Delay co-prime sampling system and frequency estimation method

Abstract

A frequency estimation method based on time-delay sampling is suitable for signal processing in an engineering scene in which leaf end timing, nondestructive testing or passive radar is difficult to deploy a high-sampling-rate or multi-sampling-rate system, and comprises the following steps: uniformly sampling signals by using two channels with the same sampling period; the time delay and the sampling period of the two sampling channels are relatively prime; carrying out Hilbert transformation preprocessing on data points of the two sampling channels, and solving aliasing frequency in cooperation with an ESPRIT algorithm; changing the initial position of the analysis data to construct a plurality of delay schemes, and repeating the step S2 to obtain equivalent sampling frequencies and equivalent aliasing frequencies under different delay schemes; according to the equivalent aliasing frequency, calculating two remainder frequencies under each delay scheme; and performing traversal solution on the congruence equation set with the equivalent sampling frequency as the modulus and the remainder frequency as the remainder by using the Chinese remainder theorem, and obtaining the target frequency in combination with the consistency criterion and the minimum estimation criterion.

Description

Technical Field

[0001] This invention relates to the field of passive sampling technology, such as signal analysis and blade tip timing vibration measurement, non-destructive testing, or direction of arrival estimation, and particularly to a frequency estimation method based on time-delay sampling. Background Technology

[0002] The well-known Shannon-Nyquist sampling theorem states that a band-limited analog signal can be perfectly reconstructed from discrete data with a sampling frequency at or above the Nyquist rate. However, as the target frequency increases, in some applications, it becomes difficult to achieve the Nyquist rate, leading to the typical problem of frequency estimation for undersampled signals.

[0003] Delay estimation is a method for estimating signal frequency by utilizing phase changes over a given time delay. This method requires the delay time to be less than the Nyquist interval to accurately identify the signal frequency. When the target frequency is high, an extremely short delay time is needed to obtain an accurate signal frequency, increasing the requirement for time accuracy and reducing the system's robustness to noise and sampling time fluctuations, thus rendering traditional delay estimation methods ineffective.

[0004] The Chinese Remainder Theorem utilizes the concept of "coprime" to recover the signal frequency from an undersampled signal. However, it requires at least two devices with coprime sampling frequencies to uniformly sample the signal, which places higher demands on the hardware. Furthermore, in passive sampling fields such as blade tip timing vibration measurement and direction of arrival estimation, the implementation of different sampling rates depends on the number and layout of sensors, which greatly increases the measurement cost. Therefore, the application of the Chinese Remainder Theorem algorithm in such passive sampling has been hindered.

[0005] The information disclosed in the background section is only for enhancing the understanding of the background of this invention, and therefore may contain information that does not constitute prior art known to those skilled in the art. Summary of the Invention

[0006] To address the shortcomings, this invention provides a frequency estimation method, system, medium, and device based on delayed sampling. From severely undersampled signals, a series of equivalent sampling frequencies and remainder frequencies are obtained based on delay estimation and the ESPRIT algorithm. Then, the robust Chinese remainder theorem algorithm is used to solve a system of congruence equations with the equivalent sampling frequencies as moduli and the remainder frequencies as remainders. The target frequency is obtained by combining the consistency criterion and the minimum estimation criterion.

[0007] A frequency estimation method based on delayed sampling is applicable to signal processing in engineering scenarios where high sampling rate or multi-sampling rate systems are difficult to deploy, such as leaf tip timing, non-destructive testing, or passive radar. The method includes:

[0008] Step S1: Use two channels with the same sampling period to uniformly sample the signal, and the time delay of the two sampling channels is coprime with the sampling period;

[0009] Step S2: Perform Hilbert transform preprocessing on the data points of the two sampling channels, and solve for the aliasing frequency using the ESPRIT algorithm;

[0010] Step S3: Change the starting position of the analysis data to construct multiple delay schemes, and repeat step S2 to obtain the equivalent sampling frequency and equivalent aliasing frequency under different delay schemes;

[0011] Step S4: Calculate two remainder frequencies for each delay scheme based on the equivalent aliasing frequency;

[0012] Step S5: Use the robust Chinese Remainder Theorem algorithm to solve the system of congruence equations with the equivalent sampling frequency as the modulus and the remainder frequency as the remainder, and combine the consistency criterion and the minimum estimation criterion to obtain the target frequency.

[0013] In the frequency estimation method based on delayed sampling, step S1 includes:

[0014] Using both sampling periods The sampling channels perform uniform sampling of the signal, with a sampling frequency of The sampling times of the two sampling channels exist. The delay is that the sampling time of the first data point of the second channel is delayed by a certain time compared to the sampling time of the first data point of the first channel. , and The relationship between them is satisfied Where M and N are a pair of coprime integers, and the sampled data vector obtained through the two sampling channels is denoted as and ,in , Where G represents the length of the vector, i.e., the length of the data snapshot. This represents the transpose operation of a vector or matrix.

