A Method for Detecting the Radiated Axial-Frequency Electromagnetic Signals of Underwater Targets

Abstract

The present invention discloses a method for detecting the radiated shaft-frequency electromagnetic signals of underwater targets, which relates to the field of signal detection. Aiming at the problem of underwater shaft-frequency electromagnetic signal detection, the present invention proposes a detection method combining stochastic resonance and ACI, and detects the shaft-frequency electromagnetic signals from two perspectives of time domain and frequency domain, and analyzes the underwater signal detection ability based on the combination of stochastic resonance and ACI. The signal detection model proposed by the present invention gives a time-frequency domain detection method for signals in a strong noise environment, and can solve the problem of underwater target magnetic detection at a certain signal-to-noise ratio.

Description

Technical Field

[0001] The present invention relates to the technical field of signal detection, and in particular to a method for detecting the radiated shaft-frequency electromagnetic signal of an underwater target. Background Art

[0002] With the continuous development of marine equipment, the importance of marine security issues has gradually increased. As a part of the characteristic signals of equipment platforms such as ships and underwater submarines, the shaft-frequency electromagnetic signal has obvious line-spectrum characteristics, slow attenuation speed, and long propagation distance. It, together with the acoustic field, magnetic field, water pressure field, etc., are all key characteristics of underwater targets, and its characteristics can be applied to the identification and positioning of ships and underwater targets. At the same time, suppressing the shaft-frequency electromagnetic signal can improve the stealth performance of ship targets.

[0003] The shaft-frequency electromagnetic signal is generally in the range of 1 - 7 Hz. In the actual marine environment, the noise components in this frequency band are relatively large. Therefore, the low signal-to-noise ratio restricts the detection range of underwater shaft-frequency electromagnetic signals. In the case of long distances, the shaft-frequency electromagnetic signals are all submerged in a large amount of noise and environmental interference. In order to detect the shaft-frequency electromagnetic signal under low signal-to-noise ratio conditions, a series of algorithms such as those based on chaotic detection and high-order zero-crossing analysis have been proposed. However, in the case of extremely low signal-to-noise ratio, the performance of these algorithms is still limited to a certain extent.

[0004] The stochastic resonance method was initially proposed by Benzi et al. to explain the problem of century glaciers, and the phenomenon that the presence of noise in a nonlinear system can increase the signal-to-noise ratio of the output signal was discovered. Then it was applied to many fields such as physics, chemistry, and biology. During the signal analysis process, noise is often considered something that is not conducive to detection. However, in some specific nonlinear systems, the presence of noise can produce a synergistic phenomenon with the nonlinear system to enhance the detection ability of weak signals.

[0005] The Adaptive Coherent Integration (ACI) algorithm is an improved algorithm of LMS (Least Mean Square). It adds a momentum term on the basis of LMS and makes it produce a signal accumulation phenomenon by adjusting the parameter range. The ACI method does not require any prior information about the signal waveform or time-varying law throughout the process and has its unique performance advantages in the field of signal detection. Therefore, the present invention proposes a method for detecting the radiated shaft-frequency electromagnetic signal of an underwater target based on the stochastic resonance method and the adaptive coherent integration algorithm. Summary of the Invention

[0006] The present invention aims to solve at least one of the technical problems existing in the prior art. For this purpose, the present invention proposes a method combining stochastic resonance and ACI. After specific selection of parameters, the shaft-frequency electromagnetic signal can be analyzed for features from both the time domain and the frequency domain, and relevant features of the signal can be extracted to achieve the purpose of improving the target detection ability.

[0007] To achieve the above object, the present invention proposes a method for detecting the radiated shaft-frequency electromagnetic signal of an underwater target, including:

[0008] S1 Obtain the noisy shaft-frequency electromagnetic signal in the ocean;

[0009] S2 Construct a preprocessing model for the noisy shaft-frequency electromagnetic signal, including decomposing and converting the noisy shaft-frequency electromagnetic signal, then performing signal filtering processing, Gaussianizing the non-Gaussian signal in the signal, and removing high-frequency noise to obtain the output signal x;

[0010] S3 Construct a bistable system stochastic resonance detection model, simulate to obtain the best matching parameters of stochastic resonance, use the best matching parameters as the reference signal, solve through the corresponding ratio to obtain the parameter values of a and b corresponding to the required signal, and calculate the required proportionality coefficient K of the input signal. The bistable system stochastic resonance detection model is expressed as:

