# Delays and reverbs (digital)

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:

1. Parameterisations of sparse unitary matrices
2. (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