Investigation of passive acoustic anemometer by use of wind noise correlation

Transcription

1 Investigation of passive acoustic anemometer by use of wind noise correlation Tetsuro MURAKAMI 1, Yoshinori YOKO, Yuya SATO 3, Hirofumi NAKAJIMA 4, Kazuhiro NAKADAI Kogakuin University, Japan 5 Honda Research Institute Japan, Japan ABSTRACT This paper examines feasibility of an acoustic anemometer by evaluating estimates of wind directions and wind velocities towards microphone array. These estimates are calculated based on a number of delay times among pair-microphones. To obtain the delay times, a low-frequency correlation of coherence function between wind noises for pair-microphones is practically utilized. This paper firstly discusses about the validity of the estimation method for wind direction and wind velocity by means of simulated wind noises to obtain some high-precision results. Secondly, the estimation accuracies of the wind directions and the wind velocities are evaluated for wind noises recorded in a practical environment. The winds are generated by a tower fan and recorded by three microphones that are closely located at each vertex of a triangle. The results for the recorded wind noises show that estimation errors are heavily dependent on the recording direction in contrast to the results for the simulated wind noises. Since these fluctuations of the errors are commonly observed in several recorded wind noises, a correction for unknown estimates is approximately calculated by employing some known estimates and their errors. As a result, the average relative error of the wind velocity was 5.4 % and the average absolute error of the wind direction was 6.64 deg. This correction method indicated the efficacy of estimating the wind directions and the wind velocities. Keywords: Anemometer, Wind noise, Correlation function I-INCE Classification of Subjects Number(s):.3 1. INTRODUCTION Wind direction (WD) and wind velocity (WV) used for the weather forecast or the damage prediction due to a typhoon are measured in a few ways, for example, a cup anemometer or a vane anemometer using the dynamics of wind itself, a hot-wire anemometer using the cooling effect of air passing on a wire, a sonic anemometer using delay time of ultrasonic sound waves between pairs of transducers (1). In the field of acoustic signal processing, there are studies about the estimation of WD and WV using wind noise. The wind noises are observed when wind blows to a diaphragm of microphones. In the estimation of WV by means of a single microphone, there are some previous studies that spectrum distribution of wind noises dependent on WV () or higher order correlations of wind noises (3) are utilized. In addition, there are studies that WV including WD is estimated using multi microphones (4,5). An anemometer based on wind noises is not active anemometer such as sonic anemometer and does not move dynamically such as cup anemometer, therefore wind noise has an advantage of estimation WV and WD. However, there are no discussions in the previous studies about any implementation of anemometer for practical use. 1 t.murakami@cc.kogakuin.ac.jp j1317@ns.kogakuin.ac.jp 3 j11346@ns.kogakuin.ac.jp 4 nakajima@cc.kogakuin.ac.jp 5 nakadai@jp.honda-ri.com 67

2 Figure 1 The relationship between wind and a pair-microphone.. PURPOSE AND APPROACH This paper aims at constructing a system of a passive acoustic anemometer by means of multi microphones. It has been found a low-frequency correlation between wind noises for closely allocated pairmicrophones, and a peak of the correlation function corresponds to not the speed of sound but the wind speed(6). Therefore, the multi microphones are closely allocated each other, and WD and WV are estimated from the delay time between microphone outputs. Section 3 explains an estimation method for WD and WV, and then Section 4 tries to test the efficacy of the estimation method by simulated wind noises. In Section 5, WD and WV are estimated based on wind noises recorded by a microphone array system. In Section 6, the estimation error occurred in Section 5 is reduced by a correction using the relationship between estimates and references. 3. ESTIMATION OF WD, WV First, the delay times, which are the differences of arrival time between microphone pairs, are estimated by the correlation of recorded wind noises. Next, a wind vector composed of WD as the direction and WV as the norm is calculated using the estimated delay times. 3.1 Estimation of wind vector Figure 1 shows the relationship between a column wind vector v and the delay time at the microphones M 1 and M. Defining the location of M 1 as a reference position, the location of M is expressed by a column vector, x 1. Here, the wind around the micprophone is expressed as the column wind vector v, which is -dimentional vector and lies in the same plane as x 1. The relationship between the delay time t and the position vector x 1 is given as t = x 1 v cos θ 1 = x 1 v = x T v v 1 v, (1) where θ 1 is the angle between x 1 and v and v is L norm of v. In Equation (1), even if x 1 and t are known, -dimentional v is non-unique. Therefore, the wind vector is estimated using three microphones M 1, M and M 3. Generalizing x 1 to x ij, assume that x ij is a position column vector indicating the position of M j referred to the position of M 1 and t ij is the delay time corresponding to x ij. Given the delay times vector t =(t 1 t 3 t 13 ) T and the microphones position matrix X =(x 1 x 3 x 13 ) T, Equation (1) is extended as t = X v v. () Now presuming that X is known, it can be considered that the wind vector ˆv is estimated using the delay time ˆt, which is also estimated from the wind noises. Using each error of the estimated delay times e = ˆt t, the relationship among the wind vector, the delay time and the microphone-array in Equation () is modeled as ˆt = X ˆv + e. (3) ˆv 68

