IEEE TRANSACTIONS ON MOBILE COMPUTING,VOL.XX,NO.XX,2020 30 Y(m)1 .Invalid Actual Trace D .Valid 6 10 False Trace Maior X(m) Detection Kegion -0.60.4 、0.20.40.6 20 2 Field of View Time(s)】 (a)Time delay curve (a)Hyperbolas generated by the time delays. 20 ·0 riginal -Smoothed Tracel 10 B 0 5 -10 Trace2 L -20 L 2.5 3 3.544.555.56 H H' X Time(s) 0 (b)Smoothed time delay curve. (b)A simplified model of the asymptotes and the trace. Fig.6:Generating Time Delay curve Fig.7:Slope Calculating. maximum valid delay Adm between the two signals should can be determined by the correlation of these time delays be calculated as equation(4): For example,in figure 5b,the threshold can be set as 0.5. 2lfs Constraint 2 can help us remove some of the noise appears △dm= C (4) in Major Detection Region.Figure 6a draws the time delay where 21 is the distance between the two microphones and curve with blue points represent the valid time delays and C is the speed of sound.For example,we set fs =44.1KHz red points represent the invalid time delays. and C=343m/s,and the distance,without loss of gener- ality,of Samsung note 8(Experiments in Section 5 are based 3.2.3 Candidate Trajectories Estimation on this type of mobile phone.)21=15cm,thus Adm is After we get a series of time delays,we need to recover the 0.15m×44100s- trace of the automobile.We utilize Major Detection Region 343m/s -=19.220 samples.We denote the delay between the segment pair as Ad.According to triangle to estimate candidate trajectories of the automobile.The inequality,the valid delay we get from cross-correlation duration of Major Detection Region is less than 3 seconds should be an integer whose absolute value Ad is less than in most situations.For example,in figure 6a the duration of Adm.That means Ad should be an integer ranging from Major Detection Region is 1.5 seconds.Since the duration is -Adm to Adm,just as figure 5a shows.We define the region short we can assume that the trace of the automobile is a where the time delays vary from Adm to -Adm(or on the line.It is known that in two dimensions,the linear trace can contrary)as Major Detection Region.For example,the Major be represented as: Detection Region in figure 5a starts at 2.5s and ends at 5s. y=mx+b. (5) In constraint 2,the threshold R,has its physical interpre- which means that we need two parameters to determine a tation.It implies that the automobile should be close enough line.The parameter m determines the slope of the line and to the mobile phone,which means the signals from the two the parameter b determines the vertical distance between corresponding segments should be similar enough.Cross- the automobile and the mobile phone. correlation is a measure of similarity of two signals.The First we try to calculate the parameter m through the larger the correlation is,the more similar the two signals time delays curve.If the time delay between the top and the will be.If the automobile is far from the mobile phone, bottom microphones is Ad at time t,the automobile should the sound made by the automobile will be too weak to locate in the hyperbolas whose foci are M1(-1,0)and dominate the signal,which means the signals from the top M2 (,0)and vertices are Vi(-zd,0)and V2(zAd,0)at and the bottom microphones are not similar enough.In this this moment.The mathematical expression of the hyperbola case,the cross-correlations of these segments are quite small. is: 12 y2 These time delays are not suitable for speed calculation. a2-2-1, (6) The threshold will change with different scenarios.And the where a=andb=√2-a2 threshold can be determined with constraint 1.Since we Figure 7a shows the hyperbolas generated by different should pay attention to Major Detection Region,we can set time delays.We can see from the figure that the hyper- the maximum cross-correlation of the boundary segments bolas look like a line.The reason is that in our scenario, in Major Detection Region as the threshold.In other words,,yl,where ,y is the coordinate of the automobile in according to constraint 1,there must exist a process in figure 4b,since l is usually shorter than 10cm and z,y are which the time delays are around Adm.The threshold usually longer than 5 meters.So we can use asymptote of Authorized licensed use limited to:Nanjing University.Downloaded on July 06,2021 at 04:35:27 UTC from IEEE Xplore.Restrictions apply.IEEE TRANSACTIONS ON MOBILE COMPUTING, VOL. XX, NO. XX, 2020 7 (a) Time delay curve. 