Spatial Sampling Artifacts of Wave Field Synthesis for the ...

Spatial Sampling Artifacts of Wave Field Synthesis for the Reproduction of Virtual Point Sources Sascha Spors and Jens Ahrens Deutsche Telekom Laboratories Quality and Usability Laboratory Technische Universität Berlin

126th AES Convention Munich 2009

Introduction Foundations of WFS Sampling Sampling Artifacts Conclusions

Motivation fs = 2000 Hz

2

2

1.5

1.5 y −> [m]

y −> [m]

fs = 500 Hz

1 0.5 0 −2

1 0.5

−1

0 x −> [m]

1

2

0 −2

−1

0 x −> [m]

1

2

[xs = [0 − 1] m, ∆x = 0.20 m]

wave field synthesis (WFS) suffers from spatial aliasing in practical implementations spatial aliasing artifacts for plane waves are well understood virtual point sources are frequently used in WFS here: detailed analysis of spatial aliasing artifacts for virtual point sources Sascha Spors and Jens Ahrens Sampling of virtual point sources

9.5.2009 1 / 16

Introduction Foundations of WFS Sampling Sampling Artifacts Conclusions

Wave Field Synthesis for Linear Arrays Rayleigh’s first integral formula provides the pressure in one half space V given the directional pressure gradient on the boundary ∂ V of that half space Z ∞ ∂ P (x, ω) = −2 P (x0 , ω) G0,2D (x − x0 , ω) dx0 −∞ ∂n

y P (x, ω) x

V

n

x0

∂V

x

Sascha Spors and Jens Ahrens Sampling of virtual point sources

x = [x y ]T x0 = [x0 0]T

9.5.2009 2 / 16

Introduction Foundations of WFS Sampling Sampling Artifacts Conclusions

Wave Field Synthesis for Linear Arrays The field of a (primary) source S (x, ω) within the area V is uniquely given by its pressure gradient on the boundary ∂ V Z ∞ ∂ P (x, ω) = −2 S (x0 , ω) G0,2D (x − x0 , ω) dx0 ∂n −∞

y P (x, ω) x

V

x = [x y ]T x0 = [x0 0]T

n

x0

∂V

x

primary source S (x, ω) Sascha Spors and Jens Ahrens Sampling of virtual point sources

9.5.2009 2 / 16

Introduction Foundations of WFS Sampling Sampling Artifacts Conclusions

Wave Field Synthesis for Linear Arrays The Green’s function can be interpreted as (secondary) source that generates the field of a desired virtual source S (x, ω) inside the listening area V . Z ∞ P (x, ω) = D (x0 , ω) G0,2D (x − x0 , ω) dx0 −∞

∂ D (x0 , ω) = −2 ∂n S (x0 , ω)

y P (x, ω) x

V

x = [x y ]T x0 = [x0 0]T

n

x0

∂V

x

virtual source S (x, ω) Sascha Spors and Jens Ahrens Sampling of virtual point sources

9.5.2009 2 / 16

Introduction Foundations of WFS Sampling Sampling Artifacts Conclusions

Secondary Source Types The explicit form of the Green’s function depends on the dimensionality two-dimensional reproduction → secondary line sources three-dimensional reproduction → secondary point sources

WFS uses secondary point sources for two-dimensional reproduction mismatch of secondary source type results in amplitude and spectral errors artifacts have no influence on sampling theorems (...but on near-field effects)

⇒ analysis of sampling artifacts based on theory of two-dimensional WFS

Sascha Spors and Jens Ahrens Sampling of virtual point sources

9.5.2009 3 / 16

Introduction Foundations of WFS Sampling Sampling Artifacts Conclusions

Spatio-temporal Frequency Domain Description Reproduced wave field using a linear distribution of secondary sources Z ∞ P (x, ω) = D (x0 , ω) G0 (x − x0 , ω) dx0 −∞

x = [x y ]T x0 = [x0 0]T

Spatial Fourier transformation Fx

P˜ (kx , y , ω) = D˜ (kx , ω) · G˜0 (kx , y , ω)

spatio-temporal frequency domain description of reproduced wave field provides insights into structure of spatial aliasing allows to identify the spatial aliasing components Sascha Spors and Jens Ahrens Sampling of virtual point sources

9.5.2009 4 / 16

Introduction Foundations of WFS Sampling Sampling Artifacts Conclusions

Example: Spectrum of Driving Function / Secondary Sources driving function

secondary source [dB] 0

2000

0

1500

−50

1500

−50

1000

−100

1000

−100

500

−150

500

−150

0 −50

0 kx −> [1/m]