[0015] In the frequency estimation method based on delayed sampling, step S2 includes:

[0016] For sampled data vectors and Perform the Hilbert transform to obtain the vector in analytical form. and The Hilbert transform includes,

[0017] Calculate the sampled data vector and Discrete Fourier Transform

[0018] (1)

[0019] in, and Representing vectors respectively and The i-th element; e represents the natural constant, ; For imaginary numbers, ; It is an integer variable that takes values ​​from 1 to G; and They represent and The nth element in the Fourier transform result

[0020] Construct a sequence of window functions of length G :

[0021] (2)

[0022] in This represents the value of the nth element in the window function sequence; Indicates to Rounding down,

[0023] calculate , and Element-wise product:

[0024] (3)

[0025] in and They represent , and The product result, obtained by iterating through n from 1 to G, yields two vectors. , ,

[0026] For vectors respectively and Performing an inverse Fourier transform yields the following results: and

[0027] (4)

[0028] in and Representing vectors respectively and The i-th element; and Representing vectors respectively and The nth element, , ,

[0029] Then the data vectors of the two sampling channels are rewritten in the following form:

[0030] (5)

[0031] (6)

[0032] Next, the ESPRIT algorithm is used to solve for different aliasing frequencies, as shown in the following expression:

[0033] (7)

[0034] in, Represents the similarity transformation matrix in the ESPRIT algorithm The kth eigenvalue, The operation for calculating the complex phase angle is defined as follows:

[0035] (8)

[0036] in, Let a be a complex number with a real part and a virtual part b.

[0037] In the frequency estimation method based on delayed sampling, step 3 includes:

[0038] By changing the starting position of the analyzed data, selecting data points 2 to N+1 from the first channel and 1 to N from the second channel, frequency estimation is performed using ESPRIT. The time delay of the first sampling point of the two data streams being analyzed is... , recorded as The aliasing frequency in this case is calculated as follows:

[0039] (9)

[0040] Next, the starting position of the analyzed data is changed, and data points 1 to N from channel 1 and data points 2 to N+1 from channel 1 are selected for frequency estimation using ESPRIT. The time delay of the first sampling point of the two data streams being analyzed is... , recorded as The aliasing frequency in this case is calculated as follows:

[0041] (10)

[0042] Different delay schemes can be constructed by selecting the starting position of the data to be analyzed. Taking the reciprocal of this delay time, we obtain the equivalent sampling frequency for this delay condition as follows:

[0043]

[0044] in, Indicates the delay time as Equivalent sampling frequency under the following conditions .

[0045] In the frequency estimation method based on delayed sampling, step S4 includes:

[0046] The formula for aliasing frequency is:

[0047] (11)

[0048] in, It is a non-negative integer such that Take the minimum value, which is the frequency. The signal at a frequency of The aliasing frequency of uniform sampling at time, , The representative frequency is The signal at a frequency of The remainder frequency of uniform sampling, according to the result of equation (11), is found to be... The following relationship must be satisfied:

[0049] (12)

[0050] Where i represents a certain integer. , , , Let represent the i-th equivalent sampling frequency, the i-th equivalent aliasing frequency, and the i-th remainder frequency, respectively. Substituting different equivalent aliasing frequencies into equation (12) yields two remainder frequencies. , , The corresponding remainder frequency is obtained as , , .

[0051] In the frequency estimation method based on delayed sampling, step S5 includes:

[0052] By selecting the equivalent sampling frequency and corresponding remainder frequency of different delay schemes, and combining them in pairs, a system of congruence equations is established:

[0053] (13)

[0054] The robust Chinese Remainder Theorem algorithm is used to solve three systems of equations to reconstruct the characteristic frequencies. ,because correspond or There are 8 possible combinations. By solving for all combinations, a series of reconstructed characteristic frequency values ​​are obtained. Among these 8 combinations, there exists only one unique combination. This makes the estimated frequency specific to Valid estimation,

[0055] If the cth Combination is In modulus If the correct remainder frequency is obtained, then the three corresponding frequency reconstruction values ​​will be obtained. Since they are relatively close, a consistency criterion is introduced to filter feature frequencies:

[0056] (14)

[0057] in, The reconstructed characteristic frequency value corresponding to the i-th congruence equation system and the c-th remainder frequency combination; This represents the average of the three characteristic frequency reconstruction values ​​under the c-th remainder frequency combination, i.e. ; The index value represents the correct combination, therefore, Characteristic frequency Valid estimation,

[0058] The minimum estimation criterion is to select the smaller of two frequency estimates that satisfy the consistency criterion, and the expression is:

[0059] (15)

[0060] in, and The index values ​​of the two combinations obtained by the consistency criterion (14) have the same minimum variance.

[0061] In the frequency estimation method based on delayed sampling, in step S4, data segments with different starting points are selected by means of a sliding window, so that the effective delay time between the two signals changes, thereby generating multiple coprime sampling frequencies.

[0062] A system for performing the method includes:

[0063] Two synchronously clock-controlled sampling channels have the same sampling frequency, but the second channel introduces a fixed time delay τ.

[0064] The signal preprocessing module is used to perform Hilbert transform on the two sampled data streams;

[0065] The frequency estimation algorithm module integrates the ESPRIT algorithm and the robust Chinese remainder theorem algorithm solver;

[0066] The delay reconstruction module dynamically generates various equivalent delays by adjusting the starting position of the data segment;

[0067] The decision module is used to execute the consistency criterion and the minimum estimation criterion, and output the final frequency estimate.

[0068] A computer storage medium including computer instructions that, when run on a computer, cause the computer to perform the method.

[0069] An electronic device, the electronic device comprising:

[0070] Memory, processor, and computer programs stored in memory and executable on the processor, wherein,

[0071] The processor implements the method when executing the program.

