A prefix‐based approach for joint Doppler and channel estimation in underwater communication

Developing a reliable and robust underwater acoustic communication system is a difficult task due to the complicated nature of the underwater channel, non‐stationary noise, and several other factors. Indeed, channel estimation or equalization presents numerous challenges in this non‐stationary, highly Doppler, multipath environment; as a result, traditional equalizers and PLL‐based methods have limited performance. Generally, communication over such time‐varying channels is accomplished via packets that contain a prefix/preamble signal for training, a payload containing the actual data, and a silent period for proper alignment. The prefix signal must be designed properly because it is used to estimate the channel and determine the start of the packet. In this paper, we propose a new prefix signal based on the hyperbolic chirp signal, which has Doppler invariance properties. These properties enable the extraction of the entire packet even under severe multipath and Doppler. Our new proposed prefix signal can accurately and efficiently characterize an underwater channel by estimating the multipath delay, amplitude, and Doppler scales. Our new method has been validated through extensive simulations using various channel models for its robustness and effectiveness under various conditions. It has also been tested on a real‐world channel.


| INTRODUCTION
Using acoustics signals as a carrier is the best way to communicate underwater, especially for long ranges.Even after research of so many decades, establishing a reliable link of a few km at a rate of a few kbps is a very hard task, much more challenging than radio frequency (RF)-based wireless air communication. 1 It is mainly because of the complex nature of acoustic channels and harsh noise characteristics. 2 Signal attenuation increases with the frequency. 3The low sound speed deteriorates the quality of communication due to the Doppler and large delay spread.Time-varying multipath due to the movement of the transmitter, receiver, and floating boundaries, debris, or scatterers make underwater channels much more difficult. 4Various multipath gives different Doppler. 57][8][9] Some good applications for the challenging UWA channel for deploying wireless sensors are given in Goyal et al. 10 Adaptive error control code for transmitting information is demonstrated in Goyal et al, 11 while Goyal et al 12 demonstrate the use of wireless sensor networks in pipeline fault detection.Wireless sensor networks also find applications in fields other than the underwater domain.The real-time critical event monitoring application is presented in Faheem et al, 13 and Faheem et al 14 provide application in healthcare.Acoustic communication is generally feasible at lower frequencies, that is, tens of kHz, with limited transmission bandwidth.A detailed review of the techniques for the synchronization task in the underwater acoustic channel is presented in Sameer Babu et al. 15 A few kHz of bandwidth is categorized as the wideband transmission, 16 resulting in the frequency selective channel. 17Underwater acoustic channels are frequency and time selective, 18 hence termed doubly selective channel. 19In wideband communication, Doppler spread mainly causes time dilation/compression of received signals and offsets the carrier frequency. 20arious methodologies handle these frequency selective channels; among them, the most effective ones are as using highly trained equalizers, 21 or by dividing complete frequency selective channels into small orthogonal flat fading channels (i.e., orthogonal frequency division multiplexing [OFDM]), 22 or using well-designed orthogonal codes for transmission, that is, code division multiple access 23 and other specialized techniques.Each technique has its merit and demerit.
Communication in the doubly selective, non-stationary channel is preferably carried out by transmitting the information in packet form. 24The packet size is generally kept less than that of the coherence time of the channel.Each packet generally contains three fields, that is, header termed prefix/preamble, silent period, and data.The header is the known signal for frame detection, synchronization, and channel estimation. 25To a large extent, the choice of prefix signal depends upon the channel, and in turn, the channel's estimation accuracy depends upon the prefix signal's properties. 26For underwater acoustic communication, the prefix signal used should be Doppler tolerant. 27Various types of prefix signal, like pseudo-random bit sequence (PRBS) and linear/hyperbolic frequency modulation (LFM/HFM), have been used in literature for channel sounding. 28Two LFM chirp signals, that is, up chirp and down chirp in tandem, are used to capture the time-varying information, that is, Doppler scale of the channel. 29Packet using 30 two signals at the head and tail end of the packet, that is, prefix and postfix, has also been suggested.The disadvantage of these methods is that both the chirp signals are separated in time.Therefore, the channels of both signals might be different.Moreover, two signals, in turn, limit the data rate.To overcome this issue, a single prefix to jointly estimate the Doppler scale and channel is proposed in Zhang et al. 31 However, it does not compromise the data rate, but the estimation error increases with the increase in Doppler.
The presence of Doppler makes channel estimation an arduous task.Since few paths dominate the UWA channel impulse response (CIR) and the majority of channel coefficients are zero, the channel estimator can use its sparsity.Taubock et al 32 investigate delay-Doppler sparsity through a basis expansion model (BEM).Recent research focuses on sparse channel estimation utilizing compressed sensing (CS) techniques to recover sparse signals from linear measurements. 33Among the CS algorithms utilized for UWA OFDM channel estimation are the matching pursuit (MP), orthogonal matching pursuit (OMP), and basis pursuit (BP). 34BP outperforms OMP in Berger et al 30 for significant Doppler spread; nevertheless, BP algorithms are more complex than OMP techniques.Compressive sensing-based methods are more effective when the channel has a sparse structure and a significant Doppler offset.In contrast, subspace-based methods are more effective when Doppler is low. 30Combining the subspace-based technique with the OMP algorithm has recently improved channel estimate. 35Multi-channel adaptive DFE-based equalizer 36 is an effective way to track and equalize.Many algorithms for estimating the channel for various scenario has been proposed in a different kind of noise, 37 in the presence of carrier and timing offset.Deep neural network framework has also been used for channel estimation and correction. 38In Liu et al, 39 a convolutional neural network-based receiver design for the UWA channel considers banded channel matrix features in a decoder-encoder CNN-type receiver.But a generic method to reliably estimate/equalize underwater channels is an open problem due to a non-stationary, highly Doppler, multipath environment.
The following are the major contributions: • We design appropriate prefix signals for the UWA communication environment.
• A single prefix-based packet structure for underwater communication has been thoroughly investigated.Further unique properties of the prefix signal assist us in developing novel and robust joint Doppler and channel estimation techniques.• The proposed approach not only estimates the channel with the presence of Doppler but also identifies Doppler scales and delay parameters for different multipath scenarios.• The algorithm operates in two stages, initially providing a coarse estimate and subsequently improving it through the utilization of lookup tables (LUTs).
• The effectiveness of the suggested frame structure and channel estimation algorithm has been proven through extensive numerical simulations conducted in both simulated and real-world settings.

