Rapid frequency measurement method based on multiple same-rate undersampling channels
By constructing a time delay for undersampled channels at the same rate and using an improved Chinese remainder theorem, the problems of high cost and remainder ambiguity in traditional multi-rate sampling are solved, achieving low-cost and high-precision frequency reconstruction, which is suitable for timing at the blade tip of aero-engines and distributed sensor signal processing.
Patent Information
- Authority / Receiving Office
- CN · China
- Patent Type
- Applications(China)
- Current Assignee / Owner
- XI AN JIAOTONG UNIV
- Filing Date
- 2025-12-31
- Publication Date
- 2026-05-05
AI Technical Summary
In passive sampling, traditional multi-rate sampling methods rely on a specific number and layout of sensors, which increases measurement costs. Furthermore, the traditional Chinese remainder theorem has remainder ambiguity when dealing with real signals with opposite frequencies, resulting in unacceptable frequency recovery errors.
A fast frequency measurement method based on multiple undersampled channels at the same rate is adopted. By constructing a non-zero and distinct delay combination through the time delay of adjacent sampling channels, the equivalent aliasing frequency is calculated using discrete Fourier transform and complex conjugate operation. Combined with the improved Chinese remainder theorem, a system of congruent equations is constructed to eliminate remainder ambiguity and reconstruct the true frequency.
It reduces hardware costs, simplifies computational complexity, and achieves efficient frequency extraction with a frequency estimation error of less than 0.1%, making it suitable for aero-engine blade timing and distributed sensing passive sampling scenarios.
Smart Images
Figure CN121978400A_ABST
Abstract
Description
Technical Field
[0001] This invention relates to the field of signal analysis and signal processing technology in aero-engine blade tip timing, distributed sensing passive or limited sampling scenarios, and particularly to a fast frequency measurement method based on multiple undersampled channels at the same rate. Background Technology
[0002] Sampling theory is the foundation of digital signal processing. The famous Shannon-Nyquist sampling theorem states that a band-limited analog signal can be perfectly reconstructed from discrete data with a sampling frequency not lower than the Nyquist rate. This has had the most profound impact on the industrial development of digital signal processing systems.
[0003] Multi-rate sampling is a technique in digital signal processing that refers to processing or transmitting signals using multiple different sampling rates within a system. Its core objective is to optimize system performance, reduce computational complexity, or adapt to the needs of different modules by flexibly adjusting the sampling rate. However, in practical applications, implementing multi-rate sampling requires multiple sampling devices with different processing rates, which places higher demands on hardware. Furthermore, in passive sampling scenarios, such as the timing vibration measurement of aero-engine blade tips, the implementation of different sampling rates depends on the specific number and layout of sensors, significantly increasing measurement costs and limiting the application and development of multi-rate sampling in passive sampling situations.
[0004] 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 delay does not meet the Nyquist interval requirement, changing the delay time between sampling channels can generate a series of aliasing frequencies, effectively achieving multi-rate sampling. Combining this with the concept of "coprime" in the Chinese Remainder Theorem, the signal frequency can be recovered from such undersampled signals. However, for real signals with opposite frequencies, the correspondence between the aliasing frequencies within the Nyquist interval and the opposite frequencies of the real signal is unknown, i.e., there is "remainder ambiguity." Directly using the Chinese Remainder Theorem to solve this introduces unacceptable errors, thus hindering the application of the traditional Chinese Remainder Theorem in real signal frequency recovery.
[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 fast frequency measurement method, system, medium, and device based on multiple undersampling channels at the same rate. It obtains two equivalent sampling frequencies and equivalent aliasing frequencies based on delayed sampling of dual channels. At the same time, to address the remainder ambiguity problem, a product-form set of congruence equations is constructed. Finally, the improved Chinese remainder theorem is used to solve the set of congruence equations to obtain the final signal frequency.
[0007] A fast frequency measurement method based on multiple undersampled channels at the same rate is proposed. This method is applicable to signal processing scenarios involving timing at the aero-engine blade tip, distributed sensing, passive or limited sampling. The method includes:
[0008] Step S1: Use L+1 channels with the same sampling period to uniformly sample the signal. By constructing the time delay between adjacent sampling channels, L sets of non-zero and distinct delay combinations can be formed.
[0009] Step S2: Perform a discrete Fourier transform on the sampled data of each channel, take the absolute value of the Fourier transform coefficients to obtain the spectrum, and determine the position index corresponding to the frequency peak.
[0010] Step S3: Select the lth position based on the position index corresponding to the frequency peak. The Fourier transform coefficients of channel l and channel l+1 are obtained and the corresponding complex numbers are extracted. The complex number corresponding to channel l is multiplied by the complex number corresponding to channel l+1 to calculate the phase angle of the intermediate variable and solve for the equivalent aliasing frequency.
