Broadband signal spectrum analysis method based on spectrum leakage suppression and improved FFT algorithm

Abstract

A broadband signal spectrum analysis method based on spectrum leakage suppression and an improved FFT algorithm comprises the following steps: setting a broadband signal model comprising a fundamental wave frequency offset signal and a plurality of broadband oscillation signals according to broadband signal characteristics; based on a Fourier transform theory, analyzing a spectrum leakage generation mechanism of the broadband signal; a Nuttall-Kaiser combination window function is designed, and is used for suppressing the spectrum leakage of the broadband signal; the FFT algorithm is improved, and side lobe attenuation is suppressed by using the convolution window property so as to reduce the influence of spectrum leakage. According to the broadband signal spectrum analysis method based on spectrum leakage suppression and the improved FFT algorithm, spectrum leakage is effectively reduced while high frequency resolution of broadband signals is guaranteed, and meanwhile the method has the advantage of being high in measurement accuracy.

Description

Technical Field

[0001] The present invention relates to the technical field of spectrum analysis, and particularly relates to a broadband signal spectrum analysis method based on spectrum leakage suppression and improved FFT algorithm. Background Art

[0002] With the rapid development and wide access of high-proportion renewable energy and high-proportion power electronic devices, the power system presents the characteristics of "double highs", namely high-proportion renewable energy and high-proportion power electronic devices. This characteristic has led to significant changes in the dynamic characteristics and operation modes of the power system, and thus triggered a series of new broadband oscillation instability phenomena. Different from traditional power system oscillations, the frequency distribution of broadband oscillations is between several hertz and several kilohertz, and is accompanied by severe noise interference, posing a great threat to the safe and stable operation of the power system.

[0003] In engineering applications, the Fast Fourier Transform (FFT) is a commonly used spectrum analysis method, which can efficiently transform a signal from the time domain to the frequency domain, thereby helping to analyze the frequency characteristics of the signal. However, the Fast Fourier Transform FFT has a significant drawback - spectrum leakage. Spectrum leakage refers to the fact that the energy of the signal fails to be concentrated at an exact frequency, but spreads to adjacent frequencies, resulting in distortion of the spectrum analysis results. This phenomenon is particularly severe when the frequency components of the signal do not exactly match the sampling points, especially when the frequency of the signal is close to half of the sampling rate, and the spectrum leakage problem becomes more obvious.

[0004] Spectrum leakage will affect the accuracy of signal measurement, resulting in inaccurate identification of the frequency components of the signal, and thus affecting the analysis and monitoring of broadband oscillations in the power system. Due to this limitation of the Fast Fourier Transform FFT method, in many applications, especially in occasions with high requirements for precise measurement and frequency analysis, reducing spectrum leakage has become an urgent problem to be solved. For this reason, some improvement measures have been adopted in the prior art, such as the application of window functions. Window functions can suppress spectrum leakage to a certain extent by changing the windowing method of the signal. However, while these methods suppress spectrum leakage, they often introduce a certain loss of frequency resolution. Therefore, how to effectively reduce spectrum leakage while ensuring a high frequency resolution is still an urgent problem to be studied in the current spectrum analysis field. Summary of the Invention

[0005] The present invention provides a broadband signal spectrum analysis method based on spectrum leakage suppression and improved FFT algorithm, which can effectively reduce spectrum leakage while ensuring a high frequency resolution of the broadband signal, and at the same time has the advantage of high measurement accuracy.

[0006] The technical solution adopted by the present invention is as follows:

[0007] A broadband signal spectrum analysis method based on spectrum leakage suppression and improved FFT algorithm includes the following steps:

[0008] Step 1: According to the characteristics of the broadband signal, a broadband signal model including a fundamental frequency offset signal and multiple broadband oscillation signals is set up;

[0009] Step 2: Based on the Fourier transform theory, analyze the generation mechanism of spectrum leakage of the broadband signal;

[0010] Step 3: Design a Nuttall-Kaiser combined window function to suppress the spectrum leakage of the broadband signal;

[0011] Step 4: Improve the FFT algorithm and use its convolution window property to suppress the sidelobe attenuation to reduce the influence of spectrum leakage.

