Weighted Pressure Matching Based on Kernel Interpolation for Sound Field Reproduction Shoichi Koyama and Kazuyuki Arikawa The University of Tokyo, Japan ABS-0197

Background Ø Sound field reproduction using loudspeakers ⎼ Recreating spatial sound for VR/AR audio, multizone system for personal audio, spatial noise cancellation Ø Two major approaches to sound field reproduct ion ⎼ Analytical approach based on integral equation • Fast and stable computation, but array geometry must be simple ⎼ Numerical approach based on minimization of squa red error • Arbitrary array geometry, but computational cost is relatively high 26 October 2022 2 Our focus because of practical applicability

Problem Formulation Ø Optimization problem to be solved ( omitted) 26 October 2022 3 : Target region Secondary source Goal: Synthesizing desired sound field inside with secondary sources (loudspeakers) AAACF3icbVDLSsNAFJ3UV62vqksXBotQQUoiRV0W3bisYB/QlDCZ3LZDZyZhZiKUkKUf4Te41bU7cevSpX/i9LGwrQcuHM65l3vvCWJGlXacbyu3srq2vpHfLGxt7+zuFfcPmipKJIEGiVgk2wFWwKiAhqaaQTuWgHnAoBUMb8d+6xGkopF40KMYuhz3Be1RgrWR/OJx4qcex3ogeRqCyrKyF/BUZudexKGPz/xiyak4E9jLxJ2REpqh7hd/vDAiCQehCcNKdVwn1t0US00Jg6zgJQpiTIa4Dx1DBeaguunkkcw+NUpo9yJpSmh7ov6dSDFXasQD0zm+WS16Y/E/r5Po3nU3pSJONAgyXdRLmK0je5yKHVIJRLORIZhIam61yQBLTLTJbm5LwDOTibuYwDJpXlTcy0r1vlqq3czSyaMjdILKyEVXqIbuUB01EEFP6AW9ojfr2Xq3PqzPaWvOms0cojlYX7/bXqDc udes(r, !) 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Pressure Matching (PM) Ø Discretize target region into control points Ø Optimization problem for PM becomes simple least-squares problem 26 October 2022 4 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 ⌦ 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 N ( L) : Target region Secondary source 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 minimize d2CL Gd udes 2 + ⌘kdk2 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 d = GHG + ⌘I 1 GHudes Transfer function matrix Driving signal vector Desired pressure vector Regularization term Closed-form solution is obtained as J Simple implementation L Fine discretization of is necessary 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 ⌦ Control points [Kirkeby+ 1996]

Idea: From Discrete To Continuous ⎼ PM is simple for implementation, but there is no guarantee that sound field in region between control points is accurately synthesized ⎼ Mode Matching / Weighted Mode Matching [Poletti 2005, Ueno+ 2019] can be u sed to synthesize continuous sound field based on expansion representation, b ut sometimes implementation is costly 26 October 2022 5 How can we take region between control points into consideration? Our idea: Incorporating sound field interpolation technique into PM • Continuous sound field estimated from measurements at control points is synthesized • Resulting algorithm is still simple for implementation

Kernel Interpolation of Sound Field Ø Problem to be solved: ⎼ is defined as reproducing kernel Hilbert space (RKHS) of functions satisfying Helmholtz equation 26 October 2022 6 Sound field interpolation problem: Estimating pressure distribution from discrete set of pressure measurements Microphone Target region AAACqXicfZFLSwMxEMfT9V3fevSyWARflF0R9SjqwYuoYGu1W8psOq3BJLskWbEu+wW8etUP5rcxra3Y+hgI/PjPP5mZTBhzpo3nveeckdGx8YnJqfz0zOzc/MLiUllHiaJYohGPVCUEjZxJLBlmOFZihSBCjtfh/XEnf/2ASrNIXpl2jDUBLcmajIKx0k2yHoQiVdlGfaHgFb1uuD/B70GB9OKivph7DhoRTQRKQzloXfW92NRSUIZRjlk+SDTGQO+hhVWLEgTqWtrtOHPXrNJwm5GyRxq3q36/kYLQui1C6xRg7vRwriP+lqsmpnlQS5mME4OSfhZqJtw1kdsZ320whdTwtgWgitleXXoHCqixnzRQJRRZPh+coB1O4ZktdB6jAhOpzTQA1RLwmNlhW8F2h/4zMtk3WvrLaB9hgj1hln7Rn1Ym+9Y+2e35w7v6CeWdor9X3L3cLRwe9fY4SVbIKlknPtknh+SUXJASoUSQF/JK3pwt59KpOLefVifXu7NMBsKhH6IN1vo= u(r) 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 {sm }M m=1 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 H 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 minimize u2H M X m=1 |u(rm) sm |2 + kuk2 H Measurements at . 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 {rm }M m=1 Kernel ridge regression (KRR) having closed-form solution 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 r2 + k2 u = 0 [Ueno+2018]