[0011] Step S4: Traverse and analyze L groups of dual-channel delayed sampling data, repeat steps S2 to S3, and obtain the corresponding equivalent aliasing frequency and equivalent sampling frequency based on the phase angle value under different delays.
[0012] Step S5: Reconstruct the true frequency from the equivalent aliasing frequency and the equivalent sampling frequency using the Chinese Remainder Theorem algorithm for real-valued signals.
[0013] In the fast frequency measurement method based on multiple undersampling channels at the same rate, step S1 includes:
[0014] Using L+1 sampling frequencies, all are The sampling channels perform uniform sampling of the signal, with corresponding sampling periods of 1000 and 10000. By constructing the time delay between adjacent sampling channels, L sets of non-zero and distinct delay combinations can be formed, where the sampling time of the first sampling point of the (l+1)th channel is delayed compared to the sampling time of the first sampling point of the lth channel. Different delay data satisfy ,in These are two coprime integers. The sampled data vector obtained from the L+1 sampling channels is denoted as... ,in , where p represents the vector length, that is, the length of the recorded data.
[0015] In the fast frequency measurement method based on multiple undersampling channels at the same rate, step S2 includes:
[0016] exist Select the 1st to pth data points for discrete Fourier transform:
[0017]
[0018] in It is the sampled signal, where k and n are the index sequences traversed from 1 to p. It is the symbol for imaginary numbers, defined as: , Indicates to The k-th data point after performing a discrete Fourier transform. This represents the Fourier transform coefficient vector of the data in the l-th channel. and to Take the absolute value. Perform the same operation on all sampled channel data to determine the index of the spectral peak location. .
[0019] In the fast frequency measurement method based on multiple undersampling channels at the same rate, step S3 includes:
[0020] Indexed by the location of the spectral peak Extract two corresponding Fourier coefficients from the Fourier coefficient vectors of the data in channel l and channel l+1. and ,Will conjugate Multiply Obtain an intermediate variable and calculate its phase angle. , divided by Obtain the l-th equivalent aliasing frequency :
[0021]
[0022] in This represents the sampling delay between channel l and channel l+1. Represents conjugate operation. To calculate the complex phase angle, its definition is as follows:
[0023]
[0024] in, Let a represent a complex number with a real part of a and an imaginary part of b.
[0025] In the fast frequency measurement method based on multiple undersampling channels at the same rate, step S4 includes:
[0026] For L sets of dual-path delayed sampling data, the L corresponding equivalent aliasing frequencies are calculated using equation (2). , .
[0027] In the fast frequency measurement method based on multiple undersampling channels at the same rate, step S5 includes:
[0028] Generate a set of remainders , , The two elements in are represented as and ,Right now .
[0029] Then calculate and Regarding the modulus The common remainder is denoted as and : , ;in for The generalized greatest common divisor of, i.e. It makes The largest real number that is pairwise coprime; The modulus is Modular operation,
[0030] Calculate the minimum circumferential distance and :
[0031]
[0032]
[0033] Where i and l represent from and an integer that can take any value from; z represents an integer that can take any value from. An integer that can take any value from the given list; Let i be the common remainder from the l-th delay scheme.
[0034] Next, calculate the offset common remainder. and :
[0035]
[0036] Then, the following system of congruence equations is established, and q is solved using the Chinese Remainder Theorem algorithm:
[0037]
[0038] Where q is the integer to be solved; The modulus is Modulo operation; , ,
[0039] Based on the solution to q, according to the equation Sure Ultimately by Reconstruct the target frequency .
[0040] In the fast frequency measurement method based on multiple undersampled channels at the same rate, the sampling frequency of each of the L+1 channels is less than twice the highest frequency of the signal under test, and is in an undersampled state.
[0041] A system for performing the method includes:
[0042] The uniform sampling module uses L+1 channels with the same sampling period to uniformly sample the signal. By constructing the time delay between adjacent sampling channels, L sets of non-zero and distinct delay combinations can be formed.
[0043] The Discrete Fourier Transform module performs a Discrete Fourier Transform on the sampled data of each channel, takes the absolute value of the Fourier Transform coefficients to obtain the spectrum, and determines the position index corresponding to the frequency peak.
[0044] The calculation module selects the lth position based on the location index corresponding to the frequency peak. The Fourier transform coefficients of channel l and channel l+1 are obtained and the corresponding complex numbers are extracted. The complex conjugate of the channel l is multiplied by the complex number of channel l+1 to calculate the phase angle of the intermediate variable. The equivalent aliasing frequency is solved. The L groups of dual-channel delayed sampling data are analyzed. Based on the phase angle values under different delays, the corresponding equivalent aliasing frequency and equivalent sampling frequency are obtained.
