TURBULENCE SIGNAL DETECTION USING SPIRAL MICROPHONE ARRAY. YiNeng Wang

Transcription

1 TURBULENCE SIGNAL DETECTION USING SPIRAL MICROPHONE ARRA ineng Wang National Taiwan Universit of Science and Technolog Department of Electronic Engineering No.43, Sec. 4, Keelung Rd., Da an Dist., Taipei Cit 06, Taiwan ABSTRACT The purpose of this project is to verif that the turbulence signal can be efficientl measured using the Fourier transform of sampled sound pressure based on the spiral microphone arra. Depending on the characteristics of microphones, microphone arra must be well configured and the frequencies of the source are restricted. Since the number of microphones is restricted, the resulting resolution of Fourier spectrum will be poor without epanding the number of sampling ponits. In this stud, we utilize the Whittaker-Shannon Sampling theorem [] to predict the transformation result. According to this theorem, the spectrum of sampled sound pressure can be obtained b erecting the spectrum of the original sound pressure. We have simulated different transmitted waves with constant or random-distributed intensities over the wavefront and their spectrums all look like turbulence. Inde Terms Microphone arra, Spiral configuration, Whittaker-Shannon sampling theorem. Fig.. Geometr configuration of the microphone arra. angular lattices to approimate the samples of the function g(, ) defined as:. INTRODUCTION Because a pressure microphone ehibits some directionalit along its main ais at short wavelengths caused principall b diffraction effects [2], we carefull arrange each microphone such that the are oriented toward the same direction, as illustrated in Fig.. Our microphones are fairl small so that the will have minimal effect on the sound field the are sampling. As microphones operating at higher frequencies, there are bound to be certain aberrations in directional response as the dimensions of the microphone case become a significant portion of the sound wavelength. Moreover, such high frequencies are inaudible to the average person [3]. In order for the application of audio recordings, we emplo signals with relativel low frequencies acoustic sound, as our sound sources. 2. WHITTAKER-SHANNON SAMPLING THEOREM To lead to the simplest wa to predict what will arise after appling Fourier transform to the sampled signals, we use rectthis draft was supported b Interactive Multimedia Laborator, NTUST. gs (, ) = comb( ) comb( )g(, ) () The sampled function gs (, ), thus consists of an arra of δ functions, spaced at intervals of width in the direction and width in the direction, as illustrated in Fig. 2. The area under each δ function is proportional to the value of the function g(, ) at that particular point in the rectangular sampling lattice. As implied b the convolution theorem, the spectrum Gs (f, f ) of g(, ), can be found b convolving the transform of comb( ) comb( ) with the transform of g(, ), or: Gs (f, f ) = F {comb( ) comb( )} G(f, f ) (2) where indicates that a two-dimensional convolution is to be performed. B using Table, we have: F {comb( ) comb( )} = comb(f ) comb( f ) (3)

2 Table. Transform pairs for some functions separable in rectangular coordinates. Fig. 2. The sampled function. F unction T ransf orm ep[ π(a b 2 2 )] ep[ π( f () 2 a 2 + f () 2 b 2 )] rect(a)rect(b) sinc(f /a) sinc(f /b) (a) (b) sinc2 (f /a) sinc 2 (f /b) δ(a, b) ep[jπ(a + b)] sgn(a)sgn(b) δ(f a/2, f b/2) ab jπf jπf comb(a)comb(b) comb(f /a)comb(f /b) ep[jπ(a b 2 2 )] ep[ (a + b )] j ep[ jπ(a + b)] 2 2 +(2πf /a) 2 +(2πf /b) 2 whereas from the propert of Dirac delta function: comb(f ) comb( f ) = δ(f n, f m ) ) (4) It follows that: n= m= n= m= G s (f, f ) = G(f n, f m ) (5) Fig. 3. Spectrum of the original function. Fig. 4. Spectrum of the sampled data (onl one period is shown for this infinitel periodic function). Evidentl the spectrum of g s (, ), can be simpl found b erecting the spectrum of g(, ) about each point (n/, m/ ) in the f -f plane as shown in Fig. 3. Since the function g(, ) is assumed to be bandlimited, its spectrum G(f, f ) is nonzero over onl a finite region R of the frequenc space. As implied b Eq. 5, the region over which the spectrum of the sampled function is nonzero can be found b constructing the region R about each point (n/, m/ ) in the frequenc plane. Notice that if and are sufficientl small, i.e., the samples are sufficientl close to each other, then the separations / and / of the various spectral islands will be great enough to assure that the adjacent regions do not overlap. However, in our case, as shown in Fig. 4, there are still some overlaps as a handful of microphones are used compared to the resolution we need. 3. SPATIAL FILTERING The recover of the original spectrum G(f, f ) from G s (f, f ) can be accomplished eactl b passing the sampled function g(, ) through a linear invariant filter that transmits the term

