Quadratic polynomial frequency modulation signal parameter estimation method

Abstract

The application discloses a quadratic polynomial frequency modulation signal parameter estimation method, and steps include: obtaining a quadratic polynomial frequency modulation signal and performing first differentiation to obtain a first signal, calculating the carrier frequency of the first signal, and then estimating the bandwidth of the quadratic polynomial frequency modulation signal by using the carrier frequency of the first signal to obtain a bandwidth estimation value; performing second differentiation on the first signal to obtain a second signal, calculating the carrier frequency of the second signal, and then estimating the frequency modulation coefficient of the quadratic polynomial frequency modulation signal by using the carrier frequency of the second signal and the bandwidth estimation value to obtain a frequency modulation coefficient estimation value; generating a target signal according to the frequency modulation coefficient estimation value, calculating the carrier frequency of a product signal obtained by multiplying the target signal and the quadratic polynomial frequency modulation signal as a starting frequency estimation value, and calculating a carrier frequency estimation value according to the bandwidth estimation value and the starting frequency estimation value. The application has high estimation performance and low engineering implementation complexity.

Description

Technical Field

[0001] This invention relates to the field of signal processing, and in particular to a method for estimating parameters of a quadratic polynomial frequency modulated signal. Background Technology

[0002] Currently, the signal forms frequently used in the radar field include linear frequency modulated (LFM) signals and some nonlinear frequency modulated (NFM) signals. Among them, nonlinear frequency modulated (NFM) signals have the important advantage of low mismatch loss, and their detection and parameter estimation have a wide range of applications in the field of radar communication, making them a hot topic in current radar communication and sonar signal processing research.

[0003] Polynomial frequency modulation (PPS) signals are a typical type of nonlinear frequency modulation signal, characterized by low intercept and high range resolution, making them widely used in radar, sonar, and other fields. For parameter estimation of quadratic polynomial frequency modulation signals, researchers have proposed many algorithms, which can be mainly divided into linear and nonlinear algorithms. Linear algorithms mainly include maximum likelihood estimation, quantization methods, rational frequency modulation Fourier transform, and parameter estimation algorithms based on linear canonical transforms. Although linear algorithms achieve better estimation performance and are free from the influence of cross terms, they have higher complexity. Nonlinear algorithms, compared to linear algorithms, have lower computational complexity and are widely used. Depending on the processing method, quadratic polynomial frequency modulation signal parameter estimation methods can be broadly categorized into two types: cubic phase function-based and third-order NPD-based. While nonlinear algorithms reduce complexity compared to linear algorithms, their complexity increases proportionally with increasing estimation accuracy. Summary of the Invention

[0004] The technical problem to be solved by this invention is to reduce the algorithm complexity of parameter estimation for quadratic polynomial frequency modulation signals.

[0005] To address the technical problems existing in the prior art, this invention provides a method for estimating parameters of a quadratic polynomial frequency modulated signal, which has high estimation performance and low engineering implementation complexity.

[0006] To solve the above-mentioned technical problems, the technical solution proposed by this invention is as follows:

[0007] A method for estimating parameters of a quadratic polynomial frequency-modulated signal includes the following steps:

[0008] S1) Obtain the quadratic polynomial frequency modulation signal and perform the first differential to obtain the first signal. After obtaining the carrier frequency of the first signal, use the carrier frequency of the first signal to estimate the bandwidth of the quadratic polynomial frequency modulation signal to obtain the bandwidth estimate.

[0009] S2) Perform a second differential on the first signal to obtain the second signal. After obtaining the carrier frequency of the second signal, use the carrier frequency and bandwidth estimate of the second signal to estimate the frequency modulation coefficient of the quadratic polynomial frequency modulation signal, and obtain the frequency modulation coefficient estimate.

[0010] S3) Generate the target signal based on the frequency modulation coefficient estimate, multiply the target signal and the quadratic polynomial frequency modulation signal to obtain the product signal, and obtain the carrier frequency of the product signal as the starting frequency estimate. Calculate the carrier frequency estimate based on the bandwidth estimate and the starting frequency estimate.

[0011] Furthermore, in step S1, the differential distance of the first differential is half the original length of the quadratic polynomial frequency modulated signal, and in step S2, the differential distance of the second differential is half the length of the first signal.

[0012] Furthermore, the expression for the target signal in step S3 is:

[0013]

[0014] In the above formula, e represents the natural constant, j represents the imaginary number, and a e and b e These are the estimated values ​​of frequency modulation coefficients a and b, respectively.

[0015] Furthermore, the first algorithm is used to obtain the carrier frequency of the first signal in step S1, which includes the following steps: multiplying the first and last conjugates of the signal to obtain a single-frequency signal, performing a Fourier transform on the single-frequency signal to obtain the spectrum, and taking half of the frequency at the position of the maximum value of the spectrum as the carrier frequency of the signal.

