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Twelfth International Congress
on Sound and Vibration
1
ON SPATIAL SAMPLING AND ALIASING IN
ACOUSTIC IMAGING
Rick Scholte, Bert Roozen and Ines Lopez
Technische Universiteit Eindhoven
P.O. Box 513
5600 MD Eindhoven, The Netherlands
r.scholte@tue.nl
Abstract In acoustic imaging practise, where two-dimensional area measurements are
performed, spatial sampling is a little addressed subject. Especially aliasing due to
high spatial sampling at short sensor distance with respect to the source of interest, is
poorly discussed in open literature. This paper will discuss the theory behind optimal
spatial sampling for Fourier acoustics and offers insights into aliasing in acoustic
images.
The discussion on spatial sampling theory will include the Nyquist sampling theorem
and the application of given theory for acoustic imaging. This will include a
transformation from time-frequency considerations in one dimension to position-
wavenumber relations in two dimensions.
Also, the function of Fourier Transforms in Planar Near-field Acoustic Holography
(PNAH) is discussed, and a formula for acoustic image resolution is given. Finally,
two-dimensional aliasing is clearly illustrated by means of an analytical example.
Using sampling theory, aliasing theory and basic knowledge of PNAH data
processing, an inequality is derived that couples physical resolution limits with spatial
sampling limitations.
INTRODUCTION
In every day acoustic imaging practice array measurements are mostly performed at a
certain distance from the source with a predefined inter-sensor distance, without
proper knowledge of the consequences of such settings on the end result. In order to
clarify these issues the following insights have been written, which are particularly
2
focused on high-resolution acoustic imaging, but can also be placed in a broader
view.
THE NYQUIST WAVEFORM FREQUENCY CRITERION
Before we start a discussion on aliasing and specific requirements of spatial sampling
and measurement conditions with respect to PNAH, spatial sampling criteria will be
explained. Also, a number of possible measurement or spatial sampling strategies will
be illustrated. This will result in a criterion based on the Nyquist theorem [2] and
shows ways to sample a sound field, such that the measurement meets this criterion
and practically makes every resolution possible.
Spatial sampling
In Acoustic Imaging practice the sound pressure field is sampled at a certain distance
from a sound source and used as input for acoustic holography. By measuring the
sound field at certain positions, the sound field is spatially sampled or discretised.
This process can be compared to sampling a continuous time signal, only in this case
a 2-dimensional area is sampled and the continuous variable that is sampled is
position instead of time.
A widely used construction for spatially sampling a sound field is a static array of
microphones, capable of both observing stationary and non-stationary sound sources.
Drawbacks of this method are the spatial resolution restrictions, following the
physical dimensions of the microphones, requirements of open space in between the
individual sensors, and the high demands for parallel signal acquisition systems. An
alternative for the full array is the application of a traverse system with a scanning
sensor. The microphone is traversed over a predefined grid, measuring sound pressure
at every single position. All positions combined make up the acoustic information of
a certain area, stored in matrix form.
Figure 1: Single sensor traverse acoustic array measurement, which fills up a matrix with
positional, complex sound pressure data. This matrix can then be plotted as shown in the image
at the right.
In order to measure and process acoustic information at a resolution higher than
possible with a static array we are forced to use a different strategy. In this case the
zh
1,1
5,5
1,5
5,1
∆x
5,55,25,1
4,24,1
3,23,1
2,52,42,32,22,1
1,51,41,31,21,1
p..pp
..pp
..pp
ppppp
ppppp
~~~
~~
~~
~~~~~
~~~~~
1 50
50
1
52
55
58
3
traverse system is used to move the microphone over a small distance, particularly
within the dimensions of the microphone itself. This procedure is illustrated in Figure
1 and also shows the way the matrix is filled and how it is transformed into an
acoustic image of the sound field at a certain distance from the source area.
In fact, this method does not provide the exact sound pressure at a single position, but
provides an average sound pressure around the given position. Also, the variation of
these small areas that partly cover their neighbors result in an image that provides the
relative changes over the full area. This procedure results in spatial acoustic
information at a very high resolution, which can be used as input to the PNAH
process that attempts to find the inverse solution for a certain source of sound.
Nyquist waveform frequency criterion
In PNAH a Fast Fourier Transform (FFT) is used to convert the spatial data into the
wavenumber domain, where a simple multiplication is used to obtain the inverse
solution. The next chapter will go into further detail on this topic. Our interest here is
the FFT and the consequences of spatial sampling properties for the wavenumber
domain.
