Direct spread spectrum layered sliding blind de-spreading system
By using a hierarchical blind despreading system, the period and starting position of the spreading pseudocode are estimated by period calculation and sliding correlation method, realizing blind despreading of direct spread spectrum system under low signal-to-noise ratio conditions, and solving the problems of complexity and low signal-to-noise ratio tolerance in the prior art.
Patent Information
- Application Number
- CN202510987428.8
- Authority / Receiving Office
- CN · China
- Patent Type
- Applications(China)
- Current Assignee / Owner
- Priority Date
- 2025-07-16
- Filing Date
- 2025-07-17
- Publication Date
- 2025-11-14
AI Technical Summary
The existing blind despreading methods of direct spread spectrum systems are complex and have low signal-to-noise ratio tolerance, making it difficult to effectively achieve blind despreading under non-cooperative conditions.
A hierarchical blind despreading system is adopted. The period calculation module estimates the period of the spreading pseudocode, the start position estimation module finds the start position of the pseudocode, and the sliding correlation method is used for blind despreading, thus eliminating the step of spreading pseudocode estimation.
It achieves simple and easy blind despreading under low signal-to-noise ratio conditions, improves the signal-to-noise ratio tolerance, simplifies the despreading process, and is suitable for non-cooperative communication environments.
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Figure CN120956294A_ABST
Abstract
Description
Technical Field
[0001] This invention relates to the field of blind despreading technology for direct spread spectrum systems, and specifically to a direct spread spectrum layered sliding blind despreading system. Background Technology
[0002] Direct spread spectrum (DSS) systems possess superior transmission characteristics such as strong anti-interference, noise immunity, and stealth, making them widely used in aircraft communication systems. DSS systems modulate the transmitted signal onto the polarity of the spreading code. For the receiving system, despreading is achieved simply by demodulating the spreading code polarity information. Spread spectrum systems require the spreading code to have good autocorrelation and cross-correlation properties. Therefore, the receiver only needs to correlate the received signal with the spreading pseudocode to obtain the transmitter's spreading code polarity information, thus achieving despreading. However, in non-cooperative spread spectrum communication, the period and pseudocode of the transmitter's spreading pseudocode are unknown. Therefore, how to achieve blind despreading in spread spectrum systems has always been a hot research topic.
[0003] Currently, the blind despreading approach for direct spread spectrum systems is as follows: First, the spreading pseudocode is precisely estimated; then, the received signal is synchronized; and finally, despreading is performed using cooperative target despreading. The starting position and period of the spreading pseudocode are confirmed by correlating the preceding and following segments of the received signal. Then, the spreading pseudocode is demodulated using non-cooperative target signal demodulation, thereby estimating the spreading pseudocode, and finally despreading using cooperative target despreading. Alternatively, Fast-ICA is used to blindly estimate the non-cooperative target pseudocode, thus achieving blind despreading. Blind despreading is also performed on Walsh code soft spread spectrum systems.
[0004] All of the above blind despreading methods first perform blind estimation of the spreading pseudocode, and then use the cooperative target method for despreading; however, this blind despreading method has drawbacks such as complexity and high signal-to-noise ratio tolerance. Summary of the Invention
[0005] To address the aforementioned problems, the purpose of this invention is to provide a direct spread spectrum layered sliding blind despreading system. This system utilizes layered blind despreading to achieve blind despreading without estimating the spreading pseudocode. The despreading method is simple and easy to implement, while also having a high signal-to-noise ratio tolerance.
[0006] The first technical solution adopted in this invention is: a direct spread spectrum layered sliding blind despreading system, including a period calculation module, a start position estimation module and a blind despreading module;
[0007] The period calculation module is used to sample and perform FFT transformation on the received signal, and calculate the spreading pseudocode period based on the minimum interval of the correlation peaks.
[0008] The starting position estimation module is used to estimate the starting position of the spreading pseudocode based on the spreading pseudocode period;
[0009] The blind despreading module is used to perform blind despreading using the sliding correlation method after estimating the starting position of the spreading pseudocode, and outputs the despreading value.
[0010] Preferably, the period calculation module is used to perform the following operations:
[0011] S11. Acquire the received signal and sample the received signal at the current time and the delayed time.
[0012] S12. Perform FFT transformation on the samples at the current time and the delayed time, then conjugate them, and then perform IFFT transformation to calculate the minimum interval of the correlation peaks at the current time and the delayed time.