[0016] The first algorithm is used to obtain the carrier frequency of the second signal in step S2 and the carrier frequency of the product signal in step S3. The algorithm includes the following steps: performing a Fourier transform on the signal to obtain the spectrum, and taking the frequency at the position of the maximum value of the spectrum as the carrier frequency of the signal.

[0017] Furthermore, the second algorithm is used to obtain the carrier frequency of the first signal in step S1, which includes the following steps: multiplying the first and last conjugates of the signal to obtain a single-frequency signal, performing differential transformation on the single-frequency signal and then performing inverse trigonometric function transformation to obtain the instantaneous frequency curve of the signal after inverse trigonometric function transformation, and then taking the statistical average of the instantaneous frequency curve and dividing it by 2 to obtain the carrier frequency of the signal.

[0018] The second algorithm is used to obtain the carrier frequency of the second signal in step S2 and the carrier frequency of the product signal in step S3. The algorithm includes the following steps: performing differential transformation on the signal and then performing inverse trigonometric function transformation to obtain the instantaneous frequency curve of the signal after inverse trigonometric function transformation, and then calculating the statistical average of the instantaneous frequency curve to obtain the carrier frequency of the signal.

[0019] Furthermore, after step S3, the following steps are also included: calculating the signal-to-noise ratio (SNR) estimate of the quadratic polynomial frequency modulated signal based on the bandwidth estimate and the starting frequency estimate corresponding to the first algorithm; if the SNR estimate is less than a preset value, outputting the bandwidth estimate, frequency modulation coefficient estimate, starting frequency estimate, and carrier frequency estimate corresponding to the first algorithm; if the SNR estimate is greater than the preset value, outputting the bandwidth estimate, frequency modulation coefficient estimate, starting frequency estimate, and carrier frequency estimate corresponding to the second algorithm.

[0020] Furthermore, calculating the signal-to-noise ratio estimate of the quadratic polynomial frequency-modulated signal based on the bandwidth estimate and the initial frequency estimate corresponding to the first algorithm includes the following steps:

[0021] Calculate the spectrum and amplitude spectrum of the quadratic polynomial frequency modulated signal, and sum the entire amplitude spectrum to obtain the total power of the signal and noise;

[0022] Based on the estimated starting frequency and bandwidth of the quadratic polynomial frequency modulated signal, the starting position and length of the signal in the spectrum are calculated.

[0023] The signal power is obtained by accumulating the corresponding amplitudes of the spectrum based on the starting position and length of the signal in the spectrum.

[0024] The noise power is obtained by subtracting the signal power from the total power. The ratio of signal power to noise power is then calculated to obtain the signal-to-noise ratio estimate.

[0025] Furthermore, the expression for the bandwidth estimate in step S1 is as follows:

[0026]

[0027] In the above formula, f c denoted as the carrier frequency of the first signal, N is the original length of the quadratic polynomial frequency-modulated signal, and m is the differential distance of the first differential.

[0028] Furthermore, the expression for the frequency modulation coefficient estimate in step S2 is as follows:

[0029]

[0030]

[0031] In the above formula, a and b are the frequency modulation coefficients, f1 is the carrier frequency of the second signal, Ts is the sampling period, N is the original length of the quadratic polynomial frequency-modulated signal, Bw is the bandwidth estimate, and t m t represents the difference distance duration of the first difference. m1 This represents the difference distance duration of the second difference.

[0032] Furthermore, the carrier frequency estimate expression in step S3 is as follows:

[0033]

[0034] In the above formula, f0 is the estimated starting frequency and Bw is the estimated bandwidth.

[0035] Compared with the prior art, the advantages of the present invention are as follows:

[0036] This invention calculates the single-carrier carrier frequencies of the first and second signals by performing first and second differential operations on the quadratic polynomial frequency-modulated signal, and then estimates the parameters of each signal accordingly. The algorithm logic is simple and easy to implement in engineering. Two methods are used for the specific calculation of the signal carrier frequency. Finally, the signal-to-noise ratio (SNR) of the quadratic polynomial frequency-modulated signal is estimated, and the parameter estimation results under the corresponding method are selected according to the estimated SNR. This enables dynamic adjustment of the algorithm, which is highly flexible. Furthermore, the parameter estimation performance does not drop sharply with the SNR at low levels, making its application range wider. Attached Figure Description

[0037] Figure 1 This is the spectrum obtained by the Fourier transform method for a single-frequency signal.

[0038] Figure 2 This is a simplified schematic diagram of the method steps in an embodiment of the present invention.

[0039] Figure 3 This is a simplified schematic diagram of step S1 in an embodiment of the present invention.

[0040] Figure 4 This is a simplified schematic diagram of step S2 in an embodiment of the present invention.

[0041] Figure 5 This is a simplified schematic diagram of step S3 in an embodiment of the present invention.

[0042] Figure 6 The graph shows the estimation performance curves of single-frequency signals with different signal-to-noise ratios for different lengths using the Fourier transform method and the instantaneous frequency statistics method.

