Research Article Open Access
Source Localization Technique for Multiple Acoustic Sources by Using Acoustic Beating Envelopes
Kyoung-Sik Choi*, Sung-Jae Heo, Tae-Wan Kim
Korea Aerospace Industries Ltd., Sacheon-City, Gyeongnam, 664-710, South Korea
*Corresponding author: Kyoung-Sik Choi, Korea Aerospace Industries Ltd., Sacheon-City, Gyeongnam, 664-710, South Korea,E-mail: @
Received: March 31, 2017; Accepted: April 11, 2017; Published: April 23, 2017
Citation: Kyoung-Sik Choi, Sung-Jae Heo, Tae-Wan Kim (2017) Source Localization Technique for Multiple Acoustic Sources by Using Acoustic Beating Envelopes. int J Adv Robot Automn 2(1):1-7.
Abstract
We propose here new technique to detect multiple acoustic source positions. This technique is more simple and cost saving method. We used several acoustic sensors to detect the exact positions of multiple acoustic sources simultaneously. Especially acoustic waves higher than audio frequency were used to avoid unnecessary acoustic noises. In this case, higher frequency means short wave length, which makes it difficult to extract the phase difference between two sensors when the distance of the two sensors exceed half of wave length. In this paper, we applied beating signal which is composed of two slightly different close frequencies. The wave length of the beating signal’s envelope which is longer than those of the orignals is controllable. Using the envelope of beating signal gives us many advantages. For example, we are free from unnecessary acoustic noises because of using higher frequency than audio frequency. Additionally, the wave length of beating signal is under control which means sensor position is not limited for the signal processing. We applied here a hypothetical simulation to prove it. by some final considerations.
Keywords: Source localization; Beating signal; Envelope; Phase delay; Acoustic wave; TDOA;
Introduction
Source localization techniques have been proposed in many areas for the various applications [1-3]. We proposed here more simple and cost saving method. The main key idea is using beating signal which is composed of two slightly different close frequencies. Especially the frequencies of two waves are higher than audio frequency. The reason why we use high frequency is low frequency (audio frequency) yields unnecessary acoustic noises and most of the mechanical noises are ranging under audio frequency. Mechanical noises could yield interferences. Because of these two critical issues, we determined to use high frequency. But high frequency means short wave length which makes a difficulty to extract exact phase difference when the distance of the two sensors is exceeding half of wave length. In order to solve this problem we used beating signal. Beating signal is composed of two slightly different close signals [4]. These two close waves make beating signal as a result of interference. The wave length of a beating signal’s envelope is longer than those of the originals. It means we can extract longer wave from high frequency waves without any transform. This gives us many advantages. For example, we are free from choosing frequency range. Even though when we have to deal with high frequency, we can extract longer wave from original signals without any transform. For these reasons we used beating signal to estimate multiple acoustic sources. In order to estimate source location we have to extract arrival time difference of the received sensor signals respectively. Time differences could be extracted from phase differences. In this paper, we extracted beating signal’s phase difference from original signals. We showed how to find a beating signal frequencies and phases and how to extract phase differences among beating signals. Finally showed how these phase differences are used finding source locations.
Fundamental Theories
Beating Signalm
Beating signal is composed of two slightly different close waves [4]. The wave length of a beating signal’s envelope is longer than those of the originals. This allows us to use high frequency.
When we estimate source location by using phase differences of received signals, we would confront some limitations. The wave length should be two times or longer than the distance between two sensors. For the reason that a wave length of a beating signal’s envelope is much longer than those of the originals, we could put the sensors far away each other. That means sensors’ locations are free for the wave length because we are able to control wave length of a beating signal. We can make beating signal by using frequency mixer or simply summing slightly different two frequencies. In case of frequency mixer (Figure 1), put the high frequency as ω h MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacqaHjpWDpaWaaSbaaSqaa8qacaWGObaapaqabaaaaa@392A@ and low frequency as ω l MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacqaHjpWDpaWaaSbaaSqaa8qacaWGSbaapaqabaaaaa@392E@ respectively. These two signals satisfy the conditions below. ω h ω l  (abs( ω h ω l )>20KHz)                                    (1) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacqaHjpWDpaWaaSbaaSqaa8qacaWGObaapaqabaGcpeGaeS4AI8Ja eqyYdC3damaaBaaaleaapeGaamiBaaWdaeqaaOWdbiaacckacaGGOa GaamyyaiaadkgacaWGZbWaaeWaa8aabaWdbiabeM8a39aadaWgaaWc baWdbiaadIgaa8aabeaak8qacqGHsislcqaHjpWDpaWaaSbaaSqaa8 qacaWGSbaapaqabaaak8qacaGLOaGaayzkaaGaeyOpa4JaaGOmaiaa icdacaWGlbGaamisaiaadQhacaGGPaGaaeiiaiaabccacaqGGaGaae iiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqG GaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabc cacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeii aiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGa GaaeikaiaabgdacaqGPaaaaa@6A1A@
Figure 1:Frequency mixer can get beating signal
For the conditions above, ω h MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacqaHjpWDpaWaaSbaaSqaa8qacaWGObaapaqabaaaaa@392A@ should be much higher than ω l MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacqaHjpWDpaWaaSbaaSqaa8qacaWGSbaapaqabaaaaa@392E@ Then we can make beating signal with these two frequencies by using frequency mixer. This can be expressed as below
cos( ω h t )×cos( ω l t )= 1 2 cos(( ω h + ω l )t)+ 1 2 cos(( ω h ω l )t)                 (2) MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qaciGGJbGaai4BaiaacohadaqadaWdaeaapeGaeqyYdC3damaaBaaa leaapeGaamiAaaWdaeqaaOWdbiaadshaaiaawIcacaGLPaaacqGHxd aTciGGJbGaai4BaiaacohadaqadaWdaeaapeGaeqyYdC3damaaBaaa leaapeGaamiBaaWdaeqaaOWdbiaadshaaiaawIcacaGLPaaacqGH9a qpdaWcaaWdaeaapeGaaGymaaWdaeaapeGaaGOmaaaaciGGJbGaai4B aiaacohacaGGOaGaaiikaiabeM8a39aadaWgaaWcbaWdbiaadIgaa8 aabeaak8qacqGHRaWkcqaHjpWDpaWaaSbaaSqaa8qacaWGSbaapaqa baGcpeGaaiykaiaadshacaGGPaGaey4kaSYaaSaaa8aabaWdbiaaig daa8aabaWdbiaaikdaaaGaci4yaiaac+gacaGGZbGaaiikaiaacIca cqaHjpWDpaWaaSbaaSqaa8qacaWGObaapaqabaGcpeGaeyOeI0Iaeq yYdC3damaaBaaaleaapeGaamiBaaWdaeqaaOWdbiaacMcacaWG0bGa aiykaiaacckacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabccaca qGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaa bccacaqGGaGaaGzaVlaabIcacaqGYaGaaeykaaaa@795C@
In this case, each wave length of ω h ω l MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacqaHjpWDpaWaaSbaaSqaa8qacaWGObaapaqabaGcpeGaeyOeI0Ia eqyYdC3damaaBaaaleaapeGaamiBaaWdaeqaaaaa@3D49@ and ω h + ω l MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacqaHjpWDpaWaaSbaaSqaa8qacaWGObaapaqabaGcpeGaey4kaSIa eqyYdC3damaaBaaaleaapeGaamiBaaWdaeqaaaaa@3D3E@ are slightly different. As a result, these two frequencies make beating signal.

