A frequency offset compensation method based on the amplitude of all-phase FFT

Abstract

The present invention relates to a frequency offset compensation method based on the amplitude of all-phase FFT, comprising the following steps: S1: windowing the signal to be estimated; S2: performing FFT on the windowed signal to obtain the maximum spectral value and the corresponding frequency index; S3: performing all-phase FFT on the signal to be estimated at the frequency index; S4: comparing the spectrum of the FFT and the spectrum of the all-phase FFT to obtain the amplitude estimation value of the signal to be estimated; S5: determining the compensation direction of the frequency compensation value; S6: determining the first compensation value; S7: after shifting the spectrum by the quantization unit of the first compensation value, determining the second compensation value and the third compensation value; S8: determining the frequency compensation value according to the first compensation value, the second compensation value and the third compensation value; S9: obtaining the estimated frequency of the signal to be estimated according to the frequency compensation value.

Description

Technical Field

[0001] The present invention relates to the field of frequency estimation, and more particularly to a frequency offset compensation method based on the amplitude of all-phase FFT. Background Art

[0002] Frequency estimation of signals is a problem that often exists in engineering applications, and accurate estimation of the frequency of signals is required in many scenarios. For example, in a frequency-modulated continuous wave (FMCW) radar ranging system, relevant distance information can be obtained based on the difference frequency signal between the transmitted wave and the reflected wave, that is, the estimated distance is obtained by estimating the frequency of the difference frequency signal. Therefore, the accuracy of the frequency estimation of the difference frequency signal directly affects the accuracy of the estimated distance, and the measurement accuracy of the difference frequency signal frequency directly determines the ranging accuracy.

[0003] Existing high-precision frequency measurement methods mainly fall into the following categories: ratio method, spectrum zooming algorithm based on FFT, and phase difference method. Common spectrum zooming methods include Zoom FFT, CZT, etc., and these methods obtain high-precision results by zooming in on the spectrum. Common ratio methods include rife, interpolation method, etc., and they use several spectral lines near the main spectral line to correct the frequency. Common phase difference methods include changing window length phase difference method, time shift phase correction method, etc. Luo Jia et al. (Frequency measurement method based on double-stage M-Rife algorithm [J]. Electronic Information Warfare Technology, 2020, 35(06): 50-53+108) perform coarse frequency estimation on the in-pulse signal through the M-Rife algorithm, then construct a conjugate signal, calculate the inter-pulse coherent sine wave sequence S, and then perform precise frequency offset estimation on the inter-pulse sequence through the M-Rife algorithm to achieve fast and high-precision frequency estimation. Professor Wang Zhaohua (Phase difference spectrum correction method based on all-phase spectrum analysis [J]. Journal of Electronics & Information Technology, 2008(02): 293-297) proposed all-phase FFT. On this basis, Huang Xiangdong proposed a phase difference spectrum correction method based on all-phase spectrum analysis. Through simple operations, the frequency and phase of the signal can be obtained, and all-phase FFT can also improve the problem of spectrum leakage. Du Haochen (Design of under-sampled frequency estimation system for beat signal of frequency-modulated laser ranging [D]. Harbin University of Science and Technology, 2019) performs FFT processing on beat signals with different sampling rates, uses the energy centroid combined with the CZT algorithm to improve the estimation accuracy of the aliased frequency, and uses the frequency offset value for phase correction and filters out the aliased frequency. According to the relationship between the aliased frequency and the sampling rate, the true frequency of the beat signal is solved, and the frequency of the beat signal can be accurately estimated under low signal-to-noise ratio.

[0004] However, existing methods all obtain high precision by increasing the amount of calculation or reduce the amount of calculation at the expense of precision. Therefore, how to reduce the amount of calculation while obtaining high-precision frequency estimation is an urgent problem to be solved by those skilled in the art. Summary of the Invention

[0005] The object of the present invention is to provide a frequency offset compensation method based on the amplitude of all-phase FFT, which has low computational complexity and high accuracy.