[0011]

[0012] Where: Γ(t) is the noise, f(x) = ax - bx 3 is the non-linear quantity, and Acosa0t is the periodic driving force;

[0013] S4 Construct an adaptive coherent accumulation algorithm model, that is, the ACI algorithm model, to perform automatic signal energy coherent accumulation. The weight vector iteration algorithm of the ACI algorithm model is:

[0014] W(n + 1) = W(n) + δ·[W(n) - W(n - 1)] + 2μ[d(n) - y(n)]·X(n)

[0015] Where the input signal is d(n) = s(n) + v(n), where v(n) represents the noise and s(n) represents the original signal; the output signal is y(n) = (W(n)) T X(n), Where W(n) is the weight vector; μ is the adaptive step factor; δ is the adaptive coherent accumulation coefficient; M is the filter order; Δ is the time delay;

[0016] S5 performs algorithm parameter optimization on the adaptive coherent accumulation algorithm model, selects the output signal-to-noise ratio as the loss function, sets a threshold to zero out the ultra-high cumulants in the output of the adaptive coherent accumulation algorithm model, and optimizes the two parameters δ and μ in the algorithm;

[0017] S6 determines the occurrence time and signal spectrum of the shaft frequency signal. Input the signal x processed in S1 and the corresponding parameter values of a and b into the bistable system stochastic resonance model, perform time-frequency domain conversion on the output result to obtain signal spectrum information; input the signal x processed in S1 and the optimized parameters of δ and μ into the ACI algorithm model to obtain the occurrence time information of the signal in the time domain.

[0018] Furthermore, set the wavelet basis function of "db4" as the decomposition and conversion algorithm, set the adaptive decomposition level, perform wavelet decomposition and conversion on the signal, input the converted signal into the discrete wavelet transform to obtain the corresponding wavelet approximation coefficients and detail components of each layer, select threshold filtering for the wavelet coefficients, and then use the reconstruction function for reconstruction to achieve the suppression of high-frequency noise. The definition formula of the wavelet coefficient is:

[0019]

[0020] where: d j,k corresponds to the detail component of the j-th layer decomposition, a j,k is the approximation coefficient, h and g are the low-pass filter and high-pass filter corresponding to the selected wavelet basis, and * represents taking the conjugate.

[0021] Furthermore, the algorithm for selecting the adaptive decomposition level is as follows:

[0022] j = [log2(fs)] - 1

[0023] where [] represents taking the integer, j represents the decomposition level, and f s is the sampling frequency of the signal.

[0024] Furthermore, the algorithm for optimizing the algorithm parameters of the adaptive coherent accumulation algorithm model is the grey wolf algorithm. Write the ACI algorithm into the fitness function of the grey wolf algorithm, select the population size as 20, the problem dimension as 2, set the upper and lower bound variables of the system and the maximum number of iterations according to the data situation, use SNR1 as the loss function of ACI, optimize the parameters of the ACI algorithm through the loss function, and send the obtained result back to the original system to update the parameters to achieve efficient signal detection. The formula of the loss function is:

[0025]

[0026] where, P s is the power of the signal, P n is the power of the noise, and SNR1 is the output signal-to-noise ratio.

[0027] Further, the Runge-Kutta method is used to analyze and solve the bistable system model, specifically as follows:

[0028]

[0029] The parameters in the above formula are defined as follows:

[0030]

[0031]

[0032]

[0033] k4 = h[a(x n + k3) - b(x n + k3) 3 + u n+2

[0034] where h is the step size, taken as 1 / fs. fs is the sampling frequency of the signal, u n is the nth sampling point of the input signal, and x n is the nth point of the output signal. a and b are two parameters controlling the output.

[0035] Further, the optimal parameters in S3 are: a0 = 1, b0 = 1, the signal frequency f0 = 0.0625 Hz, and the root mean square value of the noise σ0 = 2.47. At this time, a good matching relationship is achieved among the parameters, which is used as a reference signal for corresponding parameter matching in subsequent simulation detection.

[0036] Further, the parameter matching rule is as follows:

[0037] The first step: Obtain the parameter a1 from and take the parameter b1 close to a1.

