A time-varying underwater acoustic channel estimation method

Abstract

The present invention discloses a method for estimating a time-varying underwater acoustic channel. The method comprises the following steps: (1) using the Fourier transform method to perform time offset compensation on real underwater acoustic data; (2) performing Kronecker product structural decomposition on the real underwater acoustic channel and selecting structural parameters using the Bayesian Information Criterion; and (3) using the sparse enhanced conjugate gradient method based on known structural parameters to online estimate the time-varying channel impulse response and output a multi-pulse underwater acoustic channel estimation result. The method has the advantages of utilizing the potential structured sparsity characteristics of the real underwater acoustic channel to reduce the channel impulse response parameter estimation dimension, effectively suppressing false estimation under low signal-to-noise ratio conditions, and utilizing the adaptive recursive concept to provide good online continuous estimation capability for multi-pulse time-varying underwater acoustic channels and strong noise interference resistance.

Description

Technical Field

[0001] The present invention belongs to the field of underwater acoustic statistical signal processing, and relates to a time-varying underwater acoustic channel estimation method, in particular to a time-varying underwater acoustic channel estimation method based on structured sparsity. Background Art

[0002] Underwater acoustic channels exhibit multipath extension and time-varying characteristics, which cause signal distortion and severely impact underwater acoustic communications and underwater target detection. In recent years, many researchers have exploited the inherent sparsity of underwater acoustic channels to effectively improve the accuracy and efficiency of channel estimation. Adaptive techniques are highly adaptable to complex and rapidly varying underwater acoustic channels, while recursive approaches can effectively perform online processing and reduce device storage requirements. Therefore, adaptive sparse methods offer new strategies for estimating and tracking time-varying underwater acoustic channels, attracting widespread attention.

[0003] In recent years, in the field of echo cancellation, some scholars have proposed using low-rank ideas to reconstruct acoustic signals and use their potential repeated structures to achieve echo cancellation and enhance the acoustic signals of interest. However, they have not considered the practical application of sparse structures and adaptive technologies in the echo field. Summary of the Invention

[0004] In response to the above-mentioned existing technologies, the technical problem to be solved by the present invention is to provide a time-varying underwater acoustic channel estimation method based on structured sparsity. On the basis of real underwater acoustic signal processing, the potential structured sparsity characteristics of the underwater acoustic channel are utilized, combined with the idea of ​​adaptive recursion, and the time-varying underwater acoustic channel estimation results are updated pulse by pulse online. It still has good estimation performance under low signal-to-noise ratio conditions and has strong robustness.

[0005] To solve the above technical problems, the present invention provides a time-varying underwater acoustic channel estimation method, comprising:

[0006] Step 1: Use the estimated offset time delay to compensate the real underwater acoustic data to obtain calibrated underwater acoustic data without offset influence;

[0007] Step 2: Decompose the k-th pulse real underwater acoustic channel impulse response function h with structured sparse characteristics through Kronecker integral decomposition k Decomposed into two sub-channel components h k，1 and h k，2 , and satisfy:

[0008]

[0009] Among them, h k The length is M, the subchannel h k，1，p and h k，2，pThe lengths are M1, M2, M1>M2 and M1*M2=M; define P=rank(H k ), matrix H k It is h k Obtained by matrixing a column vector of length M1, and P≤M2;

[0010] Define M1, M2, and P as Kronecker product structure parameters, and determine P (t) All possible parameter combinations are constructed, where the superscript represents the tth group of structural parameter combinations. The subchannel estimation results corresponding to each group of parameter combinations are obtained under the kth pulse measurement data using the least squares estimation method. The normalized evaluation factor corresponding to each group of parameter combinations is calculated, and the Kronecker product structure parameters are determined using the Bayesian information criterion.