| Proposed frame structure
The acoustic underwater communication channel is very complex; therefore, both prefix and postfix, 40 as shown in Figure 1, are primarily used for channel estimation, packet extraction, synchronization, and so on.This channel usually varies rapidly. 3Hence, it limits the use of this type of packet structure.One possible way of handling this is by reducing the packet size s.t. the channel remains stationary for the entire packet duration.A single prefix-based packet, as shown in Figure 2, has been suggested.The packet contains a guard band, a prefix signal, and a payload, and its duration is to be kept less than the coherence interval of the channel.The suggested design provides better effective data as it has only one prefix signal compared to the earlier design.Therefore, it will be used for all its merits.Further, the choice of the prefix is also crucial, as this will be responsible for equalizing multipath, combating the Doppler, and performing synchronization.The packet structure, prefix signal, and the system algorithm to handle the tough underwater channel are unfolded in the following section.

| Prefix selection
The good prefix signal should have the following features: • The autocorrelation function of the prefix signal should be close to the impulse function, with a low sidelobe level (SLL) and small main lobe width.• It should be robust for Doppler-induced scaling.Ideally, it should be Doppler invariant.
• High Doppler resolution, that is, small Doppler frequency, should be resolvable.
• High temporal resolution, that is, all multipath should be resolvable.
• The compression or expansion of the prefix signal should be analytically expressed.
PRBS and LFM/HFM are prefix signals.The PRBS is very sensitive toward Doppler-based scaling.Whereas LFM and HFM signals are less sensitive to Doppler scaling, therefore, can be potential candidates for prefix signals in underwater communication.We further study these signals and their properties to understand their fit in underwater communication.

| LFM and HFM signals
LFM signal with frequency support of ½f 0 , f T is expressed as where b l is the chirp rate, f T Àf 0 T indicates the rate of change of frequency, f T denotes the highest, f 0 denotes the lowest frequency, and T is time duration.HFM signal with the same frequency and time support is given as is the chirp rate.The first role of the prefix signal is detecting the start of the packet; the received signal is correlated with the transmitted prefix, and correlation peak gives the start of the packet.The accuracy of detection depends upon the autocorrelation properties of the prefix signal.These properties can be studied using autocorrelation and wideband ambiguity function (WAF) with bandwidth (B), time duration (T), and their product, commonly known as bandwidth-time product, that is, BT, as crucial design parameters.The main performance metrics for the autocorrelation function are the width and height of the main lobe for the given energy signal.Ideally, the impulse-type autocorrelation function provides the best multipath delay resolution, that is, all arrival paths can be resolved theoretically.WAF incorporates Doppler scaling; therefore, it is used to study correlation properties in Doppler.We study these performance metrics for both prefix signals under various conditions to understand and compare their behavior.

