A frequency estimation method of bounded mapping step non-iterative interpolation

Abstract

The present application relates to the field of communication and signal processing, and particularly relates to a bounded mapping step non-iterative interpolation frequency estimation method, which comprises: performing FFT transformation on an input signal, determining the amplitude values of a first estimated frequency point and its left and right adjacent frequency points, and calculating a first frequency offset estimation value by using a bounded mapping function; performing once spectrum shift on the input signal to obtain a shifted spectrum, and determining the amplitude values of a second estimated frequency point and its left and right adjacent frequency points; calculating a second frequency offset estimation value by using the bounded mapping function again; synthesizing the two frequency offset estimation values based on the interpolation factors extracted in the two estimations to obtain a synthesized frequency offset estimation value; and finally combining the index of the first estimated frequency point to obtain the final estimated frequency of the input signal. The present application has the advantages of high precision, strong robustness and low computational complexity in a low signal-to-noise ratio environment.

Description

TECHNICAL FIELD

[0001] The present application relates to the field of communication and signal processing, and in particular to a bounded mapping step non-iterative interpolation frequency estimation method. BACKGROUND

[0002] The sinusoidal frequency estimation is a key technology in the fields of communication, radar detection and acoustic signal processing, and its estimation accuracy directly determines the effectiveness of Doppler shift compensation in wireless communication, the accuracy of target velocity estimation and target ranging and direction finding in radar. In practical engineering applications, due to the limitation of data block length and real-time requirements, the spectrum analysis method based on DFT (Discrete Fourier Transform) or FFT (Fast Fourier Transform) is widely used. The typical processing procedure is to perform FFT on the sampled signal, and then estimate the signal frequency by searching the position of the spectral peak. However, due to the fence effect, FFT can only provide spectral information at discrete frequency points, and when the real frequency of the signal falls between two discrete frequency points, directly reading the peak frequency point will introduce a large estimation error.

[0003] In order to break through the inherent resolution limit of FFT, the spectrum interpolation algorithm has become the mainstream technology route for high-precision frequency estimation. This kind of method uses the amplitude or phase information of the peak frequency point and its adjacent spectral lines to estimate the FFO (Fractional Frequency Offset) of the real frequency relative to the coarse estimated frequency point through interpolation operation.

[0004] However, the existing spectrum interpolation estimation algorithm still has obvious limitations in low signal-to-noise ratio environment, such as the algorithm proposed by Fang et al. in the literature "A New DFT-Based Frequency Estimator for Single-Tone Complex Sinusoidal Signals" (published in IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), 2014) (referred to as Fang algorithm). The model is based on the ideal assumption that there is no noise interference and the spectrum shape after zero padding satisfies a certain shape. In the low signal-to-noise ratio condition, the noise will seriously destroy the structure of the main lobe of the spectrum, causing the spectral line amplitude ratio used for interpolation to fluctuate sharply, thereby causing significant estimation bias, and even catastrophic estimation error at certain frequency points, resulting in a sharp decline in algorithm performance.

[0005] To improve the estimation accuracy in poor channels, researchers have developed various iterative interpolation algorithms, for example, the PAI-Rife algorithm proposed by Cheng et al. in the literature "Accurate Sinusoidal Frequency Estimation Algorithm for Internet of Things Based on Phase Angle Interpolation Using Frequency Shift" (published in Applied Sciences, 2022, vol. 12, article 6232), the HAQSE algorithm proposed by Serbes in the literature "Fast and Efficient Sinusoidal Frequency Estimation by Using the DFT Coefficients" (published in IEEE Transactions on Communications, 2019, vol. 67, no. 3, pp. 2333-2342), and the algorithm proposed by Fan et al. in the literature "A method for fine resolution frequency estimation from three DFT samples" (published in Signal Processing, 2018, vol. 144, pp. 52-60), which gradually approaches the true frequency value through multiple iterative calculations. Although these methods can achieve higher accuracy under certain conditions, they have high computational complexity, consume a lot of hardware implementation resources, and have the risk of convergence stability. In low SNR conditions, the iterative process is prone to divergence due to large initial value deviation, or to converge to a local extremum, which may even lead to performance degradation.