2.5 3 3.5 4 4.5 5 5.5 6 Time(s) -20 -10 0 10 20 Time Delay(sample) Original Smoothed t5 t1 t2 t3 t6 t4 t7 (b) Smoothed time delay curve. Fig. 6: Generating Time Delay curve maximum valid delay ∆dm between the two signals should be calculated as equation (4): ∆dm = 2lfs C , (4) where 2l is the distance between the two microphones and C is the speed of sound. For example, we set fs = 44.1KHz and C = 343m/s, and the distance, without loss of gener￾ality, of Samsung note 8(Experiments in Section 5 are based on this type of mobile phone.) 2l = 15cm, thus ∆dm is 0.15m×44100s −1 343m/s = 19.2 ≈ 20 samples. We denote the delay between the segment pair as ∆d. According to triangle inequality, the valid delay we get from cross-correlation should be an integer whose absolute value |∆d| is less than ∆dm. That means ∆d should be an integer ranging from −∆dm to ∆dm, just as figure 5a shows. We define the region where the time delays vary from ∆dm to −∆dm(or on the contrary) as Major Detection Region. For example, the Major Detection Region in figure 5a starts at 2.5s and ends at 5s. In constraint 2, the threshold Rs has its physical interpre￾tation. It implies that the automobile should be close enough to the mobile phone, which means the signals from the two corresponding segments should be similar enough. Cross￾correlation is a measure of similarity of two signals. The larger the correlation is, the more similar the two signals will be. If the automobile is far from the mobile phone, the sound made by the automobile will be too weak to dominate the signal, which means the signals from the top and the bottom microphones are not similar enough. In this case, the cross-correlations of these segments are quite small. These time delays are not suitable for speed calculation. The threshold will change with different scenarios. And the threshold can be determined with constraint 1. Since we should pay attention to Major Detection Region, we can set the maximum cross-correlation of the boundary segments in Major Detection Region as the threshold. In other words, according to constraint 1, there must exist a process in which the time delays are around ∆dm. The threshold (a) Hyperbolas generated by the time delays. X Y Trace1 � � � Trace2 �$ �$ �$ � �$ �) �( �' � � �$ (b) A simplified model of the asymptotes and the trace. Fig. 7: Slope Calculating. can be determined by the correlation of these time delays. For example, in figure 5b, the threshold can be set as 0.5. Constraint 2 can help us remove some of the noise appears in Major Detection Region. Figure 6a draws the time delay curve with blue points represent the valid time delays and red points represent the invalid time delays. 3.2.3 Candidate Trajectories Estimation After we get a series of time delays, we need to recover the trace of the automobile. We utilize Major Detection Region to estimate candidate trajectories of the automobile. The duration of Major Detection Region is less than 3 seconds in most situations. For example, in figure 6a the duration of Major Detection Region is 1.5 seconds. Since the duration is short we can assume that the trace of the automobile is a line. It is known that in two dimensions, the linear trace can be represented as: y = mx + b, (5) which means that we need two parameters to determine a line. The parameter m determines the slope of the line and the parameter b determines the vertical distance between the automobile and the mobile phone. First we try to calculate the parameter m through the time delays curve. If the time delay between the top and the bottom microphones is ∆d at time t, the automobile should locate in the hyperbolas whose foci are M1 (−l, 0) and M2 (l, 0) and vertices are V1 ￾ − 1 2∆d , 0
and V2 ￾ 1 2∆d , 0
at this moment. The mathematical expression of the hyperbola is: x 2 a 2 − y 2 b 2 = 1, (6) where a = ∆d 2 and b = √ l 2 − a 2. Figure 7a shows the hyperbolas generated by different time delays. We can see from the figure that the hyper￾bolas look like a line. The reason is that in our scenario, x, y  l, where x, y is the coordinate of the automobile in figure 4b, since l is usually shorter than 10cm and x, y are usually longer than 5 meters. So we can use asymptote of Authorized licensed use limited to: Nanjing University. Downloaded on July 06,2021 at 04:35:27 UTC from IEEE Xplore. Restrictions apply