[0072] Compared with the prior art, the present invention has the following advantages: The present invention achieves multi-rate sampling by changing the equivalent delay of the dual sampling channels, which allows the identification of target frequencies from signals with extremely low sampling rates. It is simple, feasible, and stable in operation, and can be used for the extraction of frequencies from undersampled signals. Attached Figure Description

[0073] Various other advantages and benefits of the present invention will become apparent to those skilled in the art upon reading the detailed description of the preferred embodiments below. The accompanying drawings are for illustrative purposes only and are not intended to limit the invention. It is obvious that the drawings described below are merely some embodiments of the invention, and those skilled in the art can obtain other drawings based on these drawings without any inventive effort. Furthermore, the same reference numerals denote the same parts throughout the drawings.

[0074] In the attached diagram:

[0075] Figure 1 This invention presents a delayed coprime sampling and frequency estimation method.

[0076] Figure 2 This is a schematic diagram of the dual-path delayed coprime sampling principle;

[0077] Figure 3 The time-domain plot of the simulated signal obtained from the two sampling channels;

[0078] Figure 4 The diagram shows three delay schemes.

[0079] The present invention will be further explained below with reference to the accompanying drawings and embodiments. Detailed Implementation

[0080] Specific embodiments of the invention will now be described in more detail with reference to the accompanying drawings. While specific embodiments of the invention are shown in the drawings, it should be understood that the invention may be implemented in various forms and should not be limited to the embodiments set forth herein. Rather, these embodiments are provided so that this invention will be thorough and complete, and will fully convey the scope of the invention to those skilled in the art.

[0081] It should be noted that certain terms are used in the specification and claims to refer to specific components. Those skilled in the art will understand that different terms may be used to refer to the same component. This specification and claims do not distinguish components based on differences in terminology, but rather on differences in function. The terms "comprising" or "including" used throughout the specification and claims are open-ended and should be interpreted as "comprising but not limited to." The following descriptions are preferred embodiments for carrying out the invention; however, these descriptions are for the purpose of understanding the general principles of the specification and are not intended to limit the scope of the invention. The scope of protection of this invention is determined by the appended claims.

[0082] To facilitate understanding of the embodiments of the present invention, further explanations and descriptions will be provided below with reference to the accompanying drawings and specific embodiments. The accompanying drawings do not constitute a limitation on the embodiments of the present invention.

[0083] like Figures 1 to 4 As shown, a frequency estimation method based on delayed sampling is applicable to signal processing in engineering scenarios where high sampling rate or multi-sampling rate systems are difficult to deploy, such as leaf tip timing, non-destructive testing, or passive radar. The method includes the following steps:

[0084] Step S1: Use two channels with the same sampling period to uniformly sample the signal, and the time delay of the two sampling channels is coprime with the sampling period;

[0085] Step S2: Perform Hilbert transform preprocessing on the data points of the two sampling channels, and solve for the aliasing frequency using the ESPRIT algorithm;

[0086] Step S3: Change the starting position of the analysis data to construct multiple delay schemes, and repeat step S2 to obtain the equivalent sampling frequency and equivalent aliasing frequency under different delay schemes;

[0087] Step S4: Calculate two remainder frequencies for each delay scheme based on the equivalent aliasing frequency;

[0088] Step S5: The robust Chinese Remainder Theorem algorithm is used to solve the system of congruence equations with the equivalent sampling frequency as the modulus and the remainder frequency as the remainder. The target frequency is obtained by combining the consistency criterion and the minimum estimation criterion. The Chinese Remainder Theorem is a fundamental theorem in number theory, which can be summarized as follows: a large number can be uniquely represented by several smaller moduli and their corresponding remainders. This idea is similar to the undersampling frequency estimation problem in signal processing. Based on the Chinese Remainder Theorem, a series of algorithms for frequency reconstruction have been developed, collectively known as the Chinese Remainder Theorem algorithms. The traditional Chinese Remainder Theorem algorithm is very sensitive to remainder noise. To overcome this problem, the robust Chinese Remainder Theorem algorithm is proposed.

[0089] In a preferred embodiment of the frequency estimation method based on delay sampling, step S1 includes:

[0090] Using both sampling periods The sampling channels perform uniform sampling of the signal, with a sampling frequency of The sampling times of the two sampling channels exist. The delay is that the sampling time of the first data point of the second channel is delayed by a certain time compared to the sampling time of the first data point of the first channel. , and The relationship between them is satisfied Where M and N are a pair of coprime integers, and the sampled data vector obtained through the two sampling channels is denoted as and ,in , Where G represents the length of the vector, i.e., the length of the data snapshot. This represents the transpose operation of a vector or matrix.

[0091] In a preferred embodiment of the frequency estimation method based on delayed sampling, step S2 includes:

[0092] For sampled data vectors and Perform the Hilbert transform to obtain the vector in analytical form. and The Hilbert transform includes,

[0093] Calculate the sampled data vector and Discrete Fourier Transform

[0094] (1)

[0095] in, and Representing vectors respectively and The i-th element; e represents the natural constant, ; For imaginary numbers, ; It is an integer variable that takes values ​​from 1 to G; and They represent and The nth element in the Fourier transform result

[0096] Construct a sequence of window functions of length G :

[0097] (2)

[0098] in This represents the value of the nth element in the window function sequence; Indicates to Rounding down,

[0099] calculate , and Element-wise product:

[0100] (3)

[0101] in and They represent , and The product result, obtained by iterating through n from 1 to G, yields two vectors. , ,

[0102] For vectors respectively and Performing an inverse Fourier transform yields the following results: and

[0103] (4)

[0104] in and Representing vectors respectively and The i-th element; and Representing vectors respectively and The nth element, , ,

[0105] Then the data vectors of the two sampling channels are rewritten in the following form:

[0106] (5)

[0107] (6)

[0108] in: and These represent the first and second data vectors constructed, respectively; Representing vectors The nth, n+1th, and n+G-1th elements in the array; Representing vectors The nth, n+1th, and n+G-1th elements in the dataset; G is the length of the data snapshot.