[0011] Step 3: Based on the Kronecker product structure parameters determined in step 2, the subchannel h is estimated by iterative update using the sparse enhanced conjugate gradient recursive method. k，1，p and h k，2，p ,get and k=k+1, k+2,…,K, where K represents the number of pulses. Then, using the Kronecker product structure assumption, the k-th pulse time-varying underwater acoustic channel estimation result is output as:

[0012]

[0013] Furthermore, the offset delay estimation method includes:

[0014] Calculate the discrete time Fourier transform DTFT of two underwater acoustic signals x[n] and y[n] of length 2N as follows:

[0015]

[0016]

[0017] The frequency index interval is 0≤f≤2N-1, T represents the time resolution, and a discrete time cross-correlation transform sequence of length 2N is constructed:

[0018]

[0019] Then the offset delay t g =m′T, where

[0020] Furthermore, the obtaining of the subchannel estimation result corresponding to each parameter combination under the k-th pulse measurement data by using the least squares estimation method includes:

[0021] Based on the Kronecker product structure assumption, the underwater acoustic receiving signal model of the kth pulse is:

[0022] Model 1:

[0023]

[0024] Or model 2:

[0025]

[0026] Among them, S k Represents the Toplitz matrix of the transmitted signal, n k represents additive noise, and Equivalently, in vector and matrix form, respectively, and Equivalent, respectively in vector and matrix form, fixed or initialized h k，2，p Right now S k，2 , and then estimate using the least squares method based on model 1 to get Then get, fix or initialize h k，1，p Right now S k，1 , and then use the least squares method to estimate Then we get the subchannel estimation result and

[0027] Furthermore, the calculation obtains the normalized evaluation factor corresponding to each parameter combination, and the Kronecker product structure parameters are determined by the Bayesian information criterion, including:

[0028] Ψ=(M1+M2)P represents the number of parameters to be estimated, and the normalized evaluation factor η(Ψ (t) ):

[0029]

[0030] Among them, h k represents the true value of the channel impulse response function of the kth pulse, and They are Ψ (t) Corresponding subchannel h k，1，p and h k，2，p The estimated value of (t) ) are substituted into the Bayesian Information Criterion formula one by one to obtain the BIC curve, which is specifically:

[0031] BIC(Ψ(t) )=Ψ (t) ln(2N)+2Nlog(η(Ψ (t) ))

[0032] Where 2N represents the length of underwater acoustic data of a pulse; the horizontal axis parameter Ψ corresponding to the minimum value of the BIC curve is (opt) Corresponding P (opt) as the Kronecker product structure parameter.

[0033] Furthermore, the iterative update of the estimated subchannel h using the sparse enhanced conjugate gradient recursive method is k，1，p and h k，2，p ,get and include:

[0034] h k，1，p The iterative update formula is:

[0035] h k，1 = h k-1，1 +W k，1 p k，1 α k，1

[0036] in, h k，1，p and h k，1 Equivalent, respectively, in matrix and vector form, W k，1 Represents an iteratively weighted diagonal matrix, with diagonal elements consisting of vectors Determine, 0<ξ1≤1 is the sparse operator characterizing the sparsity of the underwater acoustic channel, c is a constant; p k，1 and α k，1 They represent the convergence step size of the recursive search direction, which respectively control the convergence speed and computational complexity of the algorithm; where:

[0037]

[0038] p k，1 =β k，1 p k-1，1 -g k，1

[0039]

[0040] g k，1 =W k，1 (R k，1 h k-1，1 -r k，1 )

[0041] Among them, R k，1 and r k，1 denote the autocorrelation matrix and cross-correlation matrix of the system input, respectively, and

[0042]

[0043] r k，1 =λr k-1，1 +S k，2 y k

[0044] Where λ represents the forgetting factor, Determine R k，1 , r k，1 The process always includes another subchannel h k，2，p Parameter, δ is a constant to ensure that the denominator is not 0 during the calculation process;

[0045] h k，2，p The iterative update formula is:

[0046] h k，2 = h k-1，2 +W k，2 p k，2 α k，2

[0047] in, h k，2，p and h k，2 Equivalent, respectively, in matrix and vector form, W k，2 Represents an iteratively weighted diagonal matrix, with diagonal elements consisting of vectors Determine, 0<ξ2≤1 is the sparse operator characterizing the sparsity of the underwater acoustic channel, c is a constant; p k，2 and α k，2 They represent the convergence step size of the recursive search direction, which respectively control the convergence speed and computational complexity of the algorithm; where:

[0048]

[0049] p k，2 =β k，2 p k-1，2 -g k，2

[0050]

[0051] g k，2 =W k，2 (R k，2 h k-1，2 -r k，2 )

[0052] Among them, R k，2 and r k，2 denote the autocorrelation matrix and cross-correlation matrix of the system input, respectively, and

[0053]

[0054] r k，2 =λr k-1，2 + S k，1 y k

[0055] Where λ represents the forgetting factor, Determine R k，2 , r k，2 The process always includes another subchannel h k，1，p Parameter, δ is a constant to ensure that the denominator is not 0 during the calculation process;

[0056] In the recursive update process, the subchannel estimation value known at the previous moment is used as the initial value at the current moment, and the current k moment is alternately estimated. and

[0057] Furthermore, the k-th pulse in step 2 is an average of the received data of the first k pulses, which is used as a test pulse for selecting structural parameters.