[0045] The reconstruction module uses the Chinese Remainder Theorem algorithm for real-valued signals to reconstruct the true frequency from the equivalent aliasing frequency and the equivalent sampling frequency.
[0046] A computer storage medium including computer instructions that, when run on a computer, cause the computer to perform the method.
[0047] An electronic device, the electronic device comprising:
[0048] Memory, processor, and computer programs stored in memory and executable on the processor, wherein,
[0049] The processor implements the method when executing the program.
[0050] Compared with the prior art, the present invention has the following advantages: by setting up several sampling channels with the same rate and selecting a reference, the present invention traverses and analyzes the delayed sampling data of all dual-path combinations, which effectively realizes multi-rate sampling and reduces the cost of hardware equipment; at the same time, it improves the previous Chinese remainder theorem, eliminates the remainder ambiguity problem in its application, and has the advantages of simple calculation and low complexity, providing an efficient solution for extracting the frequency of undersampled signals. Attached Figure Description
[0051] 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.
[0052] In the attached diagram:
[0053] Figure 1 This is a flowchart of the fast frequency measurement method based on multiple undersampling channels at the same rate proposed in this patent;
[0054] Figure 2 A schematic diagram illustrating the process of solving for the target frequency when using the first delay scheme;
[0055] Figure 3 This is a schematic diagram illustrating the process of solving for the target frequency when choosing the second delay scheme.
[0056] The present invention will be further explained below with reference to the accompanying drawings and embodiments. Detailed Implementation
[0057] 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.
[0058] 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.
[0059] 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.
[0060] like Figures 1 to 3 As shown, a fast frequency measurement method based on multiple undersampled channels at the same rate is applicable to signal processing in scenarios such as timing at the blade tip of an aero-engine, and passive or limited sampling of distributed sensors. The method includes the following steps:
[0061] Step S1: Use L+1 channels with the same sampling period to uniformly sample the signal. By constructing the time delay between adjacent sampling channels, L sets of non-zero and distinct delay combinations can be formed.
[0062] Step S2: Perform a discrete Fourier transform on the sampled data of each channel, take the absolute value of the Fourier transform coefficients to obtain the spectrum, and determine the position index corresponding to the frequency peak.
[0063] Step S3: Select the lth position based on the position index corresponding to the frequency peak. The Fourier transform coefficients of channel l and channel l+1 are obtained and the corresponding complex numbers are extracted. The complex number corresponding to channel l is multiplied by the complex number corresponding to channel l+1 to calculate the phase angle of the intermediate variable and solve for the equivalent aliasing frequency.
[0064] Step S4: Traverse and analyze L groups of dual-channel delayed sampling data, repeat steps S2 to S3, and obtain the corresponding equivalent aliasing frequency and equivalent sampling frequency based on the phase angle value under different delays.
[0065] Step S5: Reconstruct the true frequency from the equivalent aliasing frequency and the equivalent sampling frequency using the Chinese Remainder Theorem algorithm for real-valued signals.
[0066] In a preferred embodiment of the fast frequency measurement method based on multiple undersampling channels at the same rate, step S1 includes:
[0067] Using L+1 sampling frequencies, all are The sampling channels perform uniform sampling of the signal, with corresponding sampling periods of 1000 and 10000. By constructing the time delay between adjacent sampling channels, L sets of non-zero and distinct delay combinations can be formed, where the sampling time of the first sampling point of the (l+1)th channel is delayed compared to the sampling time of the first sampling point of the lth channel. Different delay data satisfy ,in These are two coprime integers. The sampled data vector obtained from the L+1 sampling channels is denoted as... ,in , where p represents the vector length, that is, the length of the recorded data.
[0068] In a preferred embodiment of the fast frequency measurement method based on multiple undersampling channels at the same rate, step S2 includes:
[0069] exist Select the 1st to pth data points for discrete Fourier transform:
[0070]
[0071] in It is the sampled signal, where k and n are the index sequences traversed from 1 to p. It is the symbol for imaginary numbers, defined as: , Indicates to The k-th data point after performing a discrete Fourier transform. This represents the Fourier transform coefficient vector of the data in the l-th channel. and to Take the absolute value. Perform the same operation on all sampled channel data to determine the index of the spectral peak location. .
[0072] In a preferred embodiment of the fast frequency measurement method based on multiple undersampling channels at the same rate, step S3 includes:
[0073] Indexed by the location of the spectral peak Extract two corresponding Fourier coefficients from the Fourier coefficient vectors of the data in channel l and channel l+1. and ,Will conjugate Multiply Obtain an intermediate variable and calculate its phase angle. , divided by Obtain the l-th equivalent aliasing frequency :
[0074]
[0075] in This represents the sampling delay between channel l and channel l+1. Represents conjugate operation. To calculate the complex phase angle, its definition is as follows:
[0076]
[0077] in, Let a represent a complex number with a real part of a and an imaginary part of b.