[0109] Next, the ESPRIT algorithm is used to solve for different aliasing frequencies, as shown in the following expression:

[0110] (7)

[0111] in, Represents the similarity transformation matrix in the ESPRIT algorithm The kth eigenvalue, The operation for calculating the complex phase angle is defined as follows:

[0112] (8)

[0113] in, Let a be a complex number with a real part and a virtual part b.

[0114] In a preferred embodiment of the frequency estimation method based on delayed sampling, step 3 includes:

[0115] By changing the starting position of the analyzed data, selecting data points 2 to N+1 from the first channel and 1 to N from the second channel, frequency estimation is performed using ESPRIT. The time delay of the first sampling point of the two data streams being analyzed is... , recorded as The aliasing frequency in this case is calculated as follows:

[0116] (9)

[0117] Next, the starting position of the analyzed data is changed, and data points 1 to N from channel 1 and data points 2 to N+1 from channel 1 are selected for frequency estimation using ESPRIT. The time delay of the first sampling point of the two data streams being analyzed is... , recorded as The aliasing frequency in this case is calculated as follows:

[0118] (10)

[0119] Different delay schemes can be constructed by selecting the starting position of the data to be analyzed. Taking the reciprocal of this delay time, we obtain the equivalent sampling frequency for this delay condition as follows:

[0120]

[0121] in, Indicates the delay time as Equivalent sampling frequency under the following conditions .

[0122] In a preferred embodiment of the frequency estimation method based on delayed sampling, step S4 includes:

[0123] The formula for aliasing frequency is:

[0124] (11)

[0125] in, It is a non-negative integer such that Take the minimum value, which is the frequency. The signal at a frequency of The aliasing frequency of uniform sampling at time, , The representative frequency is The signal at a frequency of The remainder frequency of uniform sampling, according to the result of equation (11), is found to be... The following relationship must be satisfied:

[0126] (12)

[0127] Where i represents a certain integer. , , , Let represent the i-th equivalent sampling frequency, the i-th equivalent aliasing frequency, and the i-th remainder frequency, respectively. Substituting different equivalent aliasing frequencies into equation (12) yields two remainder frequencies. , , The corresponding remainder frequency is obtained as , , .

[0128] In a preferred embodiment of the frequency estimation method based on delayed sampling, step S5 includes:

[0129] By selecting the equivalent sampling frequency and corresponding remainder frequency of different delay schemes, and combining them in pairs, a system of congruence equations is established:

[0130] (13)

[0131] The Chinese Remainder Theorem is used to solve three systems of equations to reconstruct the characteristic frequencies. ,because correspond or There are 8 possible combinations. By solving for all combinations, a series of reconstructed characteristic frequency values ​​are obtained. Among these 8 combinations, there exists only one unique combination. This makes the estimated frequency specific to Valid estimation,

[0132] If the cth Combination is In modulus If the correct remainder frequency is obtained, then the three corresponding frequency reconstruction values ​​will be obtained. Since they are relatively close, a consistency criterion is introduced to filter feature frequencies:

[0133] (14)

[0134] in, The reconstructed characteristic frequency value corresponding to the i-th congruence equation system and the c-th remainder frequency combination; This represents the average of the three characteristic frequency reconstruction values ​​under the c-th remainder frequency combination, i.e. ; The index value represents the correct combination, therefore, Characteristic frequency Valid estimation,

[0135] The minimum estimation criterion is to select the smaller of two frequency estimates that satisfy the consistency criterion, and the expression is:

[0136] (15)

[0137] in, and The index values ​​of the two combinations obtained by the consistency criterion (14) have the same minimum variance.

[0138] In a preferred embodiment of the frequency estimation method based on delay sampling, in step S4, data segments with different starting points are selected by means of a sliding window, so that the effective delay time between the two signals changes, thereby generating multiple coprime sampling frequencies.

[0139] A system for performing the method includes:

[0140] Two synchronously clock-controlled sampling channels have the same sampling frequency, but the second channel introduces a fixed time delay τ.

[0141] The signal preprocessing module is used to perform Hilbert transform on the two sampled data streams;

[0142] The frequency estimation algorithm module integrates the ESPRIT algorithm and the Chinese remainder theorem solver;

[0143] The delay reconstruction module dynamically generates various equivalent delays by adjusting the starting position of the data segment;

[0144] The decision module is used to execute the consistency criterion and the minimum estimation criterion, and output the final frequency estimate.

[0145] A computer storage medium including computer instructions that, when run on a computer, cause the computer to perform the method.

[0146] An electronic device, the electronic device comprising:

[0147] Memory, processor, and computer programs stored in memory and executable on the processor, wherein,

[0148] The processor implements the method when executing the program.

[0149] In one embodiment, the method includes the following steps:

[0150] (1) The signal is uniformly sampled using two channels with the same sampling period. The ratio of the time delay of the two sampling channels to the sampling period is coprime. The principle is as follows: Figure 2 As shown.

[0151] In this example, the simulation signal is first generated using the following formula:

[0152] (16)

[0153] in, It is a generated simulated signal. , and These represent the frequency, amplitude, and phase of the simulated signal, respectively. For variance Gaussian white noise, satisfying the following conditions. .make , , In the simulation, the signal-to-noise ratio is... The duration is 4 seconds.

