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Clean up of source code for refs and footnotes
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Expand Up @@ -1771,81 +1771,206 @@ support driving functions from |WFS|.
Footnotes
---------

.. [#F1] Note that :math:`\sin\theta` is used here instead of :math:`\cos\theta` due to the use of another coordinate system, compare Figure 2.1 from Gumerov and Duraiswami and :numref:`fig-coordinate-system`.
.. [#F2] Note the :math:`\frac{1}{2\pi}` term is wrong in [Ahrens2012]_, eq. (3.21) and eq. (5.7) and omitted here, compare the `errata <http://www.soundfieldsynthesis.org/errata/>`_ and [SchultzSpors2014]_, eq. (24).
.. [#F3] Compare the assumptions made before (15) in [SporsZotter2013]_, which lead to the derivation of the same window function in a more explicit way.
.. [#F4] The solution mentioned by [LaxFeshbach1947]_ assumes that the listener is far away from the radiator and that the radiator is a physical source not a notional one as the secondary sources. In this case the selection criterion has to be chosen more carefully, incorporating the exact position of the listener and the virtual source. See also the `related discussion <https://github.com/sfstoolbox/sfs-documentation/issues/8>`_.
.. [#F5] Note that Williams defines the Fourier transform with transposed signs as :math:`F(\w) = \int f(t) \e{\i\w t}`. This leads also to changed signs in his definitions of the Green’s functions and field expansions.
.. [#F6] Whereby :math:`r` corresponds to :math:`|\x_0-\xs|` and :math:`\cos\varphi` to :math:`\frac{\scalarprod{\x_0-\xs}{\n_{\x_0}}}{|\x_0-\xs|}`.
.. [#F1]
Note that :math:`\sin\theta` is used here instead of :math:`\cos\theta` due
to the use of another coordinate system, compare Figure 2.1 from Gumerov and
Duraiswami and :numref:`fig-coordinate-system`.
.. [#F2]
Note the :math:`\frac{1}{2\pi}` term is wrong in [Ahrens2012]_, eq. (3.21)
and eq. (5.7) and omitted here, compare the `errata
<http://www.soundfieldsynthesis.org/errata/>`_ and [SchultzSpors2014]_, eq.
(24).
.. [#F3]
Compare the assumptions made before (15) in [SporsZotter2013]_, which lead
to the derivation of the same window function in a more explicit way.
.. [#F4]
The solution mentioned by [LaxFeshbach1947]_ assumes that the listener is
far away from the radiator and that the radiator is a physical source not a
notional one as the secondary sources. In this case the selection criterion
has to be chosen more carefully, incorporating the exact position of the
listener and the virtual source. See also the `related discussion
<https://github.com/sfstoolbox/sfs-documentation/issues/8>`_.
.. [#F5]
Note that Williams defines the Fourier transform with transposed signs as
:math:`F(\w) = \int f(t) \e{\i\w t}`. This leads also to changed signs in
his definitions of the Green’s functions and field expansions.
.. [#F6]
Whereby :math:`r` corresponds to :math:`|\x_0-\xs|` and :math:`\cos\varphi`
to :math:`\frac{\scalarprod{\x_0-\xs}{\n_{\x_0}}}{|\x_0-\xs|}`.
.. _sec-references:

References
----------

.. [AbramowitzStegun1972] Abramowitz, M. and Stegun, I. A., *Handbook of Mathematical Functions*. (National Bureau of Standards, Washington, 1972).
.. [Ahrens2012] Ahrens, J., *Analytic Methods of Sound Field Synthesis*. (Springer, New York, 2012).
.. [AhrensSpors2009a] Ahrens, J. and Spors, S., “On the Secondary Source Type Mismatch in Wave Field Synthesis Employing Circular Distributions of Loudspeakers,” in *127th Conv. Audio Eng. Soc.*, New York, NY (2009), preprint 7952, `url <http://www.aes.org/e-lib/browse.cfm?elib=15146>`__.
.. [AhrensSpors2009b] Ahrens, J. and Spors, S., “Spatial encoding and decoding of focused virtual sound sources,” in *1st Ambisonics S.*, Graz, Austria (2009), `url <http://www.deutsche-telekom-laboratories.de/~sporssas/.../AhrensSpors_AmbiSym09.pdf>`__.
.. [AhrensSpors2010] Ahrens, J. and Spors, S., “Sound Field Reproduction Using Planar and Linear Arrays of Loudspeakers,” IEEE Trans. Audio Speech Lang. Processing **18**, 2038–2050 (2010), `url <http://dx.doi.org/10.1109/TASL.2010.2041106>`__.
.. [ArfkenWeber2005] Arfken, G. B. and Weber, H. J., *Mathematical Methods for Physicists*. (Elsevier, Amsterdam, 2015).
