Filter design, linear
Especially digital
July 24, 2017 — September 18, 2020
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.
1 Relationship of discrete LTI to continuous time filters
🏗 See signal sampling.
2 Quick and dirty digital filter design
- Julius O. Smith III’s lovingly curated encyclopædia of filter tricks covers everything commonly used in audio, at the cost of eyeball-searing ugliness and impenetrable curtness. If you already know some linear systems theory, useful, otherwise not.
- Multidimensional state filtering: dsprelated’s state space filter tutorial
- Textbook: Paolo Prandoni and Martin Vetterli, Signal Processing for Communications is available online. Vetterli is smart at unexpected and enlightening perspectives; I’m a fan.
- Textbook: Antoniou has been generally recommended if you want to get hands-on ASAP. (Antoniou 2005)
- Textbook: Orfandis’ opus is free online. (Orfanidis 1996)
- Course notes/textbook: Oppenheim and Verghese, Signals, Systems, and Inference is free online.
- Numerical tours of signal processing gives python, julia and matlab tours of signal processing. Better consumed through their github repo.
- Cheat sheet: Earlevel biquad formulae crib sheet by Nigel Redmon.
- Cheat sheet: musicdsp biquad filter cookbook by Robert Bristow-Johnson.
- Cookbook: musicdsp community filter recipes musicdsp cookbook.
3 State-Variable Filters
A vacuous name; every recursive filter has state variables. Less ambiguous: Chamberlin and Zölzer filters.
Nigel Redmon, digital SVF intro.
4 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).
5 On graphs
No time to go deep on this rn, but signal processing on graphs is a thing.
6 References
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Adcock, and Hansen. 2016. “Generalized Sampling and Infinite-Dimensional Compressed Sensing.” Foundations of Computational Mathematics.
Adcock, Hansen, Roman, et al. 2014. “Generalized Sampling: Stable Reconstructions, Inverse Problems and Compressed Sensing over the Continuum.” In Advances in Imaging and Electron Physics.
Adcock, Hansen, and Roman. 2015. “The Quest for Optimal Sampling: Computationally Efficient, Structure-Exploiting Measurements for Compressed Sensing.” In Compressed Sensing and Its Applications: MATHEON Workshop 2013. Applied and Numerical Harmonic Analysis.
Alliney. 1992. “Digital Filters as Absolute Norm Regularizers.” IEEE Transactions on Signal Processing.
Antoniou. 2005. Digital signal processing: signals, systems and filters.
Berkhout, and Zaanen. 1976. “A Comparison Between Wiener Filtering, Kalman Filtering, and Deterministic Least Squares Estimation*.” Geophysical Prospecting.
Carini, Mathews, and 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.
Chamberlin. 1985. Musical applications of microprocessors.
Ephremidze, Janashia, and Lagvilava. 2007. “A New Efficient Matrix Spectral Factorization Algorithm.” In SICE Annual Conference 2007.
Geronimo, and Woerdeman. 2004. “Positive Extensions, Fejér-Riesz Factorization and Autoregressive Filters in Two Variables.” Annals of Mathematics.
Harvey, and Luati. 2014. “Filtering With Heavy Tails.” Journal of the American Statistical Association.
Hohmann. 2002. “Frequency Analysis and Synthesis Using a Gammatone Filterbank.” Acta Acustica United with Acustica.
Isufi, Loukas, Simonetto, et al. 2017. “Autoregressive Moving Average Graph Filtering.” IEEE Transactions on Signal Processing.
Koshita, Abe, and Kawamata. 2018. “Recent Advances in Variable Digital Filters.” Digital Systems.
Laroche. 2007. “On the Stability of Time-Varying Recursive Filters.” Journal of the Audio Engineering Society.
Linkwitz. 1976. “Active Crossover Networks for Noncoincident Drivers.” Journal of the Audio Engineering Society.
Marple. 1987. Digital Spectral Analysis with Applications.
Martin. 1998. “Autoregression and Irregular Sampling: Filtering.” Signal Processing.
———. 1999. “Autoregression and Irregular Sampling: Spectral Estimation.” Signal Processing.
Moon, and Stirling. 2000. Mathematical Methods and Algorithms for Signal Processing.
Moorer. 1974. “The Optimum Comb Method of Pitch Period Analysis of Continuous Digitized Speech.” IEEE Transactions on Acoustics, Speech and Signal Processing.
Murakoshi, Nishihara, and Watanabe. 1994. “Synthesis of Variable IIR Digital Filters with Complex Coefficients.” Electronics and Communications in Japan (Part III: Fundamental Electronic Science).
Narasimha, Ignjatovic, and Vaidyanathan. 2002. “Chromatic Derivative Filter Banks.” IEEE Signal Processing Letters.
Necciari, Balazs, Holighaus, et al. 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).
Nyquist. 1928. “Certain Topics in Telegraph Transmission Theory.” Transactions of the American Institute of Electrical Engineers.
Oppenheim, Schafer, and Buck. 1999. Discrete-Time Signal Processing.
Orfanidis. 1996. Introduction to Signal Processing. Prentice Hall Signal Processing Series.
Prandoni, and Vetterli. 2008. Signal processing for communications. Communication and information sciences.
Robertson, Stark, and Plumbley. 2011. “Real-Time Visual Beat Tracking Using a Comb Filter Matrix.” In Proceedings of the International Computer Music Conference 2011.
Schlecht, and Habets. 2015. “Time-Varying Feedback Matrices in Feedback Delay Networks and Their Application in Artificial Reverberation.” The Journal of the Acoustical Society of America.
Sebek. 2015. “Spectral Factorization.” In Encyclopedia of Systems and Control.
Shuman, D. I., Narang, Frossard, et al. 2013. “The Emerging Field of Signal Processing on Graphs: Extending High-Dimensional Data Analysis to Networks and Other Irregular Domains.” IEEE Signal Processing Magazine.
Shuman, David I., Vandergheynst, and Frossard. 2011. “Chebyshev Polynomial Approximation for Distributed Signal Processing.” 2011 International Conference on Distributed Computing in Sensor Systems and Workshops (DCOSS).
Smith. 2007. Introduction to Digital Filters with Audio Applications.
———. 2010. “Audio Signal Processing in Faust.” Online Tutorial: Https://Ccrma. Stanford. Edu/Jos/Aspf.
———. 2015. “Digital State-Variable Filters.”
Smith, and Michon. 2011. “Nonlinear Allpass Ladder Filters in Faust.” In Proceedings of the 14th International Conference on Digital Audio Effects (DAFx-11).
Stilson, and Smith. 1996. “Analyzing the Moog VCF with Considerations for Digital Implementation.” In.
Stoica, and Moses. 2005. Spectral Analysis of Signals.
Williams, and Lawrence. 2007. Linear State-Space Control Systems.
Wise. 2006. “The Modified Chamberlin and Zölzer Filter Structures.” In Proc. Of the 9th Int. Conference on Digital Audio Effects (DAFx-06).
Wishnick. 2014. “Time-Varying Filters for Musical Applications.” In DAFx.
Zavalishin. n.d. “The Art of VA Filter Design.”