Frequency estimation method for bounded mapping stepping non-iterative interpolation

Abstract

The invention relates to the field of communication and signal processing, in particular to a frequency estimation method for bounded mapping stepping non-iterative interpolation, which comprises the following steps of: performing FFT (Fast Fourier Transform) on an input signal, determining amplitude values of a first estimation frequency point and left and right adjacent frequency points thereof, and calculating a first frequency offset estimation value through a bounded mapping function; performing primary frequency spectrum shifting on the input signal to obtain a shifted frequency spectrum, and determining amplitude values of a second estimation frequency point and left and right adjacent frequency points; using the bounded mapping function again to calculate a second frequency offset estimation value; synthesizing the two frequency offset estimation values based on the interpolation factors extracted in the two times of estimation to obtain a synthesized frequency offset estimation value; and finally obtaining the final estimation frequency of the input signal by combining the index of the first estimation frequency point. The method has the advantages of high precision, high robustness and low calculation complexity in a low signal-to-noise ratio environment.

Description

Technical Field

[0001] This invention relates to the field of communication and signal processing, and in particular to a frequency estimation method using bounded mapping step non-iterative interpolation. Background Technology

[0002] Sinusoidal frequency estimation is a key technology in fields such as communication, radar detection, and acoustic signal processing. Its estimation accuracy directly determines the effectiveness of Doppler frequency shift compensation in wireless communication and the accuracy of radar target velocity estimation and target ranging and direction finding. In practical engineering applications, limited by data block length and real-time requirements, spectral analysis methods based on DFT (Discrete Fourier Transform) or FFT (Fast Fourier Transform) are widely used. A typical processing flow involves performing an FFT on the sampled signal and then initially estimating the signal frequency by searching for the location of spectral peaks. However, due to the picket fence effect, FFT can only provide spectral information at discrete frequency points. When the true frequency of the signal falls between two discrete frequency points, directly reading the peak frequency will introduce a large estimation error.

[0003] To overcome the inherent resolution limitations of the FFT, spectral interpolation algorithms have become the mainstream technique for high-precision frequency estimation. This type of method utilizes the amplitude or phase information of the peak frequency and its neighboring spectral lines to estimate the FFO (Fractional Frequency Offset) of the true frequency relative to the coarsely estimated frequency through interpolation operations.

[0004] However, existing spectral interpolation estimation algorithms still have significant limitations in low signal-to-noise ratio (SNR) environments. For example, the algorithm proposed by Fang et al. in their paper "A New DFT-Based Frequency Estimator for Single-Tone Complex Sinusoidal Signals" (published at the IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), 2014) (referred to as the Fang algorithm) is based on the ideal assumption of no noise interference and that the spectral shape meets a specific shape after zero-padding. Under low SNR conditions, noise can severely disrupt the structure of the main lobe of the spectrum, causing drastic fluctuations in the amplitude ratio of the spectral lines used for interpolation, thus leading to significant estimation bias, and even catastrophic estimation errors at specific frequency points, resulting in a sharp decline in algorithm performance.

[0005] To improve estimation accuracy under adverse channel conditions, researchers have developed various iterative interpolation algorithms. These include the PAI-Rife algorithm proposed by Cheng et al. in "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 "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 "A method for fineresolution frequency estimation from three DFT samples" (published in SignalProcessing, 2018, vol. 144, pp.). The algorithms proposed in (52-60) (referred to as Ip-3DFT algorithms) gradually approximate the true frequency value through multiple iterations. Although they can achieve higher accuracy under certain conditions, they have high computational complexity, consume a lot of hardware resources, and are subject to risks in convergence stability. Under low signal-to-noise ratio conditions, the iterative process is prone to divergence due to excessive deviation of the initial value, or converge to a local extremum, which leads to performance degradation.

[0006] Therefore, developing a non-iterative frequency estimation method that can achieve high accuracy, strong robustness, and low computational complexity under low signal-to-noise ratio conditions has become a pressing technical challenge in this field. Summary of the Invention

[0007] To address the shortcomings of existing technologies, this invention proposes a bounded mapping step non-iterative interpolation frequency estimation method to solve the technical problems of existing frequency estimation methods experiencing a sharp decline in estimation performance due to noise interference in low signal-to-noise ratio environments, as well as the computational complexity and hardware implementation difficulties of iterative algorithms.