[0154] Next, using both sampling frequencies... Uniform sampling is performed on the channels, with a sampling period of The sampling time of channel 2 is delayed compared to the sampling time of channel 1. At this point, the ratio of the sampling time delay to the sampling period for the two channels is: 6 and 13 are coprime. 600 samples were collected from each channel, for a total of 1200 samples. The collected data is as follows: Figure 3 As shown.

[0155] The discrete signal sample vectors acquired from the two channels are:

[0156] (17)

[0157] (2) Perform Hilbert transform preprocessing on the data points of the two sampling channels, and solve the aliasing frequency in conjunction with the ESPRIT algorithm.

[0158] In this example, for the data vector and Perform the Hilbert transform to obtain the vector in analytical form. and The specific calculation process of the Hilbert transform can be divided into the following 4 steps:

[0159] Step 1: Calculation and Discrete Fourier Transform

[0160] (1)

[0161] in, and Representing vectors respectively and The i-th element; e represents the natural constant, ; For imaginary numbers, ; It is an integer variable that takes values ​​from 1 to G; and They represent and The nth element in the Fourier transform result.

[0162] Step 2: Construct a sequence of window functions of length G. :

[0163] (2)

[0164] in This represents the value of the nth element in the window function sequence; Indicates to Rounding down.

[0165] Step 3: Calculation , and Element-wise product:

[0166] (3)

[0167] in and They represent , and The product result. By iterating through n from 1 to G, we can obtain the two data counts. , .

[0168] Step 4: [The text abruptly ends here, likely due to an incomplete sentence or a formatting error.] and Performing an inverse Fourier transform yields the following results: and

[0169] (4)

[0170] in and Representing vectors respectively and The i-th element; and Representing vectors respectively and The nth element, , .

[0171] Then the data vectors of the two sampling channels are rewritten in the following form:

[0172] (5)

[0173] (6)

[0174] in: and These represent the first and second data vectors constructed, respectively; Representing vectors The nth, n+1th, and n+G-1th elements in the array; Representing vectors The nth, n+1th, and n+G-1th elements in the data; G is the length of the data snapshot, which is taken as G=150 in this example.

[0175] Next, the ESPRIT algorithm is used to solve for different aliasing frequencies, as shown in the following expression:

[0176] (7)

[0177] in, Represents the similarity transformation matrix in the ESPRIT algorithm The kth eigenvalue, The operation for calculating the complex phase angle is defined as follows:

[0178] (8)

[0179] in, Let a be a complex number with a real part and a virtual part b.

[0180] In this example, The equivalent aliasing frequency obtained by solving is:

[0181] (18)

[0182] (3) Change the starting position of the analysis data, construct multiple delay schemes, repeat step (2), and obtain the equivalent sampling frequency and equivalent aliasing frequency under different delay schemes;

[0183] In this example, the starting position of the analyzed data is changed, and 2 to N+1 data points from channel 1 and 1 to N data points from channel 2 are selected for frequency estimation using ESPRIT. In this case, the time delay of the first sampling point of the two data streams being analyzed is... The aliasing frequency in this case is calculated as follows:

[0184] (19)

[0185] Next, the starting position of the analyzed data is changed, and data points 1 to N from channel 1 and data points 2 to N+1 from channel 1 are selected for frequency estimation using ESPRIT. In this case, the time delay of the first sampling point of the two analyzed data streams is... The aliasing frequency in this case is calculated as follows:

[0186] (20)

[0187] In summary, by selecting the starting position of the data analysis, different delay schemes can be constructed, such as... Figure 4 As shown. Taking the reciprocal of this delay time yields the equivalent sampling frequency for that delay condition:

[0188]

[0189] (4) Based on the equivalent aliasing frequency obtained in step (3), calculate two remainder frequencies for each delay scheme.

[0190] According to the following formula, there are two possible remainder frequencies for each delay scheme:

[0191] (12)

[0192] Where i represents a certain integer. . , , Let represent the i-th equivalent sampling frequency, the i-th equivalent aliasing frequency, and the i-th remainder frequency, respectively. Substituting different equivalent aliasing frequencies into this equation yields two remainder frequencies. , , The corresponding remainder frequency is obtained as , , .

[0193] In this example, because , , According to equation (29), the corresponding remainder frequency can be calculated as follows: , , .

[0194] (5) The Chinese Remainder Theorem is used to solve the system of congruence equations with equivalent sampling frequency as modulus and remainder frequency as remainder. The target frequency is obtained by combining the consistency criterion and the minimum estimation criterion.

[0195] In this example, each delay scheme has two possible remainder frequencies, which can be combined in pairs to form a total of 8 cases. The results are shown in Table 1.

[0196] Table 1 Frequency Reconstruction Results

[0197]

[0198] The variances for eight scenarios were calculated based on the consistency criterion, as shown in Table 2:

[0199] Table 2. Variance calculation results for eight combinations

[0200]

[0201] Therefore, combinations 3 and 6 initially meet the consistency criterion. Next, using the minimum estimation criterion, combination 3 is determined to be the final frequency reconstruction result, which is as follows: , , .

[0202] Finally, the average of the obtained signal frequency estimates is used to obtain the final characteristic frequency estimate:

[0203]

[0204] The estimated frequency of 754.9774 Hz is very close to the true value of 755 Hz, with an error of only 0.0226 Hz, which demonstrates the effectiveness of the method.

[0205] This invention achieves multi-rate sampling by changing the equivalent delay of dual sampling channels, allowing the identification of target frequencies from signals with extremely low sampling rates. It is simple, feasible, and computationally stable, and can be used for extracting frequencies from undersampled signals.