50

−200

f −> [Hz]

f −> [Hz]

[dB] 2000

0 −50

0 kx −> [1/m]

50

−200

[xs = [0 − 1] m, y = 1 m]

ω ω propagating for |kx | ≤ c , evanescent for |kx | > c , ω = 2π f Sascha Spors and Jens Ahrens Sampling of virtual point sources

9.5.2009 5 / 16

Introduction Foundations of WFS Sampling Sampling Artifacts Conclusions

Sampling of Secondary Source Distribution Spatial sampling of continuous secondary source distribution

y P (x, ω) x

V

n

x0 ∆x

∂V

x

virtual source S (x, ω)

∆x : spatial sampling interval Sascha Spors and Jens Ahrens Sampling of virtual point sources

9.5.2009 6 / 16

Introduction Foundations of WFS Sampling Sampling Artifacts Conclusions

Sampling of Secondary Source Distribution Continuous distribution of secondary sources

D (x0 , ω)

P (x, ω)

G˜0 (kx , y , ω)

Spatially discrete distribution of secondary sources

DS (x0 , ω)

D (x0 , ω) n ∆x

G˜0 (kx , y , ω)

PS (x, ω)

spatial sampling sampling leads to repetitions of spectrum of driving function weighted by secondary source spectrum (analog to interpolation) Sascha Spors and Jens Ahrens Sampling of virtual point sources

9.5.2009 6 / 16

Introduction Foundations of WFS Sampling Sampling Artifacts Conclusions

Spectrum of Reproduced Wave Field Qualitative illustration of spectra (magnitude) of driving function and secondary sources ω c

ω c

=k c

ω

x

c

ω

=k

x

x

G˜ (kx , y , ω)

D˜ (kx , ω)

kx

kx

exact reproduction of desired wave field without sampling spectra not strictly bandlimited in kx for given frequency ω spectral repetitions due to spatial sampling mixtures of propagating/evanescent contributions Sascha Spors and Jens Ahrens Sampling of virtual point sources

9.5.2009 7 / 16

Introduction Foundations of WFS Sampling Sampling Artifacts Conclusions

Spectrum of Reproduced Wave Field Qualitative illustration of spectra (magnitude) of driving function and secondary sources ω c

ω c

2π −∆ x

− ∆πx

=k c

ω

x

c

ω

=k

x

x

G˜ (kx , y , ω)

D˜S (kx , ω)

π ∆x

2π ∆x

kx

kx

exact reproduction of desired wave field without sampling spectra not strictly bandlimited in kx for given frequency ω spectral repetitions due to spatial sampling mixtures of propagating/evanescent contributions Sascha Spors and Jens Ahrens Sampling of virtual point sources

9.5.2009 7 / 16

Introduction Foundations of WFS Sampling Sampling Artifacts Conclusions

Theory of Secondary Source Sampling Artifacts due to the secondary source sampling can only be avoided when the spectrum of the driving function is band-limited, and the spectrum of the secondary sources is band-limited. Consequences of missing band-limitation spectral overlaps in the driving function → aliasing repetitions weighted by secondary source spectrum → reconstruction error Anti-aliasing condition (considering the propagating parts only)

fal ≤

c 2∆x

Example: ∆x = 0.20 m ⇒ fal ≈ 850 Hz Sascha Spors and Jens Ahrens Sampling of virtual point sources

9.5.2009 8 / 16

Introduction Foundations of WFS Sampling Sampling Artifacts Conclusions

Example: Sampling Artifacts without sampling (baseband)

sampling artifacts (spectral repetitions) 1

2

2

2

1.5 0.5

1

0

0.5

−0.5

1

1.5 y −> [m]

y −> [m]

1.5

0.5 1

0 −0.5

0.5

−1 −1.5

0 −1

−0.5

0 x −> [m]

0.5

1

−1

0 −1

−0.5

0 x −> [m]

0.5

1

[xs = [0 − 1] m, fs = 2000 Hz, ∆x = 0.20 m]

Sascha Spors and Jens Ahrens Sampling of virtual point sources

9.5.2009 9 / 16

−2

Introduction Foundations of WFS Sampling Sampling Artifacts Conclusions

Contributions to Reproduced Wave Field Four contributions to the reproduced spectrum can be identified

Driving Function Propagating Evanescent Secondary Propagating Propagating 1 Propagating 2 Source Evanescent Evanescent 1 Evanescent 2

evanescent contributions decay rapidly with distance to secondary sources evanescent contributions and propagating 2 occur due to spatial sampling perceptual relevance of evanescent contributions unclear propagating 1 is main contribution