Typically, the wavenumber spectrum displays all the waveform frequencies and their
specific energies that make up the spatial domain signal. For the typical waveform
displayed in Figure 2a, the dominant waveform frequencies are highlighted in the
wavenumber domain shown in Figure 2b. Every single pixel or block in Figure 2a
represents a microphone position and can be seen as discrete samples of the sound
field. For FFT it is a requirement that an equidistant grid is used. Additionally, the
row and column length should be a power of two, which allows to profit from the
numerical efficiency of the FFT algorithm.
a
kx
0
kNyquist
kNyquist
ky
b
Figure 2: Sound field measured at 32 by 32 microphone positions (a), which results in waveform
frequency spectrum or wavenumber domain (b) after the spatial domain is transformed by FFT.
In order to correctly allocate the waveform frequency components in the wavenumber
domain, it is necessary to sample the spatial domain twice as often as the highest
waveform frequency maxk present. The highest observable waveform frequency is
4
often referred to as the Nyquist (waveform) frequency, in short: Nyquistkk =max . The
sampling wavenumber is generally known as the Nyquist sampling frequency or, in
this case, Nyquist sampling wavenumber:
xkk sampleNyquist
∆==
π22 max, [rad/m]. (1)
From this equation it is clear that the Nyquist waveform frequency is directly coupled
to the sampling interval x∆ , which determines the maximum distance between the
microphone positions from a Nyquist point of view. In the next chapter the Nyquist
criterion is linked to the highest observable wavenumber under measurement
conditions.
DETERMINATION OF SPATIAL RESOLUTION IN NEAR-FIELD
ACOUSTIC HOLOGRAPHY
The matrix of complex sound pressures as a function of position and frequency,
( )ω,,,~ zyxp , is used as input to the PNAH process. This chapter will primarily
illustrate the consequences of FFT and the inverse propagator on the observability of
high frequency waveforms.
Fourier Transform in PNAH
In PNAH the Fourier Transform plays an important role in simplifying the calculation
of the inverse solution to the wave equation. This method is discussed extensively in
[3], where ( )ω,,,~hzyxp in spatial domain is 2-dimensionally transformed into
( )ω,,,~̂hyx zkkp in the wavenumber domain by means of Fourier Transform. To
calculate the reconstructed sound field at szz = , ( )ω,,,~̂hyx zkkp is multiplied by an
exponential power that is a function of distance sh zz − and ),( yx kk . Here hz is the
hologram or measurement plane and sz the source plane. Then, ( )ω,,,~̂ zkkp yx is
inverse Fourier Transformed to obtain the spatial pressure field ( )ω,,,~szyxp . The
procedure is schematically shown in Figure 3.
inverse
propagator
spatial
pre-
processor
low-pass
filter
2D
FFT
2D
iFFT( )ω,,,~
hzyxp ( )ω,,,~szyxp
( )ω,,,~̂hyx zkkp ( )ω,,,~̂
syx zkkp
wavenumber domain
Figure 3: Schematic overview of Fourier Transform functions in PNAH.
The spatial pre-processor block contains spatial processes like windowing, zero- and
border-padding functions [4], which renders the data suitable for 2-dimensional
5
Fourier Transform. In the wavenumber domain (arced gray in Figure 3),
( )ω,,,~̂hyx zkkp is multiplied by the inverse propagator. The result is then low-pass
filtered to make ( )ω,,,~̂hyx zkkp suitable for inverse 2-dimensional Fourier Transform.
From which the end result ( )ω,,,~szyxp follows. Although the use of Fourier
Transform is highly efficient and fast, there are some difficulties that will be
addressed in the next chapter.
Resolution of acoustic image
Resolution of an acoustic image is influenced by the frequency of the acoustic signal,
the dynamic range of the measurement and the distance from the hologram plane to
the source. This physical resolution limit has first been derived by Maynard et.al. [5],
( )
2
2
20
10ln
−+
=
sh zz
Dk
Rπ
.
(2)
Multiple steps in the PNAH process influence the resolution. For example, the cut-off
of the low-pass wave frequency filter, or the applied spatial window, but we will
focus on the coupling with aliasing only.
DEFINING ALIASING IN PNAH AND THE CONSEQUENCES
Aliasing
Spatial aliasing can best be explained by showing an example of a 2-dimensional
waveform, where two
waveforms of different
frequencies travel in two
different directions, see
Figure 4. In the following
illustrations dark areas
mark high sound pressure
whereas light areas mark
low sound pressure.
Figure 4: Separate wave
patterns and corresponding
wavenumber spectra; the wave
illustrated in a has the smallest
wavelength that results in a
spectrum b with energy in the
highest wavenumbers, whereas
the wave illustrated in c has the
largest wavelength and energy
in the lower wavenumbers,
illustrated in d.
a
kx
ky
kx
ky
b
c
kx
ky
d
6
The interfering wave pattern is illustrated in Figure 5a. From this waveform a 2-
dimensional FFT results in the waveform spectrum of Figure 5b. The two
components can be clearly discriminated from the image, which also show the
different directions and waveform frequencies.
a
kx
ky
b
Figure 5: Wave pattern where two waves travel in different directions (a), which results in the
wavenumber domain that clearly shows energy at particular wave frequencies in two directions.
