The Living Thing / Notebooks :

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


Arjovsky, M., Shah, A., & Bengio, Y. (2015) Unitary Evolution Recurrent Neural Networks. arXiv:1511.06464 [Cs, Stat].
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.
Hedemann, S. R.(2013) Hyperspherical Parameterization of Unitary Matrices. arXiv:1303.5904 [Quant-Ph].
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.
Jarlskog, C. (2005) A recursive parametrization of unitary matrices. Journal of Mathematical Physics, 46(10), 103508. DOI.
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].
Menzer, F., & Faller, C. (2010) Unitary Matrix Design for Diffuse Jot Reverberators.
Mhammedi, Z., Hellicar, A., Rahman, A., & Bailey, J. (2016) Efficient Orthogonal Parametrisation of Recurrent Neural Networks Using Householder Reflections. arXiv:1612.00188 [Cs].
Regalia, P., & Sanjit, M. (1989) Kronecker Products, Unitary Matrices and Signal Processing Applications. SIAM Review, 31(4), 586–613. DOI.
Schroeder, M. R.(1961) Improved Quasi-Stereophony and “Colorless” Artificial Reverberation. The Journal of the Acoustical Society of America, 33(8), 1061–1064. DOI.
Schroeder, M. R., & Logan, B. (1961) “Colorless” artificial reverberation. Audio, IRE Transactions on, AU-9(6), 209–214. DOI.
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.
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