[0006] Therefore, it has become a technical problem to be solved in the field to develop a non-iterative frequency estimation method that can have high accuracy, strong robustness and low computational complexity under low SNR conditions. SUMMARY

[0007] To overcome the shortcomings of the prior art, the present application provides a bounded mapping step non-iterative interpolation frequency estimation method to solve the technical problems of existing frequency estimation methods that are prone to sharp decline in estimation performance due to noise interference in low SNR environments, and iterative algorithms that are computationally complex and difficult to implement in hardware.

[0008] To achieve the above-mentioned purposes, the present application adopts the following technical solutions:

[0009] The application provides a frequency estimation method of bounded mapping step non-iterative interpolation, comprising the following steps:

[0010] S1, performing M-point FFT on an input signal with N points to obtain a frequency spectrum, wherein M=2N, N is an integer and N>2, preferably 8≤N≤1024; determining a first estimated frequency point corresponding to the maximum amplitude value in the frequency spectrum; obtaining the amplitude values of the first estimated frequency point, its left adjacent frequency point and its right adjacent frequency point respectively;

[0011] S2, calculating a first frequency offset estimation value based on the amplitude values obtained in S1 through a predefined bounded mapping function;

[0012] S3, determining a frequency offset, and performing a spectrum shift on the input signal based on the frequency offset to obtain a shifted spectrum;

[0013] S4, determining a second estimated frequency point corresponding to the maximum amplitude value in the shifted spectrum, and obtaining the amplitude values of the second estimated frequency point, its left adjacent frequency point and its right adjacent frequency point respectively;

[0014] S5, calculating a second frequency offset estimation value based on the amplitude values obtained in S4 through the bounded mapping function;

[0015] S6, determining a first interpolation factor through the amplitude values of the left adjacent frequency point and the right adjacent frequency point obtained in S2, determining a second interpolation factor through the amplitude values of the left adjacent frequency point and the right adjacent frequency point obtained in S4, and synthesizing the first frequency offset estimation value and the second frequency offset estimation value based on the first interpolation factor, the second interpolation factor and the frequency offset to obtain a synthesized frequency offset estimation value;

[0016] S7, obtaining the final estimated frequency of the input signal through the synthesized frequency offset estimation value and the index of the first estimated frequency point.

[0017] Further, in S2, the calculation method of the first frequency offset estimation value is:

[0018] S201, calculating a first initial ratio factor of the bounded mapping function and a second initial ratio factor :

[0019] ;

[0020] ;

[0021] wherein, is the amplitude value of the first estimated frequency point, an amplitude value of a frequency point adjacent to the left of the first estimated frequency point, an amplitude value of a frequency point adjacent to the right of the first estimated frequency point;

[0022] S202, inputting the first initial ratio factor and the second initial ratio factor into the bounded mapping function respectively to obtain a first mapped ratio factor and a second mapped ratio factor .

[0023] .

[0024] .

[0025] S203, calculating an average mapped ratio factor from the first mapped ratio factor and the second mapped ratio factor .

[0026] .

[0027] S204, converting the average mapped ratio factor to obtain a first frequency offset estimation value .

[0028] .

[0029] Further, in S6, the calculation method of the first interpolation factor is:

[0030] .

[0031] wherein, is the first interpolation factor, is an amplitude value of a frequency point adjacent to the left of the first estimated frequency point, is an amplitude value of a frequency point adjacent to the right of the first estimated frequency point.

[0032] The calculation method of the second interpolation factor is:

[0033] .

[0034] wherein, is the second interpolation factor, is an amplitude value of a frequency point adjacent to the left of the second estimated frequency point, is an amplitude value of a frequency point adjacent to the right of the second estimated frequency point.

[0035] Further, in S6, the combination of the first frequency offset estimate and the second frequency offset estimate is realized by the following equation:

[0036] ;

[0037] wherein, is the combined frequency offset estimate, is the first interpolation factor, is the second interpolation factor, is the first frequency offset estimate, is the second frequency offset estimate, is the frequency offset.