[0012] In the above Step 1, the mathematical expression of the broadband signal model is:

[0013]

[0014] In formula (1): x(t) is the broadband signal represented in the time domain, A is the amplitude of the fundamental frequency offset signal, f is the frequency of the fundamental frequency offset signal, Δf is the frequency offset of the fundamental frequency offset signal, t is time, A i is the amplitude of the broadband oscillation signal, f i is the frequency of the broadband oscillation signal, is the phase of the broadband oscillation signal, I is the total number of broadband oscillation signals, Acos(2π(f±Δf)t) represents the fundamental frequency offset signal, represents the broadband oscillation signal.

[0015] In the above Step 2, when the sampling frequency does not match the broadband signal frequency and the truncation effect on the broadband signal, both will cause spectrum leakage of the broadband signal.

[0016] Step 2 includes the following steps:

[0017] S2.1: Convert the continuous broadband signal into a discrete broadband signal model:

[0018] x[n] = Acos(2πf0nΔt + φ)(2);

[0019] In formula (2): x[n] is the discrete broadband signal, A is the signal amplitude, f0 is the signal frequency, φ is the signal initial phase, n = 1, 2, 3..., Δt is the unit time.

[0020] S2.2: The broadband signal is sampled at the sampling frequency f within the time interval [0, T] sSampling is performed, and the total number of sampling points is N. Perform a discrete Fourier transform on the discrete broadband signal represented in the time domain;

[0021]

[0022] In Equation (3): X[k] is the frequency-domain representation of the broadband signal, and k is the frequency index.

[0023] S2.3: Apply a rectangular window function w[n] to the discrete broadband signal x[n], perform a discrete Fourier transform on the truncated broadband signal x w [n], and expand it through Euler's formula;

[0024]

[0025] In Equation (4): and are both complex exponential factors in the Fourier transform.

[0026] If the sampling frequency matches the frequency of the broadband signal, i.e., f0 = m·f s / N, where m is an integer, then the exponential terms in the above two summations are integer multiples of N, and X[k] has a non-zero peak at k = m and is zero at other k values. At this time, there is no spectral leakage phenomenon; however, if the sampling frequency does not match the frequency of the broadband signal, the exponential terms in the above summations are not integer multiples of N, which will cause the energy to disperse over multiple k values, resulting in spectral leakage.

[0027] According to Equation (4), the specific form of spectral leakage depends on the choice of the rectangular window function w[n]. The direct truncation of a finite-length signal is equivalent to adding a rectangular window function to the broadband signal, and the rectangular window function w[n] can be expressed as:

[0028]

[0029] The frequency-domain expression of the rectangular window function can be obtained through the discrete-time Fourier transform, and the frequency-domain expression is the Sinc Function:

[0030]

[0031] In Equation (6): W(f) is the frequency-domain representation of the Sinc function, and T s is the time interval of the broadband signal sampling.

[0032] The frequency-domain expression X w (f) of the truncated signal is the convolution of the original signal X(f) and the frequency-domain expression W(f) of the rectangular window function, as shown in Equation (7) below:

[0033]

[0034] Step 3 includes the following steps:

[0035] Step 3.1:

[0036] 1): The Nuttall window can achieve rapid attenuation of sidelobes through adjustment coefficients:

[0037]

[0038] In Equation (8), w N (n) is the Nuttall window function, N is the total number of samples of the window function, M is the number of terms of the window function, b m is the window coefficient, and m is the m-th term of the window function.

[0039] Performing a discrete Fourier transform on the broadband signal, the spectrum of the Nuttall window function can be expressed as:

[0040]

[0041] In Equation (9), W R (f) = (sinNπf / sinπf)e -j(N-1)πf , and ω is the angular frequency.

[0042] 2): The Kaiser window controls the balance between the main lobe width and sidelobe attenuation through parameter optimization;

[0043]

[0044] In Equation (10), w k (n) is the Kaiser window function, I0 is the zero-order modified Bessel function, the parameter β controls the characteristics of the window function, Step 3.2: Performing a discrete Fourier transform on the broadband signal, the frequency-domain expression of the Kaiser