[0043] Figure 7 The graph shows the error curves of the parameter estimation results of the quadratic polynomial frequency modulation signal using the Fourier transform method and the instantaneous frequency statistics method.

[0044] Figure 8 This is a flowchart illustrating step S4 in an embodiment of the present invention.

[0045] Figure 9 The graph shows the error curves for signal-to-noise ratio estimation using windowed Fourier transform and unwindowed Fourier transform.

[0046] Figure 10 This is a detailed flowchart of the method according to an embodiment of the present invention.

[0047] Figure 11 The graph shows the error curves between the parameter estimation results of the adaptive selection of signal-to-noise ratio in this embodiment of the invention and the parameter estimation results of the Fourier transform method and the instantaneous frequency statistical method.

[0048] Figure 12 This is a normalized error curve diagram of different parameters of the quadratic polynomial frequency modulation in an embodiment of the present invention. Detailed Implementation

[0049] The present invention will be further described below with reference to the accompanying drawings and specific preferred embodiments, but this does not limit the scope of protection of the present invention.

[0050] Before introducing the specific implementation of this solution, the relevant calculation process is described as follows:

[0051] The center frequency of a single-frequency signal is also called the carrier frequency. Estimation of the center frequency, i.e., the carrier frequency, includes the Fourier transform method and the instantaneous frequency statistical method.

[0052] 1. Fourier Transform Method

[0053] Let the expression for a single-tone signal be:

[0054]

[0055] Where f c Let be the carrier frequency, i.e., the center frequency, of the signal, 0 ≤ t ≤ T, where T is the duration of the signal. Performing a Fourier transform on this signal reveals a single-peak characteristic in its spectrum. Figure 1 f c The spectrum of a signal with a frequency of 100MHz and a frequency of Fs = 4800MHz is obtained from... Figure 1 It is known that the center frequency of a single-frequency signal can be directly obtained by performing a Fourier transform on it, and then finding the location of the maximum spectral value, which is the center frequency of the signal. However, the spectral resolution of frequency determination using Fourier transform is affected by the number of Fourier transform points, and its resolution is... In other words, the longer the number of points in the Fourier transform, the lower the resolution, but the more accurate the estimated signal frequency. The more points there are, the higher the computational complexity, requiring more hardware resources, and the number of Fourier transform points is related to the actual signal length.

[0056] 2. Instantaneous frequency statistical method

[0057] By differentiating the single-tone signal in equation (1), we can obtain

[0058]

[0059] Based on the result of the inverse trigonometric function transformation of the differential signal, the instantaneous frequency curve of the single-tone signal can be obtained. Then, the statistical average of the obtained instantaneous frequency curve is taken, which is expressed as the frequency of the single-tone signal.

[0060] like Figure 2 As shown in the figure, this embodiment proposes a method for estimating parameters of a quadratic polynomial frequency-modulated signal, including the following steps:

[0061] S1) Obtain the quadratic polynomial frequency modulation signal and perform the first differential to obtain the first signal. After obtaining the carrier frequency of the first signal, use the carrier frequency of the first signal to estimate the bandwidth of the quadratic polynomial frequency modulation signal to obtain the bandwidth estimate.

[0062] S2) Perform a second differential on the first signal to obtain the second signal. After obtaining the carrier frequency of the second signal, use the carrier frequency and bandwidth estimate of the second signal to estimate the frequency modulation coefficient of the quadratic polynomial frequency modulation signal, and obtain the frequency modulation coefficient estimate.

[0063] S3) Generate the target signal based on the frequency modulation coefficient estimate, multiply the target signal and the quadratic polynomial frequency modulation signal to obtain the product signal, and obtain the carrier frequency of the product signal as the starting frequency estimate. Calculate the carrier frequency estimate based on the bandwidth estimate and the starting frequency estimate.

[0064] In step S1 of this embodiment, the derivation process of differential conversion for the quadratic polynomial frequency modulation signal is as follows:

[0065] The expression for a quadratic polynomial frequency-modulated signal is:

[0066]

[0067] Where a and b are frequency modulation coefficients, which are constants. Its instantaneous frequency is f = 3at. 2 +2bt+f0, where f0 is the signal's starting frequency. The signal bandwidth can be expressed as the difference between the signal's ending frequency and its starting frequency, which can be represented as:

[0068] Bw=(3aT 2 +2bT) (4)

[0069] Where T is the duration of the signal.

[0070] The result of taking the difference of the signal can be expressed as:

[0071]

[0072] Where t m Let m be the differential distance duration, and m be the differential distance. Since m, f0, and b are all constants, then... Also a constant, let and The initial frequency is f0 = 2t m b+3at m 2 The frequency modulation slope is 6 at m A linear frequency modulated signal.