[0078] In a preferred embodiment of the fast frequency measurement method based on multiple undersampling channels at the same rate, in step S4:
[0079] For L sets of dual-path delayed sampling data, the L corresponding equivalent aliasing frequencies are calculated using equation (2). , .
[0080] In a preferred embodiment of the fast frequency measurement method based on multiple undersampling channels at the same rate, step S5 includes:
[0081] Generate a set of remainders , , The two elements in are represented as and ,Right now .
[0082] Then calculate and Regarding the modulus The common remainder is denoted as and : , ;in for The generalized greatest common divisor of, i.e. It makes The largest real number that is pairwise coprime; The modulus is Modular operation,
[0083] Calculate the minimum circumferential distance and :
[0084]
[0085]
[0086] Where i and l represent from and an integer that can take any value from; z represents an integer that can take any value from. An integer that can take any value from the given list; Let i be the common remainder from the l-th delay scheme.
[0087] Next, calculate the offset common remainder. and :
[0088]
[0089] Then, the following system of congruence equations is established, and q is solved using the Chinese Remainder Theorem algorithm:
[0090]
[0091] Where q is the integer to be solved; The modulus is Modulo operation; , ,
[0092] Based on the solution to q, according to the equation Sure Ultimately by Reconstruct the target frequency .
[0093] In a preferred embodiment of the fast frequency measurement method based on multiple undersampled channels at the same rate, the sampling frequencies of the L+1 channels are all less than twice the highest frequency of the signal under test, and are in an undersampled state.
[0094] A system for performing the method includes:
[0095] The uniform sampling module uses L+1 channels with the same sampling period to uniformly sample the signal. By constructing the time delay between adjacent sampling channels, L sets of non-zero and distinct delay combinations can be formed.
[0096] The Discrete Fourier Transform module performs a Discrete Fourier Transform on the sampled data of each channel, takes the absolute value of the Fourier Transform coefficients to obtain the spectrum, and determines the position index corresponding to the frequency peak.
[0097] The calculation module selects the lth position based on the location index corresponding to the frequency peak. The Fourier transform coefficients of channel l and channel l+1 are obtained and the corresponding complex numbers are extracted. The complex conjugate of the channel l is multiplied by the complex number of channel l+1 to calculate the phase angle of the intermediate variable. The equivalent aliasing frequency is solved. The L groups of dual-channel delayed sampling data are analyzed. Based on the phase angle values under different delays, the corresponding equivalent aliasing frequency and equivalent sampling frequency are obtained.
[0098] The reconstruction module uses the Chinese Remainder Theorem algorithm for real-valued signals to reconstruct the true frequency from the equivalent aliasing frequency and the equivalent sampling frequency.
[0099] A computer storage medium including computer instructions that, when run on a computer, cause the computer to perform the method.
[0100] An electronic device, the electronic device comprising:
[0101] Memory, processor, and computer programs stored in memory and executable on the processor, wherein,
[0102] The processor implements the method when executing the program.
[0103] In one embodiment, a fast frequency measurement method based on multiple undersampled channels at the same rate includes the following steps:
[0104] (1) Using L+1 channels with the same sampling period to uniformly sample the signal, and by constructing the time delay between adjacent sampling channels, L sets of non-zero and distinct delay combinations can be formed.
[0105] In this example, a superimposed sinusoidal signal containing two sinusoidal components is first generated using the following formula, as shown in the following expression:
[0106]
[0107] in These represent the amplitude, frequency, and phase of the q-th signal, respectively. , The variance is Gaussian white noise, The simulated signal represents the time t. The signal parameters used in this example are shown in Table 1.
[0108] Table 1 Simulation Parameters
[0109]
[0110] The above-mentioned sinusoidal superposition signal is uniformly sampled using three channels, i.e. The sampling frequency is set to be... The corresponding period is In this example, the sampling time of channel 2 is set to be delayed compared to the sampling time of channel 1. The sampling time of the 3-channel sample is delayed compared to the sampling time of the 2-channel sample. At this point, the ratio of the sampling delay times of the two sets is: Clearly, 8 and 9 are coprime numbers. 280 samples were collected from each channel, for a total of 840 samples. The simulation lasted for 2 seconds.
[0111] In this example, the discrete signal sample vector acquired by the three channels is:
[0112]
[0113]
[0114]
[0115] (2) Perform discrete Fourier transform on the sampled data of each channel, take the absolute value of the Fourier transform coefficients to obtain the spectrum, and determine the position index corresponding to the frequency peak.
[0116] In this example, we take channel 1 as an example. Perform a discrete Fourier transform to obtain the Fourier transform coefficients. :
[0117]
[0118] in It is the sampled signal, where k and n are the index sequences traversed from 1 to p. It is the symbol for imaginary numbers, defined as: In this example , Indicates to The k-th data after performing a discrete Fourier transform. This represents the Fourier transform coefficient vector of 1-channel data. In this example, Find the absolute value, locate the index of its peak position, and then find the index position. , ,like Figure 2 As shown in the upper part.
