VECTOR:Velocity Based Temperature-field Monitoring with Distributed Acoustic Devices.144:7 signals occupy separate subcarriers in the frequency domain,they can be transmitted simultaneously.Fig.2(b) shows the modulated signal in the frequency domain. Algorithm 1:Transmitting signal generation Result:The modulated sequence zcr[n]with a length of L and a carrier frequency of fe. 1 Generate frequency domain baseband signal ZCbaseband from Eq.(3)and (4)with a length of Nzc. 2 Multiply ZCbaseband with a Hanning window with length Nic. 3 Generate a all zero sequence ZC[n]with a length of L. 42元[号--：'+Ng-]←ZCbasebandn]. 5ZCL-1:L/2+1]=ZC[1:L/2-1] 6 Perform IFFT on ZC to the time domain zcT[n]. 3.4 Signal Demodulation and ToF Acquisition On both receiving ends,the received signal with P paths can be modeled as: (5) where zcR[n]is received signal,Ai is attenuation coefficient for path i,=-2mife is the phase shift caused by the propagation of path i and ti is the ToF of path i.To get the absolute phase shift for a given path,we first perform FFT on the received signal and extract the ZC baseband sequence directly from the received signal. Then we perform cross-correlation with the conjugate transform of original baseband ZCodd[n]and ZCeven [n] to get the baseband CIR.We use zero-padding on the baseband CIR to expand the length and increase the range resolution brought by sample index,e.g.if we pad the baseband to the length 4x the original frame length,the range difference between each sampling point is 1/48000/45.2 us. To acquire the accurate ToF changes for each path,we combine the index of the cross-correlation peak and the phase of the peak.In the ideal case and without zero-padding.the coarse-grained peak position is expressed as the integer part of rifs,round(rifs),and the fine-grained phase of the peak is expressed as mod(-2mife,2),which is between 0 and 2.We can calculate the absolute phase oi=-2mrife,which considers whole turns of 2 in phase,by combining these two measurements.However,there are inevitable unknown delays for the transmitting and receiving process in low-end commercial devices.Most existing works require user to put devices on a known position to calibrate and calculate the relative distance [33,35].To allow self-calibration without user intervention,we choose to cancel the unknown delays by obtaining reciprocal measurements from both devices. For example,when device A receives signal from device B and the phase shift is oBA =-2(TBA TAR rBT)fc, where TBA is the signal propagation delay that we wish to measure,TAR is the receiving delay for device A and rBT is the transmitting delay for device B.Similarly,we have AA=-2(TAA TAR+rar)fe when device A receive its own signal.Device B can also perform two measurements of AB =-2m(TAB TBR rAr)fe and BB =-2n(TBB TBR TBT)fe.Therefore,we can use AA+BB -AB -OBA =-2(TAA TBB TAB-TBA)fe to cancel the unknown transmitting and receiving delays [39].We can further assume the distances between the speaker and microphone of same device are fixed so that TAA and TBB are known in advance.Therefore,we can measure the ToF by: TAB TBA= TAA+TBB++BB-9AB-9BA (6) 2πfe Proc.ACM Interact.Mob.Wearable Ubiquitous Technol.,Vol.6,No.3,Article 144.Publication date:September 2022.VECTOR: Velocity Based Temperature-field Monitoring with Distributed Acoustic Devices • 144:7 signals occupy separate subcarriers in the frequency domain, they can be transmitted simultaneously. Fig. 2(b) shows the modulated signal in the frequency domain. Algorithm 1: Transmitting signal generation Result: The modulated sequence 𝑧𝑐𝑇 [𝑛] with a length of 𝐿 and a carrier frequency of 𝑓𝑐 . 