| Autocorrelation function
Autocorrelation function of signal sðtÞ is defined as It is a symmetric function about τ ¼ 0, with a peak value of Rð0Þ.The width of the main lobe of RðτÞ is a measure of delay resolution.3 dB main lobe width τ 3dB can be evaluated as where τ max and τ min are the maximum and minimum value delay corresponding half-power point that can be obtained as solution of the equation RðτÞ ¼ Rð0Þ 2 , respectively.Figure 3 shows the impact of bandwidth (B) on autocorrelation for LFM and HFM signals.It can be observed from Figure 3 that, for a given signal energy, the increase in bandwidth reduces the width of the autocorrelation function's main lobe, providing better delay resolution.
For getting a better understanding, τ 3dB is evaluated for various values of bandwidth (B) normalized to the fixed center frequency.As depicted in Figure 4, the LFM signal's main lobe is narrower than that of the HFM, and the difference increases with an increase in bandwidth.
The main lobe width can be easily measured in terms of root mean square (rms) bandwidth in terms given by where Sðf Þ is the frequency response of signal.The plots of the rms bandwidth of both the LFM and the HFM signal are shown in Figure 5.For higher bandwidth, that is, 10 kHz, the LFM signal has larger rms bandwidth than that of the HFM signal, while for lower bandwidth, that is, 1 kHz, both the signals have similar rms bandwidth.Therefore, the LFM signal has a narrower main lobe than that of the HFM signal, and as bandwidth increases, the LFM signal provides better delay resolution than the HFM signal.Autocorrelation properties quantify the prefix signal performance only for the case of no or almost negligible Doppler.To analyze the impact of Doppler on correlation properties, we investigate the WAF in detail.

| WAF
Doppler is more prevalent in underwater communication, therefore, plays an important role in the prefix design.The impact of Doppler is being studied using the WAF. 41It models the Doppler effect on the wideband signal by time scaling, that is, dilation.Mathematically, WAF, χðη, τÞ, of signal sðtÞ is given by where is the τ time delay and η is the Doppler-induced scaling.It can easily be seen that χðη, τÞ ≤ χð1,0Þ: Normalized WAF is defined as χ N ðη, τÞ ¼ χðη, τÞ χð1,0Þ .To analyze the impact of Doppler on LFM and HFM signals, we have studied WAF under different conditions and scenarios.The details of the parameters and scenarios, which are chosen to get a comprehensive overview, are given in Table 1.Shape, width, and span of 3 dB WAF contours obtained by solving the equation χ N ðη, τÞ ¼ 1 2 depict the important characteristics of prefix signal.The shape tells the nature of the overall impact of the Doppler.Width quantifies delay and Doppler resolution, while span depicts the inherent Doppler tolerance.For the lower value of BT, that is, BT ¼ 1 and BT ¼ 10, the 3 dB WAF contours, as shown in Figure 6 of both LFM and HFM signals, are almost linear strips passing through the origin (η ¼ 1, τ ¼ 0), completely overlaps.On the other hand, for BT ¼ 100, the WAF contour of the LFM signal is linear, but for the HFM signal, it is curved, that is, the ridges of the contour vary slowly, which provides better insensitivity to the Doppler scale.Hence, the HFM signal has better Doppler tolerance than the LFM signal for larger BT values.
From Zoom Plot 1 of Figure 6, it can be seen that when BT ¼ 100, the 3 dB WAF contour of the LFM signal is smaller than that of the HFM signal; hence, the LFM signal incurs more signal-to-noise ratio (SNR) loss with Doppler scaling than HFM signal.Therefore, the LFM signal is more sensitive to Doppler-induced scaling than the HFM signal for higher BT values.Further, delay resolution is quantified using τ 3dB , which is measured as the distance between the points of intersection of WAF (χ N ðη, τÞ) and line η ¼ 1.In Figure 6, these points of intersection are shown by two red dots in the Zoom Plots 1 and 2. It can be observed from Zoom Plots 1 and 2 that the delay resolution for Cases II and IV is the same but better than for Cases I and III.It is due to the increase in bandwidth B, that is, from 1 to 10 kHz.
Similarly, the Doppler resolution (η 3dB ) can be obtained by measuring the distance between the point of intersection of WAF with the line τ ¼ 0, indicated by the two blue dots shown in zoomed plots.Doppler resolution for Cases II and III is approximately equal and better than that of Cases I and IV due to the increase in time duration from 1 to 10 ms.

