Anti-Multipath Underwater Acoustic Time-Delay Detection Algorithm Based on Dual-Delta Correlators

For the underwater acoustic (UAC) positioning system based on continuous waveform signal, the time-delay estimation of the Line-of-sight (Los) is not robust under the UAC strongly time-varying multipath channel. So an anti-multipath time-delay detection algorithm based on dual-delta (DD) correlators is proposed in this paper to address the issue. The algorithm uses a differential processing code-phase discriminator model to improve the traditional delay-locked-loop (DLL) time-delay detection algorithm, and combines it with the signal tracking loop adapted to the UAC channel condition to estimate time-delay. Theoretical analysis indicates that due to the reduction of local code interval and the differential processing operation, the proposed algorithm effectively mitigates the effect of time-varying multipath signals on time-delay estimation compared with conventional methods. Simulation analysis further demonstrates that the proposed algorithm can robustly detect the Los and achieve high-precision time-delay estimation under the time-varying multipath channel. The algorithm shows great potential for practical applications.


Introduction
Underwater acoustic (UAC) positioning systems accomplish the localization of underwater targets by measuring of time-delay information from target echoes.The underwater channel under shallow water environments is complex and variable, which results in the instability of traditional peak detection algorithms.Especially for the high refresh rate localization problem based on the continuous waveform signal, unstable detection of Los may lead to successive errors in the time-delay estimation results and affect the localization quality.Therefore, it is imperative from an engineering standpoint to explore a robust time-delay detection technique that can withstand multipath interference [1][2].
In the field of satellite navigation, anti-multipath techniques for the continuous waveform signal have been widely studied.Representative methods include narrow correlation technique, pulse aperture correlator (PAC) technique, and high resolution correlator (HRC) technique [3].The implementation complexity of these techniques is modest, and their resistance to multipath interference is not affected by the number of multipaths.However, they do require a high signal-tointerference ratio.Another category of techniques, mostly based on ML theory, encompasses the multipath estimation delay-locked loop (MEDLL) and its several variants [4], such as multipath mitigation techniques (MMT) and vision correlators (VCs).Their advantage is the simultaneous estimation of code and carrier phases of Los and multipath signals, and has the potential to reach the ( ) ( ) ( ) ( ) Where A is the amplitude of the Los.m is the number of multipaths.i  is the ratio of the amplitude of the first multipath signal to that of the Los.c is a pseudo-random sequence ("Kasami" sequence is used in this paper).i  Δ , i  Δ are the time-delay of the first multipath signal with respect to the Los as well as the phase deviation.nt () is the noise.And w is the carrier frequency of the signal.

Time-delay Detection Algorithm
The DLL time-delay detection model is shown in figure 1.It is mainly composed of the DLL and the carrier tracking loop (Costas loop), which realizes that the carrier match between the local carrier and the received signal, and completes the demodulation of the received signal [6].The DLL substitutes the results of the correlators into the phase discriminator to obtain the time-delay offsets, and the NCO adjusts the frequency and phase of the pseudo-code to achieve the tracking and time-delay detection of the signal.The following is a brief description of the time-delay estimation procedure.

Figure 1. DLL delay detection model.
It is assumed that the carrier tracking loop can completely strip carrier and eliminate the effect of doppler frequency to obtain the baseband received signal.The correlation integration results of three groups of correlators in the DLL can be obtained based on the signal model shown in (1).
Under the condition of the presence of multipath signals, the correlator integration results of the received signals are distorted, which results in an over-zero shift of the discriminant function.The over-zero shift is defined as the time-delay estimation error.The error expression 1 () DLL  Δ is derived from correlation integration results of the correlators under the multipath background: Where in the time-delay estimation results.In the presence of identical multipath conditions, the time-delay error will decrease by decreasing the interval d of the correlators.However, it is important to note that these errors cannot be completely eliminated.

Time-Delay Detection Algorithm Based on DD
Based on the above problems, this paper designs the DD correlators structure and an anti-multipath UAC time-delay detection algorithm based on the DD correlator is proposed.On the one hand, the idea of differential processing is introduced to estimate the Los.On the other hand, a tracking loop adapted to the UAC channel is used to track the Los under the fast-fading background and provide an accurate time-delay reference for time-delay detection.