[0008] To achieve the above objectives, the present invention adopts the following technical solution:

[0009] This invention proposes a frequency estimation method for bounded mapping step non-iterative interpolation, comprising the following steps:

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

[0011] S2. Based on the amplitude value obtained in S1, calculate the first frequency offset estimate using a predefined bounded mapping function;

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

[0013] S4. Determine the second estimated frequency point corresponding to the maximum amplitude value in the spectrum after relocation, and obtain the amplitude values ​​of the second estimated frequency point, its left adjacent frequency point, and its right adjacent frequency point.

[0014] S5. Based on the amplitude value obtained in S4, calculate the second frequency offset estimate using the bounded mapping function;

[0015] S6. Determine the first interpolation factor using the amplitude values ​​of the left and right adjacent frequency points obtained in S2. Determine the second interpolation factor using the amplitude values ​​of the left and right adjacent frequency points obtained in S4. Based on the first interpolation factor, the second interpolation factor, and the frequency offset, synthesize the first frequency offset estimate and the second frequency offset estimate to obtain the synthesized frequency offset estimate.

[0016] S7. The final estimated frequency of the input signal is obtained by using the synthesized frequency offset estimate and the index of the first estimated frequency point.

[0017] Furthermore, in S2, the method for calculating the first frequency offset estimate is as follows:

[0018] S201. Calculate the first initial ratio factor of the bounded mapping function. and the second initial ratio factor :

[0019] ;

[0020] ;

[0021] In the formula, The amplitude value at the first estimated frequency point. The amplitude value is the frequency point adjacent to the left of the first estimated frequency point. The amplitude value of the frequency point adjacent to the right of the first estimated frequency point;

[0022] 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 :

[0023] ;

[0024] ;

[0025] S203, Ratio factor after first mapping and the ratio factor after the second mapping Calculate the ratio factor after average mapping :

[0026] ;

[0027] S204, the average mapping ratio factor The transformation is performed to obtain the first frequency offset estimate. :

[0028] .

[0029] Furthermore, in S6, the first interpolation factor is calculated as follows:

[0030] ;

[0031] In the formula, The first interpolation factor, The amplitude value is the frequency point adjacent to the left of the first estimated frequency point. The amplitude value of the frequency point adjacent to the right of the first estimated frequency point;

[0032] The second interpolation factor is calculated as follows:

[0033] ;

[0034] In the formula, The second interpolation factor, The amplitude value is the frequency point adjacent to the left of the second estimated frequency point. This represents the amplitude value of the frequency point adjacent to the right of the second estimated frequency point.

[0035] Furthermore, in S6, the synthesis of the first frequency offset estimate and the second frequency offset estimate is achieved through the following formula:

[0036] ;

[0037] In the formula, This is the estimated value of the synthesized frequency shift. The first interpolation factor, The second interpolation factor, This is the estimated value of the first frequency offset. This is the second frequency offset estimate. This represents the frequency offset.

[0038] Furthermore, in S7, the formula for calculating the final estimated frequency is:

[0039] ;

[0040] in, To ultimately estimate the frequency, The index of the first estimated frequency point. This is the estimated value of the synthesized frequency shift. The sampling frequency.

[0041] Furthermore, in S4, the calculation formulas for the amplitude values ​​of the second estimated frequency point, its left adjacent frequency point, and its right adjacent frequency point are as follows:

[0042] ;

[0043] ;

[0044] ;

[0045] In the formula, For time-domain sampling point index, , For input signal, The index of the first estimated frequency point. This is the frequency offset. The amplitude value at the second estimated frequency point. The amplitude value is the frequency point adjacent to the left of the second estimated frequency point. This represents the amplitude value of the frequency point adjacent to the right of the second estimated frequency point.

[0046] Furthermore, in S3, the formula for calculating the frequency offset is:

[0047] ;

[0048] In the formula, This is the frequency offset. The error oscillation period for the first frequency estimation offset value. The sampling frequency.

