A horizontal linear array wave number extraction method and system based on spatial domain dispersion-free transformation

Abstract

The present application provides a method and system for extracting horizontal array wavenumbers based on spatial domain dispersion-eliminating transformation. The method first transforms the time domain signal received by the horizontal array into the frequency domain; secondly, transforms the obtained signal in the frequency domain from the array element domain to the wavenumber domain; then, at each frequency point, performs wavenumber axis scaling transformation on the obtained frequency-wavenumber domain signal to align the wavenumbers corresponding to the same mode at different frequency points; finally, accumulates the transformed frequency-wavenumber domain signal in the frequency domain, and the accumulated signal will have obvious multiple peaks in the wavenumber domain. Then, according to the values ​​indicated by these peaks, the estimated values ​​of the horizontal wavenumbers at different frequency points are obtained by using the inverse scaling transformation. The advantages of the present application are: wavenumber extraction can be achieved under lower signal-to-noise ratio conditions, with higher extraction accuracy, and higher robustness to the mismatch between environmental waveguide conditions and assumed conditions; horizontal wavenumber estimation can be achieved under higher frequency conditions.

Description

Technical Field

[0001] The present application belongs to the field of underwater acoustic array signal processing, marine engineering and sonar technology, and specifically relates to a horizontal line array wave number extraction method and system based on spatial domain dispersion-free transformation. Background Art

[0002] In shallow sea waveguides, due to the existence of boundary and medium inhomogeneity, the shallow sea sound field exhibits obvious multipath effect and dispersion effect. The existence of multipath effect and dispersion effect makes many algorithms based on plane wave assumption invalid. For this reason, a normal wave model that can more accurately characterize the ocean sound field environment is proposed. In shallow sea acoustics, the ocean sound field can often be represented by the superposition of several normal modes. Therefore, the sound field can be represented by normal modes and horizontal wave numbers phenomenologically. The effect of environmental parameters on the sound field is also summarized by these phenomenological variables. Therefore, if these phenomenological variables can be extracted from the observed signal, it is possible to obtain how the ocean waveguide environment affects the acoustic signal, and the extracted information can be used in the environmental adaptive underwater acoustic signal processing algorithm.

[0003] However, the existing horizontal linear array wave number extraction methods have low estimation accuracy under low signal-to-noise ratio conditions and poor robustness to the mismatch between environmental waveguide conditions and assumed conditions. Summary of the invention

[0004] The purpose of the present application is to overcome the defects of the prior art that the estimation result is not accurate under low signal-to-noise ratio conditions and the robustness to the mismatch between environmental waveguide conditions and assumed conditions is poor.

[0005] In order to achieve the above purpose, the present application proposes a horizontal line array wave number extraction method based on spatial domain dispersion elimination transformation, comprising:

[0006] Step 1: Use the horizontal line array to obtain the time domain sound pressure signal p(t,r l ),l=0,1,...,N-1, perform Fourier transform on the time domain sound pressure signal obtained by each array element, and obtain the frequency domain signal P(f,r) in the domain of each array element l ),l=0,1,...,N-1; where t represents time; r l represents the distance between the array element and the sound source; f represents the frequency; l represents the array element number; N represents the number of array elements;

[0007] Step 2: Using focused beamforming, at each frequency point, transform the frequency domain signal in the array element domain into the wave number domain;

[0008] Step 3: Using the waveguide invariant theory, the obtained frequency-wavenumber domain signal is scaled on the wavenumber axis to eliminate the horizontal wavenumber differences corresponding to different modes at different frequencies;

[0009] Step 4: Accumulate the transformed frequency-wavenumber domain signal on the frequency axis;

[0010] Step 5: The accumulated signal has multiple peaks on the wave number axis. According to the values ​​indicated by these peaks, the inverse stretching transformation is performed to obtain the horizontal wave numbers at different frequency points.

[0011] As an improvement of the above method, step 2 comprises:

[0012] The frequency domain signal P(f,r l ) to make the following transformations:

[0013]

[0014] Wherein, k represents the wave number; d represents the array element spacing; r0 represents the distance between the first array element of the linear array and the set sound source; θ represents the azimuth angle between the first array element of the linear array and the set sound source; S(f,k) represents the signal transformed from the array element domain to the wave number domain.