[0073] In step S1 of this embodiment, after obtaining the first signal, the carrier frequency of the first signal is calculated, and the bandwidth of the quadratic polynomial frequency modulated signal is estimated using the carrier frequency of the first signal. The process of obtaining the bandwidth estimate is as follows: Figure 3 As shown, the derivation process is as follows:

[0074] set up The instantaneous frequency of the signal can then be obtained as follows:

[0075] f = t m (6at+2b+3at m (6)

[0076] Let the time length of the differential signal in equation (5) be T1, and T1+t m =T, where T is the original signal duration, t m Let the differential distance time be , then the center frequency of the differential signal can be expressed as .

[0077]

[0078] Furthermore, since the bandwidth of the quadratic polynomial frequency modulation signal is B = 3aT 2 +2bT, according to the above formula, can be rewritten as:

[0079]

[0080] That is, by obtaining the center frequency f of the differential signal in equation (5) c Then the bandwidth of the quadratic polynomial frequency modulation signal can be obtained.

[0081] The above derivation also holds for discrete signals. Assume the sampling rate of the signal is Fs, i.e., the sampling period of the signal is Ts. The total number of sampling points of the signal is N = T·Fs; the length of the differential signal is N1 = T1·Fs; and the differential distance is m = t m The instantaneous frequency of a discrete differential signal can be expressed as Fs.

[0082]

[0083] The bandwidth of a discrete signal is expressed as

[0084]

[0085] That is, the bandwidth of the quadratic polynomial signal is the center frequency f of the differential signal in equation (5). cMultiply by the ratio of the length of the original signal to the length of the differential distance signal.

[0086] The signal obtained by differentiating a quadratic polynomial signal is a linear frequency modulated signal, that is, assuming the expression of the differentiated signal is:

[0087]

[0088] Multiplying the first and last conjugates of the signal yields:

[0089]

[0090] From the above equation, it can be seen that the signal after differential processing in equation (12) is a single-frequency signal with a frequency of 2f0+kT, while the center frequency of the linear frequency modulated signal in equation (11) is...

[0091]

[0092] That is, the center frequency of the single-frequency signal obtained by multiplying the first and last conjugates of the linear frequency modulated (LFM) signal is twice the center frequency of the LFM signal. The single spectral line of the LFM signal obtained by multiplying the first and last conjugates can be obtained using the Fourier transform method or the instantaneous frequency statistics method mentioned earlier, and this can be used to estimate the center frequency f of the LFM signal. c .

[0093] The specific implementation process of step S2 in this embodiment is as follows: Figure 4 As shown, the derivation process is as follows:

[0094] According to equation (5) above, since the result of differential modulation of the polynomial frequency modulation signal is Where B is a constant, The initial frequency is f0 = 2t m b+3at m 2 The frequency modulation slope is 6 at m The linear frequency modulated signal. By taking the difference again with respect to y(t), we can obtain

[0095]

[0096] Among them, t m1 The differential distance is the duration of the differential signal. From the above formula, we know that the differential signal has a frequency of -6 at. m t m1 Since it is a single-frequency signal, the differential signal y(t)y*(t+t) can be estimated using the Fourier transform method or the instantaneous frequency statistics method mentioned earlier. m1 The frequency corresponding to the single spectral line is taken as the center frequency f1 of the signal, thus:

[0097]

[0098] The estimated value of the frequency modulation coefficient 'a' can then be obtained.

[0099] Since Bw = (3aN·Ts + 2b)N·Ts in equation (10), the frequency modulation coefficient b can be expressed as:

[0100]

[0101] Where f1 is the differential signal y(t)·y*(t+t) m1 The carrier frequency is the center frequency, Ts is the sampling period, N is the original length of the quadratic polynomial frequency modulated signal, Bw is the bandwidth estimate, and t m t represents the difference distance duration of the first difference. m1 This represents the difference distance duration of the second difference.

[0102] The specific implementation process of step S3 in this embodiment is as follows: Figure 5 As shown, the derivation process is as follows:

[0103] Based on equation (3) above, the expression for the quadratic polynomial frequency modulation signal is... The frequency modulation coefficient a is estimated from step S2. e and b e Then, a quadratic polynomial frequency-modulated signal y1(t) is generated as the target signal, with the following:

[0104]

[0105] Then there is

[0106]

[0107] That is, when a=a e b = b e When Equation (18) is a single-frequency signal with a frequency of f0, the frequency of the single-frequency signal corresponding to the single spectral line in Equation (18) can be estimated by the Fourier transform method or the instantaneous frequency statistics method mentioned above, and used as the estimated starting frequency value of the quadratic polynomial frequency modulation signal.

[0108] Using the bandwidth estimate of the quadratic polynomial frequency-modulated signal obtained in step S1, and the starting frequency estimate mentioned above, the center frequency, i.e., the carrier frequency estimate, of the quadratic polynomial frequency-modulated signal can be calculated, as shown in the following expression:

[0109]

[0110] In the above formula, f0 is the estimated starting frequency and Bw is the estimated bandwidth.

