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

Duetothemultipath,Doppler,andothereffects,thenodelocationsignalshavehighprobabilityofaccesscollisionintheunderwater acousticsensornetworks(UW-ASNs),andtherefore,itcausesthesignallostandtheaccessblock;therefore,itconstrainsthe networksperformance.Inthispaper,wetakethemultilinearchirp(MLC)signalsasthelocationsignaltoimprovetheanticollision ability.InordertoincreasethedetectionefficiencyofMLC,weproposeafastefficientdetectionmethodcalledmixingchangerate-fractionalFouriertransform(MCR-FrFT).ThismethodtransformsthecombinedratesofMLCintosymmetrytriangleratesand thenseparatesthemultiusersignalsbasedonthetransformedratesbyusingFrFT.Theoreticalderivationandsimulationresults showthattheproposedmethodcandetectthelocationssignals,estimatethetimedifferenceofarrival(TDoA),reducethemultiple accessinterference,andimprovethelocationperformance.


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 TDoAbased 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 2 Mobile Information Systems 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.

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 Section 3, 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.

System Model
The system architecture of the scheme is depicted in Figure 1.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.

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 Figure 2, where  is the duration time of location signal.Nodes are  denoted by 1, 2, . . ., , +1, . . ., 2−1, 2, where the first  nodes are with positive combined slopes and the second  nodes are with negative combined slopes.Each signal is composed of two segments with two different slopes.The signal for th node is expressed as Subscripts  and  indicate the first half and the second half of duration time , respectively.And when 0 ≤  < /2, and when /2 ≤  < , where   is the carrier frequency,  is the node number,  = 1, 2, . . ., , . . ., 2,  is the amplitude of signal, and 2 is the total number of nodes in UW-ASNs.  is th node's slope within the first half of signal duration and   is the slope of th node within the second half of signal duration.The slopes of general combined chirp signal can be expressed as As shown in Figure 2, each location signal has the same bandwidth  and duration time ; thus the system produces equal time-bandwidth product  =  for all nodes.Equal  makes the nodes in the UW-ASNs have the equal position, which is important for mobile nodes.

Detection Block Diagram and Principle.
The localization signal detection problem could be described as deciding which one of the 2M possible signals was transmitted, given the received signal during the interval time of (0, ).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 local() and local(), respectively, local() for the up branch ( ≤ ) and local() for the down branch ( <  ≤ 2).

Mixing Change Rate (MCR).
where ℎ LPF () is the impulse response function of low pass filter (LPF).Assuming that th node's signal is received and  < , we get the received signal as (7) by substituting ( 2)-( 4) into (1).() is the underwater acoustic channel noise.After the MCR, the high frequency terms have been filtered out and the received signal becomes as (8) (8)

Fractional Fourier Transform (FrFT).
FrFT is a generalized Fourier transformation form and it can be regarded as the Fourier transform to th order, where  needs not be an integer; thus it can transform a function to any intermediate domain between time and frequency [18].For signal (), the FrFT has the following form: and the transform kernel is where  is the transform order,  is the angle of rotation, and  = /2.When  = 1, FrFT will degenerate into Fourier transformation, and when  = 0, FrFT is just the original signal.
Simple component LFM signal with noise has the following form: where 0 ≤  ≤  and () is white noise.The FrFT of () is 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 -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.
Figure 4 shows that the set of chirp rates is changed from two different slopes, which needs two different  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.

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.
According to ( 4) and ( 8), there are different sets of chirp rates included in the MCR results with different , ,  group: 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.
Set  = √2/, where  is the signal energy in the whole duration, and the cross-correlation of signals  takes the following form: Substituting the corresponding values of () in ( 13) and neglecting the integration over the higher frequencies, we get + cos where  =  is time-bandwidth product and − = Δ.We could see that  has an oscillatory nature as a function of , , and Δ, due to the Fresnel function form of (17).Usually, smaller cross-correlation  indicates a better performance of the system.The normalized sample values can be expressed as where  , is the cross-coherence between th and th 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  or the total number of MLC signals or the difference between chirp rates Δ, when the other two parameters are set fixed.