The other way to make beating signal is using two frequencies. This is a simpler way to make beating signal. That directly summates slightly different two waves ( ω 1 ,   ω 2 ) MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaGGOaGaeqyYdC3damaaBaaaleaapeGaaGymaaWdaeqaaOWdbiaa cYcacaGGGcGaaiiOaiabeM8a39aadaWgaaWcbaWdbiaaikdaa8aabe aak8qacaGGPaaaaa@4060@ . We can assume that each wave ( ω 1 ,  ω 2 ) MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaGGOaGaeqyYdC3damaaBaaaleaapeGaaGymaaWdaeqaaOWdbiaa cYcacaGGGcGaeqyYdC3damaaBaaaleaapeGaaGOmaaWdaeqaaOGaai ykaaaa@3F2C@ is composed of two frequencies of ω h ,   ω l MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacqaHjpWDpaWaaSbaaSqaa8qacaWGObaapaqabaGcpeGaaiilaiaa cckacaGGGcGaeqyYdC3damaaBaaaleaapeGaamiBaaWdaeqaaaaa@3F54@ These two frequencies might satisfy Eq.(1) Then we can express ω 1 ,   ω 2 MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacqaHjpWDpaWaaSbaaSqaa8qacaaIXaaapaqabaGcpeGaaiilaiaa cckacaGGGcGaeqyYdC3damaaBaaaleaapeGaaGOmaaWdaeqaaaaa@3EED@ as below. ω 1 = ω h + ω l , ω 2 = ω h ω l                    (3) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacqaHjpWDpaWaaSbaaSqaa8qacaaIXaaapaqabaGcpeGaeyypa0Ja eqyYdC3damaaBaaaleaapeGaamiAaaWdaeqaaOWdbiabgUcaRiabeM 8a39aadaWgaaWcbaWdbiaadYgaa8aabeaak8qacaGGSaGaeqyYdC3d amaaBaaaleaapeGaaGOmaaWdaeqaaOWdbiabg2da9iabeM8a39aada WgaaWcbaWdbiaadIgaa8aabeaak8qacqGHsislcqaHjpWDpaWaaSba aSqaa8qacaWGSbaapaqabaGccaqGGaGaaeiiaiaabccacaqGGaGaae iiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqG GaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeikaiaabo dacaqGPaaaaa@5B6F@
Then we can get slightly different two close waves ω 1 ,   ω 2 MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacqaHjpWDpaWaaSbaaSqaa8qacaaIXaaapaqabaGcpeGaaiilaiaa cckacaGGGcGaeqyYdC3damaaBaaaleaapeGaaGOmaaWdaeqaaaaa@3EED@ . Figure 2 shows how to make beating signal.
Figure 2:One beating signal is composed of slightly different two close waves
Envelope of Beating Signal
The next step is extracting frequency and phase of a beating signal. One method of extracting the beating signal’s envelope is to use the Hilbert Transform [8,9]. The Hilbert Transform of a real-valued time domain signal x(t) MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaqG4bGaaiikaiaabshacaGGPaaaaa@3961@ is another real-valued time domain signal denoted by x ̂ (t) MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaCbiaeaaqa aaaaaaaaWdbiaadIhaaSWdaeqabaWdbiablkWaKaaakiaacIcacaWG 0bGaaiykaaaa@3A91@ , such that z( t )=x( t )+j x ̂ (t) MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWG6bWaaeWaa8aabaWdbiaadshaaiaawIcacaGLPaaacqGH9aqp caWG4bWaaeWaa8aabaWdbiaadshaaiaawIcacaGLPaaacqGHRaWkca WGQbWdamaaxacabaWdbiaadIhaaSWdaeqabaWdbiablkWaKaaakiaa cIcacaWG0bGaaiykaaaa@44C5@ . The Hilbert transform of x(t) MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaqG4bGaaiikaiaabshacaGGPaaaaa@3961@ extending over the range <t< MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacqGHsislcqaHEisPcqGH8aapcaWG0bGaeyipaWJaeyOhIukaaa@3CE4@ is a real-valued function x ̂ (t) MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaCbiaeaaqa aaaaaaaaWdbiaadIhaaSWdaeqabaWdbiablkWaKaaakiaacIcacaWG 0bGaaiykaaaa@3A91@ defined by x ̂ ( t )=H{ x( t ) }= x(τ) π(tτ) dτ                   (4) MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaCbiaeaaqa aaaaaaaaWdbiaadIhaaSWdaeqabaWdbiablkWaKaaakmaabmaapaqa a8qacaWG0baacaGLOaGaayzkaaGaeyypa0Jaamisamaacmaapaqaa8 qacaWG4bWaaeWaa8aabaWdbiaadshaaiaawIcacaGLPaaaaiaawUha caGL9baacqGH9aqpdaGfWbqabSWdaeaapeGaeyOeI0IaeyOhIukapa qaa8qacqGHEisPa0WdaeaapeGaey4kIipaaOWaaSaaa8aabaWdbiaa dIhacaGGOaGaeqiXdqNaaiykaaWdaeaapeGaeqiWdaNaaiikaiaads hacqGHsislcqaHepaDcaGGPaaaaiaadsgacqaHepaDcaqGGaGaaeii aiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGa GaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabcca caqGGaGaaeikaiaabsdacaqGPaaaaa@664D@ x ̂ (t) MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaCbiaeaaqa aaaaaaaaWdbiaadIhaaSWdaeqabaWdbiablkWaKaaakiaacIcacaWG 0bGaaiykaaaa@3A91@ is the convolution integral of x(t) MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWG4bGaaiikaiaadshacaGGPaaaaa@3965@ and 1 πt MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qadaWccaWdaeaapeGaaGymaaWdaeaapeGaeqiWdaNaamiDaaaaaaa@39D7@ , written as x ̂ ( t )=x( t )*( 1 πt )                  (5) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaCbiaeaaqa aaaaaaaaWdbiaadIhaaSWdaeqabaWdbiablkWaKaaakmaabmaapaqa a8qacaWG0baacaGLOaGaayzkaaGaeyypa0JaamiEamaabmaapaqaa8 qacaWG0baacaGLOaGaayzkaaGaaiOkaiaacIcadaWccaWdaeaapeGa aGymaaWdaeaapeGaeqiWdaNaamiDaaaacaGGPaGaaeiiaiaabccaca qGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaa bccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaae ikaiaabwdacaqGPaaaaa@52D0@ A useful way to compute the Hilbert Transform x ̂ ( t ) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaCbiaeaaqa aaaaaaaaWdbiaadIhaaSWdaeqabaWdbiablkWaKaaakmaabmaapaqa a8qacaWG0baacaGLOaGaayzkaaaaaa@3ADF@ of x( t ) MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWG4bWaaeWaa8aabaWdbiaadshaaiaawIcacaGLPaaaaaa@39B4@ is via the analytic signal z( t )=x( t )+j x ̂ (t) MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaqG6bWaaeWaa8aabaWdbiaabshaaiaawIcacaGLPaaacqGH9aqp caWG4bWaaeWaa8aabaWdbiaadshaaiaawIcacaGLPaaacqGHRaWkca qGQbWdamaaxacabaWdbiaadIhaaSWdaeqabaWdbiablkWaKaaakiaa cIcacaWG0bGaaiykaaaa@44BF@ . One can also write z( t )=A( t ) e jθ(t) MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaqG6bWaaeWaa8aabaWdbiaabshaaiaawIcacaGLPaaacqGH9aqp caWGbbWaaeWaa8aabaWdbiaadshaaiaawIcacaGLPaaacaWGLbWdam aaCaaaleqabaWdbiaadQgacqaH4oqCcaGGOaGaamiDaiaacMcaaaaa aa@444C@ ,where A(t) MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaqGbbGaaiikaiaabshacaGGPaaaaa@392A@ denotes the envelope signal of x( t ) MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWG4bWaaeWaa8aabaWdbiaadshaaiaawIcacaGLPaaaaaa@39B4@ and θ(t) MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaqG4oGaaiikaiaabshacaGGPaaaaa@39A4@ denotes the instantaneous phase signal of x( t ) MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWG4bWaaeWaa8aabaWdbiaadshaaiaawIcacaGLPaaaaaa@39B4@ . In terms of x( t ) MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWG4bWaaeWaa8aabaWdbiaadshaaiaawIcacaGLPaaaaaa@39B4@ and x ̂ (t) MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaCbiaeaaqa aaaaaaaaWdbiaadIhaaSWdaeqabaWdbiablkWaKaaakiaacIcacaWG 0bGaaiykaaaa@3A91@ , it is clear that A( t )= [ x 2 ( t )+ x ̂ 2 (t)] 1/2                    (6) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaqGbbWaaeWaa8aabaWdbiaabshaaiaawIcacaGLPaaacqGH9aqp caGGBbGaamiEa8aadaahaaWcbeqaa8qacaaIYaaaaOWaaeWaa8aaba WdbiaabshaaiaawIcacaGLPaaacqGHRaWkpaWaaCbiaeaapeGaamiE aaWcpaqabeaapeGaeSOadqcaaOWdamaaCaaaleqabaWdbiaaikdaaa GccaGGOaGaaeiDaiaacMcacaGGDbWdamaaCaaaleqabaWdbiaaigda caGGVaGaaGOmaaaak8aacaqGGaGaaeiiaiaabccacaqGGaGaaeiiai aabccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGa aeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeikaiaabAdaca qGPaaaaa@5830@ θ( t )=tan   1 [ x ̂ ( t ) x( t ) ] MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaqG4oWaaeWaa8aabaWdbiaabshaaiaawIcacaGLPaaacqGH9aqp caWG0bGaamyyaiaad6gacaGGGcWdamaaCaaaleqabaWdbiabgkHiTi aaigdaaaGcdaWadaWdaeaapeWaaSaaa8aabaWaaCbiaeaapeGaamiE aaWcpaqabeaapeGaeSOadqcaaOWaaeWaa8aabaWdbiaadshaaiaawI cacaGLPaaaa8aabaWdbiaadIhadaqadaWdaeaapeGaamiDaaGaayjk aiaawMcaaaaaaiaawUfacaGLDbaaaaa@4BB3@ So, we can obtain envelopes by using Eq.(6).
We showed here how we can get beating signal’s envelop by using Hilbert Transform. But in many cases, received signals have lots of noises and it is not always limited to one sources. If there are many sources then it is not easy to find beating signal’s envelope. So we introduced here more general method to find frequency and phase of a beating signal. Let’s put two frequencies as ω 1 (= ω h + ω l ), MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacqaHjpWDpaWaaSbaaSqaa8qacaaIXaaapaqabaGcpeGaaiikaiab g2da9iabeM8a39aadaWgaaWcbaWdbiaadIgaa8aabeaak8qacqGHRa WkcqaHjpWDpaWaaSbaaSqaa8qacaWGSbaapaqabaGcpeGaaiykaiaa cYcaaaa@4362@ ω 1 (= ω h + ω l ), MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacqaHjpWDpaWaaSbaaSqaa8qacaaIXaaapaqabaGcpeGaaiikaiab g2da9iabeM8a39aadaWgaaWcbaWdbiaadIgaa8aabeaak8qacqGHRa WkcqaHjpWDpaWaaSbaaSqaa8qacaWGSbaapaqabaGcpeGaaiykaiaa cYcaaaa@4362@ and its correspoding phases φ( ω 1 ) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaqGgpGaaiikaiabeM8a39aadaWgaaWcbaWdbiaaigdaa8aabeaa k8qacaGGPaaaaa@3BB6@ and its correspoding phases φ( ω 1 ) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaqGgpGaaiikaiabeM8a39aadaWgaaWcbaWdbiaaigdaa8aabeaa k8qacaGGPaaaaa@3BB6@ φ( ω 2 ) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaqGgpWaaeWaa8aabaWdbiabeM8a39aadaWgaaWcbaWdbiaaikda a8aabeaaaOWdbiaawIcacaGLPaaaaaa@3C06@ respectively.These two waves are satifying the conditions of Eq.(1) and Eq.(3). Then beating signal’s frequency ω beat MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacqaHjpWDpaWaaSbaaSqaa8qacaWGIbGaamyzaiaadggacaWG0baa paqabaaaaa@3BEC@ And its corresponding phase φ( ω beat ) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaqGgpGaaiikaiabeM8a39aadaWgaaWcbaWdbiaadkgacaWGLbGa amyyaiaadshaa8aabeaak8qacaGGPaaaaa@3EAB@ can be expressed as below ω beat = ω 1 ω 2 ( ω 1 > ω 2 )                    (7) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaqGjpWdamaaBaaaleaapeGaaeOyaiaabwgacaqGHbGaaeiDaaWd aeqaaOWdbiabg2da9iaabM8apaWaaSbaaSqaa8qacaaIXaaapaqaba GcpeGaeyOeI0IaaeyYd8aadaWgaaWcbaWdbiaaikdaa8aabeaak8qa caGGOaGaaeyYd8aadaWgaaWcbaWdbiaaigdaa8aabeaak8qacqGH+a GpcaqGjpWdamaaBaaaleaapeGaaGOmaaWdaeqaaOWdbiaacMcacaqG GaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabc cacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeii aiaabccacaqGGaGaaeiiaiaabIcacaqG3aGaaeykaaaa@589B@ φ beat =φ( ω 1 ) φ( ω 2 ) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaqGgpWdamaaBaaaleaapeGaaeOyaiaabwgacaqGHbGaaeiDaaWd aeqaaOWdbiabg2da9iaabA8adaqadaWdaeaapeGaaeyYd8aadaWgaa WcbaWdbiaaigdaa8aabeaaaOWdbiaawIcacaGLPaaacqGHsislcaqG GcGaaeOXdmaabmaapaqaa8qacaqGjpWdamaaBaaaleaapeGaaGOmaa WdaeqaaaGcpeGaayjkaiaawMcaaaaa@4978@ We can express a beating sginal from Eq.