Slide 7
Slide 7 text
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
⇢ = 0
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
⇢ = 1.0
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
⇢ = 2.0
AAACwnicfZHBThsxEIadpRQItAV65LJqhIRQFe0iVDgi4NBLVSo1gJSN0Kwzmxhs72LPIoLZ1+i1vFbfpt6QVA2UjmTp0z+/PeOZtJDCUhT9agRzr+ZfLywuNZdX3rx9t7q2fmrz0nDs8Fzm5jwFi1Jo7JAgieeFQVCpxLP06qjOn92gsSLX32lUYE/BQItMcCAvJckQyCWpcqaqLlZbUTsaR/gc4gm02CROLtYaD0k/56VCTVyCtd04KqjnwJDgEqtmUlosgF/BALseNSi0PTduugo3vdIPs9z4oykcq3/fcKCsHanUOxXQ0D7N1eK/ct2Ssv2eE7ooCTV/LJSVMqQ8rCcQ9oVBTnLkAbgRvteQD8EAJz+nmSqpmvmDq2sZm9mq2UyO0X/Z4BcvfS3QAOVm2yVgBgpuKz+CQfKxpv8ZhZ4aPb1k9I8IJe6wcn/oRavQU+uU/E7jpxt8Dqc77fhTe/fbbuvgcLLdRbbBPrAtFrM9dsA+sxPWYZwV7Af7yR6C4+AyuA7sozVoTO68ZzMR3P8GjfHiRg==
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
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
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
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
s = [s1, . . . , sM ]T
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
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