[0058] Beneficial effects of the present invention: Different from traditional sparse underwater acoustic channel estimation methods, the present invention utilizes the potential structured sparse characteristics of real underwater acoustic channels and combines the adaptive recursive idea to estimate the time-varying underwater acoustic channel by pulse-by-pulse online updates, especially showing its structured sparse advantages under low signal-to-noise ratio conditions, effectively suppressing false estimates, and at the same time utilizing the adaptive recursive idea to obtain good online continuous estimation performance and anti-noise interference capability, thereby reducing the estimation dimension of the channel impulse response parameters.

[0059] The present invention focuses on the fact that underwater equipment such as transmitting transducers or hydrophones, which cannot be ignored during underwater acoustic tests, are often affected by complex ocean environments such as platform movement and waves. Therefore, it is necessary to calibrate the time offset caused by such external disturbances. The present invention utilizes the potential structured sparse characteristics of real underwater acoustic channels to decompose the high-dimensional underwater acoustic channel model into two mutually coupled sub-channel structures, and assumes that the sum of their Kronecker product forms is equivalent to the high-dimensional underwater acoustic channel. At the same time, the Bayesian information criterion and evaluation factors are used to optimally select the structural parameters. The sparse enhanced conjugate gradient recursive method has both a fast convergence speed and low computational complexity. At the same time, its adaptive recursive idea effectively realizes the alternating iterative online estimation of the mutually coupled time-varying channel substructure parameters. The present invention uses the determined structural parameters to sum and output the sub-channel online estimation results in the form of Kronecker products, outputting the multi-pulse time-varying underwater acoustic channel estimation results, and realizing effective estimation of the time-varying underwater acoustic channel under low signal-to-noise ratio conditions. BRIEF DESCRIPTION OF THE DRAWINGS

[0060] Figure 1 This is a flow chart of the structured sparse time-varying underwater acoustic channel estimation method;

[0061] Figure 2 is the Bayesian Information Criterion (BIC) curve;

[0062] Figure 3(a) is a schematic diagram of the true value of the multi-pulse underwater acoustic channel simulation;

[0063] Figure 3(b) shows the results of tracking the time-varying underwater acoustic channel using the conventional method (least squares estimation method);

[0064] FIG3( c ) is a diagram showing the results of tracking a time-varying underwater acoustic channel using the method of the present invention;

[0065] FIG3( d ) is a comparison diagram of the single pulse channel estimation results of the conventional method and the method of the present invention. DETAILED DESCRIPTION

[0066] The present invention will be further described below with reference to the accompanying drawings and embodiments.

[0067] The present invention aims to provide an online real-time pulse-by-pulse update output time-varying underwater acoustic channel estimation method. The purpose of the present invention is achieved as follows:

[0068] Step 1: Use the discrete-time Fourier transform method to perform time offset compensation on multi-pulse underwater acoustic data in the frequency domain;

[0069] Step 2: Leveraging the potential structured sparsity of the underwater acoustic channel, the high-dimensional underwater acoustic channel is decomposed into two mutually coupled low-dimensional channel structures through Kronecker decomposition. The least squares estimation method is used to obtain sub-channel estimation results based on single-pulse measurement data. The Bayesian Information Criterion and evaluation factors are then used to optimally select the structural parameters.

[0070] Step 3: Based on the determined structural parameters, the sparse enhanced conjugate gradient recursive method is used to estimate the parameters of each pulse sub-channel online, and the Kronecker product is summed and output according to the structural parameters to obtain the multi-pulse underwater acoustic channel estimation result under low signal-to-noise ratio conditions.

[0071] The following provides embodiments with reference to the accompanying drawings and parameters.

[0072] The present invention is a time-varying underwater acoustic channel estimation method based on structured sparseness, which includes two parts: time offset compensation for a real underwater acoustic channel and structured sparse modeling.