.. [Avni2013] Avni, A., Ahrens, J., Geier, M., Spors, S., Wierstorf, H., and Rafaely, B., “Spatial perception of sound fields recorded by spherical microphone arrays with varying spatial resolution,” J. Acoust. Soc. Am. **133**, 2711–2721 (2013), `url <http://dx.doi.org/10.1121/1.4795780>`__.
.. [Bracewell2000] Bracewell, R. N., *The Fourier Transform and its Applications*. (McGraw Hill, Boston, 2000).
.. [ColtonKress1983] Colton, D. and Kress, R., *Integral Equation Methods in Scattering Theory*. (Wiley, New York, 2000).
.. [ColtonKress1998] Colton, D. and Kress, R., *Inverse Acoustic and Electromagnetic Scattering Theory*, (Springer, New York, 1998).
.. [Fazi2010] Fazi, F. M., *Sound Field Reproduction*, Ph.D. dissertation, University of Southampton, Southampton, UK, 2010, `url <http://eprints.soton.ac.uk/id/eprint/158639>`__.
.. [FaziNelson2013] Fazi, F. M. and Nelson, P. A., “Sound field reproduction as an equivalent acoustical scattering problem,” J. Acoust. Soc. Am. **134**, 3721–3729 (2013), `url <http://dx.doi.org/10.1121/1.4824343>`__.
.. [GumerovDuraiswami2004] Gumerov, N. A. and Duraiswami, R., *Fast Multipole Methods for the Helmholtz Equation in Three Dimensions*. (Elsevier, Amsterdam, 2004).
.. [Hahn2015] Hahn, N. and Spors S., “Sound Field Synthesis of Virtual Cylindrical Waves Using Circular and Spherical Loudspeaker Arrays,” in *138th Conv. Audio Eng. Soc.*, Warsaw, Poland (2015), preprint 9324, `url <http://www.aes.org/e-lib/browse.cfm?elib=17748>`__.
.. [Herrin2003] Herrin, D. W., Martinus, F., Wu, T. W., and Seybert, A. F., “A New Look at the High Frequency Boundary Element and Rayleigh Integral Approximations,” SAE Technical Paper, doi:10.4271/2003-01-1451 (2003), `url <http://dx.doi.org/10.4271/2003-01-1451>`__.
.. [LaxFeshbach1947] Lax, M. and Feshbach, H., “On the Radiation Problem at High Frequencies,” J. Acoust. Soc. Am. **19**, 682–690 (1947), `url <http://dx.doi.org/10.1121/1.1916538>`__.
.. [MorseFeshbach1981] Morse, P. M. and Feshbach, H., *Methods of Theoretical Physics*. (Feshbach Publishing, Minneapolis, 1981).
.. [SchultzSpors2014] Schultz, F. and Spors, S., “Comparing Approaches to the Spherical and Planar Single Layer Potentials for Interior Sound Field Synthesis,” Acta Acust. **100**, 900–911 (2014), `url <http://dx.doi.org/10.3813/AAA.918769>`__.
.. [Schultz2016] Schultz, F., *Sound Field Synthesis for Line Source Array Applications in Large-Scale Sound Reinforcement*, doctoral thesis, Universität Rostock, Rostock, Germany, 2016, `url <http://rosdok.uni-rostock.de/resolve/urn/urn:nbn:de:gbv:28-diss2016-0078-1>`__.
.. [SporsAhrens2010] Spors, S. and Ahrens, J., “Reproduction of Focused Sources by the Spectral Division Method,” in *4th IEEE ISCCSP*, Limassol, Cyprus (2010), doi:10.1109/ISCCSP.2010.5463335, `url <http://dx.doi.org/10.1109/ISCCSP.2010.5463335>`__.
.. [SporsZotter2013] Spors, S. and Zotter, F., “Spatial Sound Synthesis with Loudspeakers,” in *Cutting Edge in Spatial Audio, EAA Winter School*, Merano, Italy (2013), pp. 32–37, `url <https://fedora.kug.ac.at/fedora/get/o:6537/bdef:Content/get>`__.
.. [Spors2011] Spors, S., Kuscher, V., and Ahrens, J., “Efficient realization of model-based rendering for 2.5-dimensional near-field compensated higher order Ambisonics,” in *IEEE WASPAA*, New Paltz, NY (2011), pp. 61–64, `url <http://dx.doi.org/10.1109/ASPAA.2011.6082325>`__.
.. [Spors2008] Spors, S., Rabenstein, R., and Ahrens, J., “The Theory of Wave Field Synthesis Revisited,” in *124th Conv. Audio Eng. Soc.*, Amsterdam, Netherlands (2008), preprint 7358i, `url <http://www.aes.org/e-lib/browse.cfm?elib=14488>`__.