3 Estimating the wind vector ˆv using the criterion of the least mean-square error, the following equation is obtained as ˆv ˆv = X+ˆt, (4) where X + is the pseudo inverse matrix of X. Taking square of L norm for both sides of the Equation (4), the equation is 1 ˆv = X+ˆt. (5) Substitute Equation (5) in Equation (4), the estimated wind vector is obtained as, The estimated wind direction ˆθ is calculated as ˆv = X+ˆt. (6) X +ˆt ˆθ =tan 1 ˆv, (7) where tan 1 is the fourth-quadrant inverse tangent function, and the range of the WD is 18 ˆθ < Estimation of delay times This section explains the estimation method of the delay times for wind noises which are utilized for the estimation of the wind vector. Wind noise recorded by the microphone pair has a correlation in coherence function (6). The delay time in sample ˆτ is calculated based on a peak of the cross-correlation function for a pair of wind noises. Defined that n is the index of discrete time, the recorded wind noises m i (n) and m j (n) are obtained from the microphones M i and M j respectively. After m i (n) and m j (n) are divided into K frames with frame length L and shift length S, each of k-th frame is expressed as m i (n, k) and m j (n, k) respectively. The estimated delay time ˆτ ij (k) in sample is expressed as ˆτ ij (k) = arg max γ(n, k), (8) n where the absolute value of the delay time τ ij (k) is assumed to be smaller than L/, and the range of L/ < ˆτ ij (k) <L/ regarding m i (n, k),m j (n, k) as a periodic signal. Note that the cross-correlation function in this paper is calculated using fast Fourier transform F and its inverse transform F 1 as, γ(n, k) =F 1 {F{m i (n, k)} F{m j (n, k)}}, (9) where {} denotes the complex conjugate of {}. The estimated delay time ˆt ij is obtained using the average of the estimate delay in sample ˆτ ij = ˆτ ij (k) as, where f s is the sampling frequency of the recorded wind noises. ˆt ij = ˆτ ij f s, (1) 4. VALIDATION OF THE ESTIMATION METHOD BY COMPUTER SIMULATION To validate the estimation method for wind vector, wind noises recorded by the microphone array are generated by a computer simulation and the wind vectors are estimated based on the wind noise signals. 69