3 (n = 0, m = 0) of Eq. 5 without distortion, whereas perfectl ecluding all other terms. 3.. Introduction The concert of linear filtering is also based on the utilization of Fourier transform for signal procession in the frequenc domain. However, here we are interested in filtering operations performed directl on the piels of an image. The linear operations consist of multipling each piel in the neighborhood b a corresponding weight and summing up the products to obtain the response at each point (, ). If the neighborhood is of size m n, mn coefficients are required. The coefficients are arranged as a matri, called a filter, mask, filter mask, kernel, template, or window. There are two similar operations: correlation and convolution. In both operations, a mask w and an image f are required. The mask w can be considered as impulse response in performing the convolution if this is a LTI sstem. The onl difference between correlation and convolution is that mask is rotated b 80 in performing the convolution. The rotation can accomplished b. build-in matlab function: rot90(w, 2), 2. the following codes [row, col ] = size(w); num = row col ; w r = zeros(row, col ); w r ( : end) = w(end : : ); The second method is just reversing the order of reading an arra. It is ver fortunate that this rotation operation can be done through the use of imfilter function b specifing the filtering mode with the conv parameter Detail operations Suppose both mask w and image f are one-dimensional arras with m and n elements, respectivel. We assume tpicall that m and n are nonnegative odd integers because the have a unique center. To perform the correlation of the two arras, we shift w in order for its last element to conincide with the origin of f (for one-dimensional case, this is just the first element of f and for two-dimensional case, it is the upper leftmost point). Notice that there are points between the two arras that do not overlap. The most common wa to solve this problem is to pad f with as man 0s as are necessar to assure that there will alwas be corresponding points for the full etension of w past f. The padded arra f is then denoted as f p and f p has m + 2(n ) = (m + n 2) + n elements since onl the last element (or the first) element of w overlaps with f for the first (or the last) operation. The response of the filter at that point is the sum of products (SOP) of the filter coefficients and the corresponding neighborhood piels in the area etended b the filter mask. Ever time the SOP operation is done, w is shifted one location to the net element and repeat the process. Proceeded in this manner, w (6) Fig. 5. Illustration of one-dimensional correlation using full option. Fig. 6. Illustration of one-dimensional correlation using same option. moves completel past f p and the process is finished. The set of values is the correlation of w with f p (denoted b w f p, not b f p w and will be eplained below). Since w has n elements, there are n products for each colored block and there are (m + n 2) + n blocks due to the fact that f p has m + 2(n ) elements as shown above. Obviousl, the resulted output g has (m+n 2) 0+ = m+n elements and is epressed mathematicall as n g[t + ] = w[τ]f p [t + τ], for t = 0 (m + n 2) τ= (7) where the inde of output arra g is denoted as t+ and starts from to m + n due to the fact that MATLAB does not allow negative indices. Compared with we know that g(t) = w f p = w(τ)f(t τ)dτ (8) imfilter(f, w, full ) (9) is an correlation operation of w on padded arra f, f p, and therefore the operation performed b imfilter is w f p instead of f p w. From the discussion so far, we know that performing correlation with a rotated filter is the same as performing convolution with the original filter. If the filter is smmetric about its center, then both options ield the same result. Notice that, had we moved w such that the starting point was with the center of the mask aligned with the origin of f. The last computation was with the center element of the mask aligned with the last element of f. Then, the padded arra f p would have m + 2((n + )/2 ) = m + n elements and the resulted output g would have m elements (the same size as f) and would be epressed as n g[t + ] = w[τ]f p [t + τ], for t = 0 (m ) τ= (0) where the inde of output arra g is denoted as t+ and starts from to m.