[0006] The present invention provides a frequency offset compensation method based on the amplitude of all-phase FFT, including the following steps:

[0007] S1: Window the signal to be estimated;

[0008] S2: Perform FFT on the windowed signal to obtain a spectral function, and extract the maximum spectral value and the corresponding frequency index from the spectral function;

[0009] S3: Perform all-phase FFT on the signal to be estimated at the frequency index to obtain an all-phase spectral function;

[0010] S4: Compare the spectral function and the all-phase spectral function to obtain an amplitude estimation value of the signal to be estimated;

[0011] S5: Determine the compensation direction of the frequency compensation value;

[0012] S6: Determine the first compensation value;

[0013] S7: After shifting the spectral function by the quantization unit of the first compensation value, determine the second compensation value and the third compensation value;

[0014] S8: Determine the frequency compensation value according to the first compensation value, the second compensation value and the third compensation value;

[0015] S9: Obtain the estimated frequency of the signal to be estimated according to the frequency compensation value.

[0016] Further, the amplitude estimation value of the signal to be estimated satisfies the following relationship:

[0017]

[0018] Wherein, is the amplitude estimation value, , represents the positive number closest to x, m is the frequency index at the maximum spectral amplitude, is the amplitude of the FFT spectrum at m, is the amplitude of the all-phase FFT spectrum at m.

[0019] Further, the compensation direction of the frequency compensation value satisfies the following relationship:

[0020] .

[0021] Further, the first compensation value Satisfy the following relationship:

[0022] , where , N is the number of sampling points of the signal to be estimated, is the amplitude at of the spectrum after FFT.

[0023] Furthermore, the second compensation value satisfies the following relationship:

[0024] , where , .

[0025] Furthermore, the third compensation value satisfies the following relationship:

[0026] , where , .

[0027] Furthermore, the first compensation value, the second compensation value and the third compensation value are all solved by the Newton iteration method.

[0028] Furthermore, the frequency compensation value satisfies the following relationship:

[0029]

[0030] where is the first compensation value, is the second compensation value, is the third compensation value.

[0031] Furthermore, the estimated frequency of the signal to be estimated satisfies the following relationship:

[0032]

[0033] where m is the frequency index corresponding to the maximum spectrum value, is the frequency compensation value, is the compensation direction of frequency compensation, , is the sampling frequency of the signal to be estimated, and N is the number of sampling points.

[0034] The frequency offset compensation method based on the amplitude of all-phase FFT in the present invention first estimates the amplitude of the signal, then uses the amplitudes of the two spectral lines at ±0.5 of the main spectral line to determine the direction of signal compensation, and then uses the spectral line with a smaller amplitude at ±0.5 of the spectral line with the largest amplitude to estimate the first compensation value; then, the frequency spectrum is shifted so that the true frequency is close to the largest spectral line, the amplitudes at ±0.5 of the largest spectral line after the frequency spectrum shift are obtained, and the second and third compensation values are estimated using this amplitude. Finally, the first, second, and third compensation values are combined to calculate the estimated frequency. By combining the first, second, and third compensation values to obtain the frequency compensation value, the present invention can greatly improve the estimation accuracy; the Newton iteration method is used to solve the first, second, and third compensation values, and a suitable starting point is selected based on the characteristics of the transcendental equation, and an accurate solution can be obtained after about five iterations, with a low computational complexity. Description of the Drawings

[0035] Figure 1 FIG. is a flowchart of the frequency offset compensation method based on the amplitude of all-phase FFT according to an embodiment of the present invention;

[0036] Figure 2 FIG. is a system framework diagram of an N-order all-phase FFT. Detailed Embodiments

[0037] The following is a detailed description of the preferred embodiments of the present invention with reference to the accompanying drawings.

[0038] Taking radar ranging as an example, the transmitted signal of the FMCW radar is:

[0039] (1)

[0040] where is the amplitude of the transmitted signal, is the initial frequency, is the slope of the linear frequency modulation, is the radar bandwidth, represents the sweep period, is the initial phase of the transmitted signal, and t is the current time. refers to the complex representation of the transmitted signal of the FMCW radar at the current time t. In the present invention, signals are all represented in complex form.