[0073] Step 1: Calculate the discrete time Fourier transform (DTFT) of two underwater acoustic signals x[n] and y[n] of length 2N as follows:

[0074]

[0075]

[0076] Where the frequency index interval is 0≤f≤2N-1, and T represents the time resolution. The discrete time 'analysis' cross-correlation transform sequence of length 2N is constructed as:

[0077]

[0078] Calculate the offset delay t g =m′T, where The real underwater acoustic data is compensated according to the estimated offset delay to obtain the calibrated underwater acoustic data without offset influence.

[0079] Step 2: Assuming that the underwater acoustic channel of the k-th pulse has a structured sparse characteristic, the original underwater acoustic channel impulse response function h is decomposed by Kronecker integral. k Decomposed into two sub-channel components h k，1 and h k，2 , and satisfy it:

[0080]

[0081] where h k Length M, subchannel h k，1，p and h k，2，p The lengths are expressed as M1 and M2 respectively, and it is always assumed here that M1>M2 and M1*M2=M.k The matrix H can be obtained by matrixing the column vector of length M1 k , P = rank (H k ), and requires P ≤ M2. Therefore, we define M1, M2, and P as Kronecker product structure parameters. Usually, the underwater acoustic receiving signal model of the k-th pulse is expressed as:

[0082]

[0083] Among them S k represents the Toplitz matrix of the transmitted signal, h k represents the channel impulse response function, n k represents additive noise. It is worth noting that Through mathematical derivation, it can be rewritten as in and Substituting in, we get:

[0084]

[0085] in, It is worth noting that the estimated It is estimating the subchannel The two are respectively the vector and matrix forms of the same physical quantity. At the same time, we find that the estimation In the process, we need to first determine h k，2，p , that is, the two sub-channels are coupled to each other, we fix or initialize h k，2，p Known S k，2 , and then the least squares method can be used to estimate

[0086] Similarly, according to the equivalent transformation of Kronecker integral decomposition, that is We can get:

[0087]

[0088] in h k，2 , S k，1 , S k，1，p The format is the same as above and will not be described here. Confirm S k，1 , further estimated by the least squares method Therefore, we realize the sub-channel under the single pulse condition of given structural parameters Right now Parameter estimation, definition of normalized evaluation factors

[0089]

[0090] Among them, h k Represents the true value of the channel impulse response function of the kth pulse.

[0091] We further determine the Kronecker product structure parameters using the Bayesian Information Criterion. The Bayesian Information Criterion formula is:

[0092] BIC(Ψ)=Ψln(2N)+2Nlog(η(Ψ))

[0093] Where Ψ=(M1+M2)P represents the number of parameters we need to estimate, 2N represents the length of underwater acoustic data of a pulse, and η(Ψ) represents the normalized evaluation factor under the condition of Ψ structural parameter. We input different structural parameter combinations, namely P (t) , where the superscript represents the t-th group of structural parameter combinations. Due to the restrictions on structural parameter combinations: M1>M2≥P, M1×M2=M, the value of Ψ is limited. Our goal of using the Bayesian information criterion is to select the structural parameter combination corresponding to the optimal normalized evaluation factor. Figure 2 As shown, by continuously updating the structural parameter combination, we can obtain the BIC curve, and the horizontal axis parameter corresponding to its minimum value is our most ideal structural parameter Ψ opt , that is, the optimal P (opt) The kth pulse is the test pulse for selecting the structural parameters by taking the average of the received data of the first k pulses, and k = 5 is usually sufficient.

[0094] Step 3: In practical applications, we usually believe that the potential structural changes of different pulses in a short time are not drastic, so there is no need to select structural parameters for each pulse. This patent approximately assumes that the structural parameters are constant under a small number of pulses, that is, the optimal structural parameters are obtained by step 2 for the subsequent online estimation process. We use the iterative update of the sparse enhanced conjugate gradient recursive method to estimate the subchannel h k，1，p and h k，2，p .

[0095] First h k，1，p The iterative update formula is:

[0096] h k，1 = h k-1，1 +W k，1 p k，1 α k，1

[0097] in, Step 2 has already explained k，1，p and h k，1 Equivalent, there is a fixed mathematical transformation between the two, which are the matrix and vector forms of the same physical quantity, W k，1 Represents an iteratively weighted diagonal matrix, with diagonal elements consisting of vectors Determine, 0<ξ1≤1 is the sparse operator to characterize the sparsity of the underwater acoustic channel, c is a constant usually taken as 10 -6 .p k，1 and α k，1 They represent the convergence step size of the recursive search direction, which control the convergence speed and computational complexity of the algorithm respectively.