[0038] The second step: Calculate the root mean square value σ1 of the noise, and calculate the scaling factor K that needs to be multiplied in the signal input stochastic resonance algorithm from

[0039] The third step: Fine-tune the parameters according to the results.

[0040] The parameters a1, b1, and σ1 are the signal parameters to be obtained.

[0041] Compared with the prior art, the beneficial effects of the present invention are:

[0042] ​​The signal detection model proposed by the present invention provides a time-frequency domain detection method for signals in a strong noise environment. The proposal of the present invention provides a solution for the detection of underwater strong noise signals and can solve the problem of underwater target magnetic detection under a certain signal-to-noise ratio. BRIEF DESCRIPTION OF THE DRAWINGS

[0043] In order to more clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the following will briefly introduce the drawings required for the description of the embodiments or the prior art. Obviously, the drawings in the following description are only some embodiments of the present invention. For those of ordinary skill in the art, without creative efforts, other drawings can also be obtained based on these drawings.

[0044] Figure 1 Schematic diagram of a signal detection method combining stochastic resonance and ACI;

[0045] Figure 2 Time domain diagram of the original signal for the simulation example;

[0046] Figure 3 Time domain diagram of the unprocessed noisy signal for the simulation example;

[0047] Figure 4 Frequency domain diagram of the unprocessed noisy signal for the simulation example;

[0048] Figure 5 Time domain diagram of the output of wavelet transform for the simulation example;

[0049] Figure 6 Output signal diagram of ACI for the simulation example;

[0050] Figure 7 Output signal diagram of stochastic resonance for the simulation example. DETAILED DESCRIPTION OF THE EMBODIMENTS

[0051] The following will clearly and completely describe the technical solutions of the present invention in conjunction with the embodiments. Obviously, the described embodiments are only some embodiments of the present invention, rather than all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those of ordinary skill in the art without creative efforts belong to the scope of protection of the present invention.

[0052] As Figure 1 shown, according to the embodiments of the present invention, the present invention proposes a signal detection method combining stochastic resonance and ACI. The main technical solutions adopted are as follows:

[0053] 1) Construct a preprocessing model for noisy shaft frequency signals

[0054] The noise in the ocean is very complex. Different from the Gaussian noise commonly used for simulation, it contains a large amount of colored noise. It has been proven that wavelet transform can Gaussianize colored noise. Considering the relatively low frequency of underwater axial-frequency electromagnetic signals, setting appropriate decomposition levels and frequency ranges can effectively denoise the signals.

[0055] Generate colored noise and add it to the sine signal, setting the signal-to-noise ratio to -25 dB. Select "db4" as the wavelet basis function. The selection of the adaptive decomposition level is as follows.

[0056] j = [log2(fs)] - 1

[0057] where [] represents taking the integer, j represents the decomposition level, and f s is the sampling frequency of the signal.

[0058] Use the "db4" basis function to transform the signal into the wavelet domain and Gaussianize the interference noise. Input the transformed signal into the discrete wavelet transform to obtain the corresponding wavelet approximation coefficients and detail components at each layer. Select threshold filtering for the wavelet coefficients and then use the reconstruction function for reconstruction to achieve the suppression of high-frequency noise.

[0059] The definition of the wavelet coefficients is as follows.

[0060]

[0061] where: d j,k corresponds to the detail component of the j-th layer decomposition, and a j,k is the approximation coefficient. h and g are the low-pass filter and high-pass filter corresponding to the selected wavelet basis, and * represents taking the conjugate.

[0062] 2) Construct a bistable system stochastic resonance detection model

[0063] The bistable system under the action of noise Γ(t) and external periodic driving force Acosa0t can be described by the following equation:

[0064]

[0065] where: Γ(t) is the noise, f(x) = ax - bx 3 is the non-linear quantity, and Acosa0t is the periodic driving force.

[0066] Due to the introduction of the non-linear force field f(x) = ax - bx 3 in the above equation, the bistable system equation is fundamentally different from the linear equation.

