# A TDoA Localization Scheme for Underwater Sensor Networks with Use of Multilinear Chirp Signals

Authors: En Cheng; Xizhou Lin; Shengli Chen; Fei Yuan
Journal: Mobile Information Systems
 (2016)

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## Abstract

Due to the multipath, Doppler, and other effects, the node location signals have high probability of access collision in the underwater acoustic sensor networks (UW-ASNs), and therefore, it causes the signal lost and the access block; therefore, it constrains the networks performance. In this paper, we take the multilinear chirp (MLC) signals as the location signal to improve the anticollision ability. In order to increase the detection efficiency of MLC, we propose a fast efficient detection method called mixing change rate-fractional Fourier transform (MCR-FrFT). This method transforms the combined rates of MLC into symmetry triangle rates and then separates the multiuser signals based on the transformed rates by using FrFT. Theoretical derivation and simulation results show that the proposed method can detect the locations signals, estimate the time difference of arrival (TDoA), reduce the multiple access interference, and improve the location performance.


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## Body

## 1. Introduction

During the last couple of years, we could observe a growing interest in underwater acoustic sensor networks (UW-ASNs). One important reason is that the UW-ASNs improve ocean exploration ability and fulfill the needs of a multitude of underwater applications, including oceanographic data collection, warning systems for natural disasters, ecological applications, military underwater surveillance, assisted navigation, and industrial applications. However, the location of the sensors needs to be determined because sensed data can only be interpreted meaningfully when referenced to the location of the sensor. Due to the absorption of electromagnetic wave, the well-known Global Position System (GPS) receivers, which may be used in terrestrial systems to accurately estimate the geographical locations of sensor nodes, could not work properly underwater [1]. Therefore, localization for UW-ASNs has been one of the major research topics since UW-ASNs started to draw the attention of the networking community in the early 2000s [2].Sensors’ self-localization is the basic and key requirement for the self-organization of UW-ASNs. It can be achieved by leveraging the low speed of sound in water to accurately determine the internode distance [3]. Localization of underwater equipment has also been an essential part of the traditional oceanographic systems where it has been established by one of two techniques: short base line (SBL) [4] or long base line (LBL) [5]. Time difference of arrival (TDoA) method is typically used in LBL systems. In TDoA-based localization, the difference in the measured time of arrivals of signals, received from a pair of reference nodes, is translated to the difference in range estimates with those reference nodes and gives rise to a hyperbola for the unknown position of the node (target node). A unique estimation of the target node position can be obtained by intersecting three such hyperbolas. However, this technique requires that reference nodes transmit at near concurrent time because of the water currents (the motion of the target node).Traditional LBL systems are designed for small deployments consisting of a few reference nodes (here called beacons), whose acoustic transmissions can be received by all submersibles within the deployment. LBL provides sufficient localization accuracy for such deployments. This is because signals from few beacons can be scheduled to occur within a short time window. Since all the target nodes are within the communication range of the beacons, these signals also arrive at near concurrent times at each target node. Therefore, the TDoA is suitable. However, as the spatial extent and the size of the system are scaled up, the number of location signals increases too. Therefore, to avoid location signals’ collision is also urgent, within the short time window.


## 2. Related Works

Acoustic-based localization was studied because of its application in underwater sensor networks localization. In [6], a TDoA-based silent positioning algorithm termed as UPS for UW-ASNs was proposed. In [7], the localization problem in sparse 3D underwater sensor networks was studied based on TDoA. In [8], a TDoA for multiple acoustic sources in reverberant environments was proposed, and the ambiguities in the TDoA estimation caused by multipath propagation and multiple sources were resolved by exploiting two TDoA constraints: the raster condition and the zero cyclic sum condition. However, these references do not consider the collisions of location signals in the target sensor.In [9], a different system where transmissions from multiple reference nodes have to be sufficiently lagged over a time epoch to avoid collisions was presented, but it needs a lot of time to localize the target sensor. In [10], the authors introduced a method for designing localization signal based on Code Division Multiple Access (CDMA), and the modulation mode is ASK; but as described in [11] about the physical layer of underwater channel, ASK modulation is not suitable for underwater channel because of high attenuation. What is more, the location signals are transmitted that are sufficiently lagged, which is the same as [9]. In [8], to avoid collisions and meet the signal concurrency requirement of traditional LBL systems simultaneously, a TDoA location scheme for the orthogonal frequency division multiplexing (OFDM) was presented; this outperforms the location schemes with traditional TDoA estimations. Unfortunately, the OFDM system is very sensitive to Doppler, which is one of the main characters of underwater channel. In [11, 12], a time-varying multichirp rate modulation for multiple access systems was used to avoid collision in terrestrial communication. Chirp signal is widely used in underwater acoustic communication [13], because it is robust to channel noise and resistant to multipath fading [14] and has low Doppler sensitivity [13].Some different methods are used to estimate the parameters of chirp signal. In [12], the method of time-delay frequency mixing (rate reduction) to convert the chirp signal with different chirp rate into MFSK signal is proposed, but it is not suited for combined chirp signal because of the change of chirp rate. In [11, 12], the detection technique of matched filter receiver (correlation method) was mentioned, but it is difficult and also needs amplitude modulation which is not suitable for underwater channel. The fractional Fourier transform (FrFT) presents best localization performance in a certain FrFT domain, which is useful for the detection and estimation of multicomponent linear frequency modulation (LFM) signals [15] and some improved algorithms based on FrFT are also proposed, such as EEMD-FrFT [16] and STFT [17]; they overcome some disadvantages such as high computation cost for combined chirp signals. In this paper, a method called mixing change rate-FrFT (MCR-FrFT) is proposed to deal with the drawback.In this paper, a TDoA location scheme using the multilinear chirp modulation signals (called MLC signals) is proposed for UW-ASNs. The rest of the paper is organized as follows: in Section3, the system model and location signals are described. In Section 4, we analyze the system with the underwater environment in TDoA estimation. In Section 5, we present our simulation results on our passive localization scheme for the multilinear chirp signals. Section 6 concludes this paper.


## 3. System Model

The system architecture of the scheme is depicted in Figure1. The white nodes are reference nodes whose locations are known, while the black node is target node which needs to be located. Arrows indicate the communication links. The system is designed to work with only one-way reference node transmissions. The reference nodes and target node are fully synchronized with each other and the target node is within the communication range of at least four reference nodes (3D network) or three reference nodes (2D network). Since the communication range and distance between reference nodes are all hundred meters’ scale, the location signals get high possibility to arrive at the same time. Therefore, avoiding collision is a key problem to node localization. We introduce the chosen location signal and the processing of the signal in the following subsections.Figure 1
Location system architecture using combined chirp signals.