Sampling frequency 100 kHz
Center frequency 20 kHz The delay resolution for various Doppler scales can also be obtained by measuring the main lobe width of WAF for the given Doppler scale η.To get a better understanding, Figure 7 plots WAF for two different values of Doppler scale factors, that is, η ¼ 1 and η ¼ 0:95 for B f c ¼ 0:5.For the Doppler scale of 0.95, the main lobe for both LFM and HFM signals is shifted; also, the delay resolution of the LFM signal becomes poorer due to the widening of the main lobe of correlation output.Further, Figure 8 plots the delay resolution, that is, τ η 3dB as a function of normalized bandwidth for η ¼ 0:95 and η ¼ 0:99, normalized bandwidth is evaluated by measuring the distance between two half-power points of the main lobe.For lower bandwidth, both LFM and HFM have similar performance; as bandwidth increases, the delay resolution of LFM tends to become poor, and HFM performs better.Another critical parameter is the peak value of the ambiguity function; with the Doppler scale, the peak value of the ambiguity function decreases.This effect is more prominent in the case of LFM than the HFM signal.It shows that the HFM signal is more robust than the LFM signal under the Doppler effect.
Wideband ambiguity function for LFM and HFM signals for various bandwidth (B) and time duration (T) values.Zoom-1 is the zoomed plot showing the delay resolution, that is, the difference between two red dots, and the Doppler scale resolution, that is, the difference between two blue dots, for Cases I, III, and IV.Zoom-2 is the zoomed plot shown for the same purpose as Zoom-1, except that it is for comparing Cases II, III, and IV.Further, the available signal processing capacity for a given Doppler scale factor η can also be quantitatively measured using The higher the value of LðηÞ, the lesser will be its Doppler tolerance; the Doppler-induced scaling reduces processing gain due to a mismatch in the transmitted and the received signal.Figure 9 plots LðηÞ for various scale factors.For the LFM signal, LðηÞ increases rapidly with a slight increase in the Doppler scale, whereas the HFM signal has marginally less loss, only 3 dB loss, with the Doppler scale changing from 0.8 to 1.2.

| Receiver operating characteristics (ROC)
The ROC curve is a standard metric used to evaluate the performance of packet detection with prefix signals.It represents the relationship between the probability of detection (PD) and the probability of false alarm (PFA).ROC under different scenarios described in Table 2 and shown in Figures 10-13 for parameters given in Table 3 for both LFM and HFM are being studied.The observations are summarized below: • On increasing the bandwidth 5 to 10 kHz, robustness to Doppler scaling for HFM improves, whereas for the LFM signal, it reduces with an increase in bandwidth.• With the decrease of T, both the LFM signal and the HFM signal performance deteriorate.It is due to reduced processing gain because of less time.It can be observed from the ROC curve that, though in the absence of the Doppler effect, that is, η ¼ 1, both the signals have equivalent performance.However, the performance of the LFM signal degrades compared to that of HFM as the Doppler effect increases.Another important result is summarized in this lemma.Proof.The proof is given in Appendix A.
The above analysis summary is given in Table 4, which shows the clear advantage of using the HFM signal over the LFM signal.A large Doppler spread characterizes the UWA channel; hence, the transmitted signal is expected to undergo a significant Doppler scaling effect.As per the above discussion, the HFM signal is more suitable under such conditions of the UWA channel.□

| DOPPLER AND DELAY ESTIMATION
As discussed, the HFM signal is Doppler insensitive for various reasons, especially for larger BT values.Therefore, the HFM signal is used as a prefix signal in our work.Estimation of Doppler, delay, and channel is essential for reliable communication.HFM signal is not only a good candidate for prefix signal but can also be used in the Doppler, delay, and channel estimation.One important property of the HFM signal is elaborated using the following Lemma.
Lemma 2. The effect of Doppler-induced scaling in an HFM pulse can be modeled as a linear time shift.Doppler scaled HFM signal, that is, sðηðt À τÞÞ can be modeled as a linear time shift in the HFM signal sðtÞ, such that where linear time shift, that is, Δt, is given by where τ is the propagation delay of the path, f 0 is the start frequency of the chirp, and η and b are the Doppler scale factor and the chirp rate, respectively.be solved to estimate η and τ.The same time support is required to avoid the impact of time variation of the channel in delay and Doppler estimation.Frequency support of the signals should cover the entire available bandwidth of the channel for its complete characterization.The best possible two HFM signals are up-chirp HFM and down-chirp HFM supported over the entire bandwidth. 31To provide the same time support and avoid channel variability, we use the linear combination of these up-chirp HFM and down-chirp HFM signals given as