(2) as below cos( ω h (t τ h ) )×cos( ω l (t τ l ) )=1/2cos(( ω h + ω l )t( ω h τ h + ω l τ l ))+1/2cos(( ω h ω l )t( ω h τ h ω l τ l ))                 (8) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qaciGGJbGaai4BaiaacohadaqadaWdaeaapeGaeqyYdC3damaaBaaa leaapeGaamiAaaWdaeqaaOWdbiaacIcacaWG0bGaeyOeI0IaeqiXdq 3damaaBaaaleaapeGaamiAaaWdaeqaaOWdbiaacMcaaiaawIcacaGL PaaacqGHxdaTciGGJbGaai4BaiaacohadaqadaWdaeaapeGaeqyYdC 3damaaBaaaleaapeGaamiBaaWdaeqaaOWdbiaacIcacaWG0bGaeyOe I0IaeqiXdq3damaaBaaaleaapeGaamiBaaWdaeqaaOWdbiaacMcaai aawIcacaGLPaaacqGH9aqpcaaIXaGaai4laiaaikdaciGGJbGaai4B aiaacohacaGGOaGaaiikaiabeM8a39aadaWgaaWcbaWdbiaadIgaa8 aabeaak8qacqGHRaWkcqaHjpWDpaWaaSbaaSqaa8qacaWGSbaapaqa baGcpeGaaiykaiaadshacqGHsislcaGGOaGaeqyYdC3damaaBaaale aapeGaamiAaaWdaeqaaOWdbiabes8a09aadaWgaaWcbaWdbiaadIga a8aabeaak8qacqGHRaWkcqaHjpWDpaWaaSbaaSqaa8qacaWGSbaapa qabaGcpeGaeqiXdq3damaaBaaaleaapeGaamiBaaWdaeqaaOWdbiaa cMcacaGGPaGaey4kaSIaaGymaiaac+cacaaIYaGaci4yaiaac+gaca GGZbGaaiikaiaacIcacqaHjpWDpaWaaSbaaSqaa8qacaWGObaapaqa baGcpeGaeyOeI0IaeqyYdC3damaaBaaaleaapeGaamiBaaWdaeqaaO WdbiaacMcacaWG0bGaeyOeI0IaaiikaiabeM8a39aadaWgaaWcbaWd biaadIgaa8aabeaak8qacqaHepaDpaWaaSbaaSqaa8qacaWGObaapa qabaGcpeGaeyOeI0IaeqyYdC3damaaBaaaleaapeGaamiBaaWdaeqa aOWdbiabes8a09aadaWgaaWcbaWdbiaadYgaa8aabeaak8qacaGGPa GaaiykaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabcca caqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiai aabccacaqGGaGaaeikaiaabIdacaqGPaaaaa@A2B7@ Then we can extract beating signal’s frequency and phase using Eq.(7) and Eq.(8). (Figure 3 shows beating signal and it’s envelope)
Figure 3:Envelope of beating signal (a) beating signal, (b) envelope for the beating signaleating signal
ω beating = ω 1 ω 2 =2 ω l                    (9) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacqaHjpWDpaWaaSbaaSqaa8qacaWGIbGaamyzaiaadggacaWG0bGa amyAaiaad6gacaWGNbaapaqabaGcpeGaeyypa0JaeqyYdC3damaaBa aaleaapeGaaGymaaWdaeqaaOWdbiabgkHiTiabeM8a39aadaWgaaWc baWdbiaaikdaa8aabeaak8qacqGH9aqpcaaIYaGaeqyYdC3damaaBa aaleaapeGaamiBaaWdaeqaaOGaaeiiaiaabccacaqGGaGaaeiiaiaa bccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaae iiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabIcacaqG 5aGaaeykaaaa@59CF@ φ beating =φ( ω 1 ) φ( ω 2 )= 2 ω l τ l MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacqaHgpGApaWaaSbaaSqaa8qacaWGIbGaamyzaiaadggacaWG0bGa amyAaiaad6gacaWGNbaapaqabaGcpeGaeyypa0JaeqOXdO2aaeWaa8 aabaWdbiabeM8a39aadaWgaaWcbaWdbiaaigdaa8aabeaaaOWdbiaa wIcacaGLPaaacqGHsislcaGGGcGaeqOXdO2aaeWaa8aabaWdbiabeM 8a39aadaWgaaWcbaWdbiaaikdaa8aabeaaaOWdbiaawIcacaGLPaaa cqGH9aqpcaGGGcGaaGOmaiabeM8a39aadaWgaaWcbaWdbiaadYgaa8 aabeaak8qacqaHepaDpaWaaSbaaSqaa8qacaWGSbaapaqabaaaaa@57C5@
Frequency and Phase of a Beating Signal
In order to estimate an acoustic source locations, the time delays between acoutic sensors signals must be calculated. The time delay is related to the phase delay of the envelopes. But as shown in Figure 4,
Figure 4:Time delay of a beating signal’s envelope
it is difficult to find the arrival time difference using the envelope in the time domain. So, we tried to find the time delays using the envelope’s phase delay in the frequency domain [7-9]. We are dealing with two beating signals p(t) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWGWbGaaiikaiaadshacaGGPaaaaa@395C@ and q(t) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWGXbGaaiikaiaadshacaGGPaaaaa@395D@ .We could think p(t) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWGWbGaaiikaiaadshacaGGPaaaaa@395C@ and q(t) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWGXbGaaiikaiaadshacaGGPaaaaa@395D@ are received signals at a different two sensors, and τ d MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacqaHepaDpaWaaSbaaSqaa8qacaWGKbaapaqabaaaaa@391D@ means the time delay between p(t) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWGWbGaaiikaiaadshacaGGPaaaaa@395C@ and q(t) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWGXbGaaiikaiaadshacaGGPaaaaa@395D@ .Then we can express it as p( t )=A{cos( ω 1 t φ 1 )+cos( ω 2 t φ 2 )}                 (10) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWGWbWaaeWaa8aabaWdbiaadshaaiaawIcacaGLPaaacqGH9aqp caWGbbGaai4EaiGacogacaGGVbGaai4Camaabmaapaqaa8qacqaHjp WDpaWaaSbaaSqaa8qacaaIXaaapaqabaGcpeGaamiDaiabgkHiTiab eA8aQ9aadaWgaaWcbaWdbiaaigdaa8aabeaaaOWdbiaawIcacaGLPa aacqGHRaWkcaWGJbGaam4BaiaadohadaqadaWdaeaapeGaeqyYdC3d amaaBaaaleaapeGaaGOmaaWdaeqaaOWdbiaadshacqGHsislcaqGgp WdamaaBaaaleaapeGaaGOmaaWdaeqaaaGcpeGaayjkaiaawMcaaiaa c2hacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaae iiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqG GaGaaeiiaiaabIcacaqGXaGaaeimaiaabMcaaaa@640E@