[0067] The bistable system equation is a non-linear stochastic differential equation, which is a special type of differential equation. The Euler method can be used to numerically solve it, but the accuracy is not high. In engineering, for solving differential equations, a numerical algorithm with relatively high accuracy is the Runge-Kutta method. By choosing to analyze and solve the bistable system model through the Runge-Kutta method, we get:

[0068]

[0069] The parameters in the above formula are defined as follows:

[0070]

[0071]

[0072]

[0073] k4 = h[a(x n + k3) - b(x n + k3) 3 + u n+2 )

[0074] where h is the step size, generally taken as 1 / fs. fs is the sampling frequency of the signal. u n is the nth sampling point of the input signal, and x n is the nth point of the output signal. a and b are two parameters that control the output.

[0075] The signal after wavelet transform processing is sent as the parameter u into the stochastic resonance method. Generally, parameters a and b are selected. Through successive iterations, the stochastic resonance output signal x is obtained. Fourier transform is performed on it to obtain the frequency-domain spectrum information of the signal.

[0076] From the simulation with different parameters of a and b, when a0 = 1, b0 = 1, the signal frequency f0 = 0.0625 Hz, and the root mean square value of the noise σ0 = 2.47, a better matching relationship is achieved among the parameters at this time. In subsequent simulation detections, this signal is used as the reference signal to perform corresponding parameter matching.

[0077] Denote the required signal parameters as a1, b1, and σ1. The matching rules are as follows:

[0078] The first step: From the parameter a1 is obtained, and the parameter b1 is taken to be close to a1;

[0079] The second step: The root mean square value σ1 of the noise is obtained, and from the scaling factor K required to multiply in the signal input to the stochastic resonance algorithm is calculated;

[0080] The third step: According to the results, the parameters are finely adjusted.

[0081] 3) Construct an Adaptive Coherent Integration (ACI) algorithm model

[0082] The Adaptive Coherent Integration algorithm is an improved algorithm of the LMS algorithm. It adds a momentum term on the basis of the LMS algorithm. When the weight coefficient of the momentum term approaches 1, it can automatically perform coherent accumulation of signal energy.

[0083] The ACI weight vector iteration formula is as follows:

[0084] W(n + 1) = W(n) + δ·[W(n) - W(n - 1)] + 2μ[d(n) - y(n)]·X(n)

[0085] Among them, the input signal is d(n) = s(n) + v(n), where v(n) represents noise and s(n) represents the original signal. The output signal is y(n) = (W(n)) T X(n), where W(n) is the weight vector; μ is the adaptive step factor; δ is the adaptive coherent accumulation coefficient; M is the filter order; Δ is the time delay.

[0086] 4) Optimize the algorithm to find the optimal parameters

[0087] In the ACI model, it is easy to find that the results vary significantly under different parameter conditions. The selection of parameters has a great impact on the output results. Using an optimization algorithm to find the optimal parameters is beneficial for more efficient signal detection.

[0088] GWO (Grey Wolf Optimization) is also known as the Grey Wolf Algorithm. The Grey Wolf Optimization Algorithm simulates the predation behavior of a grey wolf group and achieves the optimization goal based on the characteristics of the wolf pack's group cooperation. Compared with other optimization algorithms such as the Particle Swarm Optimization Algorithm and the Fruit Fly Algorithm, it has excellent global search ability and fast convergence characteristics, and can achieve a balance between local optimization and global search, showing excellent performance in many optimization problems.

[0089] The main position update formula of the algorithm is as follows:

[0090] ① Update the position of the leader wolf:

[0091] For each grey wolf, update its position as:

[0092]

[0093] where is the new position of grey wolf i, is the current position, A is a weight vector, D iis the vector distance from the gray wolf i to the leader wolf.

[0094] ② Update the positions of the follower wolves:

[0095] For each non-leader wolf j, update its position as:

[0096]

[0097] where, is the new position of the gray wolf j, is the current position, C j is the vector distance from the leader wolf to the wolf j.

[0098] ③ Update the positions of the marginal wolves:

[0099] For each marginal wolf k, update its position as:

[0100]

[0101] where, is the new position of the gray wolf k, is the current position, D k is the vector distance from the leader wolf to the wolf k.

[0102] Write the ACI model into the fitness function of the gray wolf algorithm. Select the population size as 20 and the problem dimension as 2. Set the upper and lower bound variables of the system and the maximum number of iterations according to the data situation. Take SNR1 as the loss function of ACI. Optimize the parameters δ and μ of the ACI algorithm through the loss function, and send the obtained results back to the original system to update the parameters. Achieve efficient signal detection. The loss function is:

[0103]

[0104] where, P s is the power of the signal, P n is the power of the noise, and SNR1 is the output signal-to-noise ratio.