### 3.1. Location Signal-Multilinear Chirp (MLC) Signal

Due to the match of MLC and underwater channel, the MLC is chosen as the location signal, and its time-frequency characteristic of location signals-MLC is shown as Figure2, where T is the duration time of location signal. Nodes are denoted by 1,2,…,M,M+1,…,2M-1,2M, where the first M nodes are with positive combined slopes and the second M nodes are with negative combined slopes. Each signal is composed of two segments with two different slopes. The signal for mth node is expressed as(1)Smt=Smft+Smbt.Figure 2
The time-frequency characteristic of location signals.Subscriptsf and b indicate the first half and the second half of duration time T, respectively. And (2)Smft=Aexp⁡jπμmft2·exp⁡j2πfct,when0≤t<T/2, and(3)Smbt=Aexp⁡πμmbt-T22·exp⁡j2πfc+μmf2Tt-T2,whenT/2≤t<T,where fc is the carrier frequency, m is the node number, m=1,2,…,M,…,2M, A is the amplitude of signal, and 2M is the total number of nodes in UW-ASNs. μmf is mth node’s slope within the first half of signal duration and μmb is the slope of mth node within the second half of signal duration. The slopes of general combined chirp signal can be expressed as(4)μmf=2M+1-mBM+1T,μmb=2mBM+1T,m∈1,M,(5)μmf=2m-2M-1BM+1T,μmb=2m-MBM+1T,m∈M,2M.As shown in Figure2, each location signal has the same bandwidth B and duration time T; thus the system produces equal time-bandwidth product D=BT for all nodes. Equal D makes the nodes in the UW-ASNs have the equal position, which is important for mobile nodes.


### 3.2. Detection Block Diagram and Principle

The localization signal detection problem could be described as deciding which one of the 2Mpossible signals was transmitted, given the received signal during the interval time of (0,T). The method of parameter estimation proposed in the paper is called MCR-FrFT, whose block diagram is shown as Figure 3(a). The received signal is divided into two branches. Each branch has its own local signals xlocalu(t) and xlocald(t), respectively, xlocalu(t) for the up branch m≤M and xlocald(t) for the down branch M<m≤2M.Figure 3
System block diagram. (a) The detection block diagram. (b) The block diagram of MCR.
(a) (b)


#### 3.2.1. Mixing Change Rate (MCR)

The purpose of MCR is to change the chirp rates of MLC signals into a new set of rates. It includes two parts, multiplier and low pass filter. The block diagram of MCR is shown in Figure3(b):(6)xlocalt=xlocalut=exp⁡jπBTt2·exp⁡j2πfct,xlocaldt=exp⁡-jπBTt2·exp⁡j2πfct,r0t=rft∗hLPFt,rft=rt·xlocalt,where hLPF(t) is the impulse response function of low pass filter (LPF). Assuming that mth node’s signal is received and m<M, we get the received signal as (7) by substituting (2)–(4) into (1). n(t) is the underwater acoustic channel noise. After the MCR, the high frequency terms have been filtered out and the received signal becomes as (8) and w(t)=nt·xlocalt∗hLPF(t).The frequency and time relationship of received signals after multiplier(rf) are shown in Figure 4. From the figure, mth location signal, composed of two different positive slopes, is changed into two branches of new multilinear chirp signals. The up branch m≤M has chirp rates of positive and negative relationship, while the down branch M<m≤2M is composed of another chirp rate. However, the received signal only has the up branch after being filtered in the LPF of MCR:(7)rt=Smt+nt=Aexp⁡jπμmft2·exp⁡j2πfct·rect⁡4t-T2T+Aexp⁡jπμmbt-T22·exp⁡j2πfc+μmf2Tt-T2·rect⁡4t-3T2T+nt,(8)r0t=wt+A22cos⁡πμmf-BTt2·rect⁡4t-T2T+cos⁡2πM+1-mM+1B-B2t-T2+πμmb-BTt-T22·rect⁡4t-3T2T,m≤M,cos⁡πμmf+BTt2·rect⁡4t-T2T+cos⁡2πM+1-mM+1B+12B·t-T2+πμmb+BTt-T22·rect⁡4t-3T2T,M<m≤2M.Figure 4
The time-frequency characteristic after MCR.


#### 3.2.2. Fractional Fourier Transform (FrFT)

FrFT is a generalized Fourier transformation form and it can be regarded as the Fourier transform topth order, where p needs not be an integer; thus it can transform a function to any intermediate domain between time and frequency [18]. For signal x(t), the FrFT has the following form:(9)Xpu=∫-∞+∞Kpt,uxtdt,and the transform kernel is(10)Kpt,u=1-jcotα2πexp⁡j12t2cot⁡α+12u2cot⁡α-tucsc⁡α,α≠nπ,δt-u,α=2nπ,δt+u,α=2n±1,where p is the transform order, α is the angle of rotation, and α=pπ/2. When p=1, FrFT will degenerate into Fourier transformation, and when p=0, FrFT is just the original signal.Simple component LFM signal with noise has the following form:(11)rt=A0cos⁡2πfct+πut2+nt,where 0≤t≤T and n(t) is white noise. The FrFT of r(t) is(12)Xpu=∫-∞+∞rtKpt,udt=A01-jcot⁡α2πej1/2u2cot⁡α·∫0Te2jπ1/2t2u+cot⁡α/2π+tfc-ucscα/2πdt+∫0TntKpt,udt.In (12), the LFM signal is an impulse function only in the appropriate fractional Fourier domain. The amplitude of energy aggregation of LFM signal will exhibit obvious peak by doing an appropriate p-order FrFT, while the noise, which cannot appear in energy aggregation at any fractional Fourier domain, is distributed on the whole time-frequency plane evenly.Figure4 shows that the set of chirp rates is changed from two different slopes, which needs two different p values to complete its parameter estimation, to a new set of chirp rates that the new two slopes are with same absolute value but one positive and one negative by using the MCR. At this condition, only one value needs to complete the parameter estimation, that is, one FrFT for a single branch, while two FrFT are needed for the scheme of FrFT (without MCR). Since FrFT cost more computation than the MCR module, our MCR-FrFT saves computations of system. In practical application, the discrete fractional Fourier transform (DFrFT) is usually used.