[0111] In this embodiment, the parameter estimation of the quadratic polynomial frequency-modulated signal depends on the accuracy of estimating the carrier frequency (center frequency) of the differentially differentiated linear frequency-modulated signal or single-frequency signal. The accuracy of the center frequency estimation is related to the number of Fourier transform points and the statistical length. With a fixed sampling rate, the longer the number of Fourier transform points (statistical length), the higher the frequency resolution and the higher the measurement accuracy. Furthermore, since the differential data length and differential distance satisfy N1 + m = N, meaning that a higher frequency resolution requires a larger differential signal length N1, and with the total signal length N remaining constant, a smaller differential distance m results in a higher spectral resolution for the differentially differentiated signal.

[0112] Let the carrier frequency estimation error be Δf, that is, the bandwidth can be expressed as:

[0113]

[0114] Where f ce Let m be the true value of the carrier frequency, and Δf be the estimation error of the carrier frequency. From the above formula, it can be seen that the larger the differential distance m, the larger the estimated bandwidth B. e The smaller the error, the smaller the error value. Therefore, it is necessary to find a suitable differential distance that meets the frequency resolution requirements while minimizing the measurement error. In this embodiment, the differential distance is set to m = N / 2. That is, the differential distance of the first differential in step S1 is half the original length of the quadratic polynomial frequency modulated signal, and the differential distance of the second differential in step S2 is half the length of the first signal.

[0115] In practical applications, due to estimation errors, the estimated value of the frequency modulation coefficient is not entirely consistent with the actual value. In step S3, assume aa e =d a bb e =d b Then there is

[0116]

[0117] From equation (4), we can see that the bandwidth of the signal in the above equation is Bw1 = (3d a T+2d b Given that the original signal bandwidth is Bw = (3aT + 2b)T, comparing the two equations yields...

[0118]

[0119] Assumption The combined error of the frequency modulation coefficients a and b is 1%, corresponding to Bw1 = 0.01Bw. This means the original signal bandwidth is 100MHz, and with a combined error of 1%, the bandwidth of s(t)·y(t) is 1MHz, resulting in an average error of 0.5MHz. Therefore, with a combined error of 1% for the polynomial frequency modulation coefficients, the signal start frequency measurement error is approximately 0.5% of the signal bandwidth. This demonstrates that when the estimation errors of the frequency modulation coefficients a and b are small, the above method can accurately estimate the start frequency of a quadratic polynomial frequency-modulated signal, with an error that is half the combined error of the frequency modulation coefficients a and b.

[0120] Within the range of single-frequency signal-to-noise ratio (SNR) from 5 to 25 dB, we statistically analyzed the normalized estimation error performance of the Fourier transform method and the instantaneous frequency statistical method for different Fourier transform lengths and different statistical lengths. The statistical results are as follows: Figure 6 As shown, by Figure 6 It is known that when estimating carrier frequencies using the Fourier transform method, the estimation error decreases as the Fourier transform length increases. However, its estimation accuracy does not improve with the increase of the signal-to-noise ratio (SNR). This is because increasing the SNR does not improve the spectral resolution of the signal after the Fourier transform. As long as the SNR can ensure that the spectrum has a single-peak characteristic, its error will only be related to the Fourier transform length. In contrast, the instantaneous frequency statistical method reduces its estimation error with the increase of the number of statistical points, and its accuracy can be improved with the increase of the SNR. When the SNR is greater than 12dB, the performance of the instantaneous frequency statistical method with different statistical lengths is better than that of the Fourier transform method. When the SNR is low, the performance of the Fourier transform method is generally better than that of the instantaneous frequency statistical method.

[0121] We employed both the Fourier transform method and the instantaneous frequency statistical method, performing steps S1 to S3 under different signal-to-noise ratios to estimate the parameters of the quadratic polynomial frequency-modulated signal and calculate its normalized error. The results are as follows. Figure 7 As shown, from Figure 7 It can be seen that at low signal-to-noise ratios (SNR < 12dB), the error in bandwidth and carrier frequency estimation using the Fourier transform method is smaller than that of the instantaneous frequency statistical method, and its normalization error is less than 10⁻², meeting the requirements of practical applications. When the SNR is greater than 12dB, the error in the quadratic polynomial parameters estimated using the frequency statistical method is smaller than that of the Fourier transform method. When the SNR > 15dB, its normalization error is less than 10⁻³.

[0122] Based on this characteristic, we consider dynamically outputting the estimation results according to the estimated signal-to-noise ratio. Specifically, the carrier frequency of the first signal in step S1, the carrier frequency of the second signal in step S2, and the carrier frequency of the product signal in step S3 are estimated once each using the Fourier transform method and the instantaneous frequency statistics method. After executing steps S1 to S3, we obtain the parameter estimates corresponding to the Fourier transform method and the parameter estimates corresponding to the instantaneous frequency statistics method. Then, as... Figure 8 As shown, the signal-to-noise ratio (SNR) estimate is calculated based on the parameter estimates corresponding to the Fourier transform method. Depending on the magnitude of the SNR estimate, the output parameter estimates corresponding to either the Fourier transform method or the instantaneous frequency statistics method are dynamically selected.

