Filter design, linear

Especially digital

Linear Time-Invariant (LTI) filter design is a field of signal processing, and a special case of state filtering that doesn’t necessarily involve a hidden state.

z-Transforms, bilinear transforms, Bode plots, design etc.

I am going to consider this in discrete time (i.e. for digital implementation) unless otherwise stated, because I’m implementing this in software, not with capacitors or whatever. For reasons of tradition we usually start from continuous time systems, but this is not necessarily a convenient mathematical or practical starting point for my own work.

This notebook is about designing properties of systems to given specifications, e.g. signal to noise ratios, uncertainty principles

For inference of filter parameters from data, you want system identification; and for working out the hidden states of the system given the parameters, you want the more general estimation theory in state filters.

Related, musical: delays and reverbs.

Relationship of discrete LTI to continuous time filters

🏗 See signal sampling.

Quick and dirty digital filter design

State-Variable Filters

A vacuous name; every recursive filter has state variables. Less ambiguous: Chamberlin and Zölzer filters.

Nigel Redmon, digital SVF intro.

Time-varying IIR filters

By popular acclaim, Laroche (2007) seems to be the canonical example of design rules for filters that vary over time, and Wishnick (2014) is the most popular single-channel application, which proves the effectiveness of the cytomic variable filters The latter has source code online. See also (Carini, Mathews, and Sicuranza 1999; Murakoshi, Nishihara, and Watanabe 1994; Koshita, Abe, and Kawamata 2018).

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Adcock, Ben, and Anders C. Hansen. 2016. “Generalized Sampling and Infinite-Dimensional Compressed Sensing.” Foundations of Computational Mathematics 16 (5, 5): 1263–323.
Adcock, Ben, Anders C. Hansen, and Bogdan Roman. 2015. “The Quest for Optimal Sampling: Computationally Efficient, Structure-Exploiting Measurements for Compressed Sensing.” In Compressed Sensing and Its Applications: MATHEON Workshop 2013, edited by Holger Boche, Robert Calderbank, Gitta Kutyniok, and Jan Vybíral, 143–67. Applied and Numerical Harmonic Analysis. Cham: Springer International Publishing.
Adcock, Ben, Anders Hansen, Bogdan Roman, and Gerd Teschke. 2014. “Generalized Sampling: Stable Reconstructions, Inverse Problems and Compressed Sensing over the Continuum.” In Advances in Imaging and Electron Physics, edited by Peter W. Hawkes, 182:187–279. Elsevier.
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Berkhout, A. J., and P. R. Zaanen. 1976. “A Comparison Between Wiener Filtering, Kalman Filtering, and Deterministic Least Squares Estimation*.” Geophysical Prospecting 24 (1): 141–97.
Carini, A., V. J. Mathews, and G. L. Sicuranza. 1999. “Sufficient Stability Bounds for Slowly Varying Direct-Form Recursive Linear Filters and Their Applications in Adaptive IIR Filters.” IEEE Transactions on Signal Processing 47 (9): 2561–67.
Chamberlin, Hal. 1985. Musical Applications of Microprocessors. 2nd ed. Hasbrouck Heights, N.J: Hayden Book Co.
Harvey, Andrew, and Alessandra Luati. 2014. “Filtering With Heavy Tails.” Journal of the American Statistical Association 109 (507): 1112–22.
Hohmann, V. 2002. “Frequency Analysis and Synthesis Using a Gammatone Filterbank.” Acta Acustica United with Acustica 88 (3, 3): 433–42.
Koshita, Shunsuke, Masahide Abe, and Masayuki Kawamata. 2018. “Recent Advances in Variable Digital Filters.” Digital Systems, November.
Laroche, Jean. 2007. “On the Stability of Time-Varying Recursive Filters.” Journal of the Audio Engineering Society 55 (6): 460–71.
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———. 1999. “Autoregression and Irregular Sampling: Spectral Estimation.” Signal Processing 77 (2): 139–57.
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Necciari, T., P. Balazs, N. Holighaus, and P. L. Sondergaard. 2013. “The ERBlet Transform: An Auditory-Based Time-Frequency Representation with Perfect Reconstruction.” In 2013 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), 498–502.
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Robertson, Andrew, Adam M. Stark, and Mark D. Plumbley. 2011. “Real-Time Visual Beat Tracking Using a Comb Filter Matrix.” In Proceedings of the International Computer Music Conference 2011.
Schlecht, Sebastian J., and Emanuël A. P. Habets. 2015. “Time-Varying Feedback Matrices in Feedback Delay Networks and Their Application in Artificial Reverberation.” The Journal of the Acoustical Society of America 138 (3): 1389–98.
Smith, Julius O. 2007. Introduction to Digital Filters with Audio Applications. W3K Publishing. jos/filters/filters.html.
———. 2010. “Audio Signal Processing in Faust.” Online Tutorial: Https://Ccrma. Stanford. Edu/Jos/Aspf. jos/aspf/aspf.pdf.
———. n.d. “Digital State-Variable Filters.” jos/svf/svf.pdf.
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Zavalishin, Vadim. n.d. “The Art of VA Filter Design.”