## 3.1. Location Signal-Multilinear Chirp (MLC) Signal

Due to the match of MLC and underwater channel, the MLC is chosen as the location signal, and its time-frequency characteristic of location signals-MLC is shown as Figure2, where T is the duration time of location signal. Nodes are denoted by 1,2,…,M,M+1,…,2M-1,2M, where the first M nodes are with positive combined slopes and the second M nodes are with negative combined slopes. Each signal is composed of two segments with two different slopes. The signal for mth node is expressed as(1)Smt=Smft+Smbt.Figure 2
The time-frequency characteristic of location signals.Subscriptsf and b indicate the first half and the second half of duration time T, respectively. And (2)Smft=Aexp⁡jπμmft2·exp⁡j2πfct,when0≤t<T/2, and(3)Smbt=Aexp⁡πμmbt-T22·exp⁡j2πfc+μmf2Tt-T2,whenT/2≤t<T,where fc is the carrier frequency, m is the node number, m=1,2,…,M,…,2M, A is the amplitude of signal, and 2M is the total number of nodes in UW-ASNs. μmf is mth node’s slope within the first half of signal duration and μmb is the slope of mth node within the second half of signal duration. The slopes of general combined chirp signal can be expressed as(4)μmf=2M+1-mBM+1T,μmb=2mBM+1T,m∈1,M,(5)μmf=2m-2M-1BM+1T,μmb=2m-MBM+1T,m∈M,2M.As shown in Figure2, each location signal has the same bandwidth B and duration time T; thus the system produces equal time-bandwidth product D=BT for all nodes. Equal D makes the nodes in the UW-ASNs have the equal position, which is important for mobile nodes.


## 3.2. Detection Block Diagram and Principle

The localization signal detection problem could be described as deciding which one of the 2Mpossible signals was transmitted, given the received signal during the interval time of (0,T). The method of parameter estimation proposed in the paper is called MCR-FrFT, whose block diagram is shown as Figure 3(a). The received signal is divided into two branches. Each branch has its own local signals xlocalu(t) and xlocald(t), respectively, xlocalu(t) for the up branch m≤M and xlocald(t) for the down branch M<m≤2M.Figure 3
System block diagram. (a) The detection block diagram. (b) The block diagram of MCR.
(a) (b)


### 3.2.1. Mixing Change Rate (MCR)

The purpose of MCR is to change the chirp rates of MLC signals into a new set of rates. It includes two parts, multiplier and low pass filter. The block diagram of MCR is shown in Figure3(b):(6)xlocalt=xlocalut=exp⁡jπBTt2·exp⁡j2πfct,xlocaldt=exp⁡-jπBTt2·exp⁡j2πfct,r0t=rft∗hLPFt,rft=rt·xlocalt,where hLPF(t) is the impulse response function of low pass filter (LPF). Assuming that mth node’s signal is received and m<M, we get the received signal as (7) by substituting (2)–(4) into (1). n(t) is the underwater acoustic channel noise. After the MCR, the high frequency terms have been filtered out and the received signal becomes as (8) and w(t)=nt·xlocalt∗hLPF(t).The frequency and time relationship of received signals after multiplier(rf) are shown in Figure 4. From the figure, mth location signal, composed of two different positive slopes, is changed into two branches of new multilinear chirp signals. The up branch m≤M has chirp rates of positive and negative relationship, while the down branch M<m≤2M is composed of another chirp rate. However, the received signal only has the up branch after being filtered in the LPF of MCR:(7)rt=Smt+nt=Aexp⁡jπμmft2·exp⁡j2πfct·rect⁡4t-T2T+Aexp⁡jπμmbt-T22·exp⁡j2πfc+μmf2Tt-T2·rect⁡4t-3T2T+nt,(8)r0t=wt+A22cos⁡πμmf-BTt2·rect⁡4t-T2T+cos⁡2πM+1-mM+1B-B2t-T2+πμmb-BTt-T22·rect⁡4t-3T2T,m≤M,cos⁡πμmf+BTt2·rect⁡4t-T2T+cos⁡2πM+1-mM+1B+12B·t-T2+πμmb+BTt-T22·rect⁡4t-3T2T,M<m≤2M.Figure 4
The time-frequency characteristic after MCR.


### 3.2.2. Fractional Fourier Transform (FrFT)

FrFT is a generalized Fourier transformation form and it can be regarded as the Fourier transform topth order, where p needs not be an integer; thus it can transform a function to any intermediate domain between time and frequency [18]. For signal x(t), the FrFT has the following form:(9)Xpu=∫-∞+∞Kpt,uxtdt,and the transform kernel is(10)Kpt,u=1-jcotα2πexp⁡j12t2cot⁡α+12u2cot⁡α-tucsc⁡α,α≠nπ,δt-u,α=2nπ,δt+u,α=2n±1,where p is the transform order, α is the angle of rotation, and α=pπ/2. When p=1, FrFT will degenerate into Fourier transformation, and when p=0, FrFT is just the original signal.Simple component LFM signal with noise has the following form:(11)rt=A0cos⁡2πfct+πut2+nt,where 0≤t≤T and n(t) is white noise. The FrFT of r(t) is(12)Xpu=∫-∞+∞rtKpt,udt=A01-jcot⁡α2πej1/2u2cot⁡α·∫0Te2jπ1/2t2u+cot⁡α/2π+tfc-ucscα/2πdt+∫0TntKpt,udt.In (12), the LFM signal is an impulse function only in the appropriate fractional Fourier domain. The amplitude of energy aggregation of LFM signal will exhibit obvious peak by doing an appropriate p-order FrFT, while the noise, which cannot appear in energy aggregation at any fractional Fourier domain, is distributed on the whole time-frequency plane evenly.Figure4 shows that the set of chirp rates is changed from two different slopes, which needs two different p values to complete its parameter estimation, to a new set of chirp rates that the new two slopes are with same absolute value but one positive and one negative by using the MCR. At this condition, only one value needs to complete the parameter estimation, that is, one FrFT for a single branch, while two FrFT are needed for the scheme of FrFT (without MCR). Since FrFT cost more computation than the MCR module, our MCR-FrFT saves computations of system. In practical application, the discrete fractional Fourier transform (DFrFT) is usually used.