[0119] (3) Based on the peak position obtained in (2), select the Fourier transform coefficients of channel 1 and channel 2 and extract the corresponding complex numbers. Multiply the complex conjugate of channel 1 by the complex number of channel 2, calculate the phase angle of the intermediate variable, and solve for the equivalent aliasing frequency.
[0120] In this exemplary instance, based on the index position and Extract the corresponding Fourier coefficients from the Fourier coefficient vectors of channel 1 and channel 2 data. , , and ,Will conjugate Multiply Obtain an intermediate variable and calculate its phase angle. , divided by Obtain the equivalent aliasing frequency Similarly, conjugate Multiply Obtain an intermediate variable and calculate its phase angle. , divided by Obtain the equivalent aliasing frequency Taking the first delay scheme in this example as an example, , , The corresponding equivalent aliasing frequency is calculated as follows: Figure 2 As shown.
[0121]
[0122]
[0123] (4) Analyze the two sets of dual-path delay sampling data, repeat steps (2) to (3), and obtain the corresponding equivalent aliasing frequency and equivalent sampling frequency according to the phase angle value under different delays.
[0124] In this exemplary instance, the above steps are repeated for channels 2 and 3, and the sampling delay time for this group becomes... The analysis process is the same as above; after frequency search, , The corresponding equivalent aliasing frequency is calculated as follows: Figure 3 As shown.
[0125]
[0126]
[0127] (5) The real frequency is reconstructed from the equivalent aliasing frequency and the equivalent sampling frequency using the Chinese Remainder Theorem algorithm for real-valued signals.
[0128] In this example, for the two delay combinations, corresponding remainder sets are formed, i.e. For ease of representation, The two elements in are represented as and ,Right now Then calculate. and Regarding the modulus The common remainder is denoted as and : , ;in for The generalized greatest common divisor of, i.e. It makes The largest real number that is pairwise coprime, in this example, ; The modulus is Modular operation.
[0129] Next, calculate the minimum circumferential distance. and Its specific expression is as follows:
[0130]
[0131]
[0132] Next, calculate the offset common remainder. and The specific process is as follows:
[0133]
[0134] Then, the following system of congruence equations is established, and q is solved using the Chinese Remainder Theorem algorithm:
[0135]
[0136] Where q is the integer to be solved; The modulus is Modulo operation; , .
[0137] Based on the solution to q, according to the equation Sure Ultimately by Reconstruct the target frequency .
[0138] Based on the several equivalent aliasing frequencies calculated in step (4), , , , The following section will identify signal frequencies of 824Hz and 1018Hz respectively.
[0139] (a) A real sine wave signal of 824Hz:
[0140] The remainder sets for the two delay schemes are as follows:
[0141]
[0142]
[0143] Therefore, the combination of common remainders with errors is:
[0144]
[0145]
[0146] Substitute expressions (14) and (15) into the calculation:
[0147]
[0148]
[0149] Obviously, An offset operation is required. According to equation (6):
[0150]
[0151]
[0152] Using equation (7), construct the corresponding system of congruence equations, and then solve it according to the Chinese Remainder Theorem. The solution is: ,so When confirmed Then, according to the equation The remainder frequencies after the translation are obtained by inverse solving, and are as follows: Therefore, the target frequency N reconstructed based on the above results is:
[0153]
[0154] According to equation (16), the frequency estimation error is only -0.235Hz (-0.029%), which verifies that the method has a good frequency identification effect.
[0155] (ii) A real sine wave signal of 1018Hz:
[0156] The remainder sets for the two delay schemes are as follows:
[0157]
[0158]
[0159] Therefore, the combination of common remainders with errors is:
[0160]
[0161]
[0162] Substitute expressions (14) and (15) into the calculation:
[0163]
[0164]
[0165] Obviously, No offset operation is required. According to equation (6):
[0166]
[0167]
[0168] Using equation (7), construct the corresponding system of congruence equations, and then solve it according to the Chinese Remainder Theorem. The solution is: ,so When confirmed Then, according to the equation The remainder frequencies after the translation are obtained by inverse solving, and are as follows: Therefore, the target frequency N reconstructed based on the above results is:
[0169]
[0170] According to equation (17), the frequency estimation error is only -1.035Hz (-0.102%), which verifies that the method has a good frequency identification effect.
[0171] This invention patent proposes a fast frequency measurement method based on multiple undersampled channels with the same rate. By deploying several sampling channels with the same rate and selecting a reference, the delayed sampling data of all dual-channel combinations are analyzed, which effectively realizes multi-rate sampling and reduces the cost of hardware equipment. At the same time, the previous Chinese remainder theorem is improved to eliminate the remainder ambiguity problem in its application. It has the advantages of simple calculation and low complexity, and provides an efficient solution for extracting the frequency of undersampled signals.