AAACvHicfZFNT9tAEIY3pi3gfhDgyMVqVKmqqsiuIuACQrQHLlWp1IRIcYTGm3GyzX5Yu2tEsPwfeqWn/q3+G9bBqRpoOtJKj955d2dmJ8k4MzYMfze8tSdPn61vbPrPX7x8tdXc3ukZlWuKXaq40v0EDHImsWuZ5djPNIJIOF4k049V/uIKtWFKfrOzDIcCxpKljIJ1Ui/WE3UUXjZbYTucR/AYohpapI7zy+3Gr3ikaC5QWsrBmEEUZnZYgLaMciz9ODeYAZ3CGAcOJQg0w2Lebhm8ccooSJV2R9pgrv59owBhzEwkzinATszDXCX+KzfIbXo4LJjMcouS3hdKcx5YFVSzByOmkVo+cwBUM9drQCeggVr3Q0tVErE0Q1HV0iY1pe/Hn9CNrPGzk75kqMEq/a6IQY8FXJfuC8bx+4r+Z2RyYXS0yugeYYLdYFn8oZVWJhfWBbmdRg83+Bh6H9rRfrvztdM6Oa23u0H2yGvylkTkgJyQM3JOuoSS7+QHuSU/vWNv5E09cW/1GvWdXbIU3tUdblbe9g== ⇢ = 0 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 ⇢ = 1.0 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 ⇢ = 2.0 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 ˆ r Kernel Interpolation of Sound Field Ø Closed-form solution of KRR: Ø Kernel function is derived based on plane-w ave expansion of w/ directional weighting 7 AAADEHicfZHNbhMxEMedLR8lfKXlyMUiQipf0S6qaC9IFeUAQogiNW2kOESzjjexYntXthcRLL9En4DH4ATiyhtw5wrPgHebVKSljLTSb//zn52dmbQQ3Ng4/tGIVi5cvHR59Urz6rXrN2621tYPTF5qyro0F7nupWCY4Ip1LbeC9QrNQKaCHabT3Sp/+J5pw3O1b2cFG0gYK55xCjZIw1aPTMC60m+QVDrt7+GnuCIyhaKAE/WdIxLsxGRu33tcq688foCJCJ1GUJe8rGyPEl+/GD9steNOXAc+C8kc2mgee8O1xicyymkpmbJUgDH9JC7swIG2nArmm6Q0rAA6hTHrB1QgmRm4egUe3w3KCGe5Do+yuFb/rnAgjZnJNDjrSU7nKvFfuX5ps+2B46ooLVP0uFFWCmxzXO0Tj7hm1IpZAKCah3/FdAIaqA1bX+qSyqUZXNVLm8z4ZpM8Z2FkzV4H6U3BNNhc33cE9FjCBx9WMCYPK/qfkauFMdB5xvARLvlH5t0JnWvlamFdULhpcvqCZ+HgcSd50tl8u9neeTa/7iq6je6gDZSgLbSDXqA91EUUfUE/0S/0OzqKPkdfo2/H1qgxr7mFliL6/gcoyQCx ˆ u(r) = (r)T(K + I) 1s 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 K = 2 6 4 (r1, r1) · · · (r1, rM ) . . . ... . . . (rM , r1) · · · (rM , rM ) 3 7 5 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 (r) = [(r, r1), . . . , (r, rM )]T AAAC4XicfZHNbtNAEMc35quErxSOXFZESICiyK4q6KVSBT1wqShS01aKTbTejNNV98PaHVcNKz8AJyquXHgJrvAcvA3rNEGkpYxk6ef//HdnZyYvpXAYx79a0bXrN27eWrndvnP33v0HndWH+85UlsOAG2nsYc4cSKFhgAIlHJYWmMolHOTHb5r8wQlYJ4zew2kJmWITLQrBGQZp1Hme5sq7mm7SIXWjpEdTOTboeuFnJ/vgU8XwyBV+r65HnW7cj2dBL0Myhy6Zx+5otfUtHRteKdDIJXNumMQlZp5ZFFxC3U4rByXjx2wCw4CaKXCZn/VU06dBGdPC2PBppDP17xOeKeemKg/O2Rsv5hrxX7lhhcVG5oUuKwTNzwsVlaRoaDMgOhYWOMppAMatCG+l/IhZxjGMcalKrpZ68E0t6wpXt9vpNoSWLewE6V0JlqGxL3zK7ESx0zqMYJL2GvqfUeiFMdBVxnCJUOIj1P4PXWkVemFdUNhpcnGDl2F/rZ+87K+/X+9uvZ5vd4U8Jk/IM5KQV2SLvCW7ZEA4+Uy+kx/kZ8SjT9FZ9OXcGrXmZx6RpYi+/gaA4+0b s = [s1, . . . , sM ]T 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  AAACt3icfZFLSwMxEMfT9V3fevSyWAQRKbsiPm6iHryIClaFbimz6WwbTLJrkhXrsp/Aq979Wn4bs7UV62sg8OM//2RmMmHCmTae91ZyRkbHxicmp8rTM7Nz8wuLS1c6ThXFGo15rG5C0MiZxJphhuNNohBEyPE6vD0q8tf3qDSL5aXpJtgQ0JYsYhSMlS7S5kLFq3q9cH+C34cK6cd5c7H0GrRimgqUhnLQuu57iWlkoAyjHPNykGpMgN5CG+sWJQjUjazXae6uWaXlRrGyRxq3p369kYHQuitC6xRgOvp7rhB/y9VTE+01MiaT1KCkH4WilLsmdoux3RZTSA3vWgCqmO3VpR1QQI39nKEqoRiaIStqKR3pvFwOjtGOrPDUSmcJKjCx2sgCUG0BD7n9gnawWdB/RiYHRkt/Ge0jTLBHzLNP+tPK5MA6ILtT//sGf8LVVtXfqW5fbFcODvvbnSQrZJWsE5/skgNyQs5JjVCC5Ik8kxdn32k6kdP5sDql/p1lMhTO3Tvrqt0n u Prior information on source direction can be incorporated by setting and ( w/o prior information) 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 ⇢ 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 ˆ r 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 ⇢ = 0 [Ueno+2021] 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 (r1, r2) = 1 4⇡ Z S 2 e⇢⇠T ˆ r · e jk⇠T(r1 r2)d⇠ = j0 ✓q (j⇢ˆ r k(r1 r2))T (j⇢ˆ r k(r1 r2)) ◆ Directional weighting Plane-wave function