[0013] S13. Calculate the spreading pseudocode period based on the minimum interval of the correlation peaks at the current time and the delayed time.
[0014] Preferably, the spreading pseudocode period is calculated using the following formula:
[0015] T=N est ×f sam
[0016] In the formula, T is the period of the spreading pseudocode; N est f is the minimum interval between related peaks; sam This is the sampling frequency for the received signal.
[0017] Preferably, the starting position estimation module uses the sliding correlation method to estimate the starting position of the spread spectrum pseudocode.
[0018] Preferably, the starting position estimation module is used to perform the following operations:
[0019] S21. Continuously sample the received signal to obtain w k w k+1 w k+2 ;
[0020] S22, regarding w k w k+2 Perform sliding correlation to find w k w k+2 Cases where the polarities of the related peaks are opposite;
[0021] S23, in w k w k+2 When the polarities of the correlation peaks are opposite, the sliding correlation method is used to find w. k w k+1 The extreme point of the relevant peak value is the starting position of the spread spectrum pseudocode.
[0022] Preferably, the blind despreading module is used to perform the following operations:
[0023] The received signal is shifted so that the start time of the received signal corresponds to the start position of the spread spectrum pseudocode, and the first sampled signal is taken from the start position of the spread spectrum pseudocode.
[0024] The received signal is shifted to the left to obtain a forward-shifted signal, and the received signal is shifted to the right to obtain a backward-shifted signal;
[0025] The forward-shifted signal, the backward-shifted signal, and the unshifted received signal are respectively subjected to sliding correlation with the first segment of the sampled signal to find the extreme value of the correlation peak, and output as the despread value.
[0026] Preferably, the blind despreading module is used to perform the following operations:
[0027] S31. Delay the received signal so that the start time of the received signal is aligned with the start position of the spread spectrum pseudocode;
[0028] S32. Initialize the iteration count i = 1, the corresponding shift number T_delay of the received signal = 0, and take the first sample signal w1 from the starting position of the spread spectrum pseudocode;
[0029] S33. Shift the received signal left by T_delay and right by T_delay respectively to obtain the signal w for the i-th data segment shifted forward. k′1 The signal w after shifting the i-th data segment k′2 and take the received signal itself as the i-th data signal; and take w k′1 w k′2 The i-th data signal is correlated with the first sampled signal w1, then summed, the correlation peak value is obtained and stored, and the corresponding shift value T_delay of the received signal is updated at the same time.
[0030] S34. Let i = i + 1, and repeat steps S32 and S33 to update w. k′1 w k′2 until the signal reception ends;
[0031] S35. Find the maximum value among all the stored relevant peak values and output it as the despread value.
[0032] Preferably, the relevant peak value is represented by the following formula:
[0033]
[0034] In the formula, The first sampled signal w1 and the shifted signal w k′ The relevant peak value; w1 is the first sampled signal; w k′The signal after left or right shifting the received signal; sign(·) represents the sign function.
[0035] The beneficial effects of the above technical solution are as follows:
[0036] (1) The direct spread spectrum layered sliding blind despreading system proposed in this invention utilizes the layered blind despreading method to achieve blind despreading without estimating the spread spectrum pseudocode. The despreading method is simple and easy to implement, and at the same time, the signal-to-noise ratio tolerance is high.
[0037] (2) This invention uses the uplink signal of a certain type of communication system for verification, and can realize blind despreading under low signal-to-noise ratio conditions. The despreading process is divided into three steps: first, the period of the spreading pseudocode is estimated, then the starting position of the spreading pseudocode is found by the sliding correlation method, and finally the blind despreading is performed by the sliding correlation method. Blind despreading under low signal-to-noise ratio conditions is realized, and there is no need to estimate the spreading pseudocode sequence. The algorithm is simple.
[0038] (3) The present invention utilizes the relative polarity of the received signal and the first periodic spread spectrum for despreading, thus eliminating the need for differential despreading; at the same time, the present invention uses correlation between the received signal and the first periodic signal to slide the received signal, thereby reducing the number of slides. Attached Figure Description
[0039] Figure 1 A schematic diagram of a direct spread spectrum layered sliding blind despreading system is provided as an embodiment of the present invention;
[0040] Figure 2 A schematic diagram of the spread spectrum pseudocode period calculation process provided for one embodiment of the present invention;
[0041] Figure 3 A schematic diagram of sampling for estimating the start position of the spread spectrum pseudocode, provided for one embodiment of the present invention;
[0042] Figure 4 This invention provides a correlation peak of twenty points per chip, as provided in one embodiment of the invention.