[0206]

Application Examples

[0207] In this example, the simulation signal is first generated using the following formula:

[0208] (16)

[0209] in, It is a generated simulated signal. , and These represent the frequency, amplitude, and phase of the simulated signal, respectively. For variance Gaussian white noise, satisfying the following conditions. .make , , In the simulation, the signal-to-noise ratio is... The duration is 4 seconds.

[0210] Next, using both sampling frequencies... Uniform sampling is performed on the channels, with a sampling period of The sampling time of channel 2 is delayed compared to the sampling time of channel 1. At this point, the ratio of the sampling time delay to the sampling period for the two channels is: 6 and 13 are coprime. 600 samples were collected from each channel, for a total of 1200 samples. The collected data is as follows: Figure 3 As shown.

[0211] The discrete signal sample vectors acquired from the two channels are:

[0212] (17)

[0213] After obtaining the discrete signals from the two channels, the data vector is... and Perform the Hilbert transform to obtain the vector in analytical form. and The specific calculation process of the Hilbert transform can be divided into the following 4 steps:

[0214] Step 1: Calculation and Discrete Fourier Transform

[0215] (1)

[0216] in, and Representing vectors respectively and The i-th element; e represents the natural constant, ; For imaginary numbers, ; It is an integer variable that takes values ​​from 1 to G; and They represent and The nth element in the Fourier transform result.

[0217] Step 2: Construct a sequence of window functions of length G. :

[0218] (2)

[0219] in This represents the value of the nth element in the window function sequence; Indicates to Rounding down.

[0220] Step 3: Calculation , and Element-wise product:

[0221] (3)

[0222] in and They represent , and The product result. By iterating through n from 1 to G, we can obtain the two data counts. , .

[0223] Step 4: [The text abruptly ends here, likely due to an incomplete sentence or a formatting error.] and Performing an inverse Fourier transform yields the following results: and

[0224] (4)

[0225] in and Representing vectors respectively and The i-th element; and Representing vectors respectively and The nth element, , .

[0226] Then the data vectors of the two sampling channels are rewritten in the following form:

[0227] (5)

[0228] (6)

[0229] in: and These represent the first and second data vectors constructed, respectively; Representing vectors The nth, n+1th, and n+G-1th elements in the array; Representing vectors The nth, n+1th, and n+G-1th elements in the data; G is the length of the data snapshot, which is taken as G=150 in this example.

[0230] Next, the ESPRIT algorithm is used to solve for different aliasing frequencies, as shown in the following expression:

[0231] (7)

[0232] in, Represents the similarity transformation matrix in the ESPRIT algorithm The kth eigenvalue, The operation for calculating the complex phase angle is defined as follows:

[0233] (8)

[0234] in, Let a be a complex number with a real part and a virtual part b.

[0235] In this example, The equivalent aliasing frequency obtained by solving is:

[0236] (18)

[0237] Next, by changing the starting position of the analysis data, multiple delay schemes are constructed, and the aforementioned steps are repeated to obtain the equivalent sampling frequency and equivalent aliasing frequency under different delay schemes.

[0238] In this example, the starting position of the analyzed data is changed, and 2 to N+1 data points from channel 1 and 1 to N data points from channel 2 are selected for frequency estimation using ESPRIT. In this case, the time delay of the first sampling point of the two data streams being analyzed is... The aliasing frequency in this case is calculated as follows:

[0239] (19)

[0240] Next, the starting position of the analyzed data is changed, and data points 1 to N from channel 1 and data points 2 to N+1 from channel 1 are selected for frequency estimation using ESPRIT. In this case, the time delay of the first sampling point of the two analyzed data streams is... The aliasing frequency in this case is calculated as follows:

[0241] (20)

[0242] In summary, by selecting the starting position of the data analysis, different delay schemes can be constructed, such as... Figure 4 As shown. Taking the reciprocal of this delay time yields the equivalent sampling frequency for that delay condition:

[0243]

[0244] Based on the equivalent aliasing frequency obtained from the aforementioned steps, the remainder frequency is calculated for each delay scheme.

[0245] According to the following formula, there are two possible remainder frequencies for each delay scheme:

[0246] (12)

[0247] Where i represents a certain integer. . , , Let represent the i-th equivalent sampling frequency, the i-th equivalent aliasing frequency, and the i-th remainder frequency, respectively. Substituting different equivalent aliasing frequencies into this equation yields two remainder frequencies. , , The corresponding remainder frequency is obtained as , , .

[0248] In this example, because , , According to equation (43), the corresponding remainder frequency can be calculated as follows: , , .

[0249] (5) The Chinese Remainder Theorem is used to solve the system of congruence equations with equivalent sampling frequency as modulus and remainder frequency as remainder. The target frequency is obtained by combining the consistency criterion and the minimum estimation criterion.

[0250] In this example, each delay scheme has two possible remainder frequencies, which can be combined in pairs to form a total of 8 cases. The results are shown in Table 3.

[0251] Table 3 Frequency Reconstruction Results

[0252]

[0253] The variances for the eight cases were calculated based on the consistency criterion, as shown in Table 4:

[0254] Table 4. Variance calculation results for eight combinations

[0255]

[0256] Therefore, combinations 3 and 6 initially meet the consistency criterion. Next, using the minimum estimation criterion, combination 3 is determined to be the final frequency reconstruction result, which is as follows: , , .