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 where  is the total number of propagation paths, and  = 0 corresponding to the direct path,   is the amplitude of th propagation path at receiving side, and   is the propagation delay of th propagation path.
Assuming that the transmitted signal is (), then the received signal () can be expressed as Taking the Doppler into account, let () =   (); then the received signal can be expressed as Substituting the corresponding values into (21), neglecting the third branch, and using MCR we get where 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 by th order for the fixed system, the MLC signals would not be affected by multipath properties of underwater acoustic channel.

Positioning Process.
The target node receives the location signals and determines the corresponding reference node with time (  ,   ,   ) and position (  ,   ), (  ,   ), (  ,   ), when and where the location signal was transmitted.In this system, the location signals are received at the same time so the difference between   and   is a constant; in other words, the difference between the corresponding distance   and   is a constant.Of course, |  −   | and |  −   | 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.
In Figure 6,  is the speed of sound in underwater channel in m/s.Hyperbolic curve  , is determined by nodes  and , while  , and  , are determined by ,  and ,  respectively.
For this system, the scaling parameter and the length of interval are  = (/  ) 1/2 =  1/2 /  and Δ = (  ) 1/2 =  1/2 , respectively, where  =   ⋅  is the sample number.When  =  0 is the optimal  order, then there are two different energy accumulation peaks (assuming that the coordinates are  01 and  02 ) symmetrical about /2 that are got.

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Node m

Node n
Node l (t m , x m , y m ) Communication link Reference node Unknown node

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,  = 12; the bandwidth,  = 5kHz; the signal duration time,  = 0.04 s; and the sample frequency,   = 90 kHz.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  = [0 0.01 0.013 0.017], and We take single location signal and multilocation signals (three location signals), for example.For simplicity, we only consider the first half of  (here is 6) in the system.The FrFT results are shown in Tables 1 and 2.
Node refers to location signal; node 1 is the first location signal. is the fractional power., ,  in Table 2 are th, th, th location signal. = round(rand) + 1 is 1 or 2 and  = round(rand) + 3 is 3 or 4, while  = 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  order FrFT), we can distinguish the location signal(s) in the received signal efficiently.

Simulation with Real Underwater Channel Impulse
Response.We get the underwater multipath channel impulse response, shown in Figure 7, from Wuyuan Bay in Xiamen, China.Taking the threshold as the average of all the maximum of  order FrFT response, the result of different number of location signals can be tabulated in Table 3 (0 value of (, , ) means that this location node is not selected).It should be noted that we adopt   = 64 kHz, thus getting different  values as Table 1.Tables 1, 2, and 3 indicate that the location signals, MLC signals, have the advance in avoiding collision.

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 Figure 8.
Assuming that the short time window is the same as signal duration time .From Figure 8, when the true time-delay value is larger than /2, the estimated result is unreliable because of bigger error.Comparing Figures 8(a) and 8(b), the bigger , the better the estimation result of time difference within /2.For example, when  = 0.02, the estimation range is about [0, 0.35 * ], but when  = 0.04, the estimation ranges change to about [0, 0.45 * ].In summary, Figure 8 shows that our MLC signals could be estimated successfully as long as the time delay is smaller than /2 with certain duration time T.

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 Figure 9.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.
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.

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.

Figure 1 :
Figure 1: Location system architecture using combined chirp signals.

Figure 2 :
Figure 2: The time-frequency characteristic of location signals.
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 Figure 3(b):

Figure 3 :
Figure 3: System block diagram.(a) The detection block diagram.(b) The block diagram of MCR.
of chirp rates by MCR User 1, original set User 1, up branch User 1, down branch

Figure 4 :
Figure 4: The time-frequency characteristic after MCR.

Figure 5 (
a) indicates that  decreases oscillatorily as  increases when  = 30 and Δ = 2.Figure 5(b) indicates that  increases oscillatorily as  increases when  = 100 and Δ = 2.Figure 5(c) indicates that  decreases oscillatorily as Δ increases when  = 100 and  = 30.We could select the assembling parameter according to (17) in practice.

Figure 9 :
Figure 9: Comparisons of MLC scheme and OFDM scheme.

Figure 10 :
Figure 10: Probability of successful localization with respect to localization error between MLC scheme and OFDM scheme.

Table 1 :
The single location signal. 01 and  02 are theoretical values, while   01 and   02 are measured values.For each location signal, Δ ⋅  ⋅  ⋅ sin() 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 |  −   | +  ⋅  or |  −   | +  ⋅ .

Table 3 :
The number of location signals is different.