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
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?
- Jarl05: (2005) A recursive parametrization of unitary matrices. Journal of Mathematical Physics, 46(10), 103508. DOI
- ScLo61: (1961) “Colorless” artificial reverberation. Audio, IRE Transactions On, AU-9(6), 209–214. DOI
- MHRB17: (2017) Efficient Orthogonal Parametrisation of Recurrent Neural Networks Using Householder Reflections. In PMLR (pp. 2401–2409).
- DHCS15: (2015) Efficient Synthesis of Room Acoustics via Scattering Delay Networks. IEEE/ACM Transactions on Audio, Speech, and Language Processing, 23(9), 1478–1492. DOI
- VaLa12: (2012) Fractional Delay Filters-Design and Applications. In Nonuniform Sampling: Theory and Practice. Springer Science & Business Media
- TiSu02: (2002) Generalized Euler Angle Paramterization for SU(N). Journal of Physics A: Mathematical and General, 35(48), 10467–10501. DOI
- Schr61: (1961) Improved Quasi-Stereophony and “Colorless” Artificial Reverberation. The Journal of the Acoustical Society of America, 33(8), 1061–1064. DOI
- ReSa89: (1989) Kronecker Products, Unitary Matrices and Signal Processing Applications. SIAM Review, 31(4), 586–613. DOI
- Hend74: (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
- JSDP17: (2017) Tunable Efficient Unitary Neural Networks (EUNN) and their application to RNNs. In PMLR (pp. 1733–1741).
- ArSB16: (2016) Unitary Evolution Recurrent Neural Networks. In Proceedings of the 33rd International Conference on International Conference on Machine Learning - Volume 48 (pp. 1120–1128). New York, NY, USA: JMLR.org
- MeFa10: (2010) Unitary Matrix Design for Diffuse Jot Reverberators.