[0043] Figure 5 A schematic diagram of the spread spectrum pseudocode start position estimation process provided for one embodiment of the present invention;
[0044] Figure 6 A schematic diagram of a spread spectrum blind despreading process provided for one embodiment of the present invention;
[0045] Figure 7 The result is the estimation of the spread spectrum pseudocode period;
[0046] Figure 8 The initial estimation result for the spreading pseudocode;
[0047] Figure 9This is the result of blind deamping. Detailed Implementation
[0048] The embodiments of the present invention will be further described in detail below with reference to the accompanying drawings and examples. The following detailed description of the embodiments and the accompanying drawings are used to illustrate the principles of the present invention by way of example, but should not be used to limit the scope of the present invention. That is, the present invention is not limited to the described preferred embodiments, and the scope of the present invention is defined by the claims.
[0049] In the description of this invention, it should be noted that, unless otherwise stated, "a plurality of" means two or more; the terms "first," "second," etc., are used for descriptive purposes only and should not be construed as indicating or implying relative importance; those skilled in the art can understand the specific meaning of the above terms in this invention as appropriate.
[0050] Example 1
[0051] like Figure 1 As shown, one embodiment of the present invention provides a direct spread spectrum layered sliding blind despreading system, including a period calculation module, a start position estimation module, and a blind despreading module;
[0052] The period calculation module is used to sample and perform FFT transformation on the received signal, and calculate the spreading pseudocode period based on the minimum interval of the correlation peaks.
[0053] The starting position estimation module is used to estimate the starting position of the spreading pseudocode based on the spreading pseudocode period;
[0054] The blind despreading module is used to perform blind despreading using the sliding correlation method after estimating the starting position of the spreading pseudocode, and outputs the despreading value.
[0055] (1) Periodic calculation module;
[0056] For periodically stable signals, the autocorrelation function has the same period as the signal itself. Therefore, non-cooperative parties can calculate the period of the autocorrelation peak of the received signal and thus calculate the period of the spread spectrum pseudocode. Since the signal quality at the non-cooperative receiving end is low, directly applying the autocorrelation algorithm to the received signal will cause the correlation peak to be submerged in noise. Therefore, the received signal can be delayed, and the interval between the correlation peaks of the two segments of the signal at different times can be calculated.
[0057] like Figure 2 As shown, calculating the spreading pseudocode period includes:
[0058] S11. Acquire the received signal and sample the received signal at the current time and the delayed time.
[0059] The received signal is uplink data from a certain type of communication system. The received signal is sampled at the current time t and at a delay time t+t1.
[0060] S12. Perform FFT transformation on the samples at the current time and the delayed time, then conjugate them, and then perform IFFT transformation to calculate the minimum interval of the correlation peaks at the current time and the delayed time.
[0061] By performing correlation analysis on two signals at different times using the FFT method described above, correlation peaks can be observed in the signals at the current time and the delayed time.
[0062] S13. Calculate the spreading pseudocode period based on the minimum interval of the correlation peaks at the current time and the delayed time.
[0063] The spreading pseudocode period is calculated using the following formula:
[0064] T=N est ×f sam
[0065] In the formula, T is the period of the spreading pseudocode; N est f is the minimum interval between related peaks; sam This is the sampling frequency for the received signal.
[0066] This invention utilizes FFT (Fast Fourier Transform) and IFFT (Inverse Fast Fourier Transform) to calculate the cross-correlation results of two signals, and finally obtains the spreading pseudocode period by observing the correlation peak interval of the two signals.