Proof. Given in
where s u ðtÞ is unit power normalized up-chirp HFM and s d ðtÞ is unit power normalized down-chirp HFM given as □ From Lemma 2, linear time shift Δt u between transmitted and received up-chirp HFM under the effect of Doppler scaling η is given as Similarly, linear time shift Δt d for down-chirp HFM is given as

| CHANNEL ESTIMATION ALGORITHM
Underwater channels are time-varying in nature; the output of the linear time-variant channel, that is, rðtÞ, can be given as where hðτ, tÞ is the time-varying CIR, sðtÞ is the transmitted signal, τ is the propagation path delay, and wðtÞ is the additive noise.Propagation of signal in the underwater acoustic medium is manifested by various phenomena like scattering, refraction, and reflection.The time-varying impulse response hðτ, tÞ can be modeled as hðt, τÞ ¼ where A i ðtÞ and τ i ðtÞ are the time-varying amplitude and delay corresponding to ith multipath.Let L be the number of multipath.The time-varying delay corresponding to ith multipath for the case of constant relative velocity between transmitter and receiver can be approximated as where τ i ð0Þ is fixed time delay and a i is the rate of change of ith multipath delay.Substituting ( 18) and ( 17) in ( 16), we get and η i ¼ 1 À a i .The received signal rðtÞ can be modeled as the summation of paths with propagation delay τ i with corresponding Doppler scale η i .This can be seen as a multi-scale multi-lag (MSML) type of channel, that is, different Doppler scaling and corresponding delays for each multipath.Here, the task of channel estimation boils down to the determination of three parameters, that is, the amplitude (A i ), the propagation delay(τ i ), and the Doppler scale factor (η i ) for a significant number of multipath.It can be accomplished by 2D search over each Doppler scale and delay dimension, but that will be costly in complexity.The complexity of the search algorithm can also be reduced by reducing the number of dimensions to be searched.
Again, hyperbolic time-frequency coupling of the HFM signal is used to reduce the search dimension by one order, making it a 1D search.Two-stage algorithm consisting of coarse and fine joint Doppler scale and delay estimation is formulated.The coarse estimation employs a correlation-based method, while fine estimation is based on a LUT-based search to estimate these parameters.

| Coarse Doppler estimation
The complete flow of the proposed coarse channel estimation method is shown in Figure 14.It uses a correlation-based approach to estimate the delay between the transmitted and the received signal.The peak detection algorithm uses correlator output for the detection of true peaks.The peak detection algorithm annihilates false peaks to get a true estimate.Delays associated with the estimated peaks are used to jointly estimate the associated path's Doppler scale and propagation delay.

| Correlation-based detector
Received signal rðtÞ is correlated with the transmitted up chirp, that is, s u ðtÞ.Correlator output can be expressed as where T is the duration of prefix signal and T g is the guard interval.On substituting (19) in (20), we get where w 0 ðtÞ ¼ R TþT g 0 wðtÞs u ðt À τ 0 Þdt, after rearrangement, where 11), it can be expressed as where The proposed prefix consists of a summation of up-chirp and down-chirp HFM due to the orthogonality of up-chirp and down-chirp HFM; we get putting (24) in (22), we get and in the absence of Doppler effect, from ( 17) and ( 25), we can deduce that ĝu ðτ 0 , tÞ ≈ hðτ, tÞ.Moreover, the peaks of ĝu ðτ 0 , tÞ are strictly at propagation delay τ i .Therefore, at any time instance t, the CIR, that is, hðτ,tÞ, can be estimated by observing the peaks of ĝu ðτ 0 , tÞ.However, as a result of Doppler scaling η i , as per ( 14), the position of the peaks will be shifted from τ i to Δt u,i , shifted peak position is given by Furthermore, the corresponding magnitude of the peak is given by Similarly, the output of the correlator with down-chirp HFM can be expressed as where w 0 , ðtÞ ¼ R TþT g 0 Position of peaks for ĝd ðτ 0 ,tÞ, that is, correlator output for corresponding to down-chirp HFM can be expressed as where f T is the start frequency of the down chirp, and the magnitude of the corresponding peak is given by It can be observed that in ( 25) and ( 28), the correlator output corresponding to up-chirp and down-chirp HFM are not the same, that is, ĝu ðτ 0 , tÞ ≠ ĝd ðτ 0 , tÞ, this is mainly due to Doppler.This can be more effectively visualized using the 3 dB contour of WAF (Figure 15).WAF of the proposed prefix signal consists of two ridges, concave up the ridge for the up chirp and concave down the ridge for the down chirp.For a given value of the Doppler scale, η i linear time shifts for both up and down chirps are different.It is represented by the intersection of respective contour by line η ¼ η i , as shown in Figure 15 by black dots.Hence, value of η i and τ i can be obtained by solving Equations ( 26) and ( 29) simultaneously.Therefore, the correlation peaks need to be estimated to estimate the channel, which is elaborated further.