In this case, the frequency and phase of beating signal p( t ) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWGWbWaaeWaa8aabaWdbiaadshaaiaawIcacaGLPaaaaaa@39AB@ are ω 1 ω 2 ( ω 1 > ω 2 ), MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaqGjpWdamaaBaaaleaapeGaaGymaaWdaeqaaOWdbiabgkHiTiaa bM8apaWaaSbaaSqaa8qacaaIYaaapaqabaGcpeWaaeWaa8aabaWdbi aabM8apaWaaSbaaSqaa8qacaaIXaaapaqabaGcpeGaeyOpa4JaaeyY d8aadaWgaaWcbaWdbiaaikdaa8aabeaaaOWdbiaawIcacaGLPaaaca GGSaaaaa@445C@ φ 1  φ 2 MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaqGgpWdamaaBaaaleaapeGaaGymaaWdaeqaaOWdbiabgkHiTiaa bckacaqGgpWdamaaBaaaleaapeGaaGOmaaWdaeqaaaaa@3D02@ respectively. Then, q( t )=α×p(t τ d )=α×A{cos( ω 1 ( t τ d ) φ 1 )+cos( ω 2 ( t τ d ) φ 2 )}          (11) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWGXbWaaeWaa8aabaWdbiaadshaaiaawIcacaGLPaaacqGH9aqp cqaHXoqycqGHxdaTcaWGWbGaaiikaiaadshacqGHsislcqaHepaDpa WaaSbaaSqaa8qacaWGKbaapaqabaGcpeGaaiykaiabg2da9iabeg7a HjabgEna0kaadgeacaGG7bGaci4yaiaac+gacaGGZbWaaeWaa8aaba WdbiabeM8a39aadaWgaaWcbaWdbiaaigdaa8aabeaak8qadaqadaWd aeaapeGaamiDaiabgkHiTiabes8a09aadaWgaaWcbaWdbiaadsgaa8 aabeaaaOWdbiaawIcacaGLPaaacqGHsislcqaHgpGApaWaaSbaaSqa a8qacaaIXaaapaqabaaak8qacaGLOaGaayzkaaGaey4kaSIaam4yai aad+gacaWGZbWaaeWaa8aabaWdbiabeM8a39aadaWgaaWcbaWdbiaa ikdaa8aabeaak8qadaqadaWdaeaapeGaamiDaiabgkHiTiabes8a09 aadaWgaaWcbaWdbiaadsgaa8aabeaaaOWdbiaawIcacaGLPaaacqGH sislcaqGgpWdamaaBaaaleaapeGaaGOmaaWdaeqaaaGcpeGaayjkai aawMcaaiaac2hacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabcca caqGGaGaaeiiaiaabccacaqGGaGaaeikaiaabgdacaqGXaGaaeykaa aa@7AD1@ where ω 1,  ω 2   MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaqGjpWdamaaBaaaleaapeGaaGymaiaacYcaa8aabeaak8qacaqG GcGaaeyYd8aadaWgaaWcbaWdbiaaikdaa8aabeaak8qacaGGGcaaaa@3E09@ are the frequency components composing beating signal and α is the magnitude ratio between p( t ) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWGWbWaaeWaa8aabaWdbiaadshaaiaawIcacaGLPaaaaaa@39AB@ and q( t ) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWGXbWaaeWaa8aabaWdbiaadshaaiaawIcacaGLPaaaaaa@39AC@ The cross-correlation of Eqs. (10) and (11) is derived as the following equation. R p,q ( τ ) =αE[p( t )q(t+τ)]=α R p,p ( τ τ d )              (12) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWGsbWdamaaBaaaleaapeGaamiCaiaacYcacaWGXbaapaqabaGc peWaaeWaa8aabaWdbiabes8a0bGaayjkaiaawMcaaiaacckacqGH9a qpcqaHXoqycaWGfbGaai4waiaadchadaqadaWdaeaapeGaamiDaaGa ayjkaiaawMcaaiaadghacaGGOaGaamiDaiabgUcaRiabes8a0jaacM cacaGGDbGaeyypa0JaeqySdeMaamOua8aadaWgaaWcbaWdbiaadcha caGGSaGaamiCaaWdaeqaaOWdbmaabmaapaqaa8qacqaHepaDcqGHsi slcqaHepaDpaWaaSbaaSqaa8qacaWGKbaapaqabaaak8qacaGLOaGa ayzkaaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiai aabccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGOaGa aeymaiaabkdacaqGPaaaaa@66F0@ where E[a] is the expected value of a. This equation shows that the cross-correlation of two signals can be expressed as the autocorrelation that has a time delay τd. If Eq. (12) is transformed to the frequency domain, we can obtain the cross-spectrum. Let P(ω) MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWGqbGaaiikaiabeM8a3jaacMcaaaa@3A11@ and Q(ω) MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaqGrbGaaiikaiabeM8a3jaacMcaaaa@3A10@ denote the components of the Fourier transform of p( t ) MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWGWbWaaeWaa8aabaWdbiaadshaaiaawIcacaGLPaaaaaa@39AC@ and q( t ) MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWGXbWaaeWaa8aabaWdbiaadshaaiaawIcacaGLPaaaaaa@39AD@ ;the cross-spectrum is S p,q ( ω ) = P * ( ω )×Q( ω )=| P( ω ) ||Q(ω)| e j{ φ p (ω)  φ q ( ω )}            (13) MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWGtbWdamaaBaaaleaapeGaamiCaiaacYcacaWGXbaapaqabaGc peWaaeWaa8aabaWdbiabeM8a3bGaayjkaiaawMcaaiaacckacqGH9a qpcaWGqbWdamaaCaaaleqabaWdbiaacQcaaaGcdaqadaWdaeaapeGa eqyYdChacaGLOaGaayzkaaGaey41aqRaamyuamaabmaapaqaa8qacq aHjpWDaiaawIcacaGLPaaacqGH9aqpdaabdaWdaeaapeGaamiuamaa bmaapaqaa8qacqaHjpWDaiaawIcacaGLPaaaaiaawEa7caGLiWoaca GG8bGaamyuaiaacIcacqaHjpWDcaGGPaGaaiiFaiaadwgapaWaaWba aSqabeaapeGaamOAaiaacUhacaqGgpWdamaaBaaameaapeGaamiCaa WdaeqaaSWdbiaacIcacqaHjpWDcaGGPaGaeyOeI0IaaeiOaiaabA8a paWaaSbaaWqaa8qacaWGXbaapaqabaWcpeWaaeWaa8aabaWdbiabeM 8a3bGaayjkaiaawMcaaiaac2haaaGcpaGaaeiiaiaabccacaqGGaGa aeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabccaca qGOaGaaeymaiaabodacaqGPaaaaa@75F3@ where φ p (ω) MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaqGgpWdamaaBaaaleaapeGaamiCaaWdaeqaaOWdbiaacIcacqaH jpWDcaGGPaaaaa@3BF1@ and φ q (ω) MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaqGgpWdamaaBaaaleaapeGaamyCaaWdaeqaaOWdbiaacIcacqaH jpWDcaGGPaaaaa@3BF2@ mean the phase of p( t ) MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWGWbWaaeWaa8aabaWdbiaadshaaiaawIcacaGLPaaaaaa@39AC@ and q( t ) MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWGXbWaaeWaa8aabaWdbiaadshaaiaawIcacaGLPaaaaaa@39AD@ respectively. Figure 4 shows two beating signals having some phase differences.