[0172]
Application Examples
[0173] First, a superimposed sinusoidal signal containing two sinusoidal components is generated using the following formula, as shown in the expression below:
[0174]
[0175] in These represent the amplitude, frequency, and phase of the q-th signal, respectively. , The variance is Gaussian white noise, The simulated signal represents the time t. The signal parameters used in this example are shown in Table 2.
[0176] Table 2 Simulation Parameters
[0177]
[0178] The above-mentioned sinusoidal superposition signal is uniformly sampled using three channels, i.e. The sampling frequency is set to be... The corresponding period is In this example, the sampling time of channel 2 is set to be delayed compared to the sampling time of channel 1. The sampling time of the 3-channel sample is delayed compared to the sampling time of the 2-channel sample. At this point, the ratio of the sampling delay times of the two sets is: Clearly, 8 and 9 are coprime numbers. 280 samples were collected from each channel, for a total of 840 samples. The simulation lasted for 2 seconds.
[0179] In this example, the discrete signal sample vector acquired by the three channels is:
[0180]
[0181]
[0182]
[0183] Then, a discrete Fourier transform is performed on the sampled data of each channel, the absolute value of the Fourier transform coefficients is taken to obtain the spectrum, and the position index corresponding to the frequency peak is determined.
[0184] In this example, we take channel 1 as an example. Perform a discrete Fourier transform to obtain the Fourier transform coefficients. :
[0185]
[0186] in It is the sampled signal, where k and n are the index sequences traversed from 1 to p. It is the symbol for imaginary numbers, defined as: In this example , Indicates to The k-th data after performing a discrete Fourier transform. This represents the Fourier transform coefficient vector of 1-channel data. In this example, Find the absolute value, locate the index of its peak position, and then find the index position. , ,like Figure 2 As shown in the upper part.
[0187] Based on the peak position obtained in (2), select the Fourier transform coefficients of channel 1 and channel 2 and extract the corresponding complex numbers. Multiply the complex conjugate of channel 1 by the complex number of channel 2, calculate the phase angle of the intermediate variable, and solve for the equivalent aliasing frequency.
[0188] In this exemplary instance, based on the index position and Extract the corresponding Fourier coefficients from the Fourier coefficient vectors of channel 1 and channel 2 data. , , and ,Will conjugate Multiply Obtain an intermediate variable and calculate its phase angle. , divided by Obtain the equivalent aliasing frequency Similarly, conjugate Multiply Obtain an intermediate variable and calculate its phase angle. , divided by Obtain the equivalent aliasing frequency Taking the first delay scheme in this example as an example, , , The corresponding equivalent aliasing frequency is calculated as follows: Figure 2 As shown.
[0189]
[0190]
[0191] Next, the two sets of dual-path delayed sampling data are analyzed and steps (2) to (3) are repeated. Based on the phase angle values under different delays, the corresponding equivalent aliasing frequency and equivalent sampling frequency are obtained.
[0192] In this exemplary instance, the above steps are repeated for channels 2 and 3, and the sampling delay time for this group becomes... The analysis process is the same as above; after frequency search, , The corresponding equivalent aliasing frequency is calculated as follows: Figure 3 As shown.
[0193]
[0194]
[0195] Then, the true frequency is reconstructed from the equivalent aliasing frequency and the equivalent sampling frequency using the Chinese Remainder Theorem algorithm for real-valued signals.
[0196] In this example, for the two delay combinations, corresponding remainder sets are formed, i.e. For ease of representation, The two elements in are represented as and ,Right now Then calculate. and Regarding the modulus The common remainder is denoted as and : , ;in for The generalized greatest common divisor of, i.e. It makes The largest real number that is pairwise coprime, in this example, ; The modulus is Modular operation.
[0197] Next, calculate the minimum circumferential distance. and Its specific expression is as follows:
[0198]
[0199]
[0200] Next, calculate the offset common remainder. and The specific process is as follows:
[0201]
[0202] Then, the following system of congruence equations is established, and q is solved using the Chinese Remainder Theorem algorithm:
[0203]
[0204] Where q is the integer to be solved; The modulus is Modulo operation; , .
[0205] Based on the solution to q, according to the equation Sure Ultimately by Reconstruct the target frequency :
[0206] Based on the several equivalent aliasing frequencies calculated above, , , , The following section will identify signal frequencies of 824Hz and 1018Hz respectively.
