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
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.
State-Variable Filters
A vacuous name; every recursive filter has state variables. Less ambiguous: Chamberlin and Zölzer filters.
Nigel Redmon, digital SVF intro.
Abe, T., T. Kobayashi, and S. Imai. 1995. “Harmonics Tracking and Pitch Extraction Based on Instantaneous Frequency.” In International Conference on Acoustics, Speech, and Signal Processing, 1995. ICASSP-95, 1:756–59 vol.1. https://doi.org/10.1109/ICASSP.1995.479804.
Adcock, Ben, and Anders C. Hansen. 2016. “Generalized Sampling and Infinite-Dimensional Compressed Sensing.” Foundations of Computational Mathematics 16 (5): 1263–1323. https://doi.org/10.1007/s10208-015-9276-6.
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. https://doi.org/10.1007/978-3-319-16042-9_5.
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. https://doi.org/10.1016/B978-0-12-800146-2.00004-7.
Alliney, S. 1992. “Digital Filters as Absolute Norm Regularizers.” IEEE Transactions on Signal Processing 40 (6): 1548–62. https://doi.org/10.1109/78.139258.
Antoniou, Andreas. 2005. Digital Signal Processing: Signals, Systems and Filters. New York: McGraw-Hill.
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. https://doi.org/10.1111/j.1365-2478.1976.tb00390.x.
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. https://doi.org/10.1080/01621459.2014.887011.
Hohmann, V. 2002. “Frequency Analysis and Synthesis Using a Gammatone Filterbank.” Acta Acustica United with Acustica 88 (3): 433–42.
Laroche, Jean. 2007. “On the Stability of Time-Varying Recursive Filters.” Journal of the Audio Engineering Society 55 (6): 460–71. http://www.aes.org/e-lib/browse.cfm?elib=14168.
Linkwitz, Siegfried H. 1976. “Active Crossover Networks for Noncoincident Drivers.” Journal of the Audio Engineering Society 24 (1): 2–8. http://www.aes.org/e-lib/browse.cfm?elib=2649.
Marple, S. Lawrence, Jr. 1987. Digital Spectral Analysis with Applications. http://adsabs.harvard.edu/abs/1987ph...book.....M.
Martin, R. J. 1999. “Autoregression and Irregular Sampling: Spectral Estimation.” Signal Processing 77 (2): 139–57. https://doi.org/10.1016/S0165-1684(99)00029-8.
———. 1998. “Autoregression and Irregular Sampling: Filtering.” Signal Processing 69 (3): 229–48. https://doi.org/10.1016/S0165-1684(98)00105-4.
Moon, Todd K., and Wynn C. Stirling. 2000. Mathematical Methods and Algorithms for Signal Processing. Upper Saddle River, NJ: Prentice Hall.
Moorer, J. A. 1974. “The Optimum Comb Method of Pitch Period Analysis of Continuous Digitized Speech.” IEEE Transactions on Acoustics, Speech and Signal Processing 22 (5): 330–38. https://doi.org/10.1109/TASSP.1974.1162596.
Narasimha, M. J., A. Ignjatovic, and P. P. Vaidyanathan. 2002. “Chromatic Derivative Filter Banks.” IEEE Signal Processing Letters 9 (7): 215–16. https://doi.org/10.1109/LSP.2002.801720.
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. https://doi.org/10.1109/ICASSP.2013.6637697.
Nyquist, H. 1928. “Certain Topics in Telegraph Transmission Theory.” Transactions of the American Institute of Electrical Engineers 47 (2): 617–44. https://doi.org/10.1109/T-AIEE.1928.5055024.
Oppenheim, Alan V., Ronald W. Schafer, and John R. Buck. 1999. Discrete-Time Signal Processing. 2nd ed. Upper Saddle River, N.J: Prentice Hall.
Orfanidis, Sophocles J. 1996. Introduction to Signal Processing. Prentice Hall Signal Processing Series. Englewood Cliffs, N.J: Prentice Hall. http://www.ece.rutgers.edu/~orfanidi/intro2sp/orfanidis-i2sp.pdf.
Prandoni, Paolo, and Martin Vetterli. 2008. Signal Processing for Communications. Communication and Information Sciences. Lausanne: EPFL Press.
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. https://www.eecs.qmul.ac.uk/~markp/2011/RobertsonStarkPlumbleyICMC2011_accepted.pdf.
Smith, Julius O. 2007. Introduction to Digital Filters with Audio Applications. http://www.w3k.org/books/: W3K Publishing. https://ccrma.stanford.edu/~jos/filters/filters.html.
———. 2010. “Audio Signal Processing in Faust.” Online Tutorial: Https://Ccrma. Stanford. Edu/Jos/Aspf. https://ccrma.stanford.edu/~jos/aspf/aspf.pdf.
———. n.d. “Digital State-Variable Filters.” https://ccrma.stanford.edu/~jos/svf/svf.pdf.
Smith, Julius O., and Romain Michon. 2011. “Nonlinear Allpass Ladder Filters in Faust.” In Proceedings of the 14th International Conference on Digital Audio Effects (DAFx-11), 361–64. http://recherche.ircam.fr/pub/dafx11/Papers/38_e.pdf.
Stilson, Tim, and Julius Smith. 1996. “Analyzing the Moog VCF with Considerations for Digital Implementation.” In. https://ccrma.stanford.edu/~stilti/papers/moogvcf.pdf.
Stoica, Petre, and Randolph L. Moses. 2005. Spectral Analysis of Signals. 1 edition. Upper Saddle River, N.J: Prentice Hall. http://user.it.uu.se/~ps/SAS-new.pdf.
Wise, Duane K. 2006. “The Modified Chamberlin and Zölzer Filter Structures.” In Proc. Of the 9th Int. Conference on Digital Audio Effects (DAFx-06), 2:3. http://www.dafx.ca/proceedings/papers/p_053.pdf.
Wishnick, Aaron. 2014. “Time-Varying Filters for Musical Applications.” In DAFx, 69–76. http://www.dafx14.fau.de/papers/dafx14_aaron_wishnick_time_varying_filters_for_.pdf.