Linear TimeInvariant (LTI) filter design is a field of signal processing, and a special case of state filtering that doesn’t necessarily involve a hidden state.
zTransforms, 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 processes
TBD, based on the modern summary in Mart99. But I’m way more interested in representations of systems more naturally represented with delays, and those are easier in digital discrete time than RCL circuit design, so I can’t imagine racing to get to this.
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 eyeballsearing 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 handson ASAP. (Anto05)

Textbook: Orfandis’ opus is free online. (Orfa96)

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 BristowJohnson,

Cookbook: musicdsp community filter recipes musicdsp cookbook.
StateVariable Filters
A vacuous name; every recursive filter has state variables. Less ambiguous: Chamberlin and Zölzer filters.
Nigel Redmon, digital SVF intro.
Refs
 BeZa76: A. J. Berkhout, P. R. Zaanen (1976) A Comparison Between Wiener Filtering, Kalman Filtering, and Deterministic Least Squares Estimation*. Geophysical Prospecting, 24(1), 141–197. DOI
 StSm96: Tim Stilson, Julius Smith (1996) Analyzing the Moog VCF with considerations for digital implementation.
 Smit10: Julius O. Smith (2010) Audio signal processing in Faust. Online Tutorial: Https://Ccrma. Stanford. Edu/Jos/Aspf.
 Mart98: R. J. Martin (1998) Autoregression and irregular sampling: Filtering. Signal Processing, 69(3), 229–248. DOI
 Mart99: R. J. Martin (1999) Autoregression and irregular sampling: Spectral estimation. Signal Processing, 77(2), 139–157. DOI
 Nyqu28: H. Nyquist (1928) Certain Topics in Telegraph Transmission Theory. Transactions of the American Institute of Electrical Engineers, 47(2), 617–644. DOI
 NaIV02: M. J. Narasimha, A. Ignjatovic, P. P. Vaidyanathan (2002) Chromatic derivative filter banks. IEEE Signal Processing Letters, 9(7), 215–216. DOI
 Alli92: S. Alliney (1992) Digital filters as absolute norm regularizers. IEEE Transactions on Signal Processing, 40(6), 1548–1562. DOI
 Anto05: Andreas Antoniou (2005) Digital signal processing: signals, systems and filters. New York: McGrawHill
 Marp87: S. Lawrence Marple Jr. (1987) Digital spectral analysis with applications
 Smit00: Julius O. Smith (n.d.) Digital Statevariable filters
 OpSB99: Alan V. Oppenheim, Ronald W. Schafer, John R. Buck (1999) Discretetime signal processing. Upper Saddle River, N.J: Prentice Hall
 HaLu14: Andrew Harvey, Alessandra Luati (2014) Filtering With Heavy Tails. Journal of the American Statistical Association, 109(507), 1112–1122. DOI
 Hohm02: V. Hohmann (2002) Frequency analysis and synthesis using a Gammatone filterbank. Acta Acustica United with Acustica, 88(3), 433–442.
 AbKI95: T. Abe, T. Kobayashi, S. Imai (1995) Harmonics tracking and pitch extraction based on instantaneous frequency. In International Conference on Acoustics, Speech, and Signal Processing, 1995. ICASSP95 (Vol. 1, pp. 756–759 vol.1). DOI
 Smit07: Julius O. Smith (2007) Introduction to Digital Filters with Audio Applications. http://www.w3k.org/books/: W3K Publishing
 Orfa96: Sophocles J. Orfanidis (1996) Introduction to signal processing. Englewood Cliffs, N.J: Prentice Hall
 MoSt00: Todd K. Moon, Wynn C. Stirling (2000) Mathematical methods and algorithms for signal processing. Upper Saddle River, NJ: Prentice Hall
 Cham85: Hal Chamberlin (1985) Musical applications of microprocessors. Hasbrouck Heights, N.J: Hayden Book Co.
 SmMi11: Julius O. Smith, Romain Michon (2011) Nonlinear allpass ladder filters in faust. In Proceedings of the 14th International Conference on Digital Audio Effects (DAFx11) (pp. 361–364).
 Laro07: Jean Laroche (2007) On the Stability of TimeVarying Recursive Filters. Journal of the Audio Engineering Society, 55(6), 460–471.
 RoSP11: Andrew Robertson, Adam M. Stark, Mark D. Plumbley (2011) Realtime visual beat tracking using a comb filter matrix. In Proceedings of the International Computer Music Conference 2011.
 PrVe08: Paolo Prandoni, Martin Vetterli (2008) Signal processing for communications. Lausanne: EPFL Press
 StMo05: Petre Stoica, Randolph L. Moses (2005) Spectral Analysis of Signals. Upper Saddle River, N.J: Prentice Hall
 NBHS13: T. Necciari, P. Balazs, N. Holighaus, P.L. Sondergaard (2013) The ERBlet transform: An auditorybased timefrequency representation with perfect reconstruction. In 2013 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP) (pp. 498–502). DOI
 Wise06: Duane K. Wise (2006) The modified Chamberlin and Zölzer filter structures. In Proc. of the 9th Int. Conference on Digital Audio Effects (DAFx06) (Vol. 2, p. 3).
 Moor74: J.A Moorer (1974) The optimum comb method of pitch period analysis of continuous digitized speech. IEEE Transactions on Acoustics, Speech and Signal Processing, 22(5), 330–338. DOI
 Wish14: Aaron Wishnick (2014) TimeVarying Filters for Musical Applications. In DAFx (pp. 69–76).