[0105] For the problem of excessive signal accumulation in the output of the ACI system, set a threshold to set the part exceeding the threshold to 0. This operation will cause the optimization efficiency of the gray wolf algorithm to deteriorate, but can effectively solve the problem of output divergence in the calculation.

[0106] As Figure 1-6 described, based on the above method, the present invention has carried out a verification theoretical simulation, specifically:

[0107] Simulation conditions: Set the input signal as s(t) = A * sin(2 * pi * f * t), where A = 0.1, f = 5Hz, f s = 2000 * f, and the input original signal is asFigure 1 As shown in the figure. The environmental magnetic noise component in the x-axis direction in a closed room is measured using a three-component fluxgate. The sampling rate of the measurement is 10 kHz, the sampling time is 30 s, and a part of the noise is intercepted and superimposed on the input signal. The signal after adding the noise is as shown in Figure 2 shown; the frequency domain is as shown in Figure 3 shown.

[0108] The noisy signal is input into the wavelet transform algorithm, and the wavelet basis is set to "db4". To ensure that the output signal is not distorted and considering the factor that stochastic resonance is excited by noise, the wavelet layer number is set to 5, and the noisy signal is filtered using the soft threshold method.

[0109] The parameters of the grey wolf algorithm are set. The population size is 20, the problem dimension is 2, the upper bound of the variable μ is 0.0001, the lower bound is 0.00001, the upper bound of the variable δ is 1, and the lower bound is 0.9. The number of iterations is 500 times. The grey wolf algorithm is used to optimize the parameters δ and μ of the ACI algorithm, and the loss function is SNR1. Corresponding threshold processing is performed on the output result. μ = 0.000025159 and δ = 0.9 are obtained. The result after inputting the obtained parameters into ACI is as shown in Figure 5 shown. The position where the signal occurs can be clearly seen in the time domain, and there is a relatively high detectable amplitude.

[0110] Taking a0 = 1, b0 = 1, f0 = 0.0625 Hz, and δ0 = 2.47 as the reference signal, from it is estimated that a1 = 80, and b1 = 80 is taken. The root mean square value of the input noise δ1 = 33294 is calculated, and from the scaling factor K of the input signal is calculated to be 0.005935. The input signal is multiplied by the scaling factor K, and the parameter values a = 80 and b = 80 are substituted into the model for numerical solution. The output result is as shown in Figure 6 shown. The amplitudes of the interference frequencies are all attenuated, and the component at 5.07 Hz is significantly enhanced, with good resolvability.

[0111] The preferred embodiments of the present invention disclosed above are only used to help explain the present invention. The preferred embodiments do not elaborate on all details and do not limit the invention to only the specific implementation manners. Obviously, many modifications and variations can be made according to the content of this specification. These embodiments are selected and specifically described in this specification to better explain the principle and practical application of the present invention, so that those skilled in the relevant technical field can well understand and utilize the present invention. The present invention is only limited by the claims and their full scope and equivalents.

Claims

1. A method for detecting the axial-frequency electromagnetic signal radiated by an underwater target, characterized in that, It includes the following steps: S1: Obtain the noisy axial-frequency electromagnetic signal in the ocean; S2: Construct a preprocessing model for the noisy axial-frequency electromagnetic signal, including decomposing and transforming the noisy axial-frequency electromagnetic signal, then performing signal filtering processing, Gaussianizing the non-Gaussian signals in the signal, removing high-frequency noise, and obtaining the output signal x; S3: Construct a bistable system stochastic resonance detection model, simulate to obtain the best matching parameters of stochastic resonance, use the best matching parameters as the reference signal, solve through corresponding ratios to obtain the parameter values of a and b corresponding to the required signal, and calculate the required proportionality coefficient K of the input signal. The bistable system stochastic resonance detection model is expressed as: where: Γ(t) is noise, f(x) = ax - bx 3 is a non-linear quantity, and Acos(a0t) is a periodic driving force; S4: Construct an adaptive coherent integration algorithm model, i.e., the ACI model, to perform automatic coherent integration of signal energy. The weight vector iteration algorithm of the ACI model is: W(n + 1) = W(n) + δ·[W(n) - W(n - 1)] + 2μ[d(n) - y(n)]·X(n) where the input signal is d(n) = s(n) + v(n), where v(n) represents noise and s(n) represents the original signal; the output signal is y(n) = (W(n)) T X(n), where W(n) is the weight vector; μ is the adaptive step factor; δ is the adaptive coherence accumulation coefficient; M is the filter order; Δ is the time delay; S5: Optimize the algorithm parameters of the adaptive coherent integration algorithm model, select the output signal-to-noise ratio as the loss function, set a threshold to zero the ultra-high cumulative amount in the output of the adaptive coherent integration algorithm model, and optimize the two parameters of δ and μ in the algorithm; S6: Determine the time-domain position and frequency-domain information of the axial-frequency signal, input the signal x processed in S1 and the corresponding parameter values of a and b into the bistable system stochastic resonance model, perform time-frequency domain conversion on the output result to obtain the signal spectrum information; Input the signal x processed in S1 and the optimized parameters of δ and μ into the ACI model to obtain the occurrence time information of the signal in the time domain.