## 3.2.1. Mixing Change Rate (MCR)

The purpose of MCR is to change the chirp rates of MLC signals into a new set of rates. It includes two parts, multiplier and low pass filter. The block diagram of MCR is shown in Figure3(b):(6)xlocalt=xlocalut=exp⁡jπBTt2·exp⁡j2πfct,xlocaldt=exp⁡-jπBTt2·exp⁡j2πfct,r0t=rft∗hLPFt,rft=rt·xlocalt,where hLPF(t) is the impulse response function of low pass filter (LPF). Assuming that mth node’s signal is received and m<M, we get the received signal as (7) by substituting (2)–(4) into (1). n(t) is the underwater acoustic channel noise. After the MCR, the high frequency terms have been filtered out and the received signal becomes as (8) and w(t)=nt·xlocalt∗hLPF(t).The frequency and time relationship of received signals after multiplier(rf) are shown in Figure 4. From the figure, mth location signal, composed of two different positive slopes, is changed into two branches of new multilinear chirp signals. The up branch m≤M has chirp rates of positive and negative relationship, while the down branch M<m≤2M is composed of another chirp rate. However, the received signal only has the up branch after being filtered in the LPF of MCR:(7)rt=Smt+nt=Aexp⁡jπμmft2·exp⁡j2πfct·rect⁡4t-T2T+Aexp⁡jπμmbt-T22·exp⁡j2πfc+μmf2Tt-T2·rect⁡4t-3T2T+nt,(8)r0t=wt+A22cos⁡πμmf-BTt2·rect⁡4t-T2T+cos⁡2πM+1-mM+1B-B2t-T2+πμmb-BTt-T22·rect⁡4t-3T2T,m≤M,cos⁡πμmf+BTt2·rect⁡4t-T2T+cos⁡2πM+1-mM+1B+12B·t-T2+πμmb+BTt-T22·rect⁡4t-3T2T,M<m≤2M.Figure 4
The time-frequency characteristic after MCR.


## 3.2.2. Fractional Fourier Transform (FrFT)

FrFT is a generalized Fourier transformation form and it can be regarded as the Fourier transform topth order, where p needs not be an integer; thus it can transform a function to any intermediate domain between time and frequency [18]. For signal x(t), the FrFT has the following form:(9)Xpu=∫-∞+∞Kpt,uxtdt,and the transform kernel is(10)Kpt,u=1-jcotα2πexp⁡j12t2cot⁡α+12u2cot⁡α-tucsc⁡α,α≠nπ,δt-u,α=2nπ,δt+u,α=2n±1,where p is the transform order, α is the angle of rotation, and α=pπ/2. When p=1, FrFT will degenerate into Fourier transformation, and when p=0, FrFT is just the original signal.Simple component LFM signal with noise has the following form:(11)rt=A0cos⁡2πfct+πut2+nt,where 0≤t≤T and n(t) is white noise. The FrFT of r(t) is(12)Xpu=∫-∞+∞rtKpt,udt=A01-jcot⁡α2πej1/2u2cot⁡α·∫0Te2jπ1/2t2u+cot⁡α/2π+tfc-ucscα/2πdt+∫0TntKpt,udt.In (12), the LFM signal is an impulse function only in the appropriate fractional Fourier domain. The amplitude of energy aggregation of LFM signal will exhibit obvious peak by doing an appropriate p-order FrFT, while the noise, which cannot appear in energy aggregation at any fractional Fourier domain, is distributed on the whole time-frequency plane evenly.Figure4 shows that the set of chirp rates is changed from two different slopes, which needs two different p values to complete its parameter estimation, to a new set of chirp rates that the new two slopes are with same absolute value but one positive and one negative by using the MCR. At this condition, only one value needs to complete the parameter estimation, that is, one FrFT for a single branch, while two FrFT are needed for the scheme of FrFT (without MCR). Since FrFT cost more computation than the MCR module, our MCR-FrFT saves computations of system. In practical application, the discrete fractional Fourier transform (DFrFT) is usually used.


## 4. TDoA Estimation

There are two steps of MCR-FrFT at the receiving end. In the first step, MCR takes effect to convert the chirp rates of location signal into a new set of chirp rates. Then, FrFT estimates the parameters. There are two parameters considered: chirp rate, which represents the reference node, and initial frequency, which can be used for estimation of the processing delay.


### 4.1. Collision Avoidance

One of the advantages of the paper is to avoid collision. Taking an example with three reference nodesm,l,n∈[1,M],(13)rt=Smt+Slt+Snt+nt.According to (4) and (8), there are different sets of chirp rates included in the MCR results with different m,n,l group:(14)r0t=A22cos⁡πM+1-2mBM+1Tt2+cos⁡πM+1-2lBM+1Tt2+cos⁡πM+1-2nBM+1Tt2·rect⁡4t-T2T+A22cos⁡2πM+1-mM+1B-B2t-T2-πM+1-2mBM+1Tt-T22+cos⁡2πM+1-lM+1B-B2t-T2-πM+1-2lBM+1Tt-T22+cos⁡2πM+1-nM+1B-B2t-T2-πM+1-2nBM+1Tt-T22·rect⁡4t-3T2T+wt.From (14), we could see that the performance of MLC signals mainly depends on the cross-coherence between the different location signals even by passing the MCR. Ideally, the signals should be orthogonal with zero cross-coherence to cancel the multiple access interference.SetA=2E/T, where E is the signal energy in the whole duration, and the cross-correlation of signals ρ takes the following form:(15)ρl,m=1E∫0TSl∗t·Smtdt.Substituting the corresponding values ofS(t) in (13) and neglecting the integration over the higher frequencies, we get (16)ρl,m=A48E∫0T/2cos⁡2πl-mBM+1Tt2dt+∫T/2Tcos⁡2πl-mM+1Bt-T2-2πl-mM+1TBt-T22dt.Take v=l-mB/(M+1)Tt,u=l-mB/M+1Tt-T/2; then (16) can be changed to(17)
ρ
l
,
m
=
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2
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cos
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=
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4
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Δ
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·
D
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,where D=BT is time-bandwidth product and l-m=Δn. We could see that ρ has an oscillatory nature as a function of D, M, and Δn, due to the Fresnel function form of (17). Usually, smaller cross-correlation ρ indicates a better performance of the system.The normalized sample values can be expressed as(18)ρM×M=1ρ1,2⋯ρ1,Mρ211⋯ρ2,M⋮⋮⋮ρM,1ρM,2⋯1,where ρi,j is the cross-coherence between ith and jth location signal and number 1 on the matrix diagonal represents the location signal itself. Figure 5 shows that the cross-coherence is an oscillatory function of time-bandwidth product D or the total number of MLC signals or the difference between chirp rates Δn, when the other two parameters are set fixed. Figure 5(a) indicates that ρ decreases oscillatorily as D increases when M=30 and Δn=2. Figure 5(b) indicates that ρ increases oscillatorily as M increases when D=100 and Δn=2. Figure 5(c) indicates that ρ decreases oscillatorily as Δn increases when D=100 and M=30. We could select the assembling parameter according to (17) in practice.Figure 5
Cross-coherence functionρ as a function of (a) time-bandwidth product with M=30 and Δn=2. (b) Total node numbers M with D=100 and Δn=2. (c) User difference with M=30 and D=100.
(a) (b) (c)


