TURBULENCE SIGNAL DETECTION USING SPIRAL MICROPHONE ARRAY

YiNeng Wang

National Taiwan University of Science and TechnologyDepartment of Electronic Engineering

No.43, Sec. 4, Keelung Rd., Da’an Dist., Taipei City 106, Taiwan

ABSTRACTThe purpose of this project is to verify that the turbulence sig-nal can be efficiently measured using the Fourier transformof sampled sound pressure based on the spiral microphonearray. Depending on the characteristics of microphones, mi-crophone array must be well configured and the frequenciesof the source are restricted. Since the number of microphonesis restricted, the resulting resolution of Fourier spectrum willbe poor without expanding the number of sampling ponits. Inthis study, we utilize the Whittaker-Shannon Sampling theo-rem [1] to predict the transformation result. According to thistheorem, the spectrum of sampled sound pressure can be ob-tained by erecting the spectrum of the original sound pressure.We have simulated different transmitted waves with constantor random-distributed intensities over the wavefront and theirspectrums all look like turbulence.

Index Terms— Microphone array, Spiral configuration,Whittaker-Shannon sampling theorem.

1. INTRODUCTION

Because a pressure microphone exhibits some directionalityalong its main axis at short wavelengths caused principallyby diffraction effects [2], we carefully arrange each micro-phone such that they are oriented toward the same direction,as illustrated in Fig. 1. Our microphones are fairly small sothat they will have minimal effect on the sound field they aresampling. As microphones operating at higher frequencies,there are bound to be certain aberrations in directional re-sponse as the dimensions of the microphone case become asignificant portion of the sound wavelength. Moreover, suchhigh frequencies are inaudible to the average person [3]. Inorder for the application of audio recordings, we employ sig-nals with relatively low frequencies—acoustic sound, as oursound sources.

2. WHITTAKER-SHANNON SAMPLING THEOREM

To lead to the simplest way to predict what will arise after ap-plying Fourier transform to the sampled signals, we use rect-

This draft was supported by Interactive Multimedia Laboratory, NTUST.

Fig. 1. Geometry configuration of the microphone array.

angular lattices to approximate the samples of the functiong(x, y) defined as:

gs(x, y) = comb(x

X) comb(

y

Y)g(x, y) (1)

The sampled function gs(x, y), thus consists of an array ofδ functions, spaced at intervals of width X in the x direc-tion and width Y in the y direction, as illustrated in Fig. 2.The area under each δ function is proportional to the value ofthe function g(x, y) at that particular point in the rectangularsampling lattice. As implied by the convolution theorem, thespectrum Gs(fx, fy) of g(x, y), can be found by convolvingthe transform of comb( x

X ) comb( yY ) with the transform of

g(x, y), or:

Gs(fx, fy) = F{comb(x

X) comb(

y

Y)} ⊗G(fx, fy) (2)

where ⊗ indicates that a two-dimensional convolution is tobe performed. By using Table 1, we have:

F{comb(x

X) comb(

y

Y)} =

XY comb(Xfx) comb(Y fy)

(3)

Fig. 2. The sampled function.

Fig. 3. Spectrum of the original function.

Fig. 4. Spectrum of the sampled data (only one period isshown for this infinitely periodic function).

Table 1. Transform pairs for some functions separable in rect-angular coordinates.

Function Transform

exp[−π(a2x2 + b2y2)] 1|ab| exp[−π(

f(x)2

a2 +f(y)

2

b2 )]

rect(ax)rect(by) 1|ab| sinc(fx/a) sinc(fy/b)

∧(ax) ∧ (by) 1|ab| sinc

2(fx/a) sinc2(fy/b)

δ(ax, by) 1|ab|

exp[jπ(ax+ by)] δ(fx − a/2, fy − b/2)

sgn(ax)sgn(by) ab|ab|

1jπfx

1jπfy

comb(ax)comb(by) 1|ab| comb(fx/a)comb(fy/b)

exp[jπ(a2x2 + b2y2)] j|ab| exp[−jπ(ax+ by)]

exp[−(a|x|+ b|y|)] 1|ab|

21+(2πfx/a)2

21+(2πfy/b)2

whereas from the property of Dirac delta function:

XY comb(Xfx) comb(Y fy) =∞∑

n=−∞

∞∑m=−∞

δ(fx − nX , fy − m

Y ) )(4)

It follows that:

Gs(fx, fy) =∞∑

n=−∞

∞∑m=−∞

G(fx − nX , fy − m

Y )(5)

Evidently the spectrum of gs(x, y), can be simply found byerecting the spectrum of g(x, y) about each point (n/X,m/Y )in the fx-fy plane as shown in Fig. 3. Since the functiong(x, y) is assumed to be bandlimited, its spectrum G(fx, fy)is nonzero over only a finite region R of the frequency space.As implied by Eq. 5, the region over which the spectrum ofthe sampled function is nonzero can be found by constructingthe region R about each point (n/X,m/Y ) in the frequencyplane. Notice that if X and Y are sufficiently small, i.e.,the samples are sufficiently close to each other, then the sep-arations 1/X and 1/Y of the various spectral islands willbe great enough to assure that the adjacent regions do notoverlap. However, in our case, as shown in Fig. 4, thereare still some overlaps as a handful of microphones are usedcompared to the resolution we need.