1 Generate frequency domain baseband signal 𝑍𝐶𝑏𝑎𝑠𝑒𝑏𝑎𝑛𝑑 from Eq. (3) and (4) with a length of 𝑁𝑧𝑐 . 2 Multiply 𝑍𝐶𝑏𝑎𝑠𝑒𝑏𝑎𝑛𝑑 with a Hanning window with length 𝑁𝑧𝑐 . 3 Generate a all zero sequence 𝑍𝐶c [𝑛] with a length of 𝐿. 4 𝑍𝐶c [ 𝑓𝑐𝐿 𝑓𝑠 − (𝑁𝑧𝑐−1) 2 : 𝑓𝑐𝐿 𝑓𝑠 + (𝑁𝑧𝑐−1) 2 ] ⇐ 𝑍𝐶𝑏𝑎𝑠𝑒𝑏𝑎𝑛𝑑 [𝑛]. 5 𝑍𝐶c [𝐿 − 1 : 𝐿/2 + 1] ⇐ 𝑍𝐶c∗ [1 : 𝐿/2 − 1]. 6 Perform IFFT on 𝑍𝐶c to the time domain 𝑧𝑐𝑇 [𝑛]. 3.4 Signal Demodulation and ToF Acquisition On both receiving ends, the received signal with 𝑃 paths can be modeled as: 𝑧𝑐𝑅 [𝑛] = Õ 𝑃 𝑖=1 𝐴𝑖𝑒 𝑗𝜙𝑖𝑧𝑐𝑇  𝑛 − 𝜏𝑖 𝑓𝑠  , (5) where 𝑧𝑐𝑅 [𝑛] is received signal, 𝐴𝑖 is attenuation coefficient for path 𝑖, 𝜙𝑖 = −2𝜋𝜏𝑖𝑓𝑐 is the phase shift caused by the propagation of path 𝑖 and 𝜏𝑖 is the ToF of path 𝑖. To get the absolute phase shift for a given path, we first perform FFT on the received signal and extract the ZC baseband sequence directly from the received signal. Then we perform cross-correlation with the conjugate transform of original baseband 𝑍𝐶𝑜𝑑𝑑 [𝑛] and 𝑍𝐶𝑒𝑣𝑒𝑛 [𝑛] to get the baseband CIR. We use zero-padding on the baseband CIR to expand the length and increase the range resolution brought by sample index, e.g.if we pad the baseband to the length 4× the original frame length, the range difference between each sampling point is 1/48000/4 ≈ 5.2 𝜇𝑠. To acquire the accurate ToF changes for each path, we combine the index of the cross-correlation peak and the phase of the peak. In the ideal case and without zero-padding, the coarse-grained peak position is expressed as the integer part of 𝜏𝑖𝑓𝑠 , 𝑟𝑜𝑢𝑛𝑑 (𝜏𝑖𝑓𝑠), and the fine-grained phase of the peak is expressed as 𝑚𝑜𝑑 (−2𝜋𝜏𝑖𝑓𝑐, 2𝜋), which is between 0 and 2𝜋. We can calculate the absolute phase 𝜙𝑖 = −2𝜋𝜏𝑖𝑓𝑐 , which considers whole turns of 2𝜋 in phase, by combining these two measurements. However, there are inevitable unknown delays for the transmitting and receiving process in low-end commercial devices. Most existing works require user to put devices on a known position to calibrate and calculate the relative distance [33, 35]. To allow self-calibration without user intervention, we choose to cancel the unknown delays by obtaining reciprocal measurements from both devices. For example, when device A receives signal from device B and the phase shift is 𝜙𝐵𝐴 = −2𝜋 (𝜏𝐵𝐴 + 𝜏𝐴𝑅 + 𝜏𝐵𝑇 )𝑓𝑐 , where 𝜏𝐵𝐴 is the signal propagation delay that we wish to measure, 𝜏𝐴𝑅 is the receiving delay for device A and 𝜏𝐵𝑇 is the transmitting delay for device B. Similarly, we have 𝜙𝐴𝐴 = −2𝜋 (𝜏𝐴𝐴 + 𝜏𝐴𝑅 + 𝜏𝐴𝑇 )𝑓𝑐 when device A receive its own signal. Device B can also perform two measurements of 𝜙𝐴𝐵 = −2𝜋 (𝜏𝐴𝐵 + 𝜏𝐵𝑅 + 𝜏𝐴𝑇 )𝑓𝑐 and 𝜙𝐵𝐵 = −2𝜋 (𝜏𝐵𝐵 + 𝜏𝐵𝑅 + 𝜏𝐵𝑇 )𝑓𝑐 . Therefore, we can use 𝜙𝐴𝐴 + 𝜙𝐵𝐵 − 𝜙𝐴𝐵 − 𝜙𝐵𝐴 = −2𝜋 (𝜏𝐴𝐴 + 𝜏𝐵𝐵 − 𝜏𝐴𝐵 − 𝜏𝐵𝐴)𝑓𝑐 to cancel the unknown transmitting and receiving delays [39]. We can further assume the distances between the speaker and microphone of same device are fixed so that 𝜏𝐴𝐴 and 𝜏𝐵𝐵 are known in advance. Therefore, we can measure the ToF by: 𝜏𝐴𝐵 + 𝜏𝐵𝐴 =  𝜏𝐴𝐴 + 𝜏𝐵𝐵 + 𝜙𝐴𝐴 + 𝜙𝐵𝐵 − 𝜙𝐴𝐵 − 𝜙𝐵𝐴 2𝜋 𝑓𝑐  . (6) Proc. ACM Interact. Mob. Wearable Ubiquitous Technol., Vol. 6, No. 3, Article 144. Publication date: September 2022