4 Figure Allocation of the microphone array Table 1 Estimation error for WD (deg) and WV (m/s) in the simulation WD WV Maximum of error Minimum of error Average of error Standard deviation of error Wind noises generated by a computer simulation First the microphone position matrix X is prepared by allocating microphones on microphone array, then WD and WV composing the wind vector are defined as a reference. Next, the delay time vector t is given by substituting X and v into Equation (). A wind noise recorded by the method shown in section 5 is allocated in channel 1 as a reference. The signal in channel i is the signal shifting channel 1 by the delay time in sample τ 1i corresponding to t 1i. If zero intervals in either channel are occurred due to the shifting process, the intervals of all channels are eliminated. In the simulation, 3 microphones M 1, M and M 3 in the microphone array are allocated at each vertex of a 4mm equilateral triangle shown in Figure. The direction from the center of the triangle to the microphone M 1 is defined as deg. for WD. The angle increases in a counterclockwise direction, and if the angle reaches up to 18 deg., then it becomes -18 deg. For this simulation, 1 wind vectors were generated, whose WDs were from -18 deg. to 177 deg. having an interval of 3 deg. and WVs were 3.5 m/s equally. In the case of calculating the delay time, the frame length was L = 15 = 3768 samples, the shift length was S = 11 = 48 samples and the FFT length was equal to the frame length. 4. Result of the simulation Table 1 shows the estimation error of WD and WV in the simulation. Each error in the table is shown in the absolute value. As the average error of WD was 1.5 deg. and the average relative error of WV was.6%, the estimation method was successful to indicate the efficacy in principle. Their errors are probably due to rounding from delay time t 13 to delay sample τ 1 when channels are shifting. 5. EVALUATION OF THE ESTIMATION ACCURACY IN ACTUAL ENVIRONMENT This section explains evaluation of the estimated wind vector. Wind noises are recorded by the actual microphone array. 5.1 Recording Method 3 microphones in the microphone array had the same allocation depicted in Figure 3. The microphones were omnidirectional AV-LEADER s TCM37 and the wind screen was disused in order to record the wind noises. As a wind generator, YUASA s tower fan YT-776NR was used, and the distance to microphones was adjusted so that recorded WVs were from 1. m/s to 3.5 m/s at an interval of.5 m/s, in total 6 steps of WVs. To measure a reference for WV, the averaged WV for 4 seconds was obtained by CELESTRON s Wind Guide 7

5 Figure 3 Microphone array Figure 4 Anemometer for measuring a reference of WV Figure 5 A photo of recording environment Figure 6 Angles according to the frame length L, l, the shift length S, s. 381 shown in Figure 4. WD was controlled using the microphone array placed on GSI Creos s rotating table DS1 which rotates.3 deg/s in clockwise direction. For each recording of WV, the wind noises were recorded for 1 rotation (36 deg.). Apart from the simulation, since WD changes continuously, the wind vector was estimated frame by frame basis with the frame length l and the shift length s. To obtain a reference for WD, a proportional relationship between the rotation time and the rotation angle was used. 5. Recording Environment Figure 5 shows the recording environment for wind noises. A program called pa-wavplay of MATLAB (3bit) was employed for the multi-channel recording. The output signals from the microphone array were amplified by AUDIO TECHNICA S AT-MA and saved on the computer via Roland s audio devices OCTA- CAPTURE. The sampling frequency for the recording was 44.1 khz. 5.3 Frame processing of recorded wind noises To analyze the recorded wind noises, dual frame processing was performed. Figure 6 shows the frame processing with the angles according to the frame length and the shift length. Note that the angles are depicted about 1 times larger than the actual size for clarity. Using previous frame processing for wind vector, the continuous angles in each frame were regarded as stationary signal. Processing for the wind vector divides signals into frames whose length l = 17 = 1317 samples and shift length s = 14 = samples using Hanning window. Next frame processing for delay time has a role of averaging the continuous change for the delay time in each wind noise divided by previous frame processing. In frame processing for delay time, the frame length is L = 15 = 3768 samples and the shift length is S = 11 = 48 samples, and Hanning window is used. 5.4 Results of the estimation and discussion This subsection shows the results of the estimation and discusses about the results. 71