### 4.2. The Result of Underwater Channel

The impulse response of an underwater acoustic channel is influenced by the geometry of the channel and its reflection properties, which determine the number of significant propagation paths, their relative strengths, and delays. Strictly speaking, there are infinitely many signal echoes, but those that have undergone multiple reflections and lost much of the energy can be discarded, leaving only a finite number of significant paths. At this point, the channel impulse response can be expressed as(19)ht=A0δt-τ0+∑i=1N-1Aiδt-τi,where N is the total number of propagation paths, and i=0 corresponding to the direct path, Ai is the amplitude of ith propagation path at receiving side, and τi is the propagation delay of ith propagation path.Assuming that the transmitted signal iss(t), then the received signal r(t) can be expressed as(20)rt=A0st-τ0+∑i=1N-1Aist-τi+nt=∑i=0N-1Ai·st-τi+nt.Taking the Doppler into account, let s(t)=Sm(t); then the received signal can be expressed as (21)rt=∑i=0N-1Ai·Smt-τi·exp⁡j2πεit-τi+nt=∑i=0N-1Aicos⁡2πf0t-τi+πμmft-τi2·rect⁡4t-τi-T2T+cos⁡2πf0+M+1-mM+1Bt-τi-T2+πμmbt-τi-T22·rect⁡4t-τi-3T2T·exp⁡j2πεit-τi+nt.Substituting the corresponding values into (21), neglecting the third branch, and using MCR we get(22)r0t=12∑i=1NAiXt·exp⁡j2πεit-τi+wt,where(23)Xt=cos⁡2π2M+1-mBτiM+1Tt-πM+1-2mBM+1Tt2+2πf0τ-2πM+1-mBM+1Tτi2·rect⁡4t-τi-T2T+cos⁡2πM+1-2mBT-4mBτi2M+1Tt-T2-πM+1-2mBM+1Tt-T22-2πM+1-mM+1Bτi-2πf0τi+2πmBM+1Tτi2·rect⁡4t-τi-3T2T.From the above equations, we could see that the frequencies of combined chirp signal are changed due to the Doppler, but the sets of chirp rates for different nodes are unchanged. Since our location signals are detected according to the different sets of chirp rates, which is only determined bymth order for the fixed system, the MLC signals would not be affected by multipath properties of underwater acoustic channel.


### 4.3. Positioning Process

The target node receives the location signals and determines the corresponding reference node with time(tm,tl,tn) and position (xm,ym),(xl,yl),(xn,yn), when and where the location signal was transmitted. In this system, the location signals are received at the same time so the difference between tm and tl is a constant; in other words, the difference between the corresponding distance rm and rl is a constant. Of course, rm-rn and rl-rn are constants too. Based on the property of hyperbola, the target node is localized by the intersection of three curves as shown in Figure 6.Figure 6
The hyperbola-based localization scheme with TDoA.In Figure6, c is the speed of sound in underwater channel in m/s. Hyperbolic curve Hl,n is determined by nodes l and n, while Hm,n and Hl,m are determined by m,n and l,n respectively.For this system, the scaling parameter and the length of interval areS=(T/fs)1/2=N1/2/fs and Δx=Tfs1/2=N1/2, respectively, where N=fs·T is the sample number. When p=p0 is the optimal p order, then there are two different energy accumulation peaks (assuming that the coordinates are m01 and m02) symmetrical about N/2 that are got.(24)m01=N2+Δx·fi·S·sin⁡α0,m02=N2-Δx·fi·S·sin⁡α0,where fi is the frequency of combined chirp signal with two opposite slopes at T/2 and α0=p0π/2m>N/2 or α0=2-p0π/2m<N/2. Assuming that the time difference between the first arrival location signal and the other location signal at the receiver is τ, the later location signal’s “center frequency” is changed from fi=f0+k(T/2) to fi′=f0+k(T/2-τ) within the short time window and the energy accumulation peak position is changed from m01,m02 to (m01′,m02′).According to (24), we get(25)m1-m1′+m2-m2′2=Δx·f-f′·S·sin⁡α.So time differenceτ can be expressed as(26)τ=m01+m02-m01′-m02′2·Δx·S·k·sin⁡α,where m01 and m02 are theoretical values, while m01′ and m02′ are measured values. For each location signal, Δx·S·k·sin⁡(α) is a constant; so, according to the theoretical values and measured values of position about power accumulation peak, the time difference can be got from (26). So the true value of distance difference in TDoA is rm-rn+c·τ or rl-rn+c·τ.