3. SPATIAL FILTERING

The recovery of the original spectrum G(fx, fy) from Gs(fx, fy)can be accomplished exactly by passing the sampled functiong(x, y) through a linear invariant filter that transmits the term

(n = 0,m = 0) of Eq. 5 without distortion, whereas perfectlyexcluding all other terms.

3.1. Introduction

The concert of linear filtering is also based on the utilizationof Fourier transform for signal procession in the frequencydomain. However, here we are interested in filtering opera-tions performed directly on the pixels of an image. The linearoperations consist of multiplying each pixel in the neighbor-hood by a corresponding weight and summing up the prod-ucts to obtain the response at each point (x, y). If the neigh-borhood is of size m × n, mn coefficients are required. Thecoefficients are arranged as a matrix, called a filter, mask, fil-ter mask, kernel, template, or window. There are two similaroperations: correlation and convolution. In both operations, amask w and an image f are required. The mask w can be con-sidered as impulse response in performing the convolution ifthis is a LTI system. The only difference between correlationand convolution is that mask is rotated by 180 in performingthe convolution. The rotation can accomplished by 1. build-inmatlab function: rot90(w, 2), 2. the following codes

[rowx, coly] = size(w);num = rowx ∗ coly;wr = zeros(rowx, coly);wr(1 : end) = w(end : −1 : 1);

(6)

The second method is just reversing the order of reading anarray. It is very fortunate that this rotation operation can bedone through the use of imfilter function by specifying thefiltering mode with the ’conv’ parameter.

3.2. Detail operations

Suppose both mask w and image f are one-dimensional ar-rays with m and n elements, respectively. We assume typi-cally that m and n are nonnegative odd integers because theyhave a unique center. To perform the correlation of the twoarrays, we shift w in order for its last element to conincidewith the origin of f (for one-dimensional case, this is just thefirst element of f and for two-dimensional case, it is the upperleftmost point). Notice that there are points between the twoarrays that do not overlap. The most common way to solvethis problem is to pad f with as many 0s as are necessary toassure that there will always be corresponding points for thefull extension of w past f . The padded array f is then de-noted as fp and fp has m + 2(n − 1) = (m + n − 2) + nelements since only the last element (or the first) element ofw overlaps with f for the first (or the last) operation. The re-sponse of the filter at that point is the sum of products (SOP)of the filter coefficients and the corresponding neighborhoodpixels in the area extended by the filter mask. Every time theSOP operation is done, w is shifted one location to the nextelement and repeat the process. Proceeded in this manner, w

Fig. 5. Illustration of one-dimensional correlation using ’full’option.

Fig. 6. Illustration of one-dimensional correlation using’same’ option.

moves completely past fp and the process is finished. Theset of values is the correlation of w with fp (denoted by w ∗fp, not by fp ∗ w and will be explained below). Since w hasn elements, there are n products for each colored block andthere are (m + n − 2) + n blocks due to the fact that fp hasm + 2(n − 1) elements as shown above. Obviously, the re-sulted output g has (m+n−2)−0+1 = m+n−1 elementsand is expressed mathematically as

g[t+ 1] =

n∑τ=1

w[τ ]fp[t+ τ ], for t = 0 ∼ (m+ n− 2)

(7)where the index of output array g is denoted as t+1 and startsfrom 1 to m + n − 1 due to the fact that MATLAB does notallow negative indices. Compared with

g(t) = w ∗ fp =

∞∫−∞

w(τ)f(t− τ)dτ (8)

we know that

imfilter(f, w,′ full′) (9)

is an correlation operation of w on padded array f , fp, andtherefore the operation performed by imfilter is w ∗ fp insteadof fp ∗ w.

From the discussion so far, we know that performing cor-relation with a rotated filter is the same as performing convo-lution with the original filter. If the filter is symmetric aboutits center, then both options yield the same result.

Notice that, had we moved w such that the starting pointwas with the center of the mask aligned with the origin of f .The last computation was with the center element of the maskaligned with the last element of f . Then, the padded array fpwould have m+2((n+1)/2−1) = m+n−1 elements andthe resulted output g would have m elements (the same sizeas f) and would be expressed as

g[t+ 1] =n∑

τ=1

w[τ ]fp[t+ τ ], for t = 0 ∼ (m− 1)

(10)where the index of output array g is denoted as t+1 and startsfrom 1 to m.

Table 2. Options for function imfilter

Options Description

Filtering Mode

’corr’ Filtering is done using correlation.This is the default.

’conv’ Filtering is done using convolution.

Boundary Options

’P’ The boundaries of the input imageare extended padding with a value,P(written without quotes). This isthe default, with value 0.

’replicate’ The size of the image is extendedby replicating the values in its outerborder.

