S. Haykin, Adaptive Filter Theory, 4th ed., Prentice Hall, 2002. Standard reference for LMS, NLMS, and RLS families.
S. Haykin, Adaptive Filter Theory, 5th ed., Pearson, 2014. Later edition with extensive examples and computer experiments for LMS and RLS adaptive filters.
B. Widrow and S. D. Stearns, Adaptive Signal Processing, Prentice Hall, 1985. Foundational adaptive signal processing text.
A. H. Sayed, Adaptive Filters, Wiley/IEEE Press, 2008. Modern treatment of adaptive-filter theory and analysis.
B. Farhang-Boroujeny, Adaptive Filters: Theory and Applications, 2nd ed., Wiley, 2013.
B. Hassibi, A. H. Sayed, and T. Kailath, “H∞ Optimality of the LMS Algorithm,” IEEE Transactions on Signal Processing, 44(2), 1996. Shows that LMS and normalized LMS have exact H∞/minimax interpretations, explaining a robustness property that is not visible from the usual least-squares story.
Multichannel AR, block Toeplitz systems, and block Levinson recursions¶
P. Whittle, “On the fitting of multivariate autoregressions, and the approximate canonical factorization of a spectral density matrix,” Biometrika, 1963. Classical source for multivariate autoregression and spectral factorization ideas.
R. A. Wiggins and E. A. Robinson, “Recursive solution to the multichannel filtering problem,” Journal of Geophysical Research, 1965. Early multichannel recursive filtering work often associated with the LWR family of algorithms.
T. Kailath, A. H. Sayed, and B. Hassibi, Linear Estimation, Prentice Hall, 2000. Modern reference for structured covariance, Toeplitz systems, innovations, and recursive estimation.
Multirate DSP, paraunitary systems, and matrix all-pass filters¶
P. P. Vaidyanathan, Multirate Systems and Filter Banks, Prentice Hall, 1993. Core reference for perfect reconstruction, paraunitary/orthogonal filter banks, polyphase systems, and lattice realizations.
G. Strang and T. Nguyen, Wavelets and Filter Banks, Wellesley-Cambridge Press, 1996. Accessible treatment of filter banks and wavelet connections.
P. Mehta, A. S. Bharath, K. Appaiah, R. Velmurugan, and D. Pal, “Lattice All-Pass Filter based Precoder Adaptation for MIMO Wireless Channels,” arXiv:2302.11204, 2023. Useful modern example of matrix-lattice all-pass filters as compact unitary MIMO representations. The package uses this as motivation for general matrix-lattice DSP, not as a claim to be a 5G simulator.
Model reduction, Hankel operators, Nehari, and AAK¶
Z. Nehari, “On bounded bilinear forms,” Annals of Mathematics, 65(1), 1957. Classical source for the Nehari approximation problem and its Hankel-operator interpretation.
V. M. Adamjan, D. Z. Arov, and M. G. Krein, “Analytic properties of Schmidt pairs for a Hankel operator and the generalized Schur-Takagi problem,” Mathematics of the USSR-Sbornik, 1971. Foundational AAK work connecting Hankel singular structure and rational approximation.
K. Glover, “All optimal Hankel-norm approximations of linear multivariable systems and their L∞-error bounds,” International Journal of Control, 1984. Classic control-theory source for Hankel-norm model reduction.
V. V. Peller, Hankel Operators and Their Applications, Springer, 2003. Comprehensive operator-theoretic reference for Hankel operators, Nehari-type problems, and AAK theory.
A. C. Antoulas, Approximation of Large-Scale Dynamical Systems, SIAM, 2005. Standard model-reduction reference, useful for balanced truncation, Hankel singular values, and system approximation.
Orthogonal convolutions and ML-adjacent unitary systems¶
J. Wang, Y. Chen, R. Chakraborty, and S. X. Yu, “Orthogonal Convolutional Neural Networks,” CVPR, 2020. Orthogonality constraints for convolutional layers and stability/regularization motivation.
E. M. Achour, F. Malgouyres, and F. Mamalet, “Existence, Stability and Scalability of Orthogonal Convolutional Neural Networks,” Journal of Machine Learning Research, 2022. Theoretical properties of orthogonal convolutional transforms.
J. Su, W. Byeon, and F. Huang, “Scaling-up Diverse Orthogonal Convolutional Networks by a Paraunitary Framework,” ICML, 2022. Connects orthogonal convolution layers in the spatial domain with paraunitary systems in the spectral domain.
How references map to package features¶
Package feature |
Main references |
|---|---|
Reflection/PARCOR coefficients and Schur/Pick motivation |
[Makhoul1975], [MarkelGray1976], [Pick1916], [Schur1917], [Dym1989] |
Tangential Schur and J-inner/Potapov diagnostics |
[Dym1989], [Potapov1955], [BultheelMuller1998], [HanzonOliviPeeters2010], [OliviMarmoratHanzonPeeters2003], [MarmoratHanzonOliviPeeters2002] |
Levinson-Durbin and Burg AR tools |
[Durbin1960], [Burg1967], [UlrychBishop1975], [Szego1975], [Simon2005] |
Multichannel/block Levinson AR tools |
[Whittle1963], [WigginsRobinson1965], [KailathSayedHassibi2000] |
NLMS/RLS adaptive filtering |
[WidrowHoff1960], [Macchi1995], [Haykin2002], [WidrowStearns1985], [Sayed2008] |
H∞/minimax LMS interpretation |
|
Model reduction and Hankel/AAK theory |
[Nehari1957], [AdamjanArovKrein1971], [Glover1984], [Peller2003], [Antoulas2005], [Bultheel2000], [GonnetGuettelTrefethen2013] |
Paraunitary and matrix lattice systems |
|
ML unitary convolution motivation |