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.

Contents

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 very 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.

## 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

- AbKI95
- 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.
- Alli92
- Alliney, S. (1992) Digital filters as absolute norm regularizers.
*IEEE Transactions on Signal Processing*, 40(6), 1548–1562. DOI. - Anto05
- Antoniou, A. (2005) Digital signal processing: signals, systems and filters. . New York: McGraw-Hill
- BeZa76
- 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. - Cham85
- Chamberlin, H. (1985) Musical applications of microprocessors. (2nd ed.). Hasbrouck Heights, N.J: Hayden Book Co.
- HaLu14
- Harvey, A., & Luati, A. (2014) Filtering With Heavy Tails.
*Journal of the American Statistical Association*, 109(507), 1112–1122. DOI. - Hohm02
- Hohmann, V. (2002) Frequency analysis and synthesis using a Gammatone filterbank.
*Acta Acustica United with Acustica*, 88(3), 433–442. - Laro07a
- Laroche, J. (2007a) On the Stability of Time-Varying Recursive Filters.
*Journal of the Audio Engineering Society*, 55(6), 460–471. - Laro07b
- Laroche, J. (2007b) On the Stability of Time-Varying Recursive Filters.
*Journal of the Audio Engineering Society*, 55(6), 460–471. - Marp87
- Marple, S. L., Jr. (1987) Digital spectral analysis with applications.
- Mart98
- Martin, R. J.(1998) Autoregression and irregular sampling: Filtering.
*Signal Processing*, 69(3), 229–248. DOI. - Mart99
- Martin, R. J.(1999) Autoregression and irregular sampling: Spectral estimation.
*Signal Processing*, 77(2), 139–157. DOI. - MoSt00
- Moon, T. K., & Stirling, W. C.(2000) Mathematical methods and algorithms for signal processing. . Upper Saddle River, NJ: Prentice Hall
- Moor74
- Moorer, J. . (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. - NaIV02
- Narasimha, M. J., Ignjatovic, A., & Vaidyanathan, P. P.(2002) Chromatic derivative filter banks.
*IEEE Signal Processing Letters*, 9(7), 215–216. DOI. - NBHS13
- 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.
- Nyqu28
- Nyquist, H. (1928) Certain Topics in Telegraph Transmission Theory.
*Transactions of the American Institute of Electrical Engineers*, 47(2), 617–644. DOI. - OpSB99
- Oppenheim, A. V., Schafer, R. W., & Buck, J. R.(1999) Discrete-time signal processing. (2nd ed.). Upper Saddle River, N.J: Prentice Hall
- Orfa96
- Orfanidis, S. J.(1996) Introduction to signal processing. . Englewood Cliffs, N.J: Prentice Hall
- PrVe08
- Prandoni, P., & Vetterli, M. (2008) Signal processing for communications. . Lausanne: EPFL Press
- RoSP11
- Robertson, A., Stark, A. M., & Plumbley, M. D.(2011) Real-Time Visual Beat Tracking Using a Comb Filter Matrix. In Proceedings of the International Computer Music Conference 2011.
- Smit10
- Smith III, J. O.(2010) Audio signal processing in Faust.
*Online Tutorial: Https://Ccrma. Stanford. Edu/Jos/Aspf*. - Smit07
- Smith, J. O.(2007) Introduction to Digital Filters with Audio Applications. . http://www.w3k.org/books/: W3K Publishing
- SmMi11
- Smith, J. O., & Michon, R. (2011) Nonlinear allpass ladder filters in faust. In Proceedings of the 14th International Conference on Digital Audio Effects (DAFx-11) (pp. 361–364).
- StMo05
- Stoica, P., & Moses, R. L.(2005) Spectral Analysis of Signals. (1 edition.). Upper Saddle River, N.J: Prentice Hall
- Wise06
- Wise, D. K.(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).
- Wish14
- Wishnick, A. (2014) Time-Varying Filters for Musical Applications. In DAFx (pp. 69–76).