[0098] in:

[0099]

[0100] p k，1 =γ k，1 p k-1，1 -g k，1

[0101]

[0102] g k，1 =W k，1 (R k，1 h k-1，1 -r k，1 )

[0103] where R k，1 and r k，1 denote the autocorrelation matrix and cross-correlation matrix of the system input respectively, and:

[0104]

[0105] r k，1 =λr k-1，1 + S k，2 y k

[0106] Where λ represents the forgetting factor, That is to determine R k，1 , r k，1 The process always includes another subchannel h k，2，p Parameter, δ is a constant to ensure that the denominator is not 0 during the calculation process, usually 10 -6 .

[0107] Similarly, we can deduce h k，2，p The iterative update formula is

[0108] hk，2 = h k-1，2 +W k，2 p k，2 α k，2

[0109] The parameter W k，2 , p k，2 , α k，2 , R k，2 , r k，2 , S k，1 Form and W k，1 , p k，1 , α k，1 , R k，1 , r k，1 , S k，2 The same, no longer repeated. That is to determine R k，2 , r k，2 The process always includes another subchannel h k，1，p parameter.

[0110] Therefore, in the recursive update process, h k，1，p and h k，2，p Solving is a mutually coupled process. At k=1, we initialize h 0，2，p , W 0，1 , p 0，1 , α 0，1 , R 0，1 , r 0，1 , estimated according to the above sparse enhanced conjugate gradient method Right now If h is known 1，1，p , also using the sparse enhanced conjugate gradient method to estimate Right now we will As h at time k = 2 2，2，p Initialization, estimation The same further solution estimates Notice It is h 2，2，p Initialized calibration, that is, at each moment we use the subchannel estimated at the previous moment as the initial value, and alternately estimate the current k moment and Further using the Kronecker product structure assumption, the output of the k-th pulse time-varying underwater acoustic channel estimation result (as shown in Figure 3(d)) is:

[0111]

[0112] The multi-pulse underwater acoustic channel estimation results are output online to achieve the time-varying underwater acoustic channel tracking results, as shown in Figure 3(c).

Claims

1. A time-varying underwater acoustic channel estimation method, characterized in that: include: Step 1: Use the estimated offset time delay to compensate the real underwater acoustic data to obtain calibrated underwater acoustic data without offset influence; Step 2: Decompose the k-th pulse real underwater acoustic channel impulse response function h with structured sparse characteristics through Kronecker integral decomposition k Decomposed into two sub-channel components h k，1 and h k，2 , and satisfy: Among them, h k The length is M, the subchannel h k，1，p and h k，2，p The lengths are M1, M2, M1>M2 and M1*M2=M; define P=rank(H k ), matrix H k It is h k Obtained by matrixing a column vector of length M1, and P≤M2; Define M1, M2, and P as Kronecker product structure parameters, and determine P (t) All possible parameter combinations are constructed, where the superscript represents the tth group of structural parameter combinations. The subchannel estimation results corresponding to each group of parameter combinations are obtained under the kth pulse measurement data using the least squares estimation method. The normalized evaluation factor corresponding to each group of parameter combinations is calculated, and the Kronecker product structure parameters are determined using the Bayesian information criterion. Step 3: Based on the Kronecker product structure parameters determined in step 2, the subchannel h is estimated by iterative update using the sparse enhanced conjugate gradient recursive method. k，1，p and h k，2，p ,get and K represents the number of pulses. Then, using the Kronecker product structure assumption, the k-th pulse time-varying underwater acoustic channel estimation result is output as:

