In which I think about parameterisations and implementations of audio recurrence for use in music.

A particular nook in the the linear feedback process library.

Weapons of choice:

- Parameterisations of sparse unitary matrices
- (Maybe) Composable Markov theory, as seen in Thermodynamics of life

Keywords: multichannel allpass filter.

One obvious parameterisation of unitary matrices is applying givens rotations to the identity matrix; this has \(n(n-1)\) parameters.

Note that the recurrent neural networks people have glommed on to this. (ArSB15, MHRB16, JSDP16)

## Things to try

- Generalise reverb to Gerzon m-dimensional MIMO unitary allpass
- Thiran (multi-step, SISO) allpass?

## Refs

- ArSB15
- Arjovsky, M., Shah, A., & Bengio, Y. (2015) Unitary Evolution Recurrent Neural Networks.
*arXiv:1511.06464 [Cs, Stat]*. - DHCS15
- De Sena, E., Haciihabiboglu, H., Cvetkovic, Z., & Smith, J. O.(2015) Efficient Synthesis of Room Acoustics via Scattering Delay Networks.
*IEEE/ACM Transactions on Audio, Speech, and Language Processing*, 23(9), 1478–1492. DOI. - Hede13
- Hedemann, S. R.(2013) Hyperspherical Parameterization of Unitary Matrices.
*arXiv:1303.5904 [Quant-Ph]*. - Hend74
- Hendeković, J. (1974) On parametrization of orthogonal and unitary matrices with respect to their use in the description of molecules.
*Chemical Physics Letters*, 28(2), 242–245. DOI. - Jarl05
- Jarlskog, C. (2005) A recursive parametrization of unitary matrices.
*Journal of Mathematical Physics*, 46(10), 103508. DOI. - JSDP16
- Jing, L., Shen, Y., Dubček, T., Peurifoy, J., Skirlo, S., Tegmark, M., & Soljačić, M. (2016) Tunable Efficient Unitary Neural Networks (EUNN) and their application to RNN.
*arXiv:1612.05231 [Cs, Stat]*. - MeFa10
- Menzer, F., & Faller, C. (2010) Unitary Matrix Design for Diffuse Jot Reverberators.
- MHRB16
- Mhammedi, Z., Hellicar, A., Rahman, A., & Bailey, J. (2016) Efficient Orthogonal Parametrisation of Recurrent Neural Networks Using Householder Reflections.
*arXiv:1612.00188 [Cs]*. - ReSa89
- Regalia, P., & Sanjit, M. (1989) Kronecker Products, Unitary Matrices and Signal Processing Applications.
*SIAM Review*, 31(4), 586–613. DOI. - Schr61
- Schroeder, M. R.(1961) Improved Quasi-Stereophony and “Colorless” Artificial Reverberation.
*The Journal of the Acoustical Society of America*, 33(8), 1061–1064. DOI. - ScLo61
- Schroeder, M. R., & Logan, B. (1961) “Colorless” artificial reverberation.
*Audio, IRE Transactions on*, AU-9(6), 209–214. DOI. - TiSu02
- Tilma, T., & Sudarshan, E. C. G.(2002) Generalized Euler Angle Paramterization for SU(N).
*Journal of Physics A: Mathematical and General*, 35(48), 10467–10501. DOI. - VaLa12
- Valimaki, v., & Laakso, T. I.(2012) Fractional Delay Filters-Design and Applications. In F. Marvasti (Ed.), Nonuniform Sampling: Theory and Practice. Springer Science & Business Media