Sascha Spors and Jens Ahrens Sampling of virtual point sources

9.5.2009 10 / 16

Introduction Foundations of WFS Sampling Sampling Artifacts Conclusions

Example: Contributions to Reproduced Wave Field propagating 1

propagating 2 2

2

0.01

2

1.5 1 0.5 1

0 −0.5

0.5

−1

0.005

1.5 y −> [m]

y −> [m]

1.5

1

0

0.5

−0.005

−1.5 0 −1

−0.5

0 x −> [m]

0.5

1

−2

0 −1

−0.5

0 x −> [m]

0.5

1

[xs = [0 − 1] m, fs = 2000 Hz, ∆x = 0.20 m]

Sascha Spors and Jens Ahrens Sampling of virtual point sources

9.5.2009 11 / 16

−0.01

Introduction Foundations of WFS Sampling Sampling Artifacts Conclusions

Example: Contributions to Reproduced Wave Field evanescent 1

evanescent 2 0.5

2

0.05

1.5

1

0

y −> [m]

y −> [m]

1.5

0.5

0 −1

0.1

2

1

0

0.5

−0.5

0 x −> [m]

0.5

0 −1

−0.5

1

−0.05

−0.5

0 x −> [m]

0.5

−0.1

1

[xs = [0 − 1] m, fs = 2000 Hz, ∆x = 0.20 m]

Sascha Spors and Jens Ahrens Sampling of virtual point sources

9.5.2009 11 / 16

Introduction Foundations of WFS Sampling Sampling Artifacts Conclusions

Truncation of Secondary Source Distribution model truncation by spatial windowing of driving function convolution with transformed window function in spatial frequency domain truncation artifacts (bending waves) geometric approximation y

driving function xs2 [dB] 0

2000

−20

α2

α1

x

L 2

− L2

xs 1

f −> [Hz]

1500 −40 1000 −60 500 −80

0 −50

xs 2 Sascha Spors and Jens Ahrens Sampling of virtual point sources

0 kx −> [1/m]

50

9.5.2009 12 / 16

−100

Introduction Foundations of WFS Sampling Sampling Artifacts Conclusions

Spectrum of Reproduced Wave Field for Truncated Array Qualitative illustration of spectra (magnitude) of driving function and secondary sources

=

ω

ω c

cos α1

2

kx =

kx

=

ω

c

ω c

sα co

c

kx

2π ∆x

2π −∆ x

kx

aliasing frequency ftr,al1 ≥ fal higher compared to infinitely long array

Sascha Spors and Jens Ahrens Sampling of virtual point sources

9.5.2009 13 / 16

Introduction Foundations of WFS Sampling Sampling Artifacts Conclusions

Spectrum of Reproduced Wave Field for Truncated Array Qualitative illustration of spectra (magnitude) of driving function and secondary sources

=

ω

ω c

cos α1

2

kx kx =

=

ω

c

ω c

sα co

c

kx

2π −∆ x

2π ∆x

kx

aliasing frequency ftr,al1 ≥ fal higher compared to infinitely long array increased aliasing frequency ftr,al2 ≥ ftr,al1 for distant listener positions not infinite as for the plane wave case e. g. [Spors et.al, 120th AES] Sascha Spors and Jens Ahrens Sampling of virtual point sources

9.5.2009 13 / 16

Introduction Foundations of WFS Sampling Sampling Artifacts Conclusions

Example: Truncated Secondary Source Distribution 6 5

y −> [m]

4

Aliasing frequencies

3

fal ≈ 850 Hz ftr,al1 ≈ 935 Hz ftr,al2 ≈ 1030 Hz

2 1 0 −3

−2

−1

0 x −> [m]

1

2

3

[xs = [0 − 1] m, fs = 1000 Hz, ∆x = 0.20 m, N = 16] Sascha Spors and Jens Ahrens Sampling of virtual point sources

9.5.2009 14 / 16

Introduction Foundations of WFS Sampling Sampling Artifacts Conclusions

Summary and Conclusions Main findings spatial frequency domain analysis provides interesting insights sampling artifacts are listener position dependent in practice anti-aliasing conditions for infinte/truncated arrays evanescent contributions are side effect of sampling application to focused sources [Spors et al., DAGA 2009] Further work artifacts for broadband signals extension to 2.5D reproduction (evanescent contributions!) perception of evanescent contributions

Sascha Spors and Jens Ahrens Sampling of virtual point sources

9.5.2009 15 / 16

Introduction Foundations of WFS Sampling Sampling Artifacts Conclusions

Thanks for your attention! Sascha Spors and Jens Ahrens Sampling of virtual point sources

9.5.2009 16 / 16