2. The method for estimating a time-varying underwater acoustic channel according to claim 1, wherein: Methods for estimating offset delay include: Calculate the discrete time Fourier transform DTFT of two underwater acoustic signals x[n] and y[n] of length 2N as follows: The frequency index interval is 0≤f≤2N-1, T represents the time resolution, and a discrete time cross-correlation transform sequence of length 2N is constructed: Then the offset delay t g =m′T, where 3. The method for estimating a time-varying underwater acoustic channel according to claim 1, wherein: The method of using the least squares estimation method to obtain the subchannel estimation result corresponding to each parameter combination under the k-th pulse measurement data includes: Based on the Kronecker product structure assumption, the underwater acoustic receiving signal model of the kth pulse is: Model 1: Or model 2: Among them, S k Represents the Toplitz matrix of the transmitted signal, n k represents additive noise, and Equivalently, in vector and matrix form, respectively, and Equivalent, respectively in vector and matrix form, fixed or initialized h k，2，p Right now S k，2 , and then estimate using the least squares method based on model 1 to get Then get, fix or initialize h k，1，p Right now S k，1 , and then use the least squares method to estimate Then we get the subchannel estimation result and 4. The method for estimating a time-varying underwater acoustic channel according to claim 3, wherein: The calculation obtains the normalized evaluation factor corresponding to each parameter combination, and the Kronecker product structure parameter is determined by the Bayesian information criterion, including: Ψ=(M1+M2)P represents the number of parameters to be estimated, and the normalized evaluation factor η(Ψ (t) ): Among them, h k represents the true value of the channel impulse response function of the kth pulse, and They are Ψ (t) Corresponding subchannel h k，1，p and h k，2，p The estimated value of (t) ) are substituted into the Bayesian Information Criterion formula one by one to obtain the BIC curve, which is specifically: BIC(Ψ (t) )=Ψ (t) ln(2N)+2Nlog(η(Ψ (t) )) Where 2N represents the length of underwater acoustic data of a pulse; the horizontal axis parameter Ψ corresponding to the minimum value of the BIC curve is opt Corresponding P (opt) as the Kronecker product structure parameter.

5. The method for estimating a time-varying underwater acoustic channel according to claim 1, wherein: The iterative update estimation subchannel h using the sparse enhanced conjugate gradient recursive method k,1，p and h k，2,p ,get and include: h k，1，p The iterative update formula is: h k，1 = h k-1，1 +W k，1 p k，1 a k，1 in, h k,1,p and h k，1 Equivalent, respectively, in matrix and vector form, W k,1 Represents an iteratively weighted diagonal matrix, with diagonal elements consisting of vectors Determine, 0<ξ1≤1 is the sparse operator characterizing the sparsity of the underwater acoustic channel, c is a constant; p k, 1 and α k,1 They represent the convergence step size of the recursive search direction, which respectively control the convergence speed and computational complexity of the algorithm; where: p k，1 G. β k,1 p k-1，1 -g k，1 g k，1 =W k,1 (R k，1 h k-1，1 -r k,1 ) Among them, R k,1 and r k,1 denote the autocorrelation matrix and cross-correlation matrix of the system input, respectively, and r k,1 =λr k-1，1 + S k,2 y k Where λ represents the forgetting factor, Determine R k,1 , r k,1 The process always includes another subchannel h k,2,p Parameter, δ is a constant to ensure that the denominator is not 0 during the calculation process; h k,2,p The iterative update formula is: h k，2 = h k-1，2 +W k,2 p k,2 a k,2 in, h k,2,p and h k，2 Equivalent, respectively, in matrix and vector form, W k,2 Represents an iteratively weighted diagonal matrix, with diagonal elements consisting of vectors Determine, 0<ξ2≤1 is the sparse operator characterizing the sparsity of the underwater acoustic channel, c is a constant; p k,2 and α k，2 They represent the convergence step size of the recursive search direction, which respectively control the convergence speed and computational complexity of the algorithm; where: p k，2 G. β k，2 p k-1，2 -g k，2 g k,2 =W k，2 (R k，2 h k-1，2 -r k，2 ) Among them, R k，2 and r k，2 denote the autocorrelation matrix and cross-correlation matrix of the system input, respectively, and r k，2 =λr k-1，2 + S k，1 y k Where λ represents the forgetting factor, Determine R k，2 , r k，2 The process always includes another subchannel h k，1，p Parameter, δ is a constant to ensure that the denominator is not 0 during the calculation process; In the recursive update process, the subchannel estimation value known at the previous moment is used as the initial value at the current moment, and the current k moment is alternately estimated. and 6. The method for estimating a time-varying underwater acoustic channel according to claim 1, wherein: The k-th pulse in step 2 is the average of the received data of the first k pulses, which is used as the test pulse for selecting the structural parameters.

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