[0257] Finally, the average of the obtained signal frequency estimates is used to obtain the final characteristic frequency estimate:

[0258]

[0259] The estimated frequency of 754.9774 Hz is very close to the true value of 755 Hz, with an error of only 0.0226 Hz, which demonstrates the effectiveness of the method.

[0260] This invention achieves multi-rate sampling by changing the equivalent delay of dual sampling channels, allowing the identification of target frequencies from signals with extremely low sampling rates. It is simple, feasible, and computationally stable, and can be used for extracting frequencies from undersampled signals.

[0261] Furthermore, this invention constructs multiple "virtual" non-uniform or asynchronous sampling structures without increasing the hardware sampling rate by setting up two sampling channels with the same sampling period but a specific time delay, and ensuring that the ratio of this delay to the sampling period is a coprime fraction. This delay-coprime design allows the phase difference between the two signals to contain aliasing information of the high-frequency signal at different "equivalent sampling rates," thereby overcoming the limitation of frequency ambiguity under a single low sampling rate.

[0262] Secondly, the Hilbert transform is introduced to construct an analytical signal from the original real-valued sampled data, converting the real signal into a complex analytic signal, preserving the complete instantaneous amplitude and phase information, avoiding the spectral aliasing interference caused by conjugate symmetry in traditional real signal processing, and providing high-quality input for subsequent high-precision subspace algorithms.

[0263] Next, the ESPRIT (Estimation of Signal Parameters via Rotational Invariance Techniques) algorithm is used to jointly model the two analytical signals, fully utilizing its rotational invariance properties to robustly extract aliasing frequencies from the noisy environment. Compared to traditional FFT methods, ESPRIT has super-resolution capabilities, does not require spectral peak search, and offers high computational efficiency and strong noise resistance.

[0264] Furthermore, by changing the starting positions of the two data segments through a sliding analysis window, multiple different effective delay times are dynamically generated, thereby obtaining multiple sets of distinct equivalent sampling frequencies and their corresponding aliasing frequencies. This strategy cleverly transforms "delay diversity in the time domain" into "modulus diversity in the frequency domain," providing the necessary conditions for the subsequent application of the Chinese Remainder Theorem.

[0265] Based on this, for each group of equivalent sampling frequencies, two possible remainder frequencies (i.e., the aliasing frequency itself and its complement frequency with respect to the sampling frequency) are calculated, and a system of congruence equations with the equivalent sampling frequency as the modulus and the remainder frequency as the remainder is constructed. By using the Chinese Remainder Theorem to solve all remainder combinations, multiple candidate estimates of the true frequency can be obtained.

[0266] Finally, a dual discrimination criterion is introduced: the consistency criterion is used to evaluate the consistency of the reconstruction remainders of each candidate solution under different moduli (by calculating the residual variance) and eliminate erroneous solutions caused by incorrect remainder selection; the minimum estimation criterion selects the result that best matches the physical reality (such as the minimum positive frequency or the a priori frequency band) from the remaining feasible solutions to ensure the uniqueness and accuracy of the final estimate.

[0267] In summary, this invention achieves high-precision and robust estimation of high-frequency signals at extremely low sampling rates (far lower than Nyquist rates) through a complete technology chain of "delayed coprime sampling—analytical signal construction—ESPRIT aliasing frequency estimation—multi-mode CRT reconstruction—dual-criteria optimization." This significantly reduces the dependence on high-speed ADCs and multi-sensor arrays, making it particularly suitable for engineering scenarios where it is difficult to deploy high sampling rate or multi-sampling rate systems, such as leaf tip timing, non-destructive testing, and passive radar.

[0268] Although embodiments of the present invention have been described above in conjunction with the accompanying drawings, the present invention is not limited to the specific embodiments and application fields described above. The specific embodiments described above are merely illustrative and instructive, and not restrictive. Those skilled in the art can make many other forms based on the guidance of this specification and without departing from the scope of protection of the claims of the present invention, and all of these are within the scope of protection of the present invention.

Claims

1. A frequency estimation method based on delayed sampling, characterized in that, The method is applicable to signal processing in engineering scenarios where high sampling rate or multi-sampling rate systems are difficult to deploy, such as leaf tip timing, non-destructive testing, or passive radar. The method includes the following steps: Step S1: Use two channels with the same sampling period to uniformly sample the signal, and the time delay of the two sampling channels is coprime with the sampling period; Step S2: Perform Hilbert transform preprocessing on the data points of the two sampling channels, and solve for the aliasing frequency using the ESPRIT algorithm; Step S3: Change the starting position of the analysis data to construct multiple delay schemes, and repeat step S2 to obtain the equivalent sampling frequency and equivalent aliasing frequency under different delay schemes; Step S4: Calculate two remainder frequencies for each delay scheme based on the equivalent aliasing frequency; Step S5: Use the Chinese Remainder Theorem algorithm to solve the system of congruence equations with the equivalent sampling frequency as the modulus and the remainder frequency as the remainder, and combine the consistency criterion and the minimum estimation criterion to obtain the target frequency.

2. The frequency estimation method based on delayed sampling according to claim 1, characterized in that, Preferably, step S1 includes: Using both sampling periods The sampling channels perform uniform sampling of the signal, with a sampling frequency of The sampling times of the two sampling channels exist. The delay is that the sampling time of the first data point of the second channel is delayed by a certain time compared to the sampling time of the first data point of the first channel. , and The relationship between them is satisfied Where M and N are a pair of coprime integers, and the sampled data vector obtained through the two sampling channels is denoted as and ,in , Where G represents the length of the vector, i.e., the length of the data snapshot. This represents the transpose operation of a vector or matrix.