[0067] (2) Starting position estimation module;
[0068] Find a specific position w(i) in the received signal, such that the T sampled signals following that position contain only one spreading pseudocode; the principle is as follows:
[0069] like Figure 3 As shown, a sampled signal of length 2×T is randomly selected from the received signal and divided into samples w. k with w k+1 Then, this length can contain two or three symbols; for example, the sampling w of the signal received by the receiver. k with w k+1 It can contain three symbolic information r k ,r k+1 ,r k+2 The corresponding sender is m k ,m k+1 ,m k+2 When the sampling start position w(i) is at different positions, the sampling w obtained by the sliding correlation method is different. k with w k+1The correlation peak also changes accordingly; in addition, the correlation peak changes of the two sampled signals are different when the combination of symbol information they contain is different; to ensure the completeness of the experimental results, the traversal length of the sampling start position w(i) is T. Therefore, under the condition that the period estimation of the spreading pseudocode is correct, this length must include the starting position δ of the spreading pseudocode. s .
[0070] Under non-cooperative conditions, the receiver has no knowledge of the spreading chip rate. To ensure the accuracy of the blind despreading results, the receiver sets sampling at 20 or 60 points per chip. Different sampling frequencies result in different correlation peaks for the two received signals. For example, if the data rate is 1 kb / s, the spread pseudo-code rate is 1.023 Mc / s, and the spreading pseudo-code is a 1023-bit Gold code, then the correlation peak of the receiver's sampling results corresponding to sampling at 20 points per chip is as follows: Figure 4 As shown, using a higher sampling frequency results in a higher maximum value for the correlation peak and a smaller slope value.
[0071] In addition, when the code element information r k With r k+2 When the polarities are different, the sampling start position w(i) and the spreading pseudocode start position δ s As the sample approaches the extreme value, the correlation peak continuously approaches the extreme value; the sampling w k w k+2 Performing a sliding correlation, when the correlation peak of the two is negative, it indicates that the polarities of the two spreading pseudocode segments are opposite. Only when w k w k+2 w can only be found when the polarities of the related peaks are opposite. k w k+1 The extreme point of the relevant peak, that is, only when w k w k+2 Only when the polarities of the relevant peaks are opposite can the method of finding the extreme value be used to estimate the starting position of the spread spectrum pseudocode.
[0072] like Figure 5 As shown, the estimated starting position of the spreading pseudocode includes:
[0073] S21, Reference Figure 3 The method described above involves continuously sampling the received signal to obtain w. k w k+1 w k+2 ;
[0074] S22, w at different sampling times k w k+2 Perform sliding correlation to find w k w k+2 Cases where the polarities of the related peaks are opposite;
[0075] S23, in w k w k+2 When the polarities of the correlation peaks are opposite, the sliding correlation method is used to find w. k w k+1 The extreme point of the correlation peak is the starting position δ of the spreading pseudocode. s .
[0076] (3) Blind despreading module;
[0077] Blind despreading utilizes the relative polarity of the received symbols for despreading. This is achieved when the pseudo-code period estimate T is close to the starting position δ of the spreading pseudo-code. s When estimation errors exist, they will affect the correlation peak of the received signal, thereby affecting the correctness of blind despreading. To overcome the impact of the above estimation errors, this invention proposes to use the sliding correlation method for blind despreading, that is, after estimating the starting position δ of the spreading pseudocode, the method is used. s Then, the received signal is shifted so that the start time of the received signal is aligned with the start position δ of the spreading pseudocode. s Correspondence; the start time of the received signal and the start position δ of the spreading pseudocode. s After alignment, the first sampled signal w1 (i.e., the first periodic signal of the received signal) is taken from the starting position of the spread spectrum pseudocode; the received signal is then shifted left by the corresponding shift number T_delay to obtain the forward-shifted signal.
[0078] The received signal is shifted right by the corresponding shift number T_delay to obtain the shifted signal; the forward-shifted signal, the shifted signal, and the unshifted received signal are then subjected to sliding correlation with the first sampled signal w1 to obtain the extreme values of the correlation peaks. The sign of the correlation peaks reflects the symbol information r. k′ The sign between r1 and r1; the extreme value of the correlation peak is the despread value.
[0079] like Figure 6 As shown, the blind despreading module is used to perform the following operations to obtain the despread value:
[0080] S31. Calculate the spreading pseudocode period, and estimate the starting position of the spreading pseudocode based on the spreading pseudocode period, then delay the received signal so that the starting time of the received signal is aligned with the starting position of the spreading pseudocode.
[0081] S32. Initialize the number of iterations i = 1, the corresponding shift number T_delay of the received signal = 0, and take the first sample signal w1 (i.e. the first pseudocode signal w1) from the starting position of the spread spectrum pseudocode.