Model Building
The DD correlators extends the quantity and changes the intervals of the DLL correlator sets.The DD correlators consists of two sets of correlators, and the code interval of one set of correlators is twice that of the other set.The smaller code interval is 0.1chip .The differential processing of the correlator integration results is utilized to obtain the discriminant function as: Where x IE , x IL represents the correlation results between the received signal and the early code or the late code in the xth group of correlators respectively.Under the background of single-path channel, due to the zero-point symmetry of the discriminant function, the function is characterized only by its negative semiaxis time-delay as shown in (4).
Where  is the time-delay of the Los with respect to the local signal.e  is the phase deviation.d is the code interval.The discriminant result reflects the time-delay offset of the Los relative to the local code.And the time-delay of the Los is estimated by combining the time-delay reference information.The time-delay reference information is provided by the signal tracking loop, which in this paper contains a 2nd order loop filter and NCO.Moreover the loop parameters are designed for the UAC channel environment.

Model Correction under Bandwidth-limited Condition
In practical applications, UAC receivers usually undergo filtering pre-processing to enhance their antiinterference performance.If the role of front-end filtering is simplified as the channel transfer function [4], the received multipath signal model can be modified as follows: where  denotes the convolution and the channel transfer function () ht is assumed to be an ideal low-pass filter.= or 10B , or the infinite bandwidth condition.Figure 2(a) shows that the signal energy is mainly concentrated in the main lobe range with a bandwidth of B .Moreover, the energy outside the main lobe range is not negligible.Figure 2(b) shows that the signal correlation results under different band-limit conditions are severely distorted, which affects the antimultipath performance of the algorithm.

Time-Delay Estimation Error under the Multipath Background
Similar to the analytical method of DLL time-delay error, we set the time-delay offset Where (2 ) / 4 dc Td (2 ) / 4 ec Td Comparing ( 2) and ( 6), it can be seen that when the multipath time-delay 1   is in the range of ( ) , the time-delay estimation error of the DD algorithm is smaller than that of the DLL algorithm due to the smaller code interval of the DD algorithm.In addition, the DD correlators can theoretically eliminate the inherent time-delay errors existing in the two sets of DLL correlators from each other under some specific relative time-delay.
As shown in figure 3, the multipath error envelop for both the traditional DLL algorithm and the DD algorithm are presented (Simulation conditions: dual-path model, amplitude ratio 1 0.5  = , relative time-delay

Thermal Noise Analysis of the UAC environment.
Considering only the presence of Gaussian white noise, the pseudocode time-delay error can be modeled as: Where c K is the slope at the zero-point of the discriminant function, which is not affected by noise.
c N is the time-delay error of the discriminant output result.  is the time-delay error of the discriminant output result.
Based on the random theory, the code-phase error of the discriminator in an open-loop system can be mathematically expressed as: According to the transfer function of code tracking loop delay estimation error in [7] , the tracking loop closed-loop error can be expressed as: Where L B is loop bandwidth.When tracking loop reaches steady state, the discriminator output error is considered to be very small [8]. the discriminator function can be re-expressed as: Where i w , i d denote the weight and the code interval of each set of the DD correlators respectively.{} Re is the real part of the complex signal.
Similar to the analytical method employed in (10), the noise variance in the output of the discriminator can be expressed as: By using the correlator weight and code interval of DD correlators shown in (3), and substituting ( 8), ( 10) and ( 11) into (9), the closed-loop error of code tracking loop can be expressed as: Where 0 N is energy of noise, s C is power spectral density.S is energy of signal.BW is singleside bandwidth of signal.
Figure 4 shows the comparison results of the thermal noise performance between the DLL and DD algorithm.Compared with the DLL algorithm, the thermal noise performance of DD algorithm is better with a smaller closed-loop error when the front-end bandwidth 5 / 15 BW B B = . For 1 BW B = , the loss of signal energy caused by band-limited condition affects DD algorithm more seriously, therefore the time-delay error increases slightly.Taking into account thermal noise and error caused by multipath, it can be observed that the time-delay estimation error of DD algorithm is significantly smaller than that of the DLL algorithm., the coherent integration time T for the UAC signal is 28.3ms, the loop bandwidth , and the signal-to-noise ratio (SNR) varies from 5dB to 20dB).

Dynamic Adaptation Analysis of the UAC environment.
The effective traction range of the DD correlators is ( ) , which decreases when the code interval is reduced.The time dimension of single-cycle continuous waveform signal expands or contracts due to target motion.Furthermore the presence of thermal noise and multipath signal introduces time-delay estimation error.So the predicted position of the Los under the influence of the two factors may be outside the effective traction range of the discriminator, which causes the discriminator to fail to operate properly.Especially for the UAC environment, because the speed of sound is much lower than the speed of radio, the dynamic adaptation requirement is even higher under the UAC environment.Without taking into account doppler compensation and the impact of noise, the theoretical target velocity tolerance for DD correlators with a minimum correlator interval of 1 d can be determined: Where sound is the speed of sound under the UAC environment, and N is the order of pseudocode.Figure 5 demonstrates the time-delay estimation error of the DD algorithm as well as the DLL algorithm under different dynamic conditions, with specific signal parameters shown in table 1. Timedelay estimation error of the both algorithm exhibit a gradual increase as the dynamic doppler velocity increases.Specifically, when the velocity reaches 0.4 / ms , the pseudocode tracking loop of DD algorithm becomes fully unlocked.And it is obvious that the dynamic adaptation of DLL algorithm is better than DD algorithm.Therefore, the DD algorithm sacrifices its dynamic adaptation while enhancing the anti-multipath performance.As a result, it can be concluded that this approach is specifically designed to address the issue of anti-multipath time-delay detection under the lowdynamic background.