Given exactly the same input, but now sampled at a certain rate (Figure 6a), lower
then the wavenumber of the highest wave form existent in the sample, results in an
under sampled wavenumber spectrum. The dotted square shows the Nyquist
waveform frequency and it is clear from Figure 6b that the original spectrum arced in
gray is now infinitely repeated in all directions. Because the sampling rate is too low
the neighboring spectra overlap and energy in the higher waveform is positioned in a
lower waveform. The new spectrum is shown in the dotted area, which marks the
Nyquist waveform frequency again. This effect is called spatial aliasing.
a
kx
ky
b
Figure 6: Spatial undersampling (a) of the waveform from Figure 5a, every line crossing marks a
sample or sensor position. After FFT this results in a repetitive wavenumber domain (b).
7
Inverse transforming this spectrum result in a completely different waveform as can
be seen in Figure 6b. Note that zero-padding outside the Nyquist waveform frequency
is applied to properly interpolate the resulting spatial information.
kx
ky
a b
Figure 7: The zero-padded, aliased wavenumber domain (a) results in a different waveform
compared to the original in Figure 5a after the inverse FFT is applied (b).
Influence of measurement distance and noise on sampling rate
In the previous paragraph the effect of under-sampling on a given waveform has been
clarified. Now, consider an unknown sound wave pattern that emerges from a given
object. A hologram is acquired by means of the method described in Figure 3, and the
inverse solution is determined by PNAH. In this case it is very important to
understand the various influences on the useful information about the source wave
pattern that can be measured at a certain distance. The most important issues are the
exponential decay of the evanescent waves and the dynamic range of the
measurement.
For practical cases a coupling can be made between the Nyquist criterion and the
resolution of an acoustic image based on dynamic range D, measurement distance
sh zz − and sound frequency kcf ⋅= as discussed separately in previous chapters.
From equation (2) the highest physically available wavenumber is derived and
compared to the Nyquist waveform frequency sampleNyquist kk ⋅= 5.0 , which results in
the inequality:
( ) sample
sh
kzz
Dk ⋅
−
⋅+ 5.0
20
10ln2
2f . (3)
Aliasing occurs if this inequality is true. It tells us that under certain noise conditions
the measurement distance can be too small given the chosen inter-microphone
distance. This seems contradicting, because according to this theory a microphone
with a high dynamic range should be placed further from the radiating object then a
low dynamic range microphone. The explanation is that the high dynamic range
8
microphone is capable of successfully retrieving more information about the
evanescent waves at equal distance as the low dynamic range microphone. This sets a
boundary for non-aliasing PNAH processing according to equation (3). When the
high dynamic range microphone is at the minimum distance from the source, the
microphone with lower dynamic range can still be moved closer not causing aliasing.
DISCUSSION
The use of Fourier Transforms in acoustic imaging is very beneficial in terms of
processing speed and complexity, while it provides a very accurate solution to the
inverse solution of the wave equation. On the other hand, application of the Fourier
Transform requests a thorough knowledge of sampling theory and requirements for
proper transforms.
This paper provided insight into sampling considerations and ways to obtain high
resolution sampling of acoustic fields. It is important to couple discretisation by
means of sensor positions with observability of sound waves, given a certain distance
from the source and dynamic range of the measurement. One of the most remarkable
conclusions that can be drawn from the deduced inequality is that a low quality
microphone can always be placed closer to the source than a high quality microphone
without causing aliasing, provided that the rest of the acquisition system is equal in
both cases. Future research will include error estimation of under-sampling in PNAH,
which will be derived both analytically and practically.
ACKNOWLEDGEMENT
This research is supported by the Dutch Technology Foundation (STW). Furthermore,
I would like to thank Philips Applied Technologies Eindhoven where part of this
research has been conducted.
REFERENCES
[1] R. Scholte, N.B. Roozen, “High resolution near-field acoustic holography”, 11th Intern.
congress on sound and vibration, St. Petersburg, Russia (2004) [2] H. Nyquist, “Certain Topics in Telegraph Transmission Theory”, AIEE Trans., 47, 617-
644 (1928)
[3] E.G. Williams, Fourier Acoustics. Sound Radiation and Nearfield Acoustical Holography (Academic Press, 1999)
[4] R. Scholte, N.B. Roozen, “Improved data representation in NAH applications by means
of zero-padding”, 10th Intern. congress on sound and vibration, Stockholm, Sweden
(2003)
[5] J.D. Maynard, E.G. Williams, Y. Lee, “Nearfield acoustic holography: I. Theory of
generalised holography and the development of NAH”, J.Acoust. Soc. Am., 78 (4)
(1985)
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