6 Estimated direction (deg) Estimated velocity (m/s) Estimated value True value True direction (deg) Estimted value True value True direction (deg) 6 3 (m/s) Locus of true vectors Estimated vectors -6-1 velocity Figure 7 Estimated WD (upper figure) and WV (lower figure). Solid line and chained line indicate the estimated values and the reference values respectively. Figure 8 The overview of the wind vectors in each recorded frame. Each wind vector indicates the arrival direction, and its magnitude expresses WV. A chained circle is a locus of the reference for wind vectors Effects on the estimated wind vectors by the recorded WD Figure 7 shows the estimated WV and WD which varies dependent on the recorded WD. Figure 8 shows the overview of the estimated wind vectors in each recorded frame. From these figures, it is found that the errors of the estimated WD and WV depended on the actual WD. The estimated WD and WV have a rotational symmetry at every 1 deg.: WDs have an oscillation about deg. for every 1 deg. and WVs have 3 intervals of being convex upward in every 1 deg. This symmetry seems to be caused by the triangular allocation of microphones on the microphone array. Especially, at WDs about deg., 1 deg., -1 deg., where the WD was from the front of the microphones, the slope observed in the estimated WD was the largest and the estimated WV was very close to the reference at the center of the intervals being convex upward observed in the estimated WV. On the other hand, at WDs about 6 deg., -6 deg. and -18 deg., where the WD was from between the microphones, the slope for the oscillation observed in the estimated WD was the smallest and the error of the estimated WV became larger. It is considered that the change of the estimated wind vectors in the recorded WDs was due to turbulence caused by the wind blowing to the side surface of microphones and the wind reduced WV around microphones. Figure 9 illustrates variation of WV in cases of that the recorded WDs were, 1 or -1 deg., 6, -6 or -18 deg. and others. In case of, 1 or -1 deg. (Figure 9a), as the wind blowing to the front of microphone flows along with rim of microphone, the turbulence is not occured. In case of 6, -6, or -18 deg. (Figure 9b), as the wind blowing from between microphones hits the microphone array, the turbulence is occurred. As the turbulence interfered the wind over the microphone array and reduced its WV, and also the longer delay times of the wind noises were estimated, slower WVs were obtained. In other cases, such that the allocation of the microphone array is asymmetry for the wind blowing to the microphone array (Figure 9c), a part of the wind over the microphone is strongly affected by the turbulence but some parts of the wind are unaffected. Consequently, since WVs were different each other between microphones and the delay times were larger than it should be, the false estimation was occurred such that WD changed. The cause of this false estimation could be based on the understanding such as WV is invariant around the microphone array. 7

7 (a) deg, 1 deg -1 deg (b) 6 deg, -6 deg -18 deg (c) Other cases Figure 9 The variation of WV by the recorded WD. The white arrows illustrate winds over the microphone array, the black arrows express winds at the same height as microphones. 18 Estimated direction (deg) m/s 3.5m/s True direction (deg) Estimated velocity (m/s) Average of maximum Average of all frames Average of minimum True velocity (m/s) Figure 1 The relationship between estimated WD and recorded WD. Light and dark solid lines express the estimates of recorded WVs, 1. and 3.5 m/s respectively. Figure 11 The relationship between the estimated WV and the recorded WV 5.4. Effects on the estimated wind vectors by the recorded WV Figure 1 shows the relationship between the estimated WD and the reference. The fluctuation of the estimated WVs for both 1. and 3.5 m/s were equally affected by the recorded WD, and the other recorded WVs between 1.5 and 3. m/s were also affected. On the other hand, the variance of the error for each WD was larger and the estimation accuracy was lower as WV was slower. Figure 11 shows the relationship between the estimated WV and the reference for the wind noises having WVs from 1. to 3.5 m/s. The estimated WVs for each wind noise vary widely depending on the recorded WDs as well as the estimated WDs. 3 Polygonal lines composed of the average estimated WV over all frames, the mean value of 3 maximum values for each interval of every 1 deg., and the mean value of 3 minimum values for each interval of every 1 deg., were drawn. From the figure, the mean value of 3 maximum values was larger than the reference by.1-.3 m/s in case of that the recorded WV was faster than. m/s. Therefore, the tendency that the estimated WV gets closer to the reference at deg., 1 deg. and -1 deg. can be observed at the different recorded WV. However, in case of that the recorded WV was slower than. m/s, the mean value of 3 maximum values tended to be 1 m/s larger than the reference. In addition, the slope of each line is smaller than that of the recorded WV was faster than. m/s. As shown in the above results of the estimated WD and WV, the estimated wind vectors is more accurate as the recorded WV is faster. Since the frequency bands that the wind noise is observed increases as WV is faster (), the information to calculate the correlation also increases. This leads better estimation accuracy of 73