[0038] Further, in S7, the final estimated frequency is calculated by the following equation:

[0039] ;

[0040] wherein, is the final estimated frequency, is the index of the first estimated frequency point, is the combined frequency offset estimate, is the sampling frequency.

[0041] Further, in S4, the amplitude values of the second estimated frequency point, its left neighboring frequency point and its right neighboring frequency point are calculated by the following equations, respectively:

[0042] ;

[0043] ;

[0044] ;

[0045] wherein, is the time domain sample index, , is the input signal, is the index of the first estimated frequency point, is the frequency offset, is the amplitude value of the second estimated frequency point, is the amplitude value of the left neighboring frequency point of the second estimated frequency point, is the amplitude value of the right neighboring frequency point of the second estimated frequency point.

[0046] Further, in S3, the frequency offset is calculated by the following equation:

[0047] ;

[0048] wherein, for the frequency offset, for the error oscillation period of the first frequency estimation offset value, for the sampling frequency.

[0049] Compared with the prior art, the present application has the following beneficial effects:

[0050] (1) The present application fundamentally avoids the catastrophic estimation error of the traditional interpolation method due to the instability of the ratio factor under low signal-to-noise ratio by introducing a predefined bounded mapping function to constrain the spectrum line ratio factor in a stable interval, significantly reducing the collapse threshold of the algorithm. At the same time, the step-by-step estimation strategy of virtual frequency offset and direction correction realizes two independent observations and synthesis of the signal frequency without relying on iteration, so that the estimation accuracy can approach the iterative algorithm under low signal-to-noise ratio, and closely follow the Cramer-Rao lower bound, effectively solving the contradiction between the insufficient accuracy of traditional non-iterative method and the poor robustness of iterative method.

[0051] (2) The present application adopts a completely non-iterative deterministic process, with fixed calculation steps, without complex convergence judgment and loop control logic, significantly reducing the computational complexity and time delay. This feature makes the present application very suitable for efficient and pipelined implementation on embedded hardware platforms, greatly meeting the stringent requirements of radar, communication and other systems for real-time signal processing, and providing an excellent solution for the engineering application of high-precision frequency estimation technology.

[0052] (3) Compared with the existing method, the performance of the present application is prone to deterioration when the signal frequency is close to the center of the discrete frequency grid. The present application effectively compensates for the estimation bias caused by spectral asymmetry through the comprehensive direction correction and synthesis mechanism, ensuring the reliability of the estimation method in practical applications. BRIEF DESCRIPTION OF DRAWINGS

[0053] Figure 1 for the frequency estimation mean square error comparison curve of the present application and various existing algorithms under different signal-to-noise ratios in a simulation environment.

[0054] Figure 2 for the frequency estimation mean square error comparison curve of the present application and various existing algorithms under different normalized FFO in a simulation environment.

[0055] Figure 3 for the frequency estimation mean square error comparison curve of the present application and various existing algorithms under different normalized FFO in an ultrasonic detection scene. DETAILED DESCRIPTION

[0056] With reference to the accompanying drawings, the technical solutions in the embodiments of the present application will be described clearly and completely. Obviously, the described embodiments are only a part of the embodiments of the present application, rather than all the embodiments of the present application. Based on the embodiments of the present application, all the other embodiments obtained by a person of ordinary skill in the art without creative effort belong to the protection scope of the present application.

[0057] Embodiments

[0058] The present embodiment provides a bounded mapping step non-iterative interpolation frequency estimation method, comprising the following steps:

[0059] S1, an input signal with a point number of N is obtained M-point FFT is performed to obtain the frequency spectrum ; wherein M=2N, is a time domain sampling point index, ; is a spectral line index of the frequency spectrum, ; in the present embodiment, N=512, the sampling frequency =200Hz.