[0082] S33. Shift the received signal left by T_delay and right by T_delay respectively to obtain the signal w for the forward shift of the ii-th data segment. k′1 The signal w after shifting the i-th data segmentk′2 The received signal itself is used as the i-th data segment signal (i.e., the signal when the received signal is not moved); the signal shifted forward of the i-th data segment, the signal shifted backward of the i-th data segment, and the i-th data segment signal are respectively correlated with the first sampled signal w1, then summed, the correlation peak is found and stored, and the corresponding shift number T_delay of the received signal is updated at the same time.
[0083] S34. Let i = i + 1, and repeat steps S32 and S33 to update w. k′1 w k′2 until the signal reception ends;
[0084] S35. Find the maximum value among all the stored relevant peak values and output it as the despread value.
[0085] The correlation peak, or unspread value, is expressed by the following formula:
[0086]
[0087] In the formula, The first sampled signal w1 and the shifted signal w k′ The correlation peak value, i.e., the despread value; w1 is the first sampled signal; w k′ The signal is a left-shifted or right-shifted signal; sign(·) represents the sign function.
[0088] The following simulation experiments further illustrate the layered sliding blind despreading method of the present invention.
[0089] The uplink signal (i.e. the received signal) of a certain type of communication system is obtained. The specific signal parameter settings are shown in Table 1.
[0090] Table 1 Uplink signal parameter settings for a certain type of communication system
[0091]
[0092] The uplink signal of a certain type of communication system described above is sampled, and the spreading pseudocode period is calculated. The estimation result of the spreading pseudocode period is as follows: Figure 7 As shown; by Figure 7 It can be seen that the signal correlation peak spacing is 5600 points. Based on the sampling rate of 56MHz at the receiver, the estimated spreading pseudocode period is approximately 10. -7 s, consistent with the result sent from the sender.
[0093] Next, find w k w k+2 In the case of opposite symbol polarities, then use w k with w k+1By performing sliding correlation to find the extreme values of the correlation peaks, the starting position of the spread spectrum pseudocode is derived. The estimation result of the starting position of the spread spectrum pseudocode is as follows: Figure 8 As shown; by Figure 8 It can be seen that the correlation peak value drops significantly after 720 points, that is, the extreme value of the correlation peak value is at 720 points. Therefore, it can be estimated that the starting position of the spread spectrum pseudocode is delayed by 720 points from the starting time of the received signal (i.e. the time of acquiring the received signal).
[0094] Finally, blind despreading is performed on the received signal. The attenuation of the transmitting signal can be set, and the result of the blind despreading at the receiving end is as follows: Figure 9 As shown, for ease of display, the despreading results omit the intermediate channel results; from Figure 9 The blind despreading results show that the results are consistent with the information source settings at the sending end, indicating that the blind despreading results are reliable.
[0095] In the embodiments provided in this application, it should be understood that the disclosed apparatus and methods can be implemented in other ways. For example, the apparatus embodiments described above are merely illustrative. For instance, the division of units is only a logical functional division, and in actual implementation, there may be other division methods. For example, multiple units or components may be combined or integrated into another system, or some features may be ignored or not executed. Furthermore, the coupling or direct coupling or communication connection shown or discussed may be through some interfaces; the indirect coupling or communication connection between apparatuses or units may be electrical, mechanical, or other forms.
[0096] The units described as separate components may or may not be physically separate. The components shown as units may or may not be physical units; that is, they may be located in one place or distributed across multiple network units. Some or all of the units can be selected to achieve the purpose of this embodiment according to actual needs.
[0097] In addition, the functional units in the various embodiments of the present invention can be integrated into one processing unit, or each unit can exist physically separately, or two or more units can be integrated into one unit.
[0098] If the aforementioned functions are implemented as software functional units and sold or used as independent products, they can be stored in a computer-readable storage medium. Based on this understanding, the technical solution of this invention, essentially, or the part that contributes to the prior art, or a portion of the technical solution, can be embodied in the form of a software product. This computer software product is stored in a storage medium and includes several instructions to cause a computer device (which may be a personal computer, server, or network device, etc.) to execute all or part of the steps of the methods described in the various embodiments of this invention. The aforementioned storage medium includes various media capable of storing program code, such as USB flash drives, portable hard drives, ROM, RAM, magnetic disks, or optical disks.