[0049] Compared with the prior art, the beneficial effects of the present invention are as follows:

[0050] (1) By introducing a predefined bounded mapping function, this invention constrains the spectral ratio factor, which is susceptible to noise interference, within a stable range. This fundamentally avoids the catastrophic estimation error caused by the instability of the ratio factor in traditional interpolation methods at low signal-to-noise ratios, and significantly reduces the algorithm's collapse threshold. At the same time, the step-by-step estimation strategy using virtual frequency offset and direction correction achieves two independent observations and synthesis of the signal frequency without relying on iteration. This allows the estimation accuracy to approach that of iterative algorithms at low signal-to-noise ratios and closely approach the Cramer-Rao lower bound, effectively resolving the contradiction between insufficient accuracy of traditional non-iterative methods and poor robustness of iterative methods.

[0051] (2) The present invention adopts a completely non-iterative deterministic process with fixed calculation steps, eliminating the need for complex convergence judgment and loop control logic, which significantly reduces computational complexity and time delay. This characteristic makes the present invention very suitable for efficient and pipelined implementation on embedded hardware platforms, greatly satisfying the stringent requirements of radar, communication and other systems for real-time signal processing, and providing a superior solution for the engineering application of high-precision frequency estimation technology.

[0052] (3) Compared with the shortcomings of existing methods, which are prone to performance degradation when the signal frequency is close to the center of the discrete frequency grid, this invention effectively compensates for the estimation deviation caused by spectral asymmetry by integrating direction correction and synthesis mechanisms, thus ensuring the reliability of the estimation method in practical applications. Attached Figure Description

[0053] Figure 1 The graph shows a comparison of the mean square error of frequency estimation between the embodiments of the present invention and various existing algorithms under different signal-to-noise ratio conditions in a simulation environment.

[0054] Figure 2 The graph shows a comparison of the mean square error of frequency estimation between the embodiments of the present invention and various existing algorithms under different normalized FFOs in a simulation environment.

[0055] Figure 3 The graph shows a comparison of the mean square error of frequency estimation between the embodiments of the present invention and various existing algorithms under different normalized FFOs in the context of ultrasonic detection. Detailed Implementation

[0056] The technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0057] Example

[0058] This embodiment proposes a frequency estimation method for bounded mapping step non-iterative interpolation, including the following steps:

[0059] S1, for an input signal with N points. Perform an M-point FFT to obtain its spectrum. Where M=2N, For time-domain sampling point index, ; This refers to the spectral line index of the spectrum. In this embodiment, N=512, and the sampling frequency is... =200Hz.

[0060] For the spectrum Perform peak search to determine the spectrum The first estimated frequency point corresponding to the maximum amplitude is denoted as [index of the first estimated frequency point]. And obtain the amplitude values ​​of the first estimated frequency point, its left adjacent frequency point, and its right adjacent frequency point; denoted as the amplitude of the first estimated frequency point. The amplitude value of the frequency point adjacent to the left of the first estimated frequency point is denoted as... The amplitude value of the frequency point adjacent to the right of the first estimated frequency point is denoted as... ,but:

[0061] ;

[0062] ;

[0063] .

[0064] S2, obtained based on S1 , , The first frequency offset estimate is calculated using a predefined bounded mapping function, specifically including the following sub-steps:

[0065] S201. Calculate the first initial ratio factor of the bounded mapping function. and the second initial ratio factor :

[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. Spectrum after relocation Perform a peak search to find the relocated spectrum. The second estimated frequency point corresponding to the maximum amplitude is identified, and the amplitude values ​​of the second estimated frequency point, its left adjacent frequency point, and its right adjacent frequency point are obtained; the amplitude of the second estimated frequency point is denoted as... The amplitude value of the frequency point adjacent to the left of the second estimated frequency point is denoted as... The amplitude value of the frequency point adjacent to the right of the second estimated frequency point is denoted as... ,but:

[0080] ;

[0081] ;

[0082] ;

[0083] In the formula, It is the imaginary unit.

[0084] S5, obtained based on S4 The second frequency offset estimate is calculated using the bounded mapping function. Second frequency offset estimate Compared with the first frequency offset estimate The calculation method is the same, that is, repeat the calculation process from S201 to S204.