[0041] The received signal of the FMCW radar is:

[0042] (2)

[0043] where is the amplitude of the received signal, is the initial phase of the received signal, is the time delay of the received signal relative to the transmitted signal, represents the distance between the target and the radar, is the propagation speed of electromagnetic waves.

[0044] Mix the transmitted signal and the received signal, and pass the mixed signal through a low-pass filter. Since has a small value, the term of can be ignored to obtain the beat signal :

[0045] (3)

[0046] The signal 's instantaneous frequency is obtained from the phase:

[0047] (4)

[0048] According to , the distance between the target and the radar can be deduced to be proportional to the instantaneous frequency :

[0049] (5)

[0050] To accurately estimate the frequency of the signal, as Figure 1 shown, the embodiments of the present invention provide a frequency offset compensation method based on the amplitude of the all-phase FFT, including the following steps:

[0051] S1: Window the signal to be estimated;

[0052] In this embodiment, the signal to be estimated is the beat signal of the FMCW radar , which is a single-frequency complex exponential signal. It is sampled at the sampling frequency , and the simplified signal is expressed as:

[0053] (6)

[0054] where , represents the initial phase of the simplified signal, is the sampling frequency, is the sampling point, , and N is the number of sampling points.

[0055] Window the sampled signal:

[0056] (7)

[0057] where N is the number of sampling points, is a window function, which can be a rectangular window, a Hanning window, a Hamming window, etc.

[0058] S2: For the windowed signal perform FFT to obtain a spectral function, and extract the maximum spectral value and its corresponding frequency index from the spectral function;

[0059] First, perform FFT on the windowed signal to obtain a spectral function :

[0060] (8)

[0061] where k is the spectral sampling point, ; is the Fourier transform of the window function.

[0062] If a rectangular window is added, the spectral function obtained after FFT is:

[0063] (9)

[0064] To facilitate the calculation of the signal amplitude, after normalizing the spectral function the normalized spectral function can be obtained:

[0065] (10)

[0066] S3: Perform all-phase FFT (apFFT) on the signal to be estimated at the position of the frequency index to obtain an all-phase spectral function;

[0067] As Figure 2 shown, only need to use a convolution window W with a length of (2N - 1) to weight the central sample point and a total of (2N - 1) data before and after it, with an interval of 1 between adjacent data, then perform overlapping addition on the weighted data with an interval of N to form N data, and then perform FFT with N points to obtain the all-phase spectrum analysis result. The specific principle can be referred to Chinese Patent Application CN1996986A and will not be elaborated here.

[0068] The signal after all-phase preprocessing is:

[0069] (11)

[0070] If a rectangular window is added, the obtained all-phase spectral function (apFFT) is:

[0071] (12)

[0072] After normalizing the spectrum, a normalized all-phase spectrum function is obtained:

[0073] (13)

[0074] S4: Compare the spectrum function obtained in S2 and the all-phase spectrum function obtained in S3 to obtain the amplitude estimation value of the signal to be estimated;

[0075] By comparing the spectrum function obtained in S2 and the all-phase spectrum function obtained in S3, it can be found that the normalized spectra are in a squared relationship, from which the amplitude of the signal to be estimated can be estimated :

[0076] (14)

[0077] Among them, , represents the positive number closest to x, m is the frequency index at the maximum of the spectrum amplitude, is the amplitude of the spectrum function at m, is the amplitude of the all-phase spectrum function at m.

[0078] The FFT spectrum amplitude at m is:

[0079] (15)

[0080] Among them, is the frequency compensation value, .

[0081] S5: Determine the compensation direction of the frequency compensation value;

[0082] In order to determine the compensation direction of the frequency compensation value, the parameter is introduced. Perform a single-point discrete Fourier transform on the signal (i.e., single-point DFT) to find , and use to determine :

[0083] (16) Among them = 1 indicates positive compensation, = -1 indicates negative compensation.

[0084] S6: Obtain the first compensation value;

[0085] Since the spectral line with the largest spectrum amplitude and the spectral line with a relatively large amplitude after zooming are affected by noise, there will be a situation where the frequency compensation value exceeds the domain, resulting in an overly large variance. Therefore, use the spectral line obtained by zooming To estimate the first compensation value .