[0123] Therefore, in this embodiment, step S3 is followed by step S4: calculating the signal-to-noise ratio (SNR) estimate of the quadratic polynomial frequency modulated signal based on the bandwidth estimate and the starting frequency estimate corresponding to the first algorithm (i.e., Fourier transform method). If the SNR estimate is less than a preset value, outputting the bandwidth estimate, frequency modulation coefficient estimate, starting frequency estimate, and carrier frequency estimate corresponding to the first algorithm. If the SNR estimate is greater than the preset value, outputting the bandwidth estimate, frequency modulation coefficient estimate, starting frequency estimate, and carrier frequency estimate corresponding to the second algorithm (i.e., instantaneous frequency statistics method).

[0124] Specifically, calculating the signal-to-noise ratio estimate of a quadratic polynomial frequency-modulated signal based on the bandwidth estimate and initial frequency estimate corresponding to the Fourier transform method includes the following steps:

[0125] S41) Use Fourier transform to obtain the spectrum and amplitude spectrum of the quadratic polynomial frequency modulated signal, and sum the entire amplitude spectrum to obtain the total power of the signal and noise;

[0126] S42) Based on the estimated starting frequency and bandwidth of the quadratic polynomial frequency modulated signal, calculate the starting position and length of the signal in the spectrum. Specifically, calculate the starting position of the signal in the spectrum based on the estimated starting frequency and the spectrum length, and calculate the length of the signal in the spectrum based on the estimated bandwidth and the spectrum length.

[0127] S43) Based on the starting position and length of the signal in the spectrum, the corresponding amplitudes of the spectrum are accumulated to obtain the signal power;

[0128] S44) Subtract the signal power from the total power to obtain the noise power, calculate the ratio of signal power to noise power, and obtain the signal-to-noise ratio estimate.

[0129] In this embodiment, the preset value is 10.5dB. Due to spectral leakage in Fourier transform, signal power leaks out of the band at high signal-to-noise ratios, thereby reducing estimation accuracy. Figure 9The signal-to-noise ratio estimation performance of performing Fourier transform with and without windowing was compared. Figure 9 It can be seen that when the actual signal-to-noise ratio is less than 13dB, the estimation error is less than 0.5dB regardless of whether windowing is applied. In this method, only a signal-to-noise ratio between 10-11dB is used for algorithm selection and judgment. Therefore, although the estimation error is large at high signal-to-noise ratios without windowing, it has little impact on the performance of this method.

[0130] Based on the foregoing content, the detailed process of this embodiment is as follows: Figure 10 As shown, the main steps include:

[0131] Step 1: After obtaining the quadratic polynomial frequency modulation signal, calculate the differential distance length based on the original signal length, and use this to differentially divide the original signal to obtain the linear frequency modulation signal (i.e., the first signal);

[0132] Step 2: Multiply the first and last conjugates of the linear frequency modulated signal to obtain a single-frequency signal. Then, calculate the carrier frequency of the single-frequency signal using both the Fourier transform method and the instantaneous frequency statistics method. This yields the estimated carrier frequency f of the linear frequency modulated signal for the two algorithms. c ;

[0133] Step 3: Estimate the carrier frequency f of the linear frequency modulated signal corresponding to the two algorithms. c Substituting into equation (10) respectively, the signal bandwidth estimated by the Fourier transform method and the instantaneous frequency statistics method is calculated;

[0134] Step 4: Calculate the differential distance length of the second differential based on the length of the linear frequency modulated signal, and use this distance to perform a second differential on the linear frequency modulated signal to obtain the second signal (the second signal is the previously mentioned frequency of -6at). m t m1 (single-frequency signal);

[0135] Step 5: Calculate the estimated carrier frequency of the second signal using the Fourier transform method and the instantaneous frequency statistical method respectively. At the same time, based on the estimated carrier frequency of the second signal and the signal bandwidth estimated by the Fourier transform method and the instantaneous frequency statistical method, calculate the two different frequency modulation coefficients estimated by the Fourier transform method and the instantaneous frequency statistical method respectively according to equations (15) and (16).

[0136] Step 6: Based on the two sets of different frequency modulation coefficient estimates, generate two sets of quadratic polynomial frequency modulation signals corresponding to the Fourier transform method and the instantaneous frequency statistics method respectively as target signals according to equation (17);

[0137] Step 7: Multiply the generated two sets of target signals with the original signals respectively to obtain the product signal (i.e., the single-frequency signal mentioned in formula (18) above). For the product signal corresponding to the Fourier transform method, use the Fourier transform method to estimate the carrier frequency as the starting frequency estimate. For the product signal corresponding to the instantaneous frequency statistics method, use the instantaneous frequency statistics method to estimate the carrier frequency as the starting frequency estimate f0.