TDOA(Time Difference Of Arrival)
In order to extract Time-Difference-Of-Arrival (TDOA) between two received beating signals. We should compare the arrived time difference of the two received beating signals. Put received beating signal’s phases as φ p , φ q MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacqaHgpGApaWaaSbaaSqaa8qacaWGWbaapaqabaGcpeGaaiilaiab eA8aQ9aadaWgaaWcbaWdbiaadghaa8aabeaaaaa@3CF9@ respectively then phase difference between two beating signals could be expressed as φ p φ q = φ p ( ω 1 ) φ p ( ω 2 )( φ q ( ω 1 ) φ q ( ω 2 ))= φ p ( ω 1 ) φ q ( ω 1 )( φ p ( ω 2 ) φ q ( ω 2 ))      (14) MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacqaHgpGApaWaaSbaaSqaa8qacaWGWbaapaqabaGcpeGaeyOeI0Ia eqOXdO2damaaBaaaleaapeGaamyCaaWdaeqaaOWdbiabg2da9iabeA 8aQ9aadaWgaaWcbaWdbiaadchaa8aabeaak8qadaqadaWdaeaapeGa eqyYdC3damaaBaaaleaapeGaaGymaaWdaeqaaaGcpeGaayjkaiaawM caaiabgkHiTiabeA8aQ9aadaWgaaWcbaWdbiaadchaa8aabeaak8qa daqadaWdaeaapeGaeqyYdC3damaaBaaaleaapeGaaGOmaaWdaeqaaa GcpeGaayjkaiaawMcaaiabgkHiTiaacIcacqaHgpGApaWaaSbaaSqa a8qacaWGXbaapaqabaGcpeWaaeWaa8aabaWdbiabeM8a39aadaWgaa WcbaWdbiaaigdaa8aabeaaaOWdbiaawIcacaGLPaaacqGHsislcqaH gpGApaWaaSbaaSqaa8qacaWGXbaapaqabaGcpeWaaeWaa8aabaWdbi abeM8a39aadaWgaaWcbaWdbiaaikdaa8aabeaaaOWdbiaawIcacaGL PaaacaGGPaGaeyypa0JaeqOXdO2damaaBaaaleaapeGaamiCaaWdae qaaOWdbmaabmaapaqaa8qacqaHjpWDpaWaaSbaaSqaa8qacaaIXaaa paqabaaak8qacaGLOaGaayzkaaGaeyOeI0IaeqOXdO2damaaBaaale aapeGaamyCaaWdaeqaaOWdbmaabmaapaqaa8qacqaHjpWDpaWaaSba aSqaa8qacaaIXaaapaqabaaak8qacaGLOaGaayzkaaGaeyOeI0Iaai ikaiabeA8aQ9aadaWgaaWcbaWdbiaadchaa8aabeaak8qadaqadaWd aeaapeGaeqyYdC3damaaBaaaleaapeGaaGOmaaWdaeqaaaGcpeGaay jkaiaawMcaaiabgkHiTiabeA8aQ9aadaWgaaWcbaWdbiaadghaa8aa beaak8qacaGGOaGaeqyYdC3damaaBaaaleaapeGaaGOmaaWdaeqaaO WdbiaacMcacaGGPaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqG GaGaaeikaiaabgdacaqG0aGaaeykaaaa@8C39@ Here, we could get some conclusion. Phase difference between two beating signals is a subtraction of a same frequency. This means we can get beating signal’s phase differences by crossing spectrum. It is not necessary to find beating signal envelopes. The relationship between period 'T' MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaqGNaGaaeivaiaabEcaaaa@3841@ and frequency 'f' MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaqGNaGaamOzaiaacEcaaaa@3856@ is T= 1 f MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaqGubGaeyypa0ZaaSGaa8aabaWdbiaaigdaa8aabaWdbiaadAga aaaaaa@39E9@ The time delay of acoustic beating envelope Δ t p,q (f) MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacqqHuoarcaWG0bWdamaaBaaaleaapeGaamiCaiaacYcacaWGXbaa paqabaGcpeGaaiikaiaadAgacaGGPaaaaa@3DC8@ can be obtained by multiplying the period by the ratio of phase delay. Therefore, the time delay can be expressed as following equation. Δ t p,q (f)= 1 f × φ p ( f ) φ q (f) 2π                   (15) MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacqqHuoarcaWG0bWdamaaBaaaleaapeGaamiCaiaacYcacaWGXbaa paqabaGcpeGaaiikaiaadAgacaGGPaGaeyypa0ZaaSaaa8aabaWdbi aaigdaa8aabaWdbiaadAgaaaGaey41aq7aaSaaa8aabaWdbiabeA8a Q9aadaWgaaWcbaWdbiaadchaa8aabeaak8qadaqadaWdaeaapeGaam OzaaGaayjkaiaawMcaaiabgkHiTiabeA8aQ9aadaWgaaWcbaWdbiaa dghaa8aabeaak8qacaGGOaGaamOzaiaacMcaa8aabaWdbiaaikdacq aHapaCaaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeii aiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGa GaaeiiaiaabccacaqGGaGaaeikaiaabgdacaqG1aGaaeykaaaa@5FEA@ The distance difference transferred from the impact source to the two sensors is derived by multiplying Eq.(11) by the sound velocity. (v=340m/s, @15) MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaGGOaGaamODaiabg2da9iaaiodacaaI0aGaaGimaiaab2gacaGG VaGaae4CaiaabYcacaqGGaGaaiiqaiaaigdacaaI1aWexLMBbXgBd9 gzLbvyNv2CaeHbcfgDH52zaGqbaiaa=nqicaWFPaaaaa@4AA6@
Δ r p,q = r p r q =v×Δ t p,q ( f )               (16) MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacqqHuoarcaWGYbWdamaaBaaaleaapeGaamiCaiaacYcacaWGXbaa paqabaGcpeGaeyypa0JaamOCa8aadaWgaaWcbaWdbiaadchaa8aabe aak8qacqGHsislcaWGYbWdamaaBaaaleaapeGaamyCaaWdaeqaaOWd biabg2da9iaadAhacqGHxdaTcqqHuoarcaWG0bWdamaaBaaaleaape GaamiCaiaacYcacaWGXbaapaqabaGcpeWaaeWaa8aabaWdbiaadAga aiaawIcacaGLPaaacaGGGcGaaiiOaiaabccacaqGGaGaaeiiaiaabc cacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeii aiaabccacaqGOaGaaeymaiaabAdacaqGPaaaaa@5BA2@
Triangular Method
Triangulation method was applied to point out an exact position. Figure 5 shows the triangular method for only two sensors. And figure 6 shows for more than three sensors.
Figure 5:Hyperbola line when only two sensors are used
Figure 6:Hyperbola line when more than 3 sensors are used
Figure 7:Sample of software simulation for triangulation method (a) only two sensors are used, (b) more than 3 sensors are used
In figure 7 Triangulation method looks for the hyperbola intersection point based on time-of-arrival differences (TOADs) of a burst between three different sensors [3,5,6]. Assume sensor number is N, then we can get hyperbola lines number of M= N! 2!( N2 )! = N(N1) 2                   (17) MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWGnbGaeyypa0ZaaSaaa8aabaWdbiaad6eacaGGHaaapaqaa8qa caaIYaGaaiyiamaabmaapaqaa8qacaWGobGaeyOeI0IaaGOmaaGaay jkaiaawMcaaiaacgcaaaGaeyypa0ZaaSaaa8aabaWdbiaad6eacaGG OaGaamOtaiabgkHiTiaaigdacaGGPaaapaqaa8qacaaIYaaaaiaabc cacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeii aiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGa GaaeiiaiaabIcacaqGXaGaae4naiaabMcaaaa@54D0@
Simulation and Verification
In order to verify our proposed theory by experiment, we put two acoustic emission sources and three sensors. Figure 8 shows the conditions of a hypothetical simulation.