[0086] (17)

[0087] Substitute the amplitude estimation value into Equation (17) to obtain the relationship between and the amplitude, and after simplification, we can get

[0088] (18)

[0089] When N is relatively large, it can be approximately considered that , then the formula can be further simplified to

[0090] (19)

[0091] where .

[0092] Therefore, solving the first compensation value is to solve Equation (19). Since it is a transcendental equation and the analytical solution cannot be obtained, the Newton iteration method is used to solve this equation

[0093]

[0094] (20)

[0095] Generally, after 5 iterations, an accurate solution can be obtained. The obtained after iteration , so the frequency compensation value needs to be calculated again

[0096] (21)

[0097] Calculation shows that when , the solution is not within our domain, and the solution is 0 at this time. When , use the Newton iteration method to solve iteratively

[0098] S7: After moving the normalized spectrum function by the quantization unit of the first compensation value, determine the second compensation value and the third compensation value

[0099] After analysis, it is found that when is close to 0, the finally estimated frequency value is more accurate. Therefore, after obtaining the estimated value of the frequency compensation value , move the spectrum quantization units to make the estimated frequency close to the line with the largest spectrum amplitude, and then estimate the second compensation value . After moving the spectrum quantization units, the index value of the main spectral line is , and use the spectrum zooming technique to calculate Determine the second compensation value based on the amplitudes of two spectral lines .

[0100] The amplitude of the spectral line with the larger refined spectrum amplitude obtained is:

[0101] (22)

[0102] where the second compensation value is the deviation relative to .

[0103] Simplification gives:

[0104] (23)

[0105] When N is large, it can be approximately considered that , and the formula can be further simplified to:

[0106] (24)

[0107] where .

[0108] Similar to Equation 19, this is a transcendental equation, and the Newton-Raphson method is used to solve this equation:

[0109]

[0110] (25)

[0111] The obtained after iteration , so the second compensation value needs to be calculated again:

[0112] (26)

[0113] To improve the accuracy of the frequency compensation value, use to estimate the third compensation value to estimate a more accurate signal frequency.

[0114] (27)

[0115] where the third compensation value is the deviation relative to .

[0116] Simplification gives:

[0117] (28)

[0118] When N is large, it can be approximately considered that , and the formula can be further simplified to:

[0119] (29)

[0120] Among them, .

[0121] Similar to Equation 19, this is a transcendental equation, and the Newton iteration method is used to solve this equation:

[0122]

[0123] (30)

[0124] The obtained after iteration , so it is necessary to calculate the third compensation value again:

[0125] (31)

[0126] S8: Determine the frequency compensation value according to the first compensation value, the second compensation value and the third compensation value;

[0127] Combine the first compensation value, the second compensation value and the third compensation value to obtain the frequency compensation value :

[0128] (32)

[0129] S9: Obtain the estimated frequency of the signal to be estimated according to the frequency compensation value.

[0130] The estimated frequency of the signal to be estimated is:

[0131] (33)

[0132] Among them , which is the frequency resolution.

[0133] Substitute the estimated frequency into Equation 5 to obtain the distance R.

[0134] It should be noted that although the specific steps of the frequency estimation method of the present invention are described by taking the FMCW radar ranging as an example in this embodiment, it is not limited to being applied to FMCW radar ranging, and can also be extended to other frequency estimation applications, such as power system detection, radar positioning, vibration detection and other fields.

[0135] The frequency offset compensation method based on the amplitude of all-phase FFT provided in the embodiments of the present invention first estimates the amplitude of the signal, then uses the amplitudes of the two spectral lines at ±0.5 of the main spectral line to determine the direction of signal compensation, and then uses the spectral line with a smaller amplitude at ±0.5 of the spectral line with the largest amplitude to estimate the first compensation value; then, the frequency spectrum is shifted so that the true frequency is close to the largest spectral line, and the amplitudes at ±0.5 of the largest spectral line after the frequency spectrum shift are obtained, and these amplitudes are used to estimate the second compensation value and the third compensation value. Finally, the first compensation value, the second compensation value, and the third compensation value are combined to calculate the estimated frequency. By combining the first compensation value, the second compensation value, and the third compensation value to obtain the frequency compensation value, the present invention can greatly improve the estimation accuracy; the Newton iteration method is used to solve the first compensation value, the second compensation value, and the third compensation value, and a suitable starting point is selected based on the characteristics of the transcendental equation, and an accurate solution can be obtained after about five iterations, with a low computational complexity.