Weighted Pressure Matching (WPM) Ø Transfer functions and desired sound field are estimated from those at control points: 26 October 2022 8 Pressure matching for continuous region based on kernel interpolation of sound field AAAEO3icjZJNaxQxGMezO77U8aWtXhQvwUVZrS4zUlQEoagHRcQK3baw2S6ZbGY2bDIzJBlxG3Lxk/hF/CCePSlevZuZ3S3dt9LAwJ9/fs/bPIlyzpQOgp+1unfh4qXLa1f8q9eu31jf2Ly5r7JCEtomGc/kYYQV5Sylbc00p4e5pFhEnB5Ewzfl/cEXKhXL0j09ymlX4CRlMSNYO6u3Wfv2AA2wNont8SaKhJH2IXwFS4WGOM/xKf/IIIH1QMVmz1qIOI11syI/OAhuOcfV7ePKeu8AyZKBdlFPQlt5ZQ34cpz8eHXeKYmQP+6tsL0xIYXpU2Xt8kaPlkNndD0XcY4RioWYk3nOV35Zjt5GI2gF1YGLIpyIBpicXbe176ifkULQVBOOleqEQa67BkvNCKfWR4WiOSZDnNCOkykWVHVN9VosvO+cPowz6b5Uw8o9HWGwUGokIkdWnc/fleayu06h4xddw9K80DQl40JxwaHOYPn0YJ9JSjQfOYGJZK5XSAZYYqLdA52pEomZGUz1t1SsrO+jt9SNLOlHZ33KqcQ6k48MwjIR+Kt1vyBBj0t1FsjSKejUKtAlYYIdU2tO1EqUpVN0qtxOw/kNLor9p63wWWv783Zj5/Vku2vgLrgHmiAEz8EOeAd2QRuQ2u/6ev12/Y73w/vl/fH+jtF6bRJzC8wc799/2z1ykg== ˆ gl(r) = l(r)T (Kl + I) 1 gl := zl(r)Tgl ˆ udes(r) = des(r)T Kdes + I 1 udes := zdes(r)Tudes 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 gl zl zdes : th column vector of : Interpolation filter for th secondary source : Interpolation filter for desired sound field 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 l 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 l AAACvHicfZFdSysxEIbT9atWj5+X3iwW4XA4lN2DqFdSVNAbUcFWoVtkNp2tOU2yS5IV67L/wVu98m/5b8zWVqxfA4GHd95kZjJhwpk2nvdcciYmp6ZnyrOVuflfC4tLyytNHaeKYoPGPFaXIWjkTGLDMMPxMlEIIuR4Efb2i/zFDSrNYnlu+gm2BXQlixgFY6VmEIrsML9aqno1bxDuZ/CHUCXDOL1aLj0FnZimAqWhHLRu+V5i2hkowyjHvBKkGhOgPehiy6IEgbqdDdrN3Q2rdNwoVvZI4w7U9zcyEFr3RWidAsy1/pgrxK9yrdREO+2MySQ1KOlroSjlrondYna3wxRSw/sWgCpme3XpNSigxv7QWJVQjM2QFbWUjnReqQQHaEdWeGylkwQVmFj9yQJQXQG3uf2CbvC3oJ+MTI6Mlr4z2keYYHeYZ2/0rZXJkXVEdqf+xw1+hua/mr9V2zzbrNb3htstkzWyTn4Tn2yTOjkip6RBKPlP7skDeXR2nY7Tc8Sr1SkN76ySsXBuXgA63t9O G 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 {gl(r)}L l=1 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 udes(r)