4 Table 2. Options for function imfilter Options Filtering Mode corr conv Boundar Options P replicate smmetric circular Size Options same full Description 3.3. Testing the imfilter functiom Filtering is done using correlation. This is the default. Filtering is done using convolution. The boundaries of the input image are etended padding with a value, P(written without quotes). This is the default, with value 0. The size of the image is etended b replicating the values in its outer border. The size of the image is etended b mirror-reflecting it across its border. The size of the image is etended b treating the image as one 2-D periodic function. The output is of the same size as the input. This is achieved b limiting ecursions of the center of the filter mask to points contained in the original image. This is the default. The output is of the same size as the etended(padded) image. The test of imfilter starts with the practice of EAMPLE 3.7 on page 94 [4]. First of all, lets have a look at the usage of imfilter.the IPT implements linear spatial-domain filtering using imfilter, which has the following snta: g = imf ilter(f, w, f ilteringmode, boundaroptions, sizeoptions) () After the practice, there are some important points that must be paid attention :. Although it is said that Fig.7(a) is a class double image, it is read b MATLAB and is stored as a class uint8 arra. 2. So we have to convert it into a class double image to avoid data loss caused b rounding off and filtered value outside the [0 255] range. This can be seen from Fig.7(f). a b c d e f Fig. 7. (a)original image. (b)result of using imfilter with default zero padding. (c)result with the replicate option. (d)result with the smmetric option. (e)result with the circular option. (f)result of converting the image back to class uint8 and then filtering with the replicate option. A filter of size 33 with all s was used 3. When w is an arra completel consisting of ones, the result is proportional to the arithmetic mean of the original image (240250/(3*3)= 250) in the area etended b the filter mask.as shown in the orange ring in Fig.8. In particular, if the coefficients of the mask sum to (normalization is done with w divided b 96=3*3), the result is the arithmetic mean of the original image in the area etended b the filter mask Boundar Problem Blurred results are produced on the boundaries between black (0s, zero padding) and light (s) regions. It can be understand easil through the following discussion.if zero padding were used, and the outer border is light (s). The filter mask is assumed to be of size 33 with all s. As Fig.9 shows, the sum of the nine products at different centers is different. The resulted image is darker (value is smaller) near the top (above boundar) 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 the number of microphones, we use signal processing technique to make the spectrum look more like a spiral. For eample, epanding the sampled points b zero padding and then performing Fourier transform (the sampled acoustic pressure is viewed as the intensit of an image). The reason for the two

5 Fig. 8. Variables a, b, c, d, e, and f are corresponding to figure 3. (a), (b), (c), (d), (e), and (f), respectivel. W is the filter mask with size 33. spectra showing different colors is that the spacing of each Dirac comb function is not uniform for spiral microphone configuration [5]; however, the deploment of the spectra of the sampled signal are alwas spiral regardless of the sound source is plane wave, spherical wave, or even random distribution [6]. The recover of the original spectrum G(f, f ) from G s (f, f ) can be accomplished eactl b passing the sampled function g(, ) through a linear invariant filter that transmits the term (n = 0, m = 0) of Eq. 5 without distortion, whereas perfectl ecluding all other terms. Therefore, our net objective in the future is to find an eact replica of the original data g(, ) with an FIR filter. Linear spatial-domain filtering presented here is basicall a convolution operation. Due to zero paddings, the boundar problem arises. To avoid this problem, we filter the original image with replicate option. And the problem about data loss can be solved b converting the image class to double or b normalization of the filter mask. 5. REFERENCES [] Joseph W. Goodman, Introduction to Fourier Optics, McGraw-Hill, New ork, [2] Hoffman, M.W., Microphone arra calibration for robust adaptive processing, IEEE ASSP Workshop on Applications of Signal Processing to Audio and Acoustics, pp. -4, Oct 995. [3] John Eargle, The Microphone Book: From Mono to Stereo to Surround - A Guide to Microphone Design and Application, Focal-Press, Oford, [4] Rafael C.Gonzalez, Richard E. Wooods, Steven L. Eddins Digital Image Processing Using MATLAB, Pearson Education, USA, pp , [5] Tamai,., Kagami, S., Amemia,., Sasaki,., Mizoguchi, H., and Takano, T., Circular microphone arra for robot s audition, Proc. of the IEEE Sensors, vol. 2, pp , Oct [6] Del Galdo, G., Thiergart, O., Weller, T., and Habets, E.A.P., Generating virtual microphone signals using geometrical information gathered b distributed arras, Joint Workshop on Hands-free Speech Communication and Microphone Arras (HSCMA), pp , Ma June 20. Fig. 9. Eample of boundar problem.