[0060] Peak value search is performed on the frequency spectrum to determine the first estimated frequency point corresponding to the maximum amplitude value in the frequency spectrum , the index of the first estimated frequency point is recorded as , and the amplitude values of the first estimated frequency point, the left adjacent frequency point and the right adjacent frequency point thereof are obtained; the amplitude value of the first estimated frequency point is recorded as , the amplitude value of the left adjacent frequency point of the first estimated frequency point is recorded as , and the amplitude value of the right adjacent frequency point of the first estimated frequency point is recorded as , then:

[0061] ;

[0062] ;

[0063] .

[0064] S2, based on the , , obtained in S1, a first frequency offset estimation value is calculated through a predefined bounded mapping function, specifically comprising the following sub-steps:

[0065] S201, a first initial ratio factor and a second initial ratio factor of the bounded mapping function are calculated:

[0066] ;

[0067] ;

[0068] S202, Set the first initial ratio factor and the second initial ratio factor By inputting the bounded mapping function respectively, the ratio factor after the first mapping is obtained. and the ratio factor after the second mapping :

[0069] ;

[0070] ;

[0071] S203, Ratio factor after first mapping and the ratio factor after the second mapping Calculate the ratio factor after average mapping :

[0072] ;

[0073] S204, Ratio Factor After Average Mapping The conversion is performed to obtain the first frequency offset estimate. :

[0074] .

[0075] S3. Determine the frequency offset. Based on frequency offset For input signal Perform a spectrum shift to obtain the shifted spectrum. h is the spectral line index of the relocated spectrum. In this step, the frequency offset... The calculation formula is:

[0076] ;

[0077] In the formula, The error oscillation period for the first frequency estimation offset value.

[0078] Since the error of the first frequency estimation offset is periodically oscillating, a corresponding frequency offset is applied to the input signal. Then, the corresponding spectral amplitude is selected, and the second frequency estimation offset value is calculated. The error oscillation of the second frequency estimation offset value and the first frequency estimation offset value are complementary by half a cycle, which can reduce the system error of the algorithm.

[0079] S4, searching the shifted spectrum S4, searching the shifted spectrum S4, searching the shifted spectrum S4, searching the shifted spectrum S4, searching the shifted spectrum S4, searching the shifted spectrum S4, searching the shifted spectrum

[0080] S4, searching the shifted spectrum S4, searching the shifted spectrum S4, searching the shifted spectrum

[0081] S4, searching the shifted spectrum S4, searching the shifted spectrum S4, searching the shifted spectrum

[0082] S4, searching the shifted spectrum S4, searching the shifted spectrum S4, searching the shifted spectrum

[0083] S4, searching the shifted spectrum S4, searching the shifted spectrum S4, searching the shifted spectrum

[0084] S5, based on the S5, based on the S5, based on the S5, based on the S5, based on the S5, based on the

[0085] S6, based on the S6, based on the S6, based on the S6, based on the S6, based on the S6, based on the

[0086] S6, based on the S6, based on the S6, based on the

[0087] S6, based on the S6, based on the S6, based on the S6, based on the S6, based on the S6, based on the

[0088] S6, based on the S6, based on the S6, based on the

[0089] S6, based on the S6, based on the S6, based on the S6, based on the S6, based on the S6, based on the wherein the combination of the first frequency offset estimate and the second frequency offset estimate is achieved by the following equation:

[0090] ;

[0091] S7、through the combined frequency offset estimate and the index of the first estimated frequency point , obtaining the final estimated frequency of the input signal the final estimated frequency The calculation formula of the final estimated frequency is:

[0092] .

[0093] To verify the effectiveness of the present application, the inventors have compared the frequency estimation method proposed in the above embodiment with various existing frequency estimation algorithms through computer simulation and physical experiments.

[0094] 1. Simulation analysis

[0095] In the simulation environment, the signal is set to be a real-valued sinusoidal wave, the sampling frequency is 200 kHz, the number of input signal points N is 512, the number of FFT points M is 1024, the waveform frequency range of the signal is 25 kHz~25.195 kHz, the signal-to-noise ratio range is set to be -10 dB to 0 dB, and in order to comprehensively evaluate the performance, the FFO is taken to be 11 equally spaced values in the range of [0, 0.5] for traversal. The mean square error is taken as the performance evaluation index, and it is compared with the Cramer-Rao lower bound. The simulation results are shown in Figure 1 and Figure 2 .