## 4.1. Collision Avoidance

One of the advantages of the paper is to avoid collision. Taking an example with three reference nodesm,l,n∈[1,M],(13)rt=Smt+Slt+Snt+nt.According to (4) and (8), there are different sets of chirp rates included in the MCR results with different m,n,l group:(14)r0t=A22cos⁡πM+1-2mBM+1Tt2+cos⁡πM+1-2lBM+1Tt2+cos⁡πM+1-2nBM+1Tt2·rect⁡4t-T2T+A22cos⁡2πM+1-mM+1B-B2t-T2-πM+1-2mBM+1Tt-T22+cos⁡2πM+1-lM+1B-B2t-T2-πM+1-2lBM+1Tt-T22+cos⁡2πM+1-nM+1B-B2t-T2-πM+1-2nBM+1Tt-T22·rect⁡4t-3T2T+wt.From (14), we could see that the performance of MLC signals mainly depends on the cross-coherence between the different location signals even by passing the MCR. Ideally, the signals should be orthogonal with zero cross-coherence to cancel the multiple access interference.SetA=2E/T, where E is the signal energy in the whole duration, and the cross-correlation of signals ρ takes the following form:(15)ρl,m=1E∫0TSl∗t·Smtdt.Substituting the corresponding values ofS(t) in (13) and neglecting the integration over the higher frequencies, we get (16)ρl,m=A48E∫0T/2cos⁡2πl-mBM+1Tt2dt+∫T/2Tcos⁡2πl-mM+1Bt-T2-2πl-mM+1TBt-T22dt.Take v=l-mB/(M+1)Tt,u=l-mB/M+1Tt-T/2; then (16) can be changed to(17)
ρ
l
,
m
=
A
2
4
M
+
1
l
-
m
B
T
∫
0
l
-
m
B
T
/
4
M
+
1
cos
⁡
2
π
v
2
d
v
+
∫
0
l
-
m
B
T
/
4
M
+
1
cos
⁡
2
π
l
-
m
B
T
M
+
1
u
-
2
π
u
2
d
u
=
A
2
4
M
+
1
Δ
n
·
D
∫
0
Δ
n
·
D
/
4
M
+
1
cos
⁡
2
π
t
2
d
t
+
cos
⁡
2
π
Δ
n
·
D
M
+
1
t
-
2
π
t
2
d
t
,where D=BT is time-bandwidth product and l-m=Δn. We could see that ρ has an oscillatory nature as a function of D, M, and Δn, due to the Fresnel function form of (17). Usually, smaller cross-correlation ρ indicates a better performance of the system.The normalized sample values can be expressed as(18)ρM×M=1ρ1,2⋯ρ1,Mρ211⋯ρ2,M⋮⋮⋮ρM,1ρM,2⋯1,where ρi,j is the cross-coherence between ith and jth location signal and number 1 on the matrix diagonal represents the location signal itself. Figure 5 shows that the cross-coherence is an oscillatory function of time-bandwidth product D or the total number of MLC signals or the difference between chirp rates Δn, when the other two parameters are set fixed. Figure 5(a) indicates that ρ decreases oscillatorily as D increases when M=30 and Δn=2. Figure 5(b) indicates that ρ increases oscillatorily as M increases when D=100 and Δn=2. Figure 5(c) indicates that ρ decreases oscillatorily as Δn increases when D=100 and M=30. We could select the assembling parameter according to (17) in practice.Figure 5
Cross-coherence functionρ as a function of (a) time-bandwidth product with M=30 and Δn=2. (b) Total node numbers M with D=100 and Δn=2. (c) User difference with M=30 and D=100.
(a) (b) (c)


## 4.2. The Result of Underwater Channel

The impulse response of an underwater acoustic channel is influenced by the geometry of the channel and its reflection properties, which determine the number of significant propagation paths, their relative strengths, and delays. Strictly speaking, there are infinitely many signal echoes, but those that have undergone multiple reflections and lost much of the energy can be discarded, leaving only a finite number of significant paths. At this point, the channel impulse response can be expressed as(19)ht=A0δt-τ0+∑i=1N-1Aiδt-τi,where N is the total number of propagation paths, and i=0 corresponding to the direct path, Ai is the amplitude of ith propagation path at receiving side, and τi is the propagation delay of ith propagation path.Assuming that the transmitted signal iss(t), then the received signal r(t) can be expressed as(20)rt=A0st-τ0+∑i=1N-1Aist-τi+nt=∑i=0N-1Ai·st-τi+nt.Taking the Doppler into account, let s(t)=Sm(t); then the received signal can be expressed as (21)rt=∑i=0N-1Ai·Smt-τi·exp⁡j2πεit-τi+nt=∑i=0N-1Aicos⁡2πf0t-τi+πμmft-τi2·rect⁡4t-τi-T2T+cos⁡2πf0+M+1-mM+1Bt-τi-T2+πμmbt-τi-T22·rect⁡4t-τi-3T2T·exp⁡j2πεit-τi+nt.Substituting the corresponding values into (21), neglecting the third branch, and using MCR we get(22)r0t=12∑i=1NAiXt·exp⁡j2πεit-τi+wt,where(23)Xt=cos⁡2π2M+1-mBτiM+1Tt-πM+1-2mBM+1Tt2+2πf0τ-2πM+1-mBM+1Tτi2·rect⁡4t-τi-T2T+cos⁡2πM+1-2mBT-4mBτi2M+1Tt-T2-πM+1-2mBM+1Tt-T22-2πM+1-mM+1Bτi-2πf0τi+2πmBM+1Tτi2·rect⁡4t-τi-3T2T.From the above equations, we could see that the frequencies of combined chirp signal are changed due to the Doppler, but the sets of chirp rates for different nodes are unchanged. Since our location signals are detected according to the different sets of chirp rates, which is only determined bymth order for the fixed system, the MLC signals would not be affected by multipath properties of underwater acoustic channel.


## 4.3. Positioning Process

The target node receives the location signals and determines the corresponding reference node with time(tm,tl,tn) and position (xm,ym),(xl,yl),(xn,yn), when and where the location signal was transmitted. In this system, the location signals are received at the same time so the difference between tm and tl is a constant; in other words, the difference between the corresponding distance rm and rl is a constant. Of course, rm-rn and rl-rn are constants too. Based on the property of hyperbola, the target node is localized by the intersection of three curves as shown in Figure 6.Figure 6
The hyperbola-based localization scheme with TDoA.In Figure6, c is the speed of sound in underwater channel in m/s. Hyperbolic curve Hl,n is determined by nodes l and n, while Hm,n and Hl,m are determined by m,n and l,n respectively.For this system, the scaling parameter and the length of interval areS=(T/fs)1/2=N1/2/fs and Δx=Tfs1/2=N1/2, respectively, where N=fs·T is the sample number. When p=p0 is the optimal p order, then there are two different energy accumulation peaks (assuming that the coordinates are m01 and m02) symmetrical about N/2 that are got.(24)m01=N2+Δx·fi·S·sin⁡α0,m02=N2-Δx·fi·S·sin⁡α0,where fi is the frequency of combined chirp signal with two opposite slopes at T/2 and α0=p0π/2m>N/2 or α0=2-p0π/2m<N/2. Assuming that the time difference between the first arrival location signal and the other location signal at the receiver is τ, the later location signal’s “center frequency” is changed from fi=f0+k(T/2) to fi′=f0+k(T/2-τ) within the short time window and the energy accumulation peak position is changed from m01,m02 to (m01′,m02′).According to (24), we get(25)m1-m1′+m2-m2′2=Δx·f-f′·S·sin⁡α.So time differenceτ can be expressed as(26)τ=m01+m02-m01′-m02′2·Δx·S·k·sin⁡α,where m01 and m02 are theoretical values, while m01′ and m02′ are measured values. For each location signal, Δx·S·k·sin⁡(α) is a constant; so, according to the theoretical values and measured values of position about power accumulation peak, the time difference can be got from (26). So the true value of distance difference in TDoA is rm-rn+c·τ or rl-rn+c·τ.