.. [Start1997] Start, E. W., *Direct Sound Enhancement by Wave Field Synthesis*, Ph.D. dissertation, Technische Universiteit Delft, Delft, Netherlands, 1997.
.. [Verheijen1997] Verheijen, E., *Sound Reproduction by Wave Field Synthesis*, Ph.D. dissertation, Technische Universiteit Delft, Delft, Netherlands, 1997.
.. [Wierstorf2014] Wierstorf, H., *Perceptual Assessment of Sound Field Synthesis*, Ph.D. dissertation, Technische Universität Berlin, Berlin, Germany, 2014, `url <http://dx.doi.org/10.14279/depositonce-4310>`__.
.. [Williams1999] Williams, E. G., *Fourier Acoustics*. (Academic Press, San Diego, 1999).
.. [Yon2003] Yon, S., Tanter, M., and Fink, M., “Sound focusing in rooms: The time-reversal approach,” J. Acoust. Soc. Am. **113**, 1533–1243 (2003), `url <http://dx.doi.org/10.1121/1.1543587>`__.
.. [ZotterSpors2013] Zotter, F. and Spors, S., “Is sound field control determined at all frequencies? How is it related to numerical acoustics?” in *52th Int. Conf. Audio Eng. Soc.*, Guildford, UK (2013), preprint 1.3, `url <http://www.aes.org/e-lib/browse.cfm?elib=16921>`__.
.. [AbramowitzStegun1972]
Abramowitz, M. and Stegun, I. A.,
*Handbook of Mathematical Functions*.
(National Bureau of Standards, Washington, 1972).
.. [Ahrens2012]
Ahrens, J.,
*Analytic Methods of Sound Field Synthesis*.
(Springer, New York, 2012).
.. [AhrensSpors2009a]
Ahrens, J. and Spors, S.,
“On the Secondary Source Type Mismatch in Wave Field Synthesis Employing Circular Distributions of Loudspeakers,”
in *127th Conv. Audio Eng. Soc.*, New York, NY (2009), preprint 7952,
`url <http://www.aes.org/e-lib/browse.cfm?elib=15146>`__.
.. [AhrensSpors2009b]
Ahrens, J. and Spors, S.,
“Spatial encoding and decoding of focused virtual sound sources,”
in *1st Ambisonics S.*, Graz, Austria (2009),
`url <http://www.deutsche-telekom-laboratories.de/~sporssas/.../AhrensSpors_AmbiSym09.pdf>`__.
.. [AhrensSpors2010]
Ahrens, J. and Spors, S.,
“Sound Field Reproduction Using Planar and Linear Arrays of Loudspeakers,”
IEEE Trans. Audio Speech Lang. Processing **18**, 2038–2050 (2010),
`url <http://dx.doi.org/10.1109/TASL.2010.2041106>`__.
.. [ArfkenWeber2005]
Arfken, G. B. and Weber, H. J.,
*Mathematical Methods for Physicists*.
(Elsevier, Amsterdam, 2015).
.. [Avni2013]
Avni, A., Ahrens, J., Geier, M., Spors, S., Wierstorf, H., and Rafaely, B.,
“Spatial perception of sound fields recorded by spherical microphone arrays with varying spatial resolution,”
J. Acoust. Soc. Am. **133**, 2711–2721 (2013),
`url <http://dx.doi.org/10.1121/1.4795780>`__.
.. [Bracewell2000]
Bracewell, R. N.,
*The Fourier Transform and its Applications*.
(McGraw Hill, Boston, 2000).
.. [ColtonKress1983]
Colton, D. and Kress, R.,
*Integral Equation Methods in Scattering Theory*.
(Wiley, New York, 2000).
.. [ColtonKress1998]
Colton, D. and Kress, R.,
*Inverse Acoustic and Electromagnetic Scattering Theory*,
(Springer, New York, 1998).
.. [Fazi2010]
Fazi, F. M.,
*Sound Field Reproduction*,
Ph.D. dissertation, University of Southampton, Southampton, UK, 2010,
`url <http://eprints.soton.ac.uk/id/eprint/158639>`__.
.. [FaziNelson2013]
Fazi, F. M. and Nelson, P. A.,
“Sound field reproduction as an equivalent acoustical scattering problem,”
J. Acoust. Soc. Am. **134**, 3721–3729 (2013),
`url <http://dx.doi.org/10.1121/1.4824343>`__.
.. [GumerovDuraiswami2004]
Gumerov, N. A. and Duraiswami, R.,
*Fast Multipole Methods for the Helmholtz Equation in Three Dimensions*.
(Elsevier, Amsterdam, 2004).