[0085] S6, obtained from S2 and Determine the first interpolation factor First interpolation factor The calculation formula is:

[0086] ;

[0087] Obtained from S4 and Determine the second interpolation factor Second interpolation factor The calculation formula is:

[0088] ;

[0089] Based on the first interpolation factor Second interpolation factor and frequency offset The estimated value of the first frequency offset Second frequency offset estimate The synthesized frequency shift estimate is obtained by performing synthesis. The synthesis of the first frequency offset estimate and the second frequency offset estimate is achieved by the following formula:

[0090] ;

[0091] S7. Using the synthesized frequency offset estimate and the index of the first estimated frequency point The final estimated frequency of the input signal is obtained. Final frequency estimation The calculation formula is:

[0092] .

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

[0094] 1. Simulation Analysis

[0095] In the simulation environment, the signal was set as a real-valued sine wave, the sampling frequency was 200kHz, the number of input signal points N was 512, the number of FFT points M was 1024, the waveform frequency range was 25kHz~25.195kHz, and the signal-to-noise ratio range was set to -10dB to 0dB. To comprehensively evaluate the performance, the FFO was iterated through 11 equally spaced values ​​within the range [0, 0.5]. The mean square error was used as the performance evaluation index, and it was compared with the lower bound of the Cramer-Rhodes test. The simulation results are referenced. Figure 1 and Figure 2 .

[0096] from Figure 1 As can be seen, the estimation method proposed in this invention exhibits a low collapse threshold and high robustness. During the process of the signal-to-noise ratio decreasing from 0dB to -10dB, its mean square error curve collapses later than the iterative comparison algorithm, and the error is always lower than that of the non-iterative algorithm. This proves that the bounded mapping function effectively suppresses noise interference and avoids catastrophic errors. At the same time, in the signal-to-noise ratio region above the collapse threshold, this non-iterative method can achieve or even surpass the estimation accuracy of complex iterative algorithms. This shows that the virtual offset stepping mechanism designed in this invention effectively achieves the effect of iterative convergence, thereby significantly improving efficiency while ensuring high accuracy.

[0097] like Figure 2As shown, when the signal frequency deviates from the quantization frequency grid, i.e., when the FFO changes, the mean square error curve of the frequency estimation method proposed in this invention is the flattest. Especially when the signal-to-noise ratio is as low as -3dB, its curve still closely follows the Cramer-Rhodes lower bound, proving its excellent conventional noise immunity and stability across the entire frequency offset range. When the signal-to-noise ratio further deteriorates to the extreme condition of -7dB, the performance of iterative algorithms such as PAI-Rife, HAQSE, and Ip-3DFT will fluctuate drastically or even collapse at certain FFO points. However, due to the stabilizing effect of the bounded mapping function, this invention does not exhibit such anomalies. This proves that the frequency estimation method proposed in this invention is insensitive to initial values, overcomes the convergence inconsistency problem of iterative algorithms at extremely low signal-to-noise ratios, and ensures stable performance across the entire frequency offset range from conventional to extremely low signal-to-noise ratios.

[0098] 2. Experimental verification

[0099] To verify the algorithm's performance in a real-world noise environment, this invention constructed a water tank ultrasonic detection experimental system. The system consists of a signal source driving a cylindrical transducer to emit a sinusoidal signal at a nominal frequency of 25kHz. The signal is transmitted within an anechoic water 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 was precisely adjusted to -3dB. 2000 independent repeated experiments were conducted. After preprocessing the signals captured in each experiment, frequency estimation was performed using each algorithm.

[0100] Figure 3 The performance comparison of various algorithms is presented under real-world conditions where both real noise and system errors coexist. The results show that the frequency estimation method proposed in this invention is robust overall, especially when the FFO is close to 0.5, it can still maintain a low estimation error and effectively overcome the estimation distortion problem in extreme cases.

[0101] In summary, the frequency estimation method of bounded mapping step non-iterative interpolation provided by this invention achieves estimation accuracy close to the Cramer-Rao lower bound and stable performance across the entire range in low signal-to-noise ratio environments, while also possessing lower collapse values ​​and hardware-friendly characteristics compared to iterative algorithms.

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

[0103] It should be understood that the present invention is not limited to the content already described above, and various modifications and changes can be made without departing from its scope. The scope of the present invention is limited only by the appended claims.