| Peak detection
Here, signal from the transmitter to the receiver reaches through many paths, and as discussed above, delay corresponding to each path is estimated by detecting the peaks of the ĝu ðτ 0 , tÞ.To estimate these peaks and to ascertain that they are true peaks, a two-step procedure is proposed in this work.The first step is to keep only those peaks greater than a certain threshold p Nth .The value of the threshold p Nth is decided on the targeted false alarm rate.As false peaks are due to large sidelobes, p th is determined using the SLL of the autocorrelation function of the transmitted prefix signal.Another parameter to ascertain the correct peak ensures that the time difference between consecutive/nearest two peaks has to be more than the time difference between the main lobe and sidelobe of the transmitted prefix signal, that is, N th .Therefore prominent peaks are determined by comparing the magnitude and separation between two consecutive peaks; the proposed method is described in Algorithm 1.After peak detection, the next task is to estimate the Doppler and delay using the locations of the peaks.

| Doppler and delay profile estimation
Delay and Doppler scale estimation described above is used to estimate each path's delay and Doppler jointly.Let the peak amplitude detected for the up chirp and down chirp arranged in L Â 1 vectors p u and p d , respectively, as and where p u,i and p d,i are the amplitude of ith correlator peaks corresponding to up chirp and down chirp, respectively, and t u,i and t d,i are the corresponding locations and L is number of multipath.Define Lemma 3. The Doppler scale factor η ¼ ½η 1 , η 2 ,…, η L and propagation delay τ ¼ ½τ 1 , τ 2 , …, τ L can be jointly estimated using the peak location vector t u and t d .Estimated η is given as where , I is a L Â 1 vector containing all ones, and B is the bandwidth with f c corresponding center frequency.Estimated propagation delay (τ) is given as where Proof.Given in Appendix B Estimated Doppler and delay estimated from ( 35) and ( 36) are coarse.Finer correction to the estimated values is further required to meet desired performance.□

| Fine Doppler scale correction method
It can be observed from ( 35) and ( 36) that the value of estimated Doppler scale factor η depends upon t u À t d .Therefore, the resolution of η depends upon the sampling rate.In other words, the estimated Doppler scale η can take a certain value depending upon the sampling time T s .These possible values are, η given as, As t u,i À t d,i ≉ AE nT s , hence the actual value of the Doppler scale is different from the estimated Doppler value, it can lie anywhere in the interval

!
. It shows that the quantization error is not uniform.The non-uniformity of the Doppler scale limits the joint Doppler scale and delay estimation algorithm.It may result in the raised noise floor at higher SNR.We propose a novel LUT-based method to address this issue.The process flow of the proposed algorithm is given in Algorithm 2.
Using the coarse estimate of Doppler with (38), a LUT (39) with possible values of the Doppler scale is formed.Further, the Doppler scale is always limited by relative velocity between source and receiver, that is, 2v max c , where v max is the maximum relative velocity between source and receiver, and c is the velocity of sound in water.It ascertains the upper and lower limit of the Doppler scale.
The coarse estimated Doppler scale is compared with the entries in L. Let j be the index of L corresponding to the estimated Doppler scale (η i ), that is, LðjÞ ¼ ηi .
Further, the Doppler scale values corresponding to ðj À 1Þth to ðj þ 1Þth entries of L are divided uniformly according to the targeted Doppler scale resolution desired (δη) and define G as Finally, the optimum Doppler scale value is obtained as The advantage of microgrid formulation is that the number of entries in the microgrid depends upon the desired resolution δη.Though the above approach looks computation intensive, this algorithm can run in parallel for each received path; hence, computations can be done in real time.