[0015] As an improvement of the above method, step 3 comprises:

[0016] The signal transformed from the meta-domain to the wave-number domain is translated and scaled on the wave-number axis:

[0017] S tran (f,k)=S(f,T(k))

[0018] T(k)=ω / c0-k(ω / ω0) -1 / β

[0019] Where ω represents the angular frequency of the sound source, ω = 2πf; c0 is the sound velocity at the horizontal array; ω0 is the reference angular frequency; β is the waveguide invariant; S tran (f, k) represents the result of translation and scaling transformation on the wave number axis.

[0020] As an improvement of the above method, step 4 includes:

[0021] The results of translation and scaling transformation on the wave number axis are accumulated in the frequency domain:

[0022]

[0023] Among them, S freq (k) represents the result of accumulation in the frequency domain.

[0024] As an improvement of the above method, step 5 comprises:

[0025] k em (f) = ω / c0-kpm (ω / ω0) -1 / β

[0026] Among them, k em (f) is the estimated value of the mth order horizontal wave number at frequency f; k pm is the wave number value corresponding to the mth peak extracted from the accumulated wave number diagram.

[0027] The present application also provides a horizontal line array wave number extraction system based on spatial domain dispersion-free transformation, which is implemented based on the above method, and the system includes:

[0028] A module for obtaining frequency domain signals in the array element domain is used to obtain the time domain sound pressure signal of the sound source by using the horizontal line array, and to perform Fourier transform on the time domain sound pressure signal obtained by each array element to obtain the frequency domain signal in the array element domain;

[0029] A focused beam forming module, used for transforming the frequency domain signal in the array element domain into the wave number domain at each frequency point by using focused beam forming;

[0030] The module for eliminating the horizontal wavenumber difference is used to use the waveguide invariant theory to perform a scaling transformation on the frequency-wavenumber domain signal obtained on the wavenumber axis to eliminate the horizontal wavenumber differences corresponding to different modes at different frequencies;

[0031] An accumulation module is used to accumulate the transformed signal in the frequency-wavenumber domain on the frequency axis;

[0032] The module for obtaining horizontal wavenumbers is used to utilize the multiple peaks of the accumulated signal on the wavenumber axis, and according to the values ​​indicated by these peaks, perform inverse stretching and transformation to obtain the horizontal wavenumbers at different frequency points.

[0033] Compared with the prior art, the advantages of this application are:

[0034] Compared with the existing horizontal line array wave number extraction method, the method of the present invention can realize horizontal wave number estimation under lower signal-to-noise ratio conditions, and under the same conditions, the accuracy of the estimation result is better, and the robustness to the mismatch between the environmental waveguide conditions and the assumed conditions is higher. And compared with the existing horizontal line array wave number extraction method, the method of the present invention can realize horizontal wave number estimation under higher frequency conditions. BRIEF DESCRIPTION OF THE DRAWINGS

[0035] Figure 1 Shown is a top view of the sound source and array positions in the method of the present invention;

[0036] Figure 2 It shows the energy distribution diagram of the signal received on the horizontal array into the frequency-wave number domain through Fourier transformation in the method of the present invention;

[0037] Figure 3 It is shown as the energy distribution diagram after the signal transformed into the frequency-wave number domain is translated on the wave number axis in the method of the present invention;

[0038] Figure 4 It shows the energy distribution diagram after the signal transformed into the frequency-wave number domain is translated and stretched on the wave number axis in the method of the present invention;

[0039] Figure 5 The figure shows the wave number distribution diagram obtained by accumulating the frequency using formula (3) in the method of the present invention;

[0040] Figure 6 The method of the present invention is shown as follows: Figure 5 The wave value extracted from the multi-peak structure is compared with the true value by using the inverse transformation of formula (4). The estimated value is represented by the blue dotted line, and the background is the true horizontal wave value extracted from the signal.