## 5. Experimental Results

### 5.1. MATLAB Simulation

The simulation tool MATLAB is used to evaluate the performance of MCR-FrFT. We set typical UW-ASNs parameters set as follows: the total number of nodes,M=12; the bandwidth, B=5kHz; the signal duration time, T=0.04s; and the sample frequency, fs=90kHz. White Gaussian noise is added and the signal noise ratio (SNR) is set to 0 dB. The number of underwater multipaths is 4, the delay is τ=00.010.0130.017, and the amplitude is A=10.40.30.1, respectively. Discrete FrFT is used and order p at the receiving end is given by p=2/π·arccotM+1fs/(M+1-2mB)+1 for mth node.We take single location signal and multilocation signals (three location signals), for example. For simplicity, we only consider the first half ofM (here is 6) in the system. The FrFT results are shown in Tables 1 and 2.Table 1
The single location signal.
p order Node Node 1 Node 2 Node 3 Node 4 Node 5 Node 6 p 1 = 1.0299 18.02879 5.429839 6.675838 7.805378 5.134123 5.810376 p 2 = 1.0245 8.523125 15.80866 10.10544 8.86092 6.864997 4.61941 p 3 = 1.0190 6.683892 8.061306 23.39217 8.208405 5.327722 4.381641 p 4 = 1.0136 5.311406 5.342699 8.484171 18.18579 7.588097 7.449769 p 5 = 1.0082 5.310347 6.943049 10.01721 9.669561 20.46406 7.944082 p 6 = 1.0027 7.35016 7.233013 10.42534 10.35992 8.531059 25.98142Table 2
Multilocation signal.
p order ( m, n, l) (1, 4, 5) (2, 3, 6) (1, 4, 6) p 1 = 1.0299 30.4154 22.8408 33.5639 p 2 = 1.0245 17.3811 35.5565 24.3017 p 3 = 1.0190 24.2002 33.2709 22.9931 p 4 = 1.0136 28.0625 27.1995 35.7083 p 5 = 1.0082 30.7174 23.7714 23.3232 p 6 = 1.0027 20.0376 34.6695 48.8934Node refers to location signal; node 1 is the first location signal.p is the fractional power. m,l,n in Table 2 are mth, nth, lth location signal. m=round(rand)+1 is 1 or 2 and l=round(rand)+3 is 3 or 4, while n=round(rand)+5 is 5 or 6. The numerical values in the tables are the biggest value of FrFT according to the different order power in the left of table. From Tables 1 and 2, we could see that if we set a suitable detection threshold (decision condition such as the average of all the maximum of p order FrFT), we can distinguish the location signal(s) in the received signal efficiently.


### 5.2. Simulation with Real Underwater Channel Impulse Response

We get the underwater multipath channel impulse response, shown in Figure7, from Wuyuan Bay in Xiamen, China. Taking the threshold as the average of all the maximum of p order FrFT response, the result of different number of location signals can be tabulated in Table 3 (0 value of m,n,l means that this location node is not selected). It should be noted that we adopt fs= 64 kHz, thus getting different p values as Table 1.Table 3
The number of location signals is different.
p order ( m, l, n) (3, 0, 0) (1, 6, 0) (2, 5, 6) p 1 = 1.0420 5.4391 27.3816 24.3869 p 2 = 1.0344 10.7573 10.3540 34.1121 p 3 = 1.0268 25.2653 5.8847 15.0960 p 4 = 1.0191 11.6195 5.1691 10.4770 p 5 = 1.0115 6.1462 6.5556 29.5133 p 6 = 1.0038 4.3366 21.7600 25.7836 Threshold 12.594 14.8508 25.228Figure 7
The multipath characteristic of Wuyuan Bay.Tables1, 2, and 3 indicate that the location signals, MLC signals, have the advance in avoiding collision.


### 5.3. TDoA Estimation

On the other hand, although all the nodes are synchronized, the time that the different location signals received by target node within the short time window is not considered to be the same. Estimation of the TDoA within the short time window is useful to improve the accuracy of location. The TDoA for different location signals in the short time window can be estimated by our MCR-FrFT. The simulated results are shown in Figure8.Figure 8
Time difference of arrival within the short time window. (a)T=0.02 s; (b) T=0.04 s.
(a) (b)Assuming that the short time window is the same as signal duration timeT. From Figure 8, when the true time-delay value is larger than T/2, the estimated result is unreliable because of bigger error. Comparing Figures 8(a) and 8(b), the bigger T, the better the estimation result of time difference within T/2. For example, when T=0.02, the estimation range is about [0,0.35∗T], but when T=0.04, the estimation ranges change to about [0,0.45∗T]. In summary, Figure 8 shows that our MLC signals could be estimated successfully as long as the time delay is smaller than T/2 with certain duration timeT.


### 5.4. Comparisons with OFDM Based Location Scheme

OFDM based TDoA location scheme has been used in wireless metropolitan area networks (WMANs) and it gets high performance [19]. For sensor node location in underwater channel environment, the MLC scheme and OFDM scheme are compared.Firstly, we compare the MLC scheme and OFDM scheme as shown in Figure9. It should be noted that each user of the networks occupies a subcarrier of the OFDM; therefore, this kind of multiple access could be called OFDMA. The result shows that MLC scheme always outperforms the OFDM scheme with the same total number of location signals in terms of BER with same SNR.Figure 9
Comparisons of MLC scheme and OFDM scheme.Then, similar to [19], the cumulative distribution function (CDF) for successful localization at SNR = 0 dB, 5 dB, 10 dB, and 20 dB in regard to the localization error (with unit m) is shown in Figure 10. The results reveal that our MLC scheme outperforms the OFDM scheme at low SNR and is equal to OFDM one at higher SNR. This results in the lower energy requirements of the sensors. It is an excellent characteristic for underwater sensor acoustic networks (UW-ASNs), since the mobile sensors are usually powered by battery.Figure 10
Probability of successful localization with respect to localization error between MLC scheme and OFDM scheme.


