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

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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

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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

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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

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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).

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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

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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)