Weighted Pressure Matching (WPM) Ø Original cost function is approximated as 26 October 2022 9 Pressure matching for continuous region based on kernel interpolation of sound field 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 J ⇡ Z ⌦ L X l=1 dlˆ gl(r) ˆ udes(r) 2 dr = dHWggd dHWguudes + C Term unrelated to optimization 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 Wgg = Z ⌦ ˆ g(r)⇤ ˆ g(r)Tdr Wgu = Z ⌦ ˆ g(r)⇤zdes(r)Tdr

Weighted Pressure Matching (WPM) Ø Optimal driving signal is obtained by solving 26 October 2022 10 Pressure matching for continuous region based on kernel interpolation of sound field 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 minimize d2CL dHWggd dHWguudes + ⌘kdk2 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 ˆ d = (Wgg + ⌘I) 1 Wguudes Driving signals of WPM can still be obtained in closed form with and computed in advance AAACwXicjVFdSxtBFJ2sVdP4UbWPfVkahFIkJCLVRzE+9KVUwRghG8Ldyd11yHwsM7NiHPZn+Gr7t/w3zsZETGnECwOHc8+dcz/ijDNjm83HSrD0YXlltfqxtra+sflpa3vn0qhcU+xQxZW+isEgZxI7llmOV5lGEDHHbjxql/nuDWrDlLyw4wz7AlLJEkbBeqoXxcJ1i4FL02KwVW81mpMIF4M6mcbZYLvyNxoqmguUlnIwptdqZrbvQFtGORa1KDeYAR1Bij0PJQg0fTfpuQh3PTMME6X9kzacsK8rHAhjxiL2SgH22vybK8n/5Xq5TY76jskstyjps1GS89CqsFxAOGQaqeVjD4Bq5nsN6TVooNavac4lFnMzuNJLm8QUtVp0in5kjb889TtDDVbp7y4CnQq4LfwK0mivRG8JmZwJPVok9J8wwe6wcC9ooZTJmXSG3nfTy/1G60fj4PygfnwyvW6VfCFfyTfSIofkmPwkZ6RDKFHknjyQP0E7YEEW6GdpUJnWfCZzEbgnMJLhuA== Wgg 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 Wgu