’symmetric’ The size of the image is extendedby mirror-reflecting it across itsborder.

’circular’ The size of the image is extendedby treating the image as one 2-D pe-riodic function.

Size Options

’same’ The output is of the same size as theinput. This is achieved by limitingexcursions of the center of the fil-ter mask to points contained in theoriginal image. This is the default.

’full’ The output is of the same size as theextended(padded) image.

3.3. Testing the ’imfilter’ functiom

The test of imfilter starts with the practice of EXAMPLE 3.7on page 94 [4]. First of all, lets have a look at the usageof imfilter.The IPT implements linear spatial-domain filteringusing imfilter, which has the following syntax:

g = imfilter(f, w, filteringmode, boundaryoptions, sizeoptions)

(11)After the practice, there are some important points that must

be paid attention :1. Although it is said that Fig.7(a) is a class double image,

it is read by MATLAB and is stored as a class uint8 array.2. So we have to convert it into a class double image

to avoid data loss caused by rounding off and filtered valueoutside the [0 255] range. This can be seen from Fig.7(f).

a b cd e f

Fig. 7. (a)Original image. (b)Result of using imfilter withdefault zero padding. (c)Result with the replicate option.(d)Result with the symmetric option. (e)Result with the cir-cular option. (f)Result of converting the image back to classuint8 and then filtering with the replicate option. A filter ofsize 3131 with all 1s was used

3. When w is an array completely consisting of ones, theresult is proportional to the arithmetic mean of the originalimage (240250/(31*31)= 250) in the area extended by the fil-ter mask.as shown in the orange ring in Fig.8. In particular, ifthe coefficients of the mask sum to 1 (normalization is donewith w divided by 961=31*31), the result is the arithmeticmean of the original image in the area extended by the filtermask.

3.4. Boundary Problem

Blurred results are produced on the boundaries between black(0s, zero padding) and light (1s) regions. It can be understandeasily through the following discussion.If zero padding wereused, and the outer border is light (1s). The filter mask isassumed to be of size 33 with all 1s. As Fig.9 shows, thesum of the nine products at different centers is different. Theresulted image is darker (value is smaller) near the top (aboveboundary) and the resulted image is brighter (value is larger).

4. CONCLUSION AND FUTURE RESEARCH

In spite of the poor resolution, due to the limitation of thenumber of microphones, we use signal processing techniqueto make the spectrum look more like a spiral. For example,expanding the sampled points by zero padding and then per-forming Fourier transform (the sampled acoustic pressure isviewed as the intensity of an image). The reason for the two

Fig. 8. Variables a, b, c, d, e, and f are corresponding to figure3.1 (a), (b), (c), (d), (e), and (f), respectively. W is the filtermask with size 3131.

Fig. 9. Example of boundary problem.

spectra showing different colors is that the spacing of eachDirac comb function is not uniform for spiral microphoneconfiguration [5]; however, the deployment of the spectra ofthe sampled signal are always spiral regardless of the soundsource is plane wave, spherical wave, or even random distri-bution [6]. The recovery of the original spectrum G(fx, fy)from Gs(fx, fy) can be accomplished exactly by passing thesampled function g(x, y) through a linear invariant filter thattransmits the term (n = 0,m = 0) of Eq. 5 without distor-tion, whereas perfectly excluding all other terms. Therefore,our next objective in the future is to find an exact replica of theoriginal data g(x, y) with an FIR filter. Linear spatial-domainfiltering presented here is basically a convolution operation.Due to zero paddings, the boundary problem arises. To avoidthis problem, we filter the original image with replicate op-tion. And the problem about data loss can be solved by con-verting the image class to double or by normalization of thefilter mask.

5. REFERENCES

[1] Joseph W. Goodman, Introduction to Fourier Optics,McGraw-Hill, New York, 2005.

[2] Hoffman, M.W., ”Microphone array calibration for robustadaptive processing,” IEEE ASSP Workshop on Applica-tions of Signal Processing to Audio and Acoustics, pp.11-14, Oct 1995.

[3] John Eargle, The Microphone Book: From Mono toStereo to Surround - A Guide to Microphone Design andApplication, Focal-Press, Oxford, 2004.

[4] Rafael C.Gonzalez, Richard E. Wooods, Steven L. Ed-dins Digital Image Processing Using MATLAB, PearsonEducation, USA, pp. 89-96, 2004.

[5] Tamai, Y., Kagami, S., Amemiya, Y., Sasaki, Y., Mi-zoguchi, H., and Takano, T., ”Circular microphone arrayfor robot’s audition,” Proc. of the IEEE Sensors, vol. 2,pp. 565-570, Oct. 2004.

[6] Del Galdo, G., Thiergart, O., Weller, T., and Habets,E.A.P., ”Generating virtual microphone signals using ge-ometrical information gathered by distributed arrays,”Joint Workshop on Hands-free Speech Communicationand Microphone Arrays (HSCMA), pp. 185-190 , May 302011-June 1 2011.