.. [Hahn2015]
Hahn, N. and Spors S.,
“Sound Field Synthesis of Virtual Cylindrical Waves Using Circular and Spherical Loudspeaker Arrays,”
in *138th Conv. Audio Eng. Soc.*, Warsaw, Poland (2015), preprint 9324,
`url <http://www.aes.org/e-lib/browse.cfm?elib=17748>`__.
.. [Herrin2003]
Herrin, D. W., Martinus, F., Wu, T. W., and Seybert, A. F.,
“A New Look at the High Frequency Boundary Element and Rayleigh Integral Approximations,” SAE Technical Paper,
doi:10.4271/2003-01-1451 (2003),
`url <http://dx.doi.org/10.4271/2003-01-1451>`__.
.. [LaxFeshbach1947]
Lax, M. and Feshbach, H.,
“On the Radiation Problem at High Frequencies,”
J. Acoust. Soc. Am. **19**, 682–690 (1947),
`url <http://dx.doi.org/10.1121/1.1916538>`__.
.. [MorseFeshbach1981]
Morse, P. M. and Feshbach, H.,
*Methods of Theoretical Physics*.
(Feshbach Publishing, Minneapolis, 1981).
.. [SchultzSpors2014]
Schultz, F. and Spors, S.,
“Comparing Approaches to the Spherical and Planar Single Layer Potentials for Interior Sound Field Synthesis,”
Acta Acust. **100**, 900–911 (2014),
`url <http://dx.doi.org/10.3813/AAA.918769>`__.
.. [Schultz2016]
Schultz, F.,
*Sound Field Synthesis for Line Source Array Applications in Large-Scale Sound Reinforcement*,
doctoral thesis, Universität Rostock, Rostock, Germany, 2016,
`url <http://rosdok.uni-rostock.de/resolve/urn/urn:nbn:de:gbv:28-diss2016-0078-1>`__.
.. [SporsAhrens2010]
Spors, S. and Ahrens, J.,
“Reproduction of Focused Sources by the Spectral Division Method,”
in *4th IEEE ISCCSP*, Limassol, Cyprus (2010), doi:10.1109/ISCCSP.2010.5463335,
`url <http://dx.doi.org/10.1109/ISCCSP.2010.5463335>`__.
.. [SporsZotter2013]
Spors, S. and Zotter, F.,
“Spatial Sound Synthesis with Loudspeakers,”
in *Cutting Edge in Spatial Audio, EAA Winter School*, Merano, Italy (2013), pp. 32–37,
`url <https://fedora.kug.ac.at/fedora/get/o:6537/bdef:Content/get>`__.
.. [Spors2011]
Spors, S., Kuscher, V., and Ahrens, J.,
“Efficient realization of model-based rendering for 2.5-dimensional near-field compensated higher order Ambisonics,”
in *IEEE WASPAA*, New Paltz, NY (2011), pp. 61–64,
`url <http://dx.doi.org/10.1109/ASPAA.2011.6082325>`__.
.. [Spors2008]
Spors, S., Rabenstein, R., and Ahrens, J.,
“The Theory of Wave Field Synthesis Revisited,”
in *124th Conv. Audio Eng. Soc.*, Amsterdam, Netherlands (2008), preprint 7358,
`url <http://www.aes.org/e-lib/browse.cfm?elib=14488>`__.
.. [Start1997]
Start, E. W.,
*Direct Sound Enhancement by Wave Field Synthesis*,
Ph.D. dissertation, Technische Universiteit Delft, Delft, Netherlands, 1997.
.. [Verheijen1997]
Verheijen, E.,
*Sound Reproduction by Wave Field Synthesis*,
Ph.D. dissertation, Technische Universiteit Delft, Delft, Netherlands, 1997.
.. [Wierstorf2014]
Wierstorf, H.,
*Perceptual Assessment of Sound Field Synthesis*,
Ph.D. dissertation, Technische Universität Berlin, Berlin, Germany, 2014,
`url <http://dx.doi.org/10.14279/depositonce-4310>`__.
.. [Williams1999]
Williams, E. G.,
*Fourier Acoustics*.
(Academic Press, San Diego, 1999).
.. [Yon2003]
Yon, S., Tanter, M., and Fink, M.,
“Sound focusing in rooms: The time-reversal approach,”
J. Acoust. Soc. Am. **113**, 1533–1243 (2003),
`url <http://dx.doi.org/10.1121/1.1543587>`__.
.. [ZotterSpors2013]
Zotter, F. and Spors, S.,
“Is sound field control determined at all frequencies? How is it related to numerical acoustics?”
in *52th Int. Conf. Audio Eng. Soc.*, Guildford, UK (2013), preprint 1.3,
`url <http://www.aes.org/e-lib/browse.cfm?elib=16921>`__.
.. vim: filetype=rst spell:

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