3. The frequency estimation method based on delayed sampling according to claim 2, characterized in that, Step S2 includes: For sampled data vectors and Perform the Hilbert transform to obtain the vector in analytical form. and The Hilbert transform includes, Calculate the sampled data vector and Discrete Fourier Transform (1) in, and Representing vectors respectively and The i-th element; e represents the natural constant, ; For imaginary numbers, ; It is an integer variable that takes values ​​from 1 to G; and They represent and The nth element in the Fourier transform result Construct a sequence of window functions of length G : (2) in This represents the value of the nth element in the window function sequence; Indicates to Rounding down, calculate , and Element-wise product: (3) in and They represent , and The product result, obtained by iterating through n from 1 to G, yields two vectors. , , For vectors respectively and Perform an inverse Fourier transform to obtain and ; (4) in and Representing vectors respectively and The i-th element; and Representing vectors respectively and The nth element, , , Then the data vectors of the two sampling channels are rewritten in the following form: (5) (6) Next, the ESPRIT algorithm is used to solve for different aliasing frequencies, as shown in the following expression: (7) in, Represents the similarity transformation matrix in the ESPRIT algorithm The kth eigenvalue, The operation for calculating the complex phase angle is defined as follows: (8) in, Let a be a complex number with a real part and a virtual part b.

4. The frequency estimation method based on delayed sampling according to claim 3, characterized in that, Step 3 includes: By changing the starting position of the analyzed data, selecting data points 2 to N+1 from the first channel and 1 to N from the second channel, frequency estimation is performed using ESPRIT. The time delay of the first sampling point of the two data streams being analyzed is... , recorded as The aliasing frequency in this case is calculated as follows: (9) Next, the starting position of the analyzed data is changed, and data points 1 to N from channel 1 and data points 2 to N+1 from channel 1 are selected for frequency estimation using ESPRIT. The time delay of the first sampling point of the two data streams being analyzed is... , recorded as The aliasing frequency in this case is calculated as follows: (10) Different delay schemes can be constructed by selecting the starting position of the data to be analyzed. Taking the reciprocal of this delay time, we obtain the equivalent sampling frequency for this delay condition as follows: ； in, Indicates the delay time as Equivalent sampling frequency under the following conditions .

5. The frequency estimation method based on delayed sampling according to claim 1, characterized in that, Step S4 includes: The formula for aliasing frequency is: (11) in, It is a non-negative integer such that Take the minimum value, which is the frequency. The signal at a frequency of The aliasing frequency of uniform sampling at time, , The representative frequency is The signal at a frequency of The remainder frequency of uniform sampling, according to the result of equation (11), is found to be... The following relationship must be satisfied: (12) Where i represents a certain integer. , , , Let represent the i-th equivalent sampling frequency, the i-th equivalent aliasing frequency, and the i-th remainder frequency, respectively. Substituting different equivalent aliasing frequencies into equation (12) yields two remainder frequencies. , , The corresponding remainder frequency is obtained as , , .

6. The frequency estimation method based on delayed sampling according to claim 1, characterized in that, Step S5 includes: By selecting the equivalent sampling frequency and corresponding remainder frequency of different delay schemes, and combining them in pairs, a system of congruence equations is established: (13) The robust Chinese Remainder Theorem algorithm is used to solve three systems of equations to reconstruct the characteristic frequencies. ,because correspond or There are 8 possible combinations. By solving for all combinations, a series of reconstructed characteristic frequency values ​​are obtained. Among these 8 combinations, there exists only one unique combination. This makes the estimated frequency specific to Valid estimation, If the cth Combination is In modulus If the correct remainder frequency is obtained, then the three corresponding frequency reconstruction values ​​will be obtained. Since they are relatively close, a consistency criterion is introduced to filter feature frequencies: (14) in, The reconstructed characteristic frequency value corresponding to the i-th congruence equation system and the c-th remainder frequency combination; This represents the average of the three characteristic frequency reconstruction values ​​under the c-th remainder frequency combination, i.e. ; The index value represents the correct combination, therefore, Characteristic frequency Valid estimation, The minimum estimation criterion is to select the smaller of two frequency estimates that satisfy the consistency criterion, and the expression is: (15) in, and The index values ​​of the two combinations obtained by the consistency criterion (14) have the same minimum variance.

7. The frequency estimation method based on delayed sampling according to claim 1, characterized in that, In step S4, data segments with different starting points are selected by using a sliding window method, so that the effective delay time between the two signals changes, thereby generating multiple coprime sampling frequencies.

8. A system for performing the method as described in any one of claims 1-7, characterized in that, It includes: Two synchronously clock-controlled sampling channels have the same sampling frequency, but the second channel introduces a fixed time delay τ. The signal preprocessing module is used to perform Hilbert transform on the two sampled data streams; The frequency estimation algorithm module integrates the ESPRIT algorithm and the robust Chinese remainder theorem algorithm solver; The delay reconstruction module dynamically generates various equivalent delays by adjusting the start position of the data segment. The decision module is used to execute the consistency criterion and the minimum estimation criterion, and output the final frequency estimate.

9. A computer storage medium, characterized in that, The storage medium includes computer instructions that, when executed on a computer, cause the computer to perform the method as described in any one of claims 1-7.

10. An electronic device, characterized in that, The electronic device includes: Memory, processor, and computer programs stored in memory and executable on the processor, wherein, When the processor executes the program, it implements the method as described in any one of claims 1-7.

Images