| Channel coefficient estimation
After delay and Doppler estimation, the next task is to estimate the amplitude of the respective path.The amplitude of the respective path is estimated by correlating the transmitted prefix with the resampled received signal s rr ðtÞ, resampled by estimated Doppler scale, where ηi is that path's estimated Doppler scale factor.The amplitude of that particular path can be estimated as follows: where R rs ðτÞ is the correlation of the resampled received signal with the transmitted prefix as τi is the estimated propagation delay of that multipath, associated with the peak of R rs ðτÞ.

| Limitation of coarse Doppler scale and delay estimation
To assess the effect of quantization on Doppler scale estimation, we simulated signals with Doppler scaling ranging from AE20%, with other parameters given in Table 5 under no-noise condition.Figure 16 demonstrates that for the wavelet-based technique, 31 the quantization error is non-uniform and increases as the value of the Doppler scale to be estimated increases.For the proposed approach, however, the error in the estimated Doppler scale is bounded by the LUT resolution.

| Performance of the proposed algorithm for the delay and the Doppler scale estimation with SNR
Performance of delay and Doppler estimation algorithm has been evaluated for parameters given in Table 6.All results are obtained by keeping desired Doppler grid resolution, that is, δη ¼ 1e À6 .The proposed algorithm has been evaluated for four Doppler scale values: η ¼ 0:995,0:99,1:001, and 1.01.1e 4 Monte Carlo simulations are used for the performance evaluation.Estimation error for the Doppler scale factor (η i ), propagation delay (τ i ), and the channel amplitude are evaluated by using the following relationships: where ψð:Þ i is the estimation error.The Doppler scale and delay estimation results have been shown in Figures 17 and 18, respectively.The proposed algorithm is compared with the wavelet-based channel estimation suggested in Zhang et al. 31 It can be observed from the figures that the performances of the proposed Doppler scale estimation method outperform the method proposed in Zhang et al 31 by orders of magnitude, except for the Doppler scale value of η ¼ 1:01; in this case, the performance improvement is marginal at low SNR.Similarly, the delay estimation using the proposed method also performs significantly better than that of Zhang et al. 31 The performance improvement is mainly governed by the better representation resolution of the proposed method in both delay and the Doppler domain.The Doppler estimation error at η ¼ 0:995 is best, which provides a much better estimate of the channel as shown in Figure 19 for the given Doppler scale resolution of δη ¼ 1e À6 .Better Doppler scale and delay estimation have resulted in more accurate channel estimation.The floor in the error function is due to the finite resolution of the Doppler and delay estimation characteristics of the proposed method.

| Performance of channel estimation with SNR
Further channel estimation performance has been evaluated for BELLHOP 42 channel simulated on MATLAB, as shown in Figure 20, with parameters given in Table 7.The performance of the Doppler scale and delay estimation algorithm is given in Figures 21 and 22, respectively, which shows the performance of the proposed algorithm is superior to where b l ¼ f T Àf 0 T .b l is the chirp rate parameter; it indicates the rate of variation of frequency of the LFM signal with time.f T and f 0 are the instantaneous frequency at time t ¼ T and 0, respectively.T is the duration of LFM signal, bandwidth B of LFM signal is given by B ¼ jf T À f 0 j.Considering channel consisting single path with Doppler scale factor η and with delay of τ, resulting signal is given by After Doppler-based scaling: Instantaneous frequency of both lðtÞ and l η ðtÞ is given by and respectively.s η ðtÞ has initial instantaneous frequency of ηf 0 at t ¼ τ and at t ¼ T η þ τ instantaneous frequency is ηf T .If the LFM signal is Doppler scale invariant, it should result in a linear time shift on subjecting to Doppler-induced scaling, that is, On substituting (A3) and (A4) in (A5), we get It shows that the value of Δt l obtained varies with time.Hence, no fixed value of Δt l exists for the LFM signal.
where b ¼ f T Àf 0 f T f 0 T .b is the chirp rate parameter, it indicates the rate of variation of frequency of HFM signal with time.f T and f 0 are the instantaneous frequency at time t ¼ T and 0, respectively.T is the duration of HFM signal, and bandwidth B of HFM signal is given by B ¼ jf T À f 0 j.
Considering channel consisting single path with Doppler scale factor η and with delay of τ, resulting signal is given by respectively.s η ðtÞ has initial instantaneous frequency of ηf 0 at t ¼ τ, and at t ¼ T η þ τ, instantaneous frequency is ηf T .Effective chirp rate factor, that is, b η is given by which is essentially the same as that of the chirp rate of sðtÞ.The chirp rate governs the shape of the time-frequency relationship.Hence, the HFM pulse retains the shape of the time-frequency curve even after Doppler-induced scaling.
As a result that it is invariant to Doppler-induced scaling as shown in Figure A1.Since the HFM chirp signal is Doppler invariant, it means that Doppler-induced scaling with factor η of the HFM signal will result in a linear time shift ΔT.It will satisfy the following relationship: Putting (A9) and (A10) in Equation (A12).Value of Δt for up chirp is given by Similarly, for down-chirp value of Δt is given by It indicates that delaying sðtÞ by Δt will result in a signal that is identical to sðηtÞ.