[0207] (a) A real sine wave signal of 824Hz:
[0208] The remainder sets for the two delay schemes are as follows:
[0209]
[0210]
[0211] Therefore, the combination of common remainders with errors is:
[0212]
[0213]
[0214] Substitute expressions (14) and (15) into the calculation:
[0215]
[0216]
[0217] Obviously, An offset operation is required. According to equation (6):
[0218]
[0219]
[0220] Using equation (4), construct the corresponding system of congruence equations, and then solve it according to the Chinese Remainder Theorem. The solution is: ,so When confirmed Then, according to the equation The remainder frequencies after the translation are obtained by inverse solving, and are as follows: Therefore, the target frequency N reconstructed based on the above results is:
[0221]
[0222] According to equation (16), the frequency estimation error is only -0.235Hz (-0.029%), which verifies that the method has a good frequency identification effect.
[0223] (ii) A real sine wave signal of 1018Hz:
[0224] The remainder sets for the two delay schemes are as follows:
[0225]
[0226]
[0227] Therefore, the combination of common remainders with errors is:
[0228]
[0229]
[0230] Substitute expressions (14) and (15) into the calculation:
[0231]
[0232]
[0233] Obviously, No offset operation is required. According to equation (6):
[0234]
[0235]
[0236] Using equation (7), construct the corresponding system of congruence equations, and then solve it according to the Chinese Remainder Theorem. The solution is: ,so When confirmed Then, according to the equation The remainder frequencies after the translation are obtained by inverse solving, and are as follows: Therefore, the target frequency N reconstructed based on the above results is:
[0237]
[0238] According to equation (17), the frequency estimation error is only -1.035Hz (-0.102%), which verifies that the method has a good frequency identification effect.
[0239] The present invention provides a fast frequency measurement method based on multiple undersampled channels with the same rate. By deploying several sampling channels with the same rate and selecting a reference, the delayed sampling data of all dual-channel combinations are analyzed, which effectively realizes multi-rate sampling and reduces the cost of hardware equipment. At the same time, the method improves the previous Chinese remainder theorem, eliminates the remainder ambiguity problem in its application, and has the advantages of simple calculation and low complexity, providing an efficient solution for extracting the frequency of undersampled signals.
[0240] Furthermore, this invention sets up multiple sampling channels with the same sampling rate but a precisely known time delay. Without changing the sampling frequency, the system generates multiple "virtual" different sampling rates by utilizing the relative delay between channels, thus forming multiple different aliasing modes in the frequency domain. This design avoids the dependence of traditional multi-rate sampling on multiple sets of asynchronous sampling hardware, significantly reducing system cost and synchronization complexity. It is particularly suitable for passive or constrained sampling scenarios such as aero-engine blade timing and distributed sensing. Furthermore, this invention introduces the complex conjugate multiplication method to extract the phase difference of the same frequency component between two channels and accurately calculates the equivalent aliasing frequency by combining the known delay. This process is highly robust to noise and does not require complex spectral interpolation or phase unwrapping. Most importantly, in the frequency reconstruction stage, addressing the problem that the aliasing remainders of real signals cannot uniquely correspond to the true frequency due to the symmetry of positive and negative frequencies (i.e., "remainder ambiguity"), this invention proposes an improved Chinese Remainder Theorem algorithm based on the minimum circumferential distance criterion and offset correction. By analyzing the geometric relationship of each remainder on the unit circle, it determines whether a ±π offset needs to be applied to the phase remainders to restore the correct congruence relationship, thereby constructing an unambiguous set of congruence equations. This mechanism fundamentally solves the failure problem of traditional CRTs in real signal frequency estimation, enabling high-precision and high-reliability reconstruction of the original frequency even under severe undersampling conditions (such as a sampling rate of only 1 / 5 or even lower than the signal frequency), with a measured error of less than 0.1%. In summary, this invention achieves an organic unity of hardware simplification, algorithm robustness, and accuracy improvement, providing a practical and feasible technical path for rapid and low-cost frequency measurement of high-frequency real signals.
[0241] 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 fast frequency measurement method based on multiple undersampling channels at the same rate, characterized in that, The method is applicable to signal processing in scenarios involving timing at the blade tip of aero-engines, passive or limited sampling of distributed sensing, and includes the following steps: Step S1: Use L+1 channels with the same sampling period to uniformly sample the signal. By constructing the time delay between adjacent sampling channels, L sets of non-zero and distinct delay combinations can be formed. Step S2: Perform a discrete Fourier transform on the sampled data of each channel, take the absolute value of the Fourier transform coefficients to obtain the spectrum, and determine the position index corresponding to the frequency peak. Step S3: Select the lth position based on the position index corresponding to the frequency peak. The Fourier transform coefficients of channel l and channel l+1 are obtained and the corresponding complex numbers are extracted. The complex number corresponding to channel l is multiplied by the complex number corresponding to channel l+1 to calculate the phase angle of the intermediate variable and solve for the equivalent aliasing frequency. Step S4: Traverse and analyze L groups of dual-channel delayed sampling data, repeat steps S2 to S3, and obtain the corresponding equivalent aliasing frequency and equivalent sampling frequency based on the phase angle value under different delays. Step S5: Reconstruct the true frequency from the equivalent aliasing frequency and the equivalent sampling frequency using the Chinese Remainder Theorem algorithm for real-valued signals.