[0096] From Figure 1 it can be seen that the estimation method proposed in the present application exhibits a low collapse threshold and high robustness. In the process of reducing the signal-to-noise ratio from 0 dB to -10 dB, the mean square error curve of the present application collapses later than the iterative comparison algorithm, and the error is always lower than that of the non-iterative algorithm, which proves that the bounded mapping function effectively suppresses noise interference and avoids catastrophic error. At the same time, in the signal-to-noise ratio region higher than the collapse threshold, the non-iterative method of the present application can achieve or even exceed the estimation accuracy of the complex iterative algorithm, which shows that the virtual offset stepping mechanism designed in the present application equivalently realizes the effect of iterative convergence, thereby significantly improving the efficiency on the premise of ensuring high accuracy.

[0097] As Figure 2As shown, when the signal frequency deviates from the quantization frequency grid, that is, FFO changes, the mean square error curve of the frequency estimation method proposed by the application is the flattest, especially when the signal-to-noise ratio is as low as-3dB, the curve can always be the closest to the Cramer-Rao lower bound, proving the excellent conventional anti-noise performance and stability in the full frequency offset range. When the signal-to-noise ratio is further deteriorated to the extreme condition of-7dB, the performance of PAI-Rife, HAQSE, Ip-3DFT and other iterative algorithms will fluctuate sharply at certain FFO points or even collapse, while the application does not appear such abnormal points due to the stabilizing effect of the bounded mapping function. This proves that the frequency estimation method proposed by the application is not sensitive to the initial value, overcomes the inconsistent convergence problem of the iterative algorithm under extremely low signal-to-noise ratio, and ensures stable performance in the full frequency offset range from the conventional to the extremely low signal-to-noise ratio.

[0098] 2. Experimental verification

[0099] To verify the algorithm performance in a real noise environment, the application builds an ultrasonic detection experimental system in a water tank. The system is driven by a signal source to drive a cylindrical transducer to emit a sinusoidal signal at a nominal frequency of 25 kHz; the signal is transmitted in an anechoic tank to simulate an ideal underwater acoustic channel, and is captured by a differential spherical hydrophone with a receiving sensitivity of-198.3dB. During the experiment, the signal-to-noise ratio of the received signal is accurately adjusted to-3dB. 2000 independent repeated experiments are performed, and after preprocessing the captured signal of each experiment, the frequency estimation is performed using each algorithm respectively.

[0100] Figure 3 The performance of each algorithm is compared in the actual environment where real noise and system error coexist. The results show that the frequency estimation method proposed by the application is stable overall, especially when FFO is close to 0.5, it can still maintain a low estimation error, effectively overcoming the estimation distortion problem in extreme conditions.

[0101] In summary, the bounded mapping step-by-step non-iterative interpolation frequency estimation method provided by the application not only realizes the estimation accuracy close to the Cramer-Rao lower bound and the stable performance in the full range under low signal-to-noise ratio environment, but also has the collapse value lower than the iterative algorithm and the characteristics friendly to hardware implementation.

[0102] The specific embodiments of the application enable those skilled in the art to understand or implement the application. Various modifications to these embodiments will be apparent to those skilled in the art, and the general principles defined herein can be implemented in other embodiments without departing from the spirit or scope of the application.

[0103] It should be understood that the application is not limited to what has been described hereinabove and that various modifications and changes can be made without departing from the scope of the application. The scope of the application is limited only by the claims that follow.