APPENDIX B
HFM signal is Doppler invariant, from ( 26) and ( 29), the following relationship can be derived: where τ is a L Â 1 vector of propagation delay and Doppler scale factors L multipaths, respectively, 1 is as L Â 1 vector containing all ones, τ ¼ ½τ 1 , τ 2 ,…, τ L , η ¼ ½η 1 , η 2 ,…, η L , and From ( 35) and (36) Doppler scaling factor, that is, η and channel delay profile, that is, τ can be estimated as follows: where where f c and B is the center frequency and bandwidth, respectively.Equation (B4) can be written as where

1 FF
I G U R E 4 Autocorrelation plot for HFM and LFM signals.I G U R E 3 Impact of increasing BT by changing the bandwidth of signal.
F I G U R E 5 rms bandwidth comparison of LFM and HFM signals.

F I G U R E 7
Impact of Doppler scaling on LFM and HFM signals at B f c .

F
I G U R E 8 Delay resolution of LFM and HFM signals under Doppler effect.dB) BT = 100, B = 10kHz , T = 10ms LFM HFM F I G U R E 9 SNR loss for Doppler-induced scaling for BT ¼ 100.

T A B L E 2 1 F 1 F 1 F
Cases of ROC simulation.Case I B ¼ 5 kHz, T ¼ 5 ms Case II B ¼ 5 kHz, T ¼ 10 ms Case III B ¼ 10 kHz, T ¼ 5 ms Case IV B ¼ 10 kHz, T ¼ 10 ms I G U R E 1 0 ROC curve for LFM and HFM of various Doppler scale values-Case I.I G U R E 1 1 ROC curve for LFM and HFM of various Doppler scale values-Case II.F I G U R E 1 2 ROC curve for LFM and HFM of various Doppler scale values-Case III.I G U R E 1 3 ROC curve for LFM and HFM of various Doppler scale values-Case IV.T A B L E 3 Simulation parameters.

Lemma 1 .
HFM signals are invariant to Doppler-induced scaling due to its hyperbolic-type time-frequency coupling, whereas LFM signals are affected by Doppler-induced scaling.
U R E 1 5 3 dB contour of WAF for proposed prefix signal, black dots represent the intersection of η ¼ η i with respective curve.

F I G U R E 2 2 F 1 |
Performance of the delay estimation with SNR.A1, wavelet-based algorithm31 ; A2, proposed algorithm.I G U R E 2 3 Performance of the proposed algorithm for channel estimation with SNR.A1, wavelet-based algorithm31 ; A2, proposed algorithm.Impact of Doppler on LFM signal Let lðtÞ be the transmitted LFM signal given by lðtÞ

A. 2 |
Impact of Doppler on HFM signalLet sðtÞ be the transmitted HFM signal given bysðtÞ ¼ cos À2π b logð1 À bf 0 tÞ 0 ≤ t ≤ T, based scaling: Instantaneous frequency of both sðtÞ and s η ðtÞ is given by

F
I G U R E A 1 Scale invariance of HFM pulse.

f
ηI ðtÞ ¼ f I ðt À ΔtÞ: Time shifts Δt corresponding to different HFM signals with the same time and frequency support can simultaneously T A B L E 4 Comparison of LFM and HFM signals.
Appendix A. This relation can be used to estimate the delay and Doppler scale induced, that is, τ and η.
Simulation parameters for channel estimation.
I G U R E 2 0 Simulated channel geometry.