# Filter design, linear

Especially digital

July 24, 2017 — September 18, 2020

dynamical systems
Hilbert space
signal processing
statistics
time series

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.

## 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 .

## 5 On graphs

No time to go deep on this rn, but signal processing on graphs is a thing.

## 6 References

Abe, Kobayashi, and Imai. 1995. In International Conference on Acoustics, Speech, and Signal Processing, 1995. ICASSP-95.
Adcock, and Hansen. 2016. Foundations of Computational Mathematics.
Adcock, Hansen, Roman, et al. 2014. In Advances in Imaging and Electron Physics.
Adcock, Hansen, and Roman. 2015. In Compressed Sensing and Its Applications: MATHEON Workshop 2013. Applied and Numerical Harmonic Analysis.
Alliney. 1992. IEEE Transactions on Signal Processing.
Antoniou. 2005. Digital signal processing: signals, systems and filters.
Berkhout, and Zaanen. 1976. Geophysical Prospecting.
Carini, Mathews, and Sicuranza. 1999. IEEE Transactions on Signal Processing.
Chamberlin. 1985. Musical applications of microprocessors.
Ephremidze, Janashia, and Lagvilava. 2007. In SICE Annual Conference 2007.
Geronimo, and Woerdeman. 2004. Annals of Mathematics.
Harvey, and Luati. 2014. 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. IEEE Transactions on Signal Processing.
Koshita, Abe, and Kawamata. 2018. Digital Systems.
Laroche. 2007. Journal of the Audio Engineering Society.
Linkwitz. 1976. Journal of the Audio Engineering Society.
Marple. 1987. Digital Spectral Analysis with Applications.
Martin. 1998. Signal Processing.
———. 1999. Signal Processing.
Moon, and Stirling. 2000. Mathematical Methods and Algorithms for Signal Processing.
Moorer. 1974. IEEE Transactions on Acoustics, Speech and Signal Processing.
Murakoshi, Nishihara, and Watanabe. 1994. Electronics and Communications in Japan (Part III: Fundamental Electronic Science).
Narasimha, Ignjatovic, and Vaidyanathan. 2002. IEEE Signal Processing Letters.
Necciari, Balazs, Holighaus, et al. 2013. In 2013 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP).
Nyquist. 1928. 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. In Proceedings of the International Computer Music Conference 2011.
Schlecht, and Habets. 2015. The Journal of the Acoustical Society of America.
Sebek. 2015. In Encyclopedia of Systems and Control.
Shuman, D. I., Narang, Frossard, et al. 2013. IEEE Signal Processing Magazine.
Shuman, David I., Vandergheynst, and Frossard. 2011. 2011 International Conference on Distributed Computing in Sensor Systems and Workshops (DCOSS).
———. 2010. Online Tutorial: Https://Ccrma. Stanford. Edu/Jos/Aspf.
———. n.d.
Smith, and Michon. 2011. In Proceedings of the 14th International Conference on Digital Audio Effects (DAFx-11).
Stilson, and Smith. 1996. In.
Stoica, and Moses. 2005. Spectral Analysis of Signals.
Wise. 2006. In Proc. Of the 9th Int. Conference on Digital Audio Effects (DAFx-06).
Wishnick. 2014. In DAFx.
Zavalishin. n.d.