Claims

1. A method of frequency estimation by bounded mapping step non-iterative interpolation, characterized in that, The method comprises the following steps: S1, performing M-point FFT on an input signal with N points to obtain a spectrum, wherein M=2N, N is an integer and N>2; determining a first estimated frequency point corresponding to a maximum amplitude value in the spectrum; obtaining amplitude values of the first estimated frequency point, a left adjacent frequency point and a right adjacent frequency point thereof; S2, calculating a first frequency offset estimation value based on the amplitude values obtained in S1 through a predefined bounded mapping function; The calculation method of the first frequency offset estimation value is: S201、calculating a first initial ratio factor of the bounded mapping function and a second initial ratio factor : ； ； wherein is the amplitude value of the first estimated frequency point, is the amplitude value of the left neighboring frequency point of the first estimated frequency point, is the amplitude value of the right neighboring frequency point of the first estimated frequency point; S202、inputting the first initial ratio factor and the second initial ratio factor to obtain a first mapped ratio factor and a second mapped ratio factor ​ ； ； S203、calculating a first mapped ratio factor and a second mapped ratio factor calculating an average mapped ratio factor : ； S204、to the average mapped ratio factor performing a conversion to obtain a first frequency offset estimate : ； S3, determining a frequency offset, and performing a spectrum shift on the input signal based on the frequency offset to obtain a shifted spectrum; S4, determining a second estimated frequency point corresponding to a maximum amplitude value in the shifted spectrum, and obtaining amplitude values of the second estimated frequency point, a left adjacent frequency point and a right adjacent frequency point thereof; S5, calculating a second frequency offset estimation value based on the amplitude values obtained in S4 through the bounded mapping function; S6, determining a first interpolation factor through the amplitude value of the left adjacent frequency point and the amplitude value of the right adjacent frequency point obtained in S2, determining a second interpolation factor through the amplitude value of the left adjacent frequency point and the amplitude value of the right adjacent frequency point obtained in S4, and synthesizing the first frequency offset estimation value and the second frequency offset estimation value based on the first interpolation factor, the second interpolation factor and the frequency offset to obtain a synthesized frequency offset estimation value; S7, obtaining a final estimated frequency of the input signal through the synthesized frequency offset estimation value and an index of the first estimated frequency point.

2. The method of claim 1, wherein the bounded mapping step non-iterative interpolation frequency estimation method is characterized by, In S6, the calculation method of the first interpolation factor is: ； wherein is a first interpolation factor, is an amplitude value of a frequency point adjacent to the left of the first estimated frequency point, is an amplitude value of a frequency point adjacent to the right of the first estimated frequency point; The calculation method of the second interpolation factor is: ； wherein is a second interpolation factor, is an amplitude value of a left neighboring frequency point of the second estimated frequency point, is an amplitude value of a right neighboring frequency point of the second estimated frequency point.

3. The method of claim 1, wherein the bounded mapping step non-iterative interpolation frequency estimation method is characterized by, In S6, the synthesis of the first frequency offset estimation value and the second frequency offset estimation value is realized through the following formula: ； wherein is a synthesized frequency offset estimate, is a first interpolation factor, is a second interpolation factor, is a first frequency offset estimate, is a second frequency offset estimate, is a frequency offset.

4. The method of claim 1, wherein the bounded mapping step non-iterative interpolation frequency estimation method is characterized by, In S7, the calculation formula of the final estimated frequency is: ； wherein, is the final estimated frequency, is an index of the first estimated frequency point, is the synthesized frequency offset estimate value, is the sampling frequency.

5. The method of claim 1, wherein the bounded mapping step non-iterative interpolation frequency estimation method is characterized by, In S4, the calculation formulas of the amplitude values of the second estimated frequency point, the left adjacent frequency point and the right adjacent frequency point thereof are respectively: ； ； ； In the formula, is a time domain sampling point index, , is an input signal, is an index of a first estimated frequency point, is a frequency offset, is an amplitude value of a second estimated frequency point, is an amplitude value of a frequency point adjacent to the left of the second estimated frequency point, is an amplitude value of a frequency point adjacent to the right of the second estimated frequency point.

6. The method of claim 1, wherein the bounded mapping step non-iterative interpolation frequency estimation method is characterized by, In S3, the calculation formula of the frequency offset is: ； wherein is a frequency offset, is an error oscillator period of the first frequency estimation offset value, is a sampling frequency.

Images