8 the delay time. 6. CORRECTION OF ESTIMATE WIND VECTOR There was a fluctuation of the error in the estimated wind vector according to the recorded WD. This error of the estimated wind vector is corrected based on the fluctuation. 6.1 Correction method In section 5, the wind vector was estimated by the microphone array. At the same time, the reference of the wind vector was measured by the anemometer and the rotating table having an constant angular velocity. The estimated wind vector (referred to as "the estimated vector" lator on) and the reference are defined as u and u true respectively. The correction vector c(u) for u is defined as c(u) =u true u. (11) However, the reference is unavailable when the wind vector is estimated using only microphone array. Defining the estimated vector as w whose reference is unavailable, the correction vector c(w) is also unavailable. To avoid this, a dataset consisted of u and u true is prepared in advance, and then the estimated vector is corrected by calculating an approximation of the correction vector ĉ(w). The approximation of the correction vector ĉ(w) is obtained as ĉ(w) = 1 R(w) u R(w) where R(w) is defined as a set of u within a radius r from w, such as c(u), (1) R(w) ={u U; r u w }. (13) 6. Correction for the estimated vector and the evaluation New wind noises whose WV is.5 m/s are recorded as the estimated vectors w, but the reference w true is also measured in order to investigate the error after the correction. The wind noise recorded in Section 5 is utilized as the estimated vector u whose reference is available. To indicate how much the error is reduced before and after the correction, Error reduction ratio is defined as (Error reduction ratio) = 1 (Error after correction) (Error before correction). (14) Figure 1 shows the Error reduction ratio in WD and WV according to the radius r. The Error reduction ratio reached a maximum 39 % in WV at r =.18 and76%inwdatr =.13. Figure 13 shows the estimated WV and WD after the correction at r =.18. Table shows the estimation error before and after the correction. The correction was effective for WV whose error was large before the correction, and the averaged error for WV after the correction was.135 m/s. On the other hand, the correction was ineffective for WD whose error was small before the correction, and the averaged error for WD after the correction was 6.64 deg. By applying the correction to the estimated vector, the error of WD was reduced from 1.8 deg. to 6.64 deg. and the relative error of WV was reduced from 1.9 % to 5.4 % for the reference of.5 m/s. Consequently, WD and WV were well estimated towards the reference, and the efficacy of the correction was observed. 7. CONCLUSION WD and WV were estimated by means of small size microphone array and the acoustical anemometer was designed. As the fluctuation of WD and WV due to the form of microphone was observed, it was corrected to reduce the errors of the estimates. For high-precise estimation, the design of microphone array and the 74

9 Error reduction ratio Velocity Direction r Figure 1 The relationship between Error reduction ratio and radius r. Solid line and chained line express WV and WD respectively. Estimated velocity (m/s) Estimated direction (deg) 3.5 Estimated value 1.5 True value Corrected value True direction (deg) Estimted value True value Corrected value True direction (deg) Figure 13 The estimated WV (upper figure) and WD (lower figure) after correction. Solid line and chained line express the estimated values and the reference values respectively. Table Estimation error for WD (deg) and WV (m/s) before and after correction Before After WD WV WD WV Maximum of error Minimum of error Average of error Standard deviation of error improvement of the correction method should be reconsidered as future works so that the whole system could have less errors. Moreover, for the implementation of a practical anemometer, concerning the real time processing and expanding the microphone array system to 3-dimentional would be important tasks. REFERENCES 1. Ito M. The Story of the Wind I ( I). GIHODO SHUPPAN; Japan p Sugano K, Chiba N. Development of a Microphone-based Wind Velocity Sensor and Its Application to Real-time Animation of a Tree Swaying in Real World Wind. The Journal of the Society for Art and Science; Japan 7. p Fujita Y, Ota M, Takakuwa Y. A Principle for Measurement Method of Wind Speed Using Higher Order Correlation Information between Wind Noise and Wind Speed. Acoust. Soc. Jap Spring Meeting, Japan 1998(1). p Bass, Henry E. and Raspet, Richard and Messer, John O. Experimental determination of wind speed and direction using a three microphone array. J. Acoust. Soc. Am., 97; p Godin, Oleg A. and Irisov, Vladimir G. and Charnotskii, Mikhail I. Passive acoustic measurements of wind velocity and sound speed in air. J. Acoust. Soc. Am., 135; 14. p

10 6. Sakata N, Murakami T, Nakajima H, Nakadai. K. Wind-induced noise reduction by linear beamforming using a -channel microphone. The 8th Workshop on Circuits and Systems; 3-4 August 15; Himeji, Japan 15. p