[0138] Step 8: Based on the bandwidth estimates obtained by the Fourier transform method and the instantaneous frequency statistical method, and the initial frequency estimate f0, estimate the carrier frequency f of the quadratic polynomial frequency-modulated signal corresponding to the Fourier transform method and the instantaneous frequency statistical method, respectively. c0 ;

[0139] Step 9: Estimate the signal-to-noise ratio of the quadratic polynomial frequency modulated signal based on the bandwidth estimated by the Fourier transform method and the signal start frequency;

[0140] Step 10: Based on the estimated signal-to-noise ratio, select the parameter estimation result using either the Fourier transform method or the instantaneous frequency statistical method.

[0141] Steps 1 to 3 above are the specific steps of step S1 mentioned earlier; steps 4 to 5 are the specific steps of step S2 mentioned earlier; steps 6 to 8 are the specific steps of step S3 mentioned earlier; and steps 9 to 10 are the specific steps of step S4 mentioned earlier. The signal-to-noise ratio estimation steps in step 9 are as follows:

[0142] Step 9.1: Use Fourier transform to obtain the amplitude spectrum and frequency spectrum of the quadratic polynomial frequency modulated signal, and sum the entire amplitude spectrum to express the total power of the signal and noise;

[0143] Step 9.2: Calculate the starting position of the quadratic polynomial frequency modulated signal in the spectrum based on the estimated starting frequency and the spectral length of the quadratic polynomial frequency modulated signal;

[0144] Step 9.3: Calculate the length of the quadratic polynomial frequency modulated signal in the spectrum based on the bandwidth estimate and the spectral length of the quadratic polynomial frequency modulated signal;

[0145] Step 9.4: Based on the calculated starting position and length, accumulate the amplitude of the quadratic polynomial frequency modulated signal spectrum to represent the signal power;

[0146] Step 9.5: Subtract the signal power from the total spectrum power to obtain the noise power. Compare the signal power with the noise power to obtain an estimate of the signal-to-noise ratio.

[0147] based on Figure 10 The error curve obtained by simulating the process is compared with the parameter estimation obtained by using only the Fourier transform method and the instantaneous frequency statistical method, as shown in the figure below. Figure 11As shown, adaptively selecting the parameter estimates for the appropriate algorithm based on the signal-to-noise ratio (SNR) yields optimal performance. Specifically, its performance is consistent with the Fourier transform method at low SNR and with the instantaneous frequency statistical method at high SNR. If the application scenario remains constant (low or high SNR), a fixed Fourier transform method and instantaneous frequency statistical method can be selected to simplify the computation and reduce the algorithm's complexity. However, in environments with significant SNR variations, a more suitable approach is to use... Figure 10 By using a process that adaptively selects the signal-to-noise ratio, the estimation performance of the system can be guaranteed.

[0148] Figure 12 Based on Figure 10 The normalized error curves of different parameters of the quadratic polynomial frequency modulation estimated by the process show that the estimation errors of the quadratic coefficients a and b are consistent, between 10⁻² and 10⁻³. The bandwidth estimation has the best estimation performance, with a maximum normalized error of only 10⁻³ and a minimum of 10⁻⁴. The carrier frequency estimation is between the quadratic polynomial coefficients and the bandwidth estimation. This is because the carrier frequency estimation is obtained by further estimating the polynomial coefficients and the bandwidth estimation values.

[0149] Through the theoretical derivation and simulation analysis described above, it can be seen that the quadratic polynomial frequency modulation signal parameter estimation method in this embodiment can selectively output the optimal parameter estimates for different scenarios, exhibiting high flexibility and high estimation performance, thus meeting the needs of engineering applications. Furthermore, its principle is simple, and its engineering implementation complexity is low.

[0150] The above description is merely a preferred embodiment of the present invention and is not intended to limit the invention in any way. Although the present invention has been disclosed above with reference to preferred embodiments, it is not intended to limit the invention. Therefore, any simple modifications, equivalent changes, and alterations made to the above embodiments based on the technical essence of the present invention without departing from the scope of the present invention should fall within the protection scope of the present invention.

Claims

1. A method of quadratic polynomial frequency modulated signal parameter estimation, characterized by, The method comprises the following steps: S1) obtaining a quadratic polynomial frequency modulation signal and performing first differentiation to obtain a first signal, calculating the carrier frequency of the first signal, and then estimating the bandwidth of the quadratic polynomial frequency modulation signal using the carrier frequency of the first signal to obtain a bandwidth estimation value; S2) performing second differentiation on the first signal to obtain a second signal, calculating the carrier frequency of the second signal, and then estimating the frequency modulation coefficient of the quadratic polynomial frequency modulation signal using the carrier frequency of the second signal and the bandwidth estimation value to obtain a frequency modulation coefficient estimation value; S3) generating a target signal according to the frequency modulation coefficient estimation value, multiplying the target signal and the quadratic polynomial frequency modulation signal to obtain a product signal, calculating the carrier frequency of the product signal as a starting frequency estimation value, and calculating a carrier frequency estimation value according to the bandwidth estimation value and the starting frequency estimation value.