Weighted Pressure Matching (WPM) Ø When uniform weighting is used for all kernel functions Ø Simplified form of WPM: 26 October 2022 11 AAACvnicfZFdSxtBFIYna+tHrN+XvRkahFIk7IqoN0JQL7wpWmg0kA1ydnI2GZyPZWZWjMv+CW/bC/+W/8bZmJRGaw8MPLznnTnnzEkywa0Lw6daMPfh4/zC4lJ9+dPK6tr6xual1blh2GZaaNNJwKLgCtuOO4GdzCDIROBVcnNS5a9u0Viu1U83yrAnYaB4yhk4L3ViM9T0iIbX642wGY6DvoVoAg0yiYvrjdpj3Ncsl6gcE2BtNwoz1yvAOM4ElvU4t5gBu4EBdj0qkGh7xbjhkm57pU9TbfxRjo7Vv28UIK0dycQ7JbihfZ2rxH/lurlLD3sFV1nuULGXQmkuqNO0mp72uUHmxMgDMMN9r5QNwQBz/o9mqiRyZoaiqmVsast6PT5FP7LB7146z9CA0+ZbEYMZSLgr/RcM4p2K/mfkamr09J7RP8Ilv8ey+EPvWrmaWqfkdxq93uBbuNxtRvvNvR97jdbxZLuL5DP5Qr6SiByQFjkjF6RNGBHkgfwiv4NWkAYy0C/WoDa5s0VmIrh7BlKo30o= ⇢ = 0 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 zl(r)T = zdes(r)T = (r)T (K + I) 1 := z(r)T 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 ˆ d = arg min d2CL Gd udes H W Gd udes = GHW G + ⌘I 1 GHW udes 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 W = Z ⌦ z(r)⇤z(r)Tdr • Simple implementation as PM because difference is only in • Equivalent to PM when setting AAACvHicfZFNaxsxEIblTdq42y87PeayxBRKKWa3hCSnEpoecgl1oP4ArzGz8qyjWNIukjbUXfY/5Nqe8rf6b6r1R6jtOAOCh3deaWY0UcqZNr7/t+Ls7D57vld94b589frN21p9v6OTTFFs04QnqheBRs4ktg0zHHupQhARx240OS/z3VtUmiXyh5mmOBAwlixmFIyVOmEk8m4xrDX8pj8LbxOCBTTIIlrDeuU+HCU0EygN5aB1P/BTM8hBGUY5Fm6YaUyBTmCMfYsSBOpBPmu38N5bZeTFibJHGm+m/n8jB6H1VETWKcBc6/VcKT6W62cmPh3kTKaZQUnnheKMeybxytm9EVNIDZ9aAKqY7dWj16CAGvtDK1UisTJDXtZSOtaF64bf0I6s8NJK31NUYBL1MQ9BjQX8LOwXjMNPJT1lZHJptLTNaB9hgv3CIn+grVYml9Yl2Z0G6xvchM7nZnDcPLo6apx9XWy3Sg7IIflAAnJCzsgFaZE2oeSG3JHf5I/zxRk5E0fMrU5lcecdWQnn9h9gHt9e W 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 W = I

Experiments Ø Setting ⎼ 2D free field ⎼ Target region : square of 1.0 m x 1.0 m ⎼ 12 loudspeakers along square of 2.0 m x 2.0 m ⎼ 16 control points regularly placed over ⎼ Desired field: plane wave (direction π/4 rad) ⎼ Methods: PM and WPM (uniform/directional) ⎼ Evaluation measure: 26 October 2022 12 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 ⌦ 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 ⌦ 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 SDR(!) = R ⌦ |udes(r, !)|2dr R ⌦ |usyn(r, !) udes(r, !)|2dr Synthesized sound field Loudspeaker Control points 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 ⌦ 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 ⇢ = 5.0

Result: Frequency vs. SDR Ø SDR between 100 – 800 Hz 26 October 2022 13 WPM outperformed PM particularly at high frequencies (Uniform) Good Bad

Result: Pressure and Error Distribution Ø At 450 Hz 26 October 2022 14 PM WPM (uniform) WPM (directional) Pressure Error SDR: 11.9 dB SDR: 17.3 dB SDR: 18.3 dB

Conclusion Ø Weighted pressure matching for sound field reproduction ⎼ PM incorporated with kernel interpolation of sound field ⎼ Weighting matrices defined with integration of kernel functions are used to approximate cost function ⎼ Implementation is still simple owing to closed-form solution ⎼ Outperformed (standard) PM in experiments, especially at high frequencies 26 October 2022 15 Paper on comparing weighted pressure/mode matching will be published on Journal of the Audio Engineering Society, Special Issue on Spatial Audio! Koyama, Kimura, and Ueno, “Weighted Pressure and Mode Matching for Sound Field Reproducti on: Theoretical and Experimental Comparisons,” Journal of the Audio Engineering Society, 2022. ( in press) Paper: https://doi.org/10.17743/jaes.2022.0058 Codes: https://www.sh01.org/MeshRIR/ (included in example codes of MeshRIR dataset)