## 5.1. MATLAB Simulation

The simulation tool MATLAB is used to evaluate the performance of MCR-FrFT. We set typical UW-ASNs parameters set as follows: the total number of nodes,M=12; the bandwidth, B=5kHz; the signal duration time, T=0.04s; and the sample frequency, fs=90kHz. White Gaussian noise is added and the signal noise ratio (SNR) is set to 0 dB. The number of underwater multipaths is 4, the delay is τ=00.010.0130.017, and the amplitude is A=10.40.30.1, respectively. Discrete FrFT is used and order p at the receiving end is given by p=2/π·arccotM+1fs/(M+1-2mB)+1 for mth node.We take single location signal and multilocation signals (three location signals), for example. For simplicity, we only consider the first half ofM (here is 6) in the system. The FrFT results are shown in Tables 1 and 2.Table 1
The single location signal.
p order Node Node 1 Node 2 Node 3 Node 4 Node 5 Node 6 p 1 = 1.0299 18.02879 5.429839 6.675838 7.805378 5.134123 5.810376 p 2 = 1.0245 8.523125 15.80866 10.10544 8.86092 6.864997 4.61941 p 3 = 1.0190 6.683892 8.061306 23.39217 8.208405 5.327722 4.381641 p 4 = 1.0136 5.311406 5.342699 8.484171 18.18579 7.588097 7.449769 p 5 = 1.0082 5.310347 6.943049 10.01721 9.669561 20.46406 7.944082 p 6 = 1.0027 7.35016 7.233013 10.42534 10.35992 8.531059 25.98142Table 2
Multilocation signal.
p order ( m, n, l) (1, 4, 5) (2, 3, 6) (1, 4, 6) p 1 = 1.0299 30.4154 22.8408 33.5639 p 2 = 1.0245 17.3811 35.5565 24.3017 p 3 = 1.0190 24.2002 33.2709 22.9931 p 4 = 1.0136 28.0625 27.1995 35.7083 p 5 = 1.0082 30.7174 23.7714 23.3232 p 6 = 1.0027 20.0376 34.6695 48.8934Node refers to location signal; node 1 is the first location signal.p is the fractional power. m,l,n in Table 2 are mth, nth, lth location signal. m=round(rand)+1 is 1 or 2 and l=round(rand)+3 is 3 or 4, while n=round(rand)+5 is 5 or 6. The numerical values in the tables are the biggest value of FrFT according to the different order power in the left of table. From Tables 1 and 2, we could see that if we set a suitable detection threshold (decision condition such as the average of all the maximum of p order FrFT), we can distinguish the location signal(s) in the received signal efficiently.


## 5.2. Simulation with Real Underwater Channel Impulse Response

We get the underwater multipath channel impulse response, shown in Figure7, from Wuyuan Bay in Xiamen, China. Taking the threshold as the average of all the maximum of p order FrFT response, the result of different number of location signals can be tabulated in Table 3 (0 value of m,n,l means that this location node is not selected). It should be noted that we adopt fs= 64 kHz, thus getting different p values as Table 1.Table 3
The number of location signals is different.
p order ( m, l, n) (3, 0, 0) (1, 6, 0) (2, 5, 6) p 1 = 1.0420 5.4391 27.3816 24.3869 p 2 = 1.0344 10.7573 10.3540 34.1121 p 3 = 1.0268 25.2653 5.8847 15.0960 p 4 = 1.0191 11.6195 5.1691 10.4770 p 5 = 1.0115 6.1462 6.5556 29.5133 p 6 = 1.0038 4.3366 21.7600 25.7836 Threshold 12.594 14.8508 25.228Figure 7
The multipath characteristic of Wuyuan Bay.Tables1, 2, and 3 indicate that the location signals, MLC signals, have the advance in avoiding collision.


## 5.3. TDoA Estimation

On the other hand, although all the nodes are synchronized, the time that the different location signals received by target node within the short time window is not considered to be the same. Estimation of the TDoA within the short time window is useful to improve the accuracy of location. The TDoA for different location signals in the short time window can be estimated by our MCR-FrFT. The simulated results are shown in Figure8.Figure 8
Time difference of arrival within the short time window. (a)T=0.02 s; (b) T=0.04 s.
(a) (b)Assuming that the short time window is the same as signal duration timeT. From Figure 8, when the true time-delay value is larger than T/2, the estimated result is unreliable because of bigger error. Comparing Figures 8(a) and 8(b), the bigger T, the better the estimation result of time difference within T/2. For example, when T=0.02, the estimation range is about [0,0.35∗T], but when T=0.04, the estimation ranges change to about [0,0.45∗T]. In summary, Figure 8 shows that our MLC signals could be estimated successfully as long as the time delay is smaller than T/2 with certain duration timeT.


## 5.4. Comparisons with OFDM Based Location Scheme

OFDM based TDoA location scheme has been used in wireless metropolitan area networks (WMANs) and it gets high performance [19]. For sensor node location in underwater channel environment, the MLC scheme and OFDM scheme are compared.Firstly, we compare the MLC scheme and OFDM scheme as shown in Figure9. It should be noted that each user of the networks occupies a subcarrier of the OFDM; therefore, this kind of multiple access could be called OFDMA. The result shows that MLC scheme always outperforms the OFDM scheme with the same total number of location signals in terms of BER with same SNR.Figure 9
Comparisons of MLC scheme and OFDM scheme.Then, similar to [19], the cumulative distribution function (CDF) for successful localization at SNR = 0 dB, 5 dB, 10 dB, and 20 dB in regard to the localization error (with unit m) is shown in Figure 10. The results reveal that our MLC scheme outperforms the OFDM scheme at low SNR and is equal to OFDM one at higher SNR. This results in the lower energy requirements of the sensors. It is an excellent characteristic for underwater sensor acoustic networks (UW-ASNs), since the mobile sensors are usually powered by battery.Figure 10
Probability of successful localization with respect to localization error between MLC scheme and OFDM scheme.


## 6. Conclusion

In this paper, we proposed a TDoA location scheme for UW-ASNs. The key idea is that we took the MLC as the location signal to avoid collision and proposed a new detection method MCR-FrFT. Compared to the complicated matching filter, the FrFT make the detection simple and efficient. Due to the time-frequency analysis ability of FrFT, we could estimate the TDoA and thus locate the sensor nodes in UW-ASNs. Both the detection diagram and theoretical derivation were given. Experimental results demonstrated that the MLC signals can avoid collision as location signals in TDoA scheme. Also the corresponding detection method MCR-FrFT could estimate the TDoA between different location signals within a short time window and locate the nodes.



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*Source: 2098927-2016-10-27.xml*