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 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 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. (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 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.

## 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: McGraw-Hill - Marp87: S. Lawrence Marple Jr. (1987)
*Digital spectral analysis with applications* - Smit00: Julius O. Smith (n.d.) Digital State-variable filters
- OpSB99: Alan V. Oppenheim, Ronald W. Schafer, John R. Buck (1999)
*Discrete-time 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. ICASSP-95 (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 (DAFx-11) (pp. 361–364).
- Laro07: Jean Laroche (2007) On the Stability of Time-Varying Recursive Filters.
*Journal of the Audio Engineering Society*, 55(6), 460–471. - RoSP11: Andrew Robertson, Adam M. Stark, Mark D. Plumbley (2011) Real-time 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 auditory-based time-frequency 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 (DAFx-06) (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) Time-Varying Filters for Musical Applications. In DAFx (pp. 69–76).