2. The fast frequency measurement method based on multiple undersampling channels at the same rate according to claim 1, characterized in that, Preferably, step S1 includes: Using L+1 sampling frequencies, all are The sampling channels perform uniform sampling of the signal, with corresponding sampling periods of 1000 and 10000. By constructing the time delay between adjacent sampling channels, L sets of non-zero and distinct delay combinations can be formed, where the sampling time of the first sampling point of the (l+1)th channel is delayed compared to the sampling time of the first sampling point of the lth channel. Different delay data satisfy ,in These are two coprime integers. The sampled data vector obtained from the L+1 sampling channels is denoted as... ,in , where p represents the vector length, that is, the length of the recorded data.
3. The fast frequency measurement method based on multiple undersampling channels at the same rate according to claim 2, characterized in that, Step S2 includes: exist Select the 1st to pth data points for discrete Fourier transform: ； in It is the sampled signal, where k and n are the index sequences traversed from 1 to p. It is the symbol for imaginary numbers, defined as: , Indicates to The k-th data point after performing a discrete Fourier transform. This represents the Fourier transform coefficient vector of the data in the l-th channel. and to Take the absolute value, perform the same operation on all sampled channel data, and determine the index of the location of the spectral peak. .
4. The fast frequency measurement method based on multiple undersampling channels at the same rate according to claim 1, characterized in that, Step S3 includes: Indexed by the location of the spectral peak Extract two corresponding Fourier coefficients from the Fourier coefficient vectors of the data in channel l and channel l+1. and ,Will conjugate Multiply Obtain an intermediate variable and calculate its phase angle. , divided by Obtain the l-th equivalent aliasing frequency : ； in This represents the sampling delay between channel l and channel l+1. Represents conjugate operation. To calculate the complex phase angle, its definition is as follows: ； in, Let a represent a complex number with a real part of a and an imaginary part of b.
5. The fast frequency measurement method based on multiple undersampling channels at the same rate according to claim 4, characterized in that, In step S4: For L sets of dual-path delayed sampling data, the L corresponding equivalent aliasing frequencies are calculated using equation (2). , .
6. The fast frequency measurement method based on multiple undersampling channels at the same rate according to claim 5, characterized in that, Step S5 includes: Generate a set of remainders , , The two elements in are represented as and ,Right now ; Then calculate and Regarding the modulus The common remainder is denoted as and : , ;in for The generalized greatest common divisor, i.e. It makes The largest real number that is pairwise coprime; The modulus is Modulo operation, Calculate the minimum circumferential distance and : ； ； Where i and l represent from and an integer that can take any value from; z represents an integer that can take any value from. An integer that can take any value from the given list; Let i be the common remainder from the l-th delay scheme. Next, calculate the offset common remainder. and : ； Then, the following system of congruence equations is established, and q is solved using the Chinese Remainder Theorem algorithm: ； Where q is the integer to be solved; The modulus is Modulo operation; , , Based on the solution to q, according to the equation Sure Ultimately by Reconstruct the target frequency .
7. The fast frequency measurement method based on multiple undersampling channels at the same rate according to claim 1, characterized in that, The sampling frequencies of the L+1 channels are all less than twice the highest frequency of the signal under test, indicating that they are in an undersampled state.
8. A system for performing the method as described in any one of claims 1-7, characterized in that, It includes: The uniform sampling module uses L+1 channels with the same sampling period to uniformly sample the signal. By constructing the time delay between adjacent sampling channels, L sets of non-zero and distinct delay combinations can be formed. The Discrete Fourier Transform module performs a Discrete Fourier Transform on the sampled data of each channel, takes the absolute value of the Fourier Transform coefficients to obtain the spectrum, and determines the position index corresponding to the frequency peak. The calculation module selects the lth position based on the location index corresponding to the frequency peak. The Fourier transform coefficients of channel l and channel l+1 are obtained and the corresponding complex numbers are extracted. The complex conjugate of the channel l is multiplied by the complex number of channel l+1 to calculate the phase angle of the intermediate variable. The equivalent aliasing frequency is solved. The L groups of dual-channel delayed sampling data are analyzed. Based on the phase angle values under different delays, the corresponding equivalent aliasing frequency and equivalent sampling frequency are obtained. The reconstruction module uses the Chinese Remainder Theorem algorithm for real-valued signals to reconstruct the true frequency from the equivalent aliasing frequency and the equivalent sampling frequency.
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.