[0041] Figure 7 The figure shows the comparison results of the relative errors of horizontal wave numbers corresponding to different modes extracted by the method of the present invention under different signal-to-noise ratio conditions and the warping transformation method;

[0042] Figure 8 The figure shows the comparison results of the relative standard deviation of horizontal wave numbers corresponding to different modes extracted by the method of the present invention under different signal-to-noise ratio conditions and the warping transformation method;

[0043] Fig. 9 Shown is the result of extracting the wave number structure using the method of the present invention under higher frequency conditions;

[0044] Fig.10 The figure shows the result of extracting the wave number structure using the warping transformation method under higher frequency conditions in the method of the present invention;

[0045] Fig.11 Shown is the result of extracting the wave number structure using the method of the present invention under the waveguide condition with a thermocline;

[0046] Fig.12 It shows the result of extracting the wave number structure using the warping transformation method under the waveguide condition with a thermocline in the method of the present invention;

[0047] Fig.13 Shown is a flow chart of the horizontal array wave number extraction method based on spatial domain dispersion-free transformation. DETAILED DESCRIPTION

[0048] The technical solution of the present application is described in detail below with reference to the accompanying drawings.

[0049] The present invention proposes a method and system for extracting horizontal array wavenumbers based on spatial domain dispersion-eliminating transformation. The method first transforms the horizontal array signal from the time domain to the frequency domain; then, using focused beamforming, at each frequency point, the array element domain signal is transformed to the wavenumber domain; then, using the waveguide invariant theory, the obtained frequency-wavenumber domain signal is scaled on the wavenumber axis to eliminate the differences in horizontal wavenumbers corresponding to different modes at different frequencies; finally, the transformed signal in the frequency-wavenumber domain is accumulated on the frequency axis, and the accumulated signal will have multiple peaks on the wavenumber axis, and then, based on the values ​​indicated by these peaks, the horizontal wavenumbers at different frequencies can be obtained by inversely scaling and transforming it back. Figure 1 Shown is a top view of the sound source and array positions in the method of the present invention.

[0050] The method of the present invention is applicable to cooperative sound sources with a broadband phase spectrum that changes slowly, or to situations where the phase spectrum of the sound source is known.

[0051] Compared with the warping transform method, the method of the present invention can achieve wave number extraction under lower signal-to-noise ratio conditions and higher estimation accuracy, and does not require prior knowledge of the environmental parameters of the waveguide. However, like the warping transform and other methods that utilize dispersion effects, the method of the present invention requires a slowly varying phase spectrum of the sound source, or a known phase spectrum of the sound source.

[0052] Example 1

[0053] like Fig.13 As shown, the method of the present invention provides a horizontal linear array wave number extraction method based on spatial domain dispersion-free transformation, and the method comprises the following steps:

[0054] Step 1: Use the horizontal line array to obtain the time domain sound pressure signal p(t,r l ),l=0,1,...,N-1, where t represents time and r l Represents the distance between the array element and the sound source. Perform Fourier transform on the time domain sound pressure signal obtained by each array element to obtain the frequency domain signal P(f,r l ),l=0,1,...,N-1; f represents frequency; l represents array element number.

[0055] Step 2: P(f,r) is the frequency domain signal in the matrix domain. l ) to make the following transformations:

[0056]

[0057] Wherein, k represents the wave number; N represents the number of array elements; d represents the array element spacing; r0 represents the distance between the first array element of the linear array and the set sound source; θ represents the azimuth angle between the first array element of the linear array and the set sound source; S(f,k) represents the signal transformed from the array element domain to the wave number domain.

[0058] Step 3: Perform translation and scaling transformations on the wave number axis for the obtained results:

[0059] S tran (f,k)=S(f,T(k)), (2)

[0060] T(k)=ω / c0-k(ω / ω0) -1 / β

[0061] Among them, ω represents the angular frequency of the sound source, ω = 2πf; c0 is the sound velocity at the horizontal array, and ω0 is the reference angular frequency, and β is the waveguide invariant, which can be taken as 1 for most shallow sea waveguides.

[0062] Step 4: Accumulate the results in the frequency domain.

[0063]

[0064] Step 5: Extract the wave number corresponding to the peak value from the result accumulated in the frequency domain, and then use formula (4) to inversely transform the obtained wave number:

[0065] k em (f) = ω / c0-k pm (ω / ω0) -1 / β (4)

[0066] Among them, k em (f) is the estimated value of the mth order horizontal wave number at frequency f, and k pm is the wave number value corresponding to the mth peak extracted from the accumulated wave number diagram.