[0136] To verify the performance of the present invention, the results of FFT, CZT, zero-padding method, and rife algorithm are compared with the present invention. In the experiment, the sampling frequency , the amplitude of the transmitted signal , the initial frequency , the radar bandwidth , the sweep period , the slope of the linear frequency modulation , the initial phase . Then the signal transmitted by the radar

[0137] is:

[0138] Ignoring the signal attenuation during transmission, assuming the radar is at a distance of from the object, then the time delay . Then the signal received by the radar is:

[0139]

[0140] The transmitted signal and the received signal are subjected to a mixing operation, and the mixed signal is passed through a low-pass filter to obtain a beat signal:

[0141]

[0142] To verify the performance, noise is added to the signal, then the beat signal becomes:

[0143]

[0144] The added noise is Gaussian white noise, and the performance of each algorithm will be compared under the condition that the signal-to-noise ratio is in [-14dB, 14dB]. The set , 。The experiment was simulated through Matlab, and 20,000 independent Monte Carlo simulation experiments were carried out. The experimental results were used to compare the performance by calculating the arithmetic mean and variance.

[0145] The obtained experimental results are shown in Table 1 and Table 2. Among them, the refinement multiple of CZT is 256 times, the data length of the zero-padding method is 4096 points, and the data length of the remaining algorithms is 1024 points. It can be seen from the above table that starting from -4dB, the mean value of the frequency offset compensation method based on the all-phase FFT amplitude has been comprehensively better than the compared algorithms. Starting from -4dB, the variance of the frequency offset compensation method based on the all-phase FFT amplitude has also been comprehensively better than the compared algorithms. The frequency offset compensation method based on the all-phase FFT amplitude requires one 1024-point FFT, a single-point apFFT, and 4 single-point FFTs. When solving the compensation value, it generally ends after 5 iterations, and the overall computational amount is also less than that of CZT. In summary, when the signal-to-noise ratio is higher than -4dB, the variance and mean value of the frequency offset compensation method based on the all-phase FFT amplitude are more excellent than other algorithms.

[0146] Table 1 Estimated frequency mean values (Hz) obtained by simulation under different signal-to-noise ratios

[0147] SNR (dB) FFT Zero-padding method CZT rife The present invention -14 5164.6114 5138.8241 5136.7139 5142.7558 5136.5705 -12 5164.6975 5140.1447 5136.7725 5140.3440 5136.5660 -10 5164.7065 5141.0859 5136.7971 5138.3949 5136.6540 -8 5164.7065 5141.7089 5136.7541 5137.2528 5136.6691 -6 5164.7065 5141.9818 5136.7695 5136.8107 5136.7157 -4 5164.7065 5142.0441 5136.7953 5136.7670 5136.7675 -2 5164.7065 5142.0543 5136.7803 5136.7535 5136.7555 0 5164.7065 5142.0543 5136.7805 5136.7576 5136.7672 2 5164.7065 5142.0543 5136.7718 5136.7614 5136.7642 4 5164.7065 5142.0543 5136.7909 5136.7635 5136.7687 6 5164.7065 5142.0543 5136.8401 5136.7669 5136.7682 8 5164.7065 5142.0543 5136.9174 5136.7631 5136.7639 10 5164.7065 5142.0543 5137.0075 5136.7637 5136.7638 12 5164.7065 5142.0543 5137.0970 5136.7651 5136.7663 14 5164.7065 5142.0543 5137.1577 5136.7676 5136.7679

[0148] Table 2 Estimated frequency variances (Hz) obtained by simulation under different signal-to-noise ratios