2. The method of claim 1, wherein, The differentiation distance of the first differentiation in step S1 is half of the original length of the quadratic polynomial frequency modulation signal, and the differentiation distance of the second differentiation in step S2 is half of the length of the first signal.

3. The method of claim 1, wherein, The expression of the target signal in step S3 is: In the above formula, e represents a natural constant, j represents an imaginary number, a e and b e are estimated values of the frequency modulation coefficient a and the frequency modulation coefficient b, respectively.

4. The method of claim 1, wherein, The first algorithm is used to calculate the carrier frequency of the first signal in step S1, which comprises the following steps: multiplying the signal at the beginning and the end to obtain a single-frequency signal, performing Fourier transform on the single-frequency signal to obtain a frequency spectrum, and taking half of the frequency at the position of the maximum value of the frequency spectrum as the carrier frequency of the signal; The first algorithm is used to calculate the carrier frequency of the second signal in step S2 and the carrier frequency of the product signal in step S3, which comprises the following steps: performing Fourier transform on the signal to obtain a frequency spectrum, and taking the frequency at the position of the maximum value of the frequency spectrum as the carrier frequency of the signal.

5. The method of claim 4, wherein, The second algorithm is used to calculate the carrier frequency of the first signal in step S1, which comprises the following steps: multiplying the signal at the beginning and the end to obtain a single-frequency signal, performing differentiation on the single-frequency signal, and then performing inverse trigonometric function transformation, obtaining the instantaneous frequency curve of the signal after inverse trigonometric function transformation, and then calculating the statistical average of the instantaneous frequency curve and dividing by 2 to obtain the carrier frequency of the signal; The second algorithm is used to calculate the carrier frequency of the second signal in step S2 and the carrier frequency of the product signal in step S3, which comprises the following steps: performing differentiation on the signal, and then performing inverse trigonometric function transformation, obtaining the instantaneous frequency curve of the signal after inverse trigonometric function transformation, and then calculating the statistical average of the instantaneous frequency curve to obtain the carrier frequency of the signal.

6. The method of claim 5, wherein the step of estimating the parameters of the quadratic polynomial frequency modulated signal is characterized by, The following steps are further included after step S3: calculating the signal-to-noise ratio estimation value of the quadratic polynomial frequency modulation signal according to the bandwidth estimation value and the starting frequency estimation value corresponding to the first algorithm, if the signal-to-noise ratio estimation value is less than a preset value, outputting the bandwidth estimation value, the frequency modulation coefficient estimation value, the starting frequency estimation value and the carrier frequency estimation value corresponding to the first algorithm, and if the signal-to-noise ratio estimation value is greater than the preset value, outputting the bandwidth estimation value, the frequency modulation coefficient estimation value, the starting frequency estimation value and the carrier frequency estimation value corresponding to the second algorithm.

7. The method of claim 6, wherein the step of estimating the parameters of the quadratic polynomial frequency modulated signal is characterized by, The signal-to-noise ratio estimation value of the quadratic polynomial frequency modulation signal is calculated according to the bandwidth estimation value and the starting frequency estimation value corresponding to the first algorithm, which comprises the following steps: calculating the frequency spectrum and amplitude spectrum of the quadratic polynomial frequency modulation signal, and accumulating the entire amplitude spectrum to obtain the total power of the signal and noise; calculating the starting position and length of the signal in the frequency spectrum according to the starting frequency estimation value and the bandwidth estimation value of the quadratic polynomial frequency modulation signal; According to the initial position and length of the signal in the frequency spectrum, the corresponding amplitudes of the frequency spectrum are accumulated to obtain a signal power; The total power is subtracted from the signal power to obtain a noise power, and a signal-to-noise ratio estimation value is calculated by calculating the ratio of the signal power to the noise power.

8. The method of claim 1, wherein, The bandwidth estimation value expression in step S1 is as follows: In the above formula, f c is the carrier frequency of the first signal, N is the original length of the quadratic polynomial frequency modulation signal, and m is the differential distance of the first difference.

9. The method of claim 1, wherein, The frequency modulation coefficient estimation value expression in step S2 is as follows: In the above formula, a and b are frequency modulation coefficients, f1 is the carrier frequency of the second signal, Ts is the sampling period, N is the original length of the quadratic polynomial frequency modulation signal, Bw is the bandwidth estimation value, t m is the differential distance duration of the first difference m1 is the differential distance duration of the second difference.

10. The method of claim 1, wherein, The carrier frequency estimation value expression in step S3 is as follows: In the above formula, f0 is the initial frequency estimation value, and Bw is the bandwidth estimation value.

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