[0067] The simulation test was carried out using the method of the present invention. The simulation parameters were as follows: the waveguide environment was a 100m deep Pekeris waveguide, the upper seawater sound velocity was 1540m / s, and the density was 1g / cm 3 The sound velocity of the lower layer is 1600m and the density is 1.6g / cm 3 , the seabed attenuation coefficient is 0.1dB / λ, where λ represents the wavelength of the sound wave and the sound source position r a =10km,θ a =15°, the array is placed on the seabed, the array element spacing is 5m, Figure 1-4 In order to clearly demonstrate the process of eliminating dispersion by the method of the present invention, a large aperture array with an array length of 2 km is used without adding noise. However, the method of the present invention can be applied to arrays of any aperture. Figure 2-6 The frequency range of the sound is set to 300-400Hz with a frequency interval of 1Hz, and Figure 7 , Figure 8 , Fig.11 , Fig.12 The mid-frequency range is set to 100-200 Hz with a frequency interval of 0.05 Hz. Fig. 9 , Fig.10 The mid-frequency range is set to the higher frequency band of 400-600Hz, and the frequency interval is still 0.05Hz, where Fig.11 and Fig.12 The waveguide environment used is an ocean waveguide containing a temperature jump layer.

[0068] Step 1: Use an array with 401 elements and 5 m spacing in this example, and select the element closest to the sound source as the reference element. Perform a time-domain Fourier transform on the acoustic signal recorded by the array and transform the signal to the frequency domain to obtain P(f,r l ),l=0,1,...,400.

[0069] Step 2: According to formula (1), the frequency domain signal P(f,r l ) is transformed into the wave number domain, and we get S(f,θ,r0,k). When (θ,r0)=(θ a ,r a ), we can get Figure 2 The results shown are the distribution of S on the fk domain.

[0070] Step 3: According to formula (2), S(f,θ,r0,k) obtained in step 2 is translated and scaled on the wave number axis. In this example, the reference angular frequency ω0 is 400 Hz, and the sound speed c0 is 1540 m / s. The result after translation transformation is as follows: Figure 3 As shown, the result after translation and scaling transformation is as follows Figure 4 shown.

[0071] Step 4: Perform frequency domain accumulation on the signal after translation and scaling transformation according to formula (3), and we can get Figure 5 The multi-peak structure shown in Figure 5 The wave numbers corresponding to different peaks are extracted from the waveform, and the estimated values ​​of the horizontal wave numbers of different modes at different frequencies are obtained according to formula (4). The estimated values ​​are plotted on the horizontal wave number structure diagram extracted from the signal with blue dotted lines, and the result is: Figure 6 Results shown.

[0072] Step 5: Add Gaussian white noise with signal-to-noise ratios of -15dB, -10dB, -5d, 0dB, 5dB, and 10d to the array element domain in turn, repeat steps 2, 3, and 4 one hundred times, and count the relative errors and relative standard deviations of all wavenumber estimates under different signal-to-noise ratios, and compare them with the statistical results of the warping method. The comparison results are displayed in Figure 7 and Figure 8In the figure, the warping method cannot estimate the horizontal wave value below 0 dB under the given simulation conditions, so the results are not shown.

[0073] Step 6: When the sound signal frequency band is 400-600 Hz, repeat steps 1, 2, 3, and 4 to obtain the estimation result at a higher frequency. The result of the method of the present invention is shown in Fig. 9 , the results of the warping transformation method are shown in Fig.10 .

[0074] Step 7: Repeat steps 1, 2, 3, and 4 under the waveguide condition with the temperature jump layer to obtain the estimated result under the condition deviating from the assumed waveguide environment. The result of the method of the present invention is shown in Fig.11 , the results of the warping transformation method are shown in Fig.12 .

[0075] Example 2

[0076] The present application also provides a horizontal line array wave number extraction system based on spatial domain dispersion-free transformation, which is implemented based on the above method, and the system includes:

[0077] A module for obtaining frequency domain signals in the array element domain is used to obtain the time domain sound pressure signal of the sound source by using the horizontal line array, and to perform Fourier transform on the time domain sound pressure signal obtained by each array element to obtain the frequency domain signal in the array element domain;

[0078] A focused beam forming module, used for transforming the frequency domain signal in the array element domain into the wave number domain at each frequency point by using focused beam forming;

[0079] The module for eliminating the horizontal wavenumber difference is used to use the waveguide invariant theory to perform a scaling transformation on the frequency-wavenumber domain signal obtained on the wavenumber axis to eliminate the horizontal wavenumber differences corresponding to different modes at different frequencies;

[0080] An accumulation module is used to accumulate the transformed signal in the frequency-wavenumber domain on the frequency axis;

[0081] The module for obtaining horizontal wavenumbers is used to utilize the multiple peaks of the accumulated signal on the wavenumber axis, and according to the values ​​indicated by these peaks, perform inverse stretching and transformation to obtain the horizontal wavenumbers at different frequency points.