The coordinates of two acoustic emission sources are E1(250,350) and E2(350,250). The coordinates of three acoustic sensors are S1(0,0), S2(600,0) and S3(300,600) respectively. We can derive two beating signals from the frequencies of the two acoustic emission sources. The first beating signal is composed of 25KHz, 25.5KHz, and the second beating signal is composed of 27KHz and 27.5KHz. Figure 9 shows received signals from S1, S2 and S3. Figure 10 shows the result of cross power spectrum of the received signals. In order to find phase differences between beating signals depending on Eq.(14) we should find phase differences of the same frequencies first. The phase differences of the same frequencies for the first beating signal (E1) are as below
Figure 8:Simulation conditions which motion box area is 600mm x 600mm, red circle denotes the emission sources
Figure 9:Measured signals for each acoustic sensors, (a) S1, (b) S2, (c) S3
Figure 10:The result of cross power spectrum (a) S1&S2 (b) S1&S3, (c) S2&S3. Blue line denote magnitude, red line denote phase
φ 1,2 ( 25KHz )=0.594, MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacqaHgpGApaWaaSbaaSqaa8qacaaIXaGaaiilaiaaikdaa8aabeaa k8qadaqadaWdaeaapeGaaGOmaiaaiwdacaWGlbGaamisaiaadQhaai aawIcacaGLPaaacqGH9aqpcaaIWaGaaiOlaiaaiwdacaaI5aGaaGin aiaacYcaaaa@458F@
φ 1,3 ( 25KHz )=0.0, MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacqaHgpGApaWaaSbaaSqaa8qacaaIXaGaaiilaiaaiodaa8aabeaa k8qadaqadaWdaeaapeGaaGOmaiaaiwdacaWGlbGaamisaiaadQhaai aawIcacaGLPaaacqGH9aqpcaaIWaGaaiOlaiaaicdacaGGSaaaaa@440A@
φ 2,3 ( 25KHz )=0.594, MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacqaHgpGApaWaaSbaaSqaa8qacaaIYaGaaiilaiaaiodaa8aabeaa k8qadaqadaWdaeaapeGaaGOmaiaaiwdacaWGlbGaamisaiaadQhaai aawIcacaGLPaaacqGH9aqpcqGHsislcaaIWaGaaiOlaiaaiwdacaaI 5aGaaGinaiaacYcaaaa@467E@
φ 1,2 ( 25.5KHz )=0.0, MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacqaHgpGApaWaaSbaaSqaa8qacaaIXaGaaiilaiaaikdaa8aabeaa k8qadaqadaWdaeaapeGaaGOmaiaaiwdacaGGUaGaaGynaiaadUeaca WGibGaamOEaaGaayjkaiaawMcaaiabg2da9iaaicdacaGGUaGaaGim aiaacYcaaaa@457A@
φ 1,3 ( 25.5KHz )=0.0, MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacqaHgpGApaWaaSbaaSqaa8qacaaIXaGaaiilaiaaiodaa8aabeaa k8qadaqadaWdaeaapeGaaGOmaiaaiwdacaGGUaGaaGynaiaadUeaca WGibGaamOEaaGaayjkaiaawMcaaiabg2da9iaaicdacaGGUaGaaGim aiaacYcaaaa@457B@
φ 2,3 ( 25.5KHz )=1.615 MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacqaHgpGApaWaaSbaaSqaa8qacaaIYaGaaiilaiaaiodaa8aabeaa k8qadaqadaWdaeaapeGaaGOmaiaaiwdacaGGUaGaaGynaiaadUeaca WGibGaamOEaaGaayjkaiaawMcaaiabg2da9iaaigdacaGGUaGaaGOn aiaaigdacaaI1aaaaa@464D@
and then we can extract phase difference for the two beating signals by using Eq.(14)
φ 1,2 ( 25.5KHz ) φ 1,2 ( 25KHz )=0.594 MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacqaHgpGApaWaaSbaaSqaa8qacaaIXaGaaiilaiaaikdaa8aabeaa k8qadaqadaWdaeaapeGaaGOmaiaaiwdacaGGUaGaaGynaiaadUeaca WGibGaamOEaaGaayjkaiaawMcaaiabgkHiTiabeA8aQ9aadaWgaaWc baWdbiaaigdacaGGSaGaaGOmaaWdaeqaaOWdbmaabmaapaqaa8qaca aIYaGaaGynaiaadUeacaWGibGaamOEaaGaayjkaiaawMcaaiabg2da 9iabgkHiTiaaicdacaGGUaGaaGynaiaaiMdacaaI0aaaaa@5241@
φ 1,3 ( 25.5KHz ) φ 1,3 ( 25KHz )=1.615 MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacqaHgpGApaWaaSbaaSqaa8qacaaIXaGaaiilaiaaiodaa8aabeaa k8qadaqadaWdaeaapeGaaGOmaiaaiwdacaGGUaGaaGynaiaadUeaca WGibGaamOEaaGaayjkaiaawMcaaiabgkHiTiabeA8aQ9aadaWgaaWc baWdbiaaigdacaGGSaGaaG4maaWdaeqaaOWdbmaabmaapaqaa8qaca aIYaGaaGynaiaadUeacaWGibGaamOEaaGaayjkaiaawMcaaiabg2da 9iaaigdacaGGUaGaaGOnaiaaigdacaaI1aaaaa@5151@
φ 2,3 ( 25.5Hz ) φ 2,3 ( 25KHz )=2.209 MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacqaHgpGApaWaaSbaaSqaa8qacaaIYaGaaiilaiaaiodaa8aabeaa k8qadaqadaWdaeaapeGaaGOmaiaaiwdacaGGUaGaaGynaiaadIeaca WG6baacaGLOaGaayzkaaGaeyOeI0IaeqOXdO2damaaBaaaleaapeGa aGOmaiaacYcacaaIZaaapaqabaGcpeWaaeWaa8aabaWdbiaaikdaca aI1aGaam4saiaadIeacaWG6baacaGLOaGaayzkaaGaeyypa0JaaGOm aiaac6cacaaIYaGaaGimaiaaiMdaaaa@5083@
Time delays for the first beating signal is as below
Δ t 1,2 ( 0.5KHz )=0.000189 MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacqqHuoarcaWG0bWdamaaBaaaleaapeGaaGymaiaacYcacaaIYaaa paqabaGcpeWaaeWaa8aabaWdbiaaicdacaGGUaGaaGynaiaadUeaca WGibGaamOEaaGaayjkaiaawMcaaiabg2da9iabgkHiTiaaicdacaGG UaGaaGimaiaaicdacaaIWaGaaGymaiaaiIdacaaI5aaaaa@494C@
Δ t 1,3 ( 0.5KHz )=0.000514 MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacqqHuoarcaWG0bWdamaaBaaaleaapeGaaGymaiaacYcacaaIZaaa paqabaGcpeWaaeWaa8aabaWdbiaaicdacaGGUaGaaGynaiaadUeaca WGibGaamOEaaGaayjkaiaawMcaaiabg2da9iaaicdacaGGUaGaaGim aiaaicdacaaIWaGaaGynaiaaigdacaaI0aaaaa@4858@
Δ t 2,3 ( 0.5KHz )=0.000703 MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacqqHuoarcaWG0bWdamaaBaaaleaapeGaaGOmaiaacYcacaaIZaaa paqabaGcpeWaaeWaa8aabaWdbiaaicdacaGGUaGaaGynaiaadUeaca WGibGaamOEaaGaayjkaiaawMcaaiabg2da9iaaicdacaGGUaGaaGim aiaaicdacaaIWaGaaG4naiaaicdacaaIZaaaaa@4859@
Then the delay distance can be obtained by multiplying sound velocity.