Numerical Simulation
In this section, the UAC time-varying multipath channel model is used in order to validate the universal adaptability of the algorithm.Specifically, the traditional matched filter(MF) algorithm, the traditional DLL algorithm, and the DD algorithm are used to detect time-delay.The performance of the algorithm in terms of anti-multipath capabilities is then compared and analyzed.
During the simulation, the signal parameters are consistent with table 1.In order to characterize the time-varying features of multipath signals, the amplitude of multipath and time-delay are set to have randomness.Figure 6    2.06e-5 5.86e-6 RMSE_2(s) 9.57e-5 9.69e-5 8.80e-6 Simulation results show that the DD algorithm can robustly detect the Los and has higher timedelay estimation accuracy under the multipath background of multipath amplitude ratio and timevarying relative delay.On the contrary, the MF algorithm is susceptible to the influence of timevarying multipath, resulting in significant errors in time-delay detection results due to misjudgment in peak detection.Similarly, the DLL algorithm experiences a shift in detection results from the Los position due to the effect of multipath signal "pulling", leading to larger jitter errors.Therefore, the DD algorithm has higher robustness and better anti-multipath performance for the Los time-delay detection problem under strong time-varying multipath background.

Conclusion
For the UAC positioning system based on continuous waveform signal, this paper addresses the issue that the Los time-delay estimation is not robust under the UAC strongly time-varying multipath channel.The proposed algorithm introduces the idea of differential processing to improve DLL timedelay detection model, and utilizes 2 sets of correlators with smaller code interval to design the differential phase discriminator model.Furthermore, the signal tracking loop adapted to the UAC background is combined to obtain robust time-delay estimation results under the multipath background.Theoretical analysis shows that, the DD algorithm effectively reduces the impact of time-varying multipath signals on time-delay estimation due to the reduction of code interval and the differential processing operation, and the anti-multipath performance is better than that of traditional time-delay detection algorithms.However, in order to adapt to the UAC time-varying multipath channel background, the optimization of the structural parameters leads to the fact that DD algorithm is only applicable to the time-delay detection under the low dynamic background.Finally, the theoretical simulation verifies the robustness of the DD algorithm under strongly time-varying multipath channel background and the higher accuracy of time-delay estimation.

Figure 2 .
Figure 2. Comparison of band-limited signal correlation results.

Figure 2
Figure 2 shows the time-frequency domain comparison of the received signals passing through the low-pass filters with bandwidth BW B= or 10B , or the infinite bandwidth condition.Figure2(a) shows that the signal energy is mainly concentrated in the main lobe range with a bandwidth of B .Moreover, the energy outside the main lobe range is not negligible.Figure2(b)shows that the signal correlation results under different band-limit conditions are severely distorted, which affects the antimultipath performance of the algorithm.

_ ( ) 0
DDtau shift  = of the DD correlators under the multipath background with infinite bandwidth.The DD correlators time-delay estimation error is obtained as:

.
It can be observed that under any identical bandwidth conditions, the peaks of the multipath error envelop is significantly smaller for the DD algorithm compared to the DLL algorithm for This verifies the conclusion that the proposed algorithm exhibits superior anti-multipath performance over the DLL algorithm.

Figure 4 .
Figure 4. Comparison of thermal noise error analysis(The bandwidth 1 / 5 / 15 BW B B B =, the coherent integration time T for the UAC signal is 28.3ms, the loop bandwidth illustrates Power-Delay-Profile(PDP) Evolution for two types of channels: the amplitude of multipath is time-varying, and the amplitude of multipath is time-varying with doppler velocity.(a)Time-varying amplitude.(b)Time-varying amplitude with doppler velocity.

Figure 7 .
Figure 7. Time-delay estimation results under the time-varying multipath background.

Table 1 .
UAC Signal Parameters.Figure 5. Time-delay Estimation Error under Dynamic Background.

Table 2 .
The time-delay estimation results of MF, DLL, and DD algorithm under different time-varying multipath backgrounds are shown in figure7, and the time-delay estimation accuracy are shown in table2.Time-Delay Estimation Accuracy.