[0082] The present application may also provide a computer device, comprising: at least one processor, a memory, at least one network interface and a user interface. The various components in the device are coupled together through a bus system. It is understood that the bus system is used to achieve connection and communication between these components. In addition to the data bus, the bus system also includes a power bus, a control bus and a status signal bus.

[0083] The user interface may include a display, a keyboard or a pointing device, such as a mouse, a trackball, a touch pad or a touch screen.

[0084] It is understood that the memory in the embodiments disclosed in the present application may be a volatile memory or a non-volatile memory, or may include both volatile and non-volatile memories. Among them, the non-volatile memory may be a read-only memory (ROM), a programmable read-only memory (PROM), an erasable programmable read-only memory (EPROM), an electrically erasable programmable read-only memory (EEPROM), or a flash memory. The volatile memory may be a random access memory (RAM), which is used as an external cache. By way of example and not limitation, many forms of RAM are available, such as static random access memory (SRAM), dynamic random access memory (DRAM), synchronous dynamic random access memory (SDRAM), double data rate synchronous dynamic random access memory (DDRSDRAM), enhanced synchronous dynamic random access memory (ESDRAM), synchronous link dynamic random access memory (SLDRAM), and direct memory bus random access memory (DRRAM). The memories described herein are intended to include, but are not limited to, these and any other suitable types of memories.

[0085] In some embodiments, the memory stores the following elements, executable modules or data structures, or a subset thereof, or an extended set thereof: an operating system and applications.

[0086] The operating system includes various system programs, such as a framework layer, a core library layer, a driver layer, etc., which are used to implement various basic services and process hardware-based tasks. The application includes various application programs, such as a media player (Media Player), a browser (Browser), etc., which are used to implement various application services. The program for implementing the method of the embodiment of the present disclosure can be included in the application.

[0087] In the above embodiment, the processor may also call a program or instruction stored in the memory, specifically, a program or instruction stored in an application program, and is used to:

[0088] Execute the steps of the above method.

[0089] The above method can be applied to a processor or implemented by a processor. The processor may be an integrated circuit chip with signal processing capabilities. In the implementation process, each step of the above method can be completed by an integrated logic circuit of hardware in the processor or an instruction in the form of software. The above processor may be a general processor, a digital signal processor (Digital Signal Processor, DSP), an application specific integrated circuit (Application Specific Integrated Circuit, ASIC), a field programmable gate array (Field Programmable Gate Array, FPGA) or other programmable logic devices, discrete gates or transistor logic devices, discrete hardware components. The above-disclosed methods, steps and logic block diagrams can be implemented or executed. The general processor may be a microprocessor or the processor may also be any conventional processor, etc. The steps of the above-disclosed method can be directly embodied as a hardware decoding processor to execute, or the hardware and software modules in the decoding processor are combined to execute. The software module can be located in a mature storage medium in the field such as a random access memory, a flash memory, a read-only memory, a programmable read-only memory or an electrically erasable programmable memory, a register, etc. The storage medium is located in the memory, and the processor reads the information in the memory and completes the steps of the above method in combination with its hardware.

[0090] It is understood that the embodiments described in the present application can be implemented by hardware, software, firmware, middleware, microcode or a combination thereof. For hardware implementation, the processing unit can be implemented in one or more application specific integrated circuits (ASIC), digital signal processors (DSP), digital signal processing devices (DSPD), programmable logic devices (PLD), field programmable gate arrays (FPGA), general-purpose processors, controllers, microcontrollers, microprocessors, other electronic units for performing the functions described in the present application or a combination thereof.

[0091] For software implementation, the technology of the present application can be implemented by executing the functional modules (such as procedures, functions, etc.) of the present application. The software code can be stored in a memory and executed by a processor. The memory can be implemented in the processor or outside the processor.

[0092] The present application may also provide a non-volatile storage medium for storing a computer program. When the computer program is executed by a processor, each step in the above method embodiment can be implemented.