Δ r 1,2 =0.06426m MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacqqHuoarcaWGYbWdamaaBaaaleaapeGaaGymaiaacYcacaaIYaaa paqabaGcpeGaeyypa0JaeyOeI0IaaGimaiaac6cacaaIWaGaaGOnai aaisdacaaIYaGaaGOnaiaab2gaaaa@4311@
Δ r 1,3 =0.17476m MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacqqHuoarcaWGYbWdamaaBaaaleaapeGaaGymaiaacYcacaaIZaaa paqabaGcpeGaeyypa0JaaGimaiaac6cacaaIXaGaaG4naiaaisdaca aI3aGaaGOnaiaab2gaaaa@422C@
Δ r 2,3 =0.23902m MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacqqHuoarcaWGYbWdamaaBaaaleaapeGaaGOmaiaacYcacaaIZaaa paqabaGcpeGaeyypa0JaaGimaiaac6cacaaIYaGaaG4maiaaiMdaca aIWaGaaGOmaiaab2gaaaa@4224@
The second beating signals are as below(E2)
φ 1,2 ( 27KHz )=0.0 MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacqaHgpGApaWaaSbaaSqaa8qacaaIXaGaaiilaiaaikdaa8aabeaa k8qadaqadaWdaeaapeGaaGOmaiaaiEdacaWGlbGaamisaiaadQhaai aawIcacaGLPaaacqGH9aqpcaaIWaGaaiOlaiaaicdaaaa@435B@
φ 1,3 ( 27KHz )=0.0 MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacqaHgpGApaWaaSbaaSqaa8qacaaIXaGaaiilaiaaiodaa8aabeaa k8qadaqadaWdaeaapeGaaGOmaiaaiEdacaWGlbGaamisaiaadQhaai aawIcacaGLPaaacqGH9aqpcaaIWaGaaiOlaiaaicdaaaa@435C@
φ 2,3 ( 27KHz )=0.0 MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacqaHgpGApaWaaSbaaSqaa8qacaaIYaGaaiilaiaaiodaa8aabeaa k8qadaqadaWdaeaapeGaaGOmaiaaiEdacaWGlbGaamisaiaadQhaai aawIcacaGLPaaacqGH9aqpcaaIWaGaaiOlaiaaicdaaaa@435D@
φ 1,2 ( 27.5KHz )=0.701 MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacqaHgpGApaWaaSbaaSqaa8qacaaIXaGaaiilaiaaikdaa8aabeaa k8qadaqadaWdaeaapeGaaGOmaiaaiEdacaGGUaGaaGynaiaadUeaca WGibGaamOEaaGaayjkaiaawMcaaiabg2da9iaaicdacaGGUaGaaG4n aiaaicdacaaIXaaaaa@4648@
φ 1,3 ( 27.5KHz )=0.713 MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacqaHgpGApaWaaSbaaSqaa8qacaaIXaGaaiilaiaaiodaa8aabeaa k8qadaqadaWdaeaapeGaaGOmaiaaiEdacaGGUaGaaGynaiaadUeaca WGibGaamOEaaGaayjkaiaawMcaaiabg2da9iaaicdacaGGUaGaaG4n aiaaigdacaaIZaaaaa@464C@
φ 2,3 ( 27.5KHz )=0.021 MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacqaHgpGApaWaaSbaaSqaa8qacaaIYaGaaiilaiaaiodaa8aabeaa k8qadaqadaWdaeaapeGaaGOmaiaaiEdacaGGUaGaaGynaiaadUeaca WGibGaamOEaaGaayjkaiaawMcaaiabg2da9iaaicdacaGGUaGaaGim aiaaikdacaaIXaaaaa@4645@
For the second beating signal
φ 1,2 ( 27,500Hz ) φ 1,2 ( 27,000Hz )=0.701 MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacqaHgpGApaWaaSbaaSqaa8qacaaIXaGaaiilaiaaikdaa8aabeaa k8qadaqadaWdaeaapeGaaGOmaiaaiEdacaGGSaGaaGynaiaaicdaca aIWaGaamisaiaadQhaaiaawIcacaGLPaaacqGHsislcqaHgpGApaWa aSbaaSqaa8qacaaIXaGaaiilaiaaikdaa8aabeaak8qadaqadaWdae aapeGaaGOmaiaaiEdacaGGSaGaaGimaiaaicdacaaIWaGaamisaiaa dQhaaiaawIcacaGLPaaacqGH9aqpcaaIWaGaaiOlaiaaiEdacaaIWa GaaGymaaaa@53FE@
φ 1,3 ( 27,500Hz ) φ 1,3 ( 27,000Hz )=0.713 MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacqaHgpGApaWaaSbaaSqaa8qacaaIXaGaaiilaiaaiodaa8aabeaa k8qadaqadaWdaeaapeGaaGOmaiaaiEdacaGGSaGaaGynaiaaicdaca aIWaGaamisaiaadQhaaiaawIcacaGLPaaacqGHsislcqaHgpGApaWa aSbaaSqaa8qacaaIXaGaaiilaiaaiodaa8aabeaak8qadaqadaWdae aapeGaaGOmaiaaiEdacaGGSaGaaGimaiaaicdacaaIWaGaamisaiaa dQhaaiaawIcacaGLPaaacqGH9aqpcaaIWaGaaiOlaiaaiEdacaaIXa GaaG4maaaa@5403@
φ 2,3 ( 27,500Hz ) φ 2,3 ( 27,000Hz )=0.012 MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacqaHgpGApaWaaSbaaSqaa8qacaaIYaGaaiilaiaaiodaa8aabeaa k8qadaqadaWdaeaapeGaaGOmaiaaiEdacaGGSaGaaGynaiaaicdaca aIWaGaamisaiaadQhaaiaawIcacaGLPaaacqGHsislcqaHgpGApaWa aSbaaSqaa8qacaaIYaGaaiilaiaaiodaa8aabeaak8qadaqadaWdae aapeGaaGOmaiaaiEdacaGGSaGaaGimaiaaicdacaaIWaGaamisaiaa dQhaaiaawIcacaGLPaaacqGH9aqpcaaIWaGaaiOlaiaaicdacaaIXa GaaGOmaaaa@53FD@
time delays for the second beating signal is
Δ t 1,2 ( 0.5KHz )=0.000223 MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacqqHuoarcaWG0bWdamaaBaaaleaapeGaaGymaiaacYcacaaIYaaa paqabaGcpeWaaeWaa8aabaWdbiaaicdacaGGUaGaaGynaiaadUeaca WGibGaamOEaaGaayjkaiaawMcaaiabg2da9iaaicdacaGGUaGaaGim aiaaicdacaaIWaGaaGOmaiaaikdacaaIZaaaaa@4854@
Δ t 1,3 ( 0.5KHz )=0.000227 MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacqqHuoarcaWG0bWdamaaBaaaleaapeGaaGymaiaacYcacaaIZaaa paqabaGcpeWaaeWaa8aabaWdbiaaicdacaGGUaGaaGynaiaadUeaca WGibGaamOEaaGaayjkaiaawMcaaiabg2da9iaaicdacaGGUaGaaGim aiaaicdacaaIWaGaaGOmaiaaikdacaaI3aaaaa@4859@
Δ t 2,3 ( 0.5KHz )=0.000004 MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacqqHuoarcaWG0bWdamaaBaaaleaapeGaaGOmaiaacYcacaaIZaaa paqabaGcpeWaaeWaa8aabaWdbiaaicdacaGGUaGaaGynaiaadUeaca WGibGaamOEaaGaayjkaiaawMcaaiabg2da9iaaicdacaGGUaGaaGim aiaaicdacaaIWaGaaGimaiaaicdacaaI0aaaaa@4853@
Then the delay distance can be obtained by multiplying sound velocity.
Δ r 1,2 =0.07582m MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacqqHuoarcaWGYbWdamaaBaaaleaapeGaaGymaiaacYcacaaIYaaa paqabaGcpeGaeyypa0JaaGimaiaac6cacaaIWaGaaG4naiaaiwdaca aI4aGaaGOmaiaab2gaaaa@4228@
Δ r 1,3 =0.07718m MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacqqHuoarcaWGYbWdamaaBaaaleaapeGaaGymaiaacYcacaaIZaaa paqabaGcpeGaeyypa0JaaGimaiaac6cacaaIWaGaaG4naiaaiEdaca aIXaGaaGioaiaab2gaaaa@422A@
Δ r 2,3 =0.00136m MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacqqHuoarcaWGYbWdamaaBaaaleaapeGaaGOmaiaacYcacaaIZaaa paqabaGcpeGaeyypa0JaaGimaiaac6cacaaIWaGaaGimaiaaigdaca aIZaGaaGOnaiaab2gaaaa@421E@
Figure 11 shows the hypothetical simulation result. We were able to verify our proposal. This is one of the best solutions to find exact source location for the two acoustic beating signals or more.
Figure 11:Source localization test result
Conclusion
We proposed here a localization technique for multiple acoustic sources using acoustic beating envelopes of which wave length are longer than audio frequency. Then we used a hypothetical simulation to prove it. The reason why we used high frequencies is not to make unnecessary acoustic noises. Additional purpose for using high frequency is to avoid unnecessary mechanical noise interferences. Because most mechanical noises are ranging under 20 KHz. We also newly proposed here how to use beating signal’s envelope. Wave length of a beating signal is much longer than those of the originals. For this reason we are free from sensor locations. We were able to get several advantages by using acoustic beating signal.
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