Claims

1. A frequency estimation method for bounded mapping step non-iterative interpolation, characterized in that, Includes the following steps: S1. Perform an M-point FFT on the input signal with N points to obtain the spectrum, where M=2N, N is an integer and N>2; determine the first estimated frequency point corresponding to the maximum amplitude value in the spectrum; obtain the amplitude values ​​of the first estimated frequency point, its left adjacent frequency point, and its right adjacent frequency point; S2. Based on the amplitude value obtained in S1, calculate the first frequency offset estimate using a predefined bounded mapping function; S3. Determine the frequency offset, and perform a spectrum shift on the input signal based on the frequency offset to obtain the shifted spectrum; S4. Determine the second estimated frequency point corresponding to the maximum amplitude value in the spectrum after relocation, and obtain the amplitude values ​​of the second estimated frequency point, its left adjacent frequency point, and its right adjacent frequency point. S5. Based on the amplitude value obtained in S4, calculate the second frequency offset estimate using the bounded mapping function; S6. Determine the first interpolation factor using the amplitude values ​​of the left and right adjacent frequency points obtained in S2. Determine the second interpolation factor using the amplitude values ​​of the left and right adjacent frequency points obtained in S4. Based on the first interpolation factor, the second interpolation factor, and the frequency offset, synthesize the first frequency offset estimate and the second frequency offset estimate to obtain the synthesized frequency offset estimate. S7. The final estimated frequency of the input signal is obtained by using the synthesized frequency offset estimate and the index of the first estimated frequency point.

2. The frequency estimation method for bounded mapping step non-iterative interpolation according to claim 1, characterized in that, In S2, the method for calculating the first frequency offset estimate is as follows: S201. Calculate the first initial ratio factor of the bounded mapping function. and the second initial ratio factor : ； ； In the formula, The amplitude value at the first estimated frequency point. The amplitude value is the frequency point adjacent to the left of the first estimated frequency point. The amplitude value of the frequency point adjacent to the right of the first estimated frequency point; 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 : ； ； S203, Ratio factor after first mapping and the ratio factor after the second mapping Calculate the ratio factor after average mapping : ； S204, the average mapping ratio factor The transformation is performed to obtain the first frequency offset estimate. : 。 3. The frequency estimation method for bounded mapping step non-iterative interpolation according to claim 1, characterized in that, In S6, the first interpolation factor is calculated as follows: ； In the formula, The first interpolation factor, The amplitude value is the frequency point adjacent to the left of the first estimated frequency point. The amplitude value of the frequency point adjacent to the right of the first estimated frequency point; The second interpolation factor is calculated as follows: ； In the formula, The second interpolation factor, The amplitude value is the frequency point adjacent to the left of the second estimated frequency point. This represents the amplitude value of the frequency point adjacent to the right of the second estimated frequency point.

4. The frequency estimation method for bounded mapping step non-iterative interpolation according to claim 1, characterized in that, In S6, the synthesis of the first frequency offset estimate and the second frequency offset estimate is achieved through the following formula: ； In the formula, This is the estimated value of the synthesized frequency shift. The first interpolation factor, The second interpolation factor, This is the estimated value of the first frequency offset. This is the second frequency offset estimate. This represents the frequency offset.

5. The frequency estimation method for bounded mapping step non-iterative interpolation according to claim 1, characterized in that, In S7, the formula for calculating the final estimated frequency is: ； in, To ultimately estimate the frequency, The index of the first estimated frequency point. This is the estimated value of the synthesized frequency shift. The sampling frequency.

6. The frequency estimation method for bounded mapping step non-iterative interpolation according to claim 1, characterized in that, In S4, the calculation formulas for the amplitude values ​​of the second estimated frequency point, its left adjacent frequency point, and its right adjacent frequency point are as follows: ； ； ； In the formula, For time-domain sampling point index, , For input signal, The index of the first estimated frequency point. This is the frequency offset. The amplitude value at the second estimated frequency point. The amplitude value is the frequency point adjacent to the left of the second estimated frequency point. This represents the amplitude value of the frequency point adjacent to the right of the second estimated frequency point.

7. The frequency estimation method for bounded mapping step non-iterative interpolation according to claim 1, characterized in that, In S3, the formula for calculating the frequency offset is: ； In the formula, This is the frequency offset. The error oscillation period for the first frequency estimation offset value. The sampling frequency.

Images