2. The method for detecting the axial-frequency electromagnetic signal radiated by an underwater target according to claim 1, characterized in that, Set the wavelet basis function of "db4” as the decomposition and transformation algorithm, set the adaptive decomposition level, perform wavelet decomposition and transformation on the signal, input the transformed signal into the discrete wavelet transform to obtain the corresponding wavelet approximation coefficients and detail components of each layer, select threshold filtering for the wavelet coefficients, and then use the reconstruction function for reconstruction to achieve the suppression of high-frequency noise. The definition formula of the wavelet coefficients is: where: d j,k corresponds to the detail component decomposed at the j-th level, a j,k is the approximation coefficient, h and g are the low-pass filter and high-pass filter corresponding to the selected wavelet basis, and * denotes taking the conjugate.

3. The method for detecting the axial-frequency electromagnetic signal radiated by an underwater target according to claim 2, characterized in that, The algorithm for selecting the adaptive decomposition level is as follows: j = [log2(fs)] - 1 where [] represents rounding, j represents the number of decomposition levels, and f s is the sampling frequency of the signal.

4. The method for detecting the axial-frequency electromagnetic signal radiated by an underwater target according to claim 1, characterized in that, The algorithm for optimizing the algorithm parameters of the adaptive coherent integration algorithm model is the grey wolf algorithm. Write the ACI algorithm into the fitness function of the grey wolf algorithm, select the population size as 20, the problem dimension as 2, set the upper and lower bound variables of the system and the maximum number of iterations according to the data situation, use SNR1 as the loss function of the ACI, optimize the parameters of the ACI algorithm through the loss function, and send the obtained results back to the original system to update the parameters to achieve efficient signal detection. The loss function formula is: Among them, P s is the power of the signal, P n is the power of the noise, and SNR1 is the output signal-to-noise ratio.

5. The method for detecting the axial-frequency electromagnetic signal radiated by an underwater target according to claim 1, characterized in that, Use the Runge-Kutta method to analyze and solve the bistable system model. Specifically: The definitions of the parameters in the above formula are as follows: k4 = h[a(x n + k3) - b(x n + k3) 3 + u n+2 ​ where h is the step size, taken as 1 / fs, fs being the sampling frequency of the signal, u n is the n-th sampling point of the input signal, x n is the n-th point of the output signal, and a and b are two parameters controlling the output.

6. The method for detecting the axial-frequency electromagnetic signal radiated by an underwater target according to claim 1, characterized in that, The best parameters in S3 are: a0 = 1, b0 = 1, the signal frequency f0 = 0.0625Hz, the root mean square value of the noise σ0 = 2.

47. At this time, a good matching relationship is achieved among the parameters, which is used as the reference signal and corresponding matching is performed on the parameters in subsequent simulation detections.

7. The underwater target radiated shaft-frequency electromagnetic signal detection method according to claim 6, characterized in that, The parameter matching rule is: Step 1: From obtain parameter a1, and take parameter b1 close to a1; Step 2: Calculate the root mean square value σ1 of the noise, and calculate the proportionality factor K to be multiplied in the signal input stochastic resonance algorithm from ; The third step: Fine-tune the parameters according to the results; The parameters a1, b1, and σ1 are the required signal parameters.

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