[0093] Finally, it should be noted that the above embodiments are only used to illustrate the technical solution of the present application and are not intended to limit it. Although the present application is described in detail with reference to the embodiments, a person skilled in the art should understand that any modification or equivalent replacement of the technical solution of the present application does not depart from the spirit and scope of the technical solution of the present application and should be included in the scope of the claims of the present application.

Claims

1. A method for extracting horizontal linear array wavenumber based on spatial domain dispersion-free transformation, comprising: Step 1: Use the horizontal line array to obtain the time domain sound pressure signal p(t,r l ),l=0,1,...,N-1, perform Fourier transform on the time domain sound pressure signal obtained by each array element, and obtain the frequency domain signal P(f,r) in the domain of each array element l ),l=0,1,...,N-1; where t represents time; r l represents the distance between the array element and the sound source; f represents the frequency; l represents the array element number; N represents the number of array elements; Step 2: Using focused beamforming, at each frequency point, transform the frequency domain signal in the array element domain into the wave number domain; Step 3: Using the waveguide invariant theory, the obtained frequency-wavenumber domain signal is scaled on the wavenumber axis to eliminate the horizontal wavenumber differences corresponding to different modes at different frequencies; Step 4: Accumulate the transformed frequency-wavenumber domain signal on the frequency axis; Step 5: The accumulated signal has multiple peaks on the wave number axis. According to the values ​​indicated by these peaks, the inverse stretching transformation is performed to obtain the horizontal wave numbers at different frequency points.

2. The horizontal line array wave number extraction method based on spatial domain dispersion elimination transformation according to claim 1 is characterized in that: The step 2 comprises: The frequency domain signal P(f,r l ) to make the following transformations: Wherein, k represents the wave number; d represents the array element spacing; r0 represents the distance between the first array element of the linear array and the set sound source; θ represents the azimuth angle between the first array element of the linear array and the set sound source; S(f,k) represents the signal transformed from the array element domain to the wave number domain.

3. The horizontal line array wave number extraction method based on spatial domain dispersion elimination transformation according to claim 2 is characterized in that: The step 3 comprises: The signal transformed from the meta-domain to the wave-number domain is translated and scaled on the wave-number axis: S tran (f,k)=S(f,T(k)) T(k)=ω / c0-k(ω / ω0) -1 / β Where ω represents the angular frequency of the sound source, ω = 2πf; c0 is the sound velocity at the horizontal array; ω0 is the reference angular frequency; β is the waveguide invariant; S tran (f, k) represents the result of translation and scaling transformation on the wave number axis.

4. The horizontal line array wave number extraction method based on spatial domain dispersion elimination transformation according to claim 3 is characterized in that: The step 4 comprises: The results of translation and scaling transformation on the wave number axis are accumulated in the frequency domain: Among them, S freq (k) represents the result of accumulation in the frequency domain.

5. The horizontal line array wave number extraction method based on spatial domain dispersion elimination transformation according to claim 4 is characterized in that: The step 5 comprises: k em (f)=ω / c0-k pm (ω / ω0) -1 / β Among them, k em (f) is the estimated value of the mth order horizontal wave number at frequency f; k pm is the wave number value corresponding to the mth peak extracted from the accumulated wave number diagram.

6. A horizontal line array wave number extraction system based on spatial domain dispersion-free transformation, implemented based on any method described in claims 1-5, characterized in that: The system comprises: A frequency domain signal acquisition module in the array element domain is used to obtain the time domain sound pressure signal of the sound source by using the horizontal line array, and perform Fourier transform on the time domain sound pressure signal obtained by each array element to obtain the frequency domain signal in the array element domain; A focused beam forming module, used for transforming the frequency domain signal in the array element domain into the wave number domain at each frequency point by using focused beam forming; The module for eliminating the horizontal wavenumber difference is used to use the waveguide invariant theory to perform a scaling transformation on the frequency-wavenumber domain signal obtained on the wavenumber axis to eliminate the horizontal wavenumber differences corresponding to different modes at different frequencies; An accumulation module, used for accumulating the transformed frequency-wavenumber domain signal on the frequency axis; and The module for obtaining horizontal wavenumbers is used to utilize the multiple peaks of the accumulated signal on the wavenumber axis, and according to the values ​​indicated by these peaks, perform inverse stretching and transformation to obtain the horizontal wavenumbers at different frequency points.

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