[0099] The above description is merely a specific embodiment of the present invention, but the scope of protection of the present invention is not limited thereto. Any variations or substitutions that can be easily conceived by those skilled in the art within the technical scope disclosed in the present invention should be included within the scope of protection of the present invention. Therefore, the scope of protection of the present invention should be determined by the scope of the claims.
Claims
1. A direct spread spectrum layered sliding blind despreading system, characterized in that, It includes a period calculation module, a starting position estimation module, and a blind despreading module; The period calculation module is used to sample and perform FFT transformation on the received signal, and calculate the spreading pseudocode period based on the minimum interval of the correlation peaks. The starting position estimation module is used to estimate the starting position of the spreading pseudocode based on the spreading pseudocode period; The blind despreading module is used to perform blind despreading using the sliding correlation method after estimating the starting position of the spreading pseudocode, and outputs the despreading value.
2. The direct spread spectrum layered sliding blind despreading system according to claim 1, characterized in that, The period calculation module is used to perform the following operations: S11. Acquire the received signal and sample the received signal at the current time and the delayed time. S12. Perform FFT transformation on the samples at the current time and the delayed time, then conjugate them, and then perform IFFT transformation to calculate the minimum interval of the correlation peaks at the current time and the delayed time. S13. Calculate the spreading pseudocode period based on the minimum interval of the correlation peaks at the current time and the delayed time.
3. The direct spread spectrum layered sliding blind despreading system according to claim 2, characterized in that, The spreading pseudocode period is calculated using the following formula: T=N est ×f sam In the formula, T is the period of the spreading pseudocode; N est f is the minimum interval between related peaks; sam This is the sampling frequency for the received signal.
4. The direct spread spectrum layered sliding blind despreading system according to claim 1, characterized in that, The starting position estimation module uses the sliding correlation method to estimate the starting position of the spread spectrum pseudocode.
5. The direct spread spectrum layered sliding blind despreading system according to claim 4, characterized in that, The starting position estimation module is used to perform the following operations: S21. Continuously sample the received signal to obtain w k w k+1 w k+2 ; S22, regarding w k w k+2 Perform sliding correlation to find w k w k+2 Cases where the polarities of the related peaks are opposite; S23, in w k w k+2 When the polarities of the correlation peaks are opposite, the sliding correlation method is used to find w. k w k+1 The extreme point of the relevant peak value is the starting position of the spread spectrum pseudocode.
6. The direct spread spectrum layered sliding blind despreading system according to claim 1, characterized in that, The blind despreading module is used to perform the following operations: The received signal is shifted so that the start time of the received signal corresponds to the start position of the spread spectrum pseudocode, and the first sampled signal is taken from the start position of the spread spectrum pseudocode. The received signal is shifted to the left to obtain a forward-shifted signal, and the received signal is shifted to the right to obtain a backward-shifted signal; The forward-shifted signal, the backward-shifted signal, and the unshifted received signal are respectively subjected to sliding correlation with the first segment of the sampled signal to find the extreme value of the correlation peak, and output as the despread value.
7. The direct spread spectrum layered sliding blind despreading system according to claim 6, characterized in that, The blind despreading module is used to perform the following operations: S31. Delay the received signal so that the start time of the received signal is aligned with the start position of the spread spectrum pseudocode; S32. Initialize the iteration count i = 1, the corresponding shift number T_delay of the received signal = 0, and take the first sample signal w1 from the starting position of the spread spectrum pseudocode; S33. Shift the received signal left by T_delay and right by T_delay respectively to obtain the signal w for the i-th data segment shifted forward. k′1 The signal w after shifting the i-th data segment k′2 and take the received signal itself as the i-th data signal; and take w k′1 w k′2 The i-th data signal is correlated with the first sampled signal w1, then summed, the correlation peak value is obtained and stored, and the corresponding shift value T_delay of the received signal is updated at the same time. S34. Let i = i + 1, and repeat steps S32 and S33 to update w. k′1 w k′2 until the signal reception ends; S35. Find the maximum value among all the stored relevant peak values and output it as the despread value.
8. The direct spread spectrum layered sliding blind despreading system according to claim 7, characterized in that, The relevant peak value is expressed by the following formula: In the formula, The first sampled signal w1 and the shifted signal w k′ The relevant peak value; w1 is the first sampled signal; w k′ The signal after left or right shifting the received signal; sign(·) represents the sign function.