The Living Thing / Notebooks :

Filter design, linear

especially digital

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

z-Transforms, bilinear transforms (blech!), 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

State-Variable Filters

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

Nigel Redmon, digital SVF intro.

Heavy-tailed noise

If your noise is heavy tailed, what works? I suspect this causes problems even if it is white noise, but haven’t looked into it. TBD.

Refs, sundry

Abe, T., Kobayashi, T., & Imai, S. (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.
Alliney, S. (1992) Digital filters as absolute norm regularizers. IEEE Transactions on Signal Processing, 40(6), 1548–1562. DOI.
Antoniou, A. (2005) Digital signal processing: signals, systems and filters. . New York: McGraw-Hill
Berkhout, A. J., & Zaanen, P. R.(1976) A Comparison Between Wiener Filtering, Kalman Filtering, and Deterministic Least Squares Estimation*. Geophysical Prospecting, 24(1), 141–197. DOI.
Chamberlin, H. (1985) Musical applications of microprocessors. (2nd ed.). Hasbrouck Heights, N.J: Hayden Book Co.
Harvey, A., & Luati, A. (2014) Filtering With Heavy Tails. Journal of the American Statistical Association, 109(507), 1112–1122. DOI.
Hohmann, V. (2002) Frequency analysis and synthesis using a Gammatone filterbank. Acta Acustica United with Acustica, 88(3), 433–442.
Laroche, J. (2007a) On the Stability of Time-Varying Recursive Filters. Journal of the Audio Engineering Society, 55(6), 460–471.
Laroche, J. (2007b) On the Stability of Time-Varying Recursive Filters. Journal of the Audio Engineering Society, 55(6), 460–471.
Marple, S. L., Jr. (1987) Digital spectral analysis with applications.
Martin, R. J.(1998) Autoregression and irregular sampling: Filtering. Signal Processing, 69(3), 229–248. DOI.
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Necciari, T., Balazs, P., Holighaus, N., & Sondergaard, P. L.(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.
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Smith III, J. O.(2010) Audio signal processing in Faust. Online Tutorial: Https://Ccrma. Stanford. Edu/Jos/Aspf.
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Wishnick, A. (2014) Time-Varying Filters for Musical Applications. In DAFx (pp. 69–76).