[0149] SNR (dB) FFT Zero-padding method CZT rife The present invention -14 3.0713 8.0079 5.6498 19.7561 6.4889 -12 0.9061 6.3060 4.4609 15.3754 4.9979 -10 0.0000 4.5825 3.5548 10.7403 3.8390 -8 0.0000 2.7760 2.8220 6.5390 2.9099 -6 0.0000 1.2794 2.2687 3.4938 2.2792 -4 0.0000 0.0000 1.8286 2.0153 1.7931 -2 0.0000 0.0000 1.4813 1.5591 1.4132 0 0.0000 0.0000 1.2068 1.2295 1.1092 2 0.0000 0.0000 1.0101 0.9827 0.8865 4 0.0000 0.0000 0.8565 0.7792 0.7060 6 0.0000 0.0000 0.7477 0.6217 0.5603 8 0.0000 0.0000 0.6489 0.4867 0.4405 10 0.0000 0.0000 0.5508 0.3917 0.3522 12 0.0000 0.0000 0.4199 0.3087 0.2790 14 0.0000 0.0000 0.2855 0.2475 0.2232

[0150] The above-mentioned are only the preferred embodiments of the present invention, and are not used to limit the scope of the present invention. Various changes can be made to the above embodiments of the present invention. That is, all simple, equivalent changes and modifications made according to the claims and the content of the specification of the present invention application shall fall within the protection scope of the claims of the present invention patent. The content not described in detail in the present invention is all conventional technical content.

Claims

1. A frequency offset compensation method based on the amplitude of all-phase FFT, characterized in that It includes the following steps: S1: Window the signal to be estimated; S2: Perform FFT on the windowed signal to obtain a spectral function, and extract the maximum spectral value and the corresponding frequency index from the spectral function; S3: Perform all-phase FFT on the signal to be estimated at the frequency index to obtain an all-phase spectral function; S4: Compare the spectral function and the all-phase spectral function to obtain an amplitude estimation value of the signal to be estimated; S5: Determine the compensation direction of the frequency compensation value; S6: Determine the first compensation value; S7: After shifting the spectral function by the first compensation value quantization unit, determine the second compensation value and the third compensation value; S8: Determine the frequency compensation value according to the first compensation value, the second compensation value and the third compensation value; S9: Obtain the estimated frequency of the signal to be estimated according to the frequency compensation value; The amplitude estimation value of the signal to be estimated satisfies the following relational expression: ， Among them, is the amplitude estimation value, , represents the positive number closest to x, is the instantaneous frequency of the signal to be estimated, is the sampling frequency of the signal to be estimated, and m is the frequency index at the maximum spectral amplitude, is the amplitude of the spectrum after FFT at m, is the amplitude of the spectrum after all-phase FFT at m; The compensation direction of the frequency compensation value satisfies the following relational expression: ； Wherein, is the compensation direction of the frequency compensation value; The first compensation value satisfies the following relational expression: , where is the first compensation value, , N is the number of sampling points of the signal to be estimated, is the amplitude of the spectrum after FFT at position.

2. The frequency offset compensation method based on the all-phase FFT amplitude according to claim 1, wherein, The second compensation value satisfies the following relational expression: , wherein, is the second compensation value, , .

3. The frequency offset compensation method based on the all-phase FFT amplitude according to claim 1, wherein The third compensation value satisfies the following relational expression: , where, is the third compensation value, , .

4. The frequency offset compensation method based on the all-phase FFT amplitude according to claim 1, wherein The first compensation value, the second compensation value and the third compensation value are all solved by the Newton iteration method.

5. The frequency offset compensation method based on the all-phase FFT amplitude according to claim 1, characterized in that The frequency compensation value Satisfies the following relational expression: ， Among them, is the first compensation value, is the second compensation value, is the third compensation value.

6. The frequency offset compensation method based on the all-phase FFT amplitude according to claim 1, wherein The estimated frequency of the signal to be estimated satisfies the following relational expression: ， wherein, is the estimated frequency of the signal to be estimated, m is the frequency index corresponding to the maximum spectral value, is the frequency compensation value, is the compensation direction of the frequency compensation, , is the sampling frequency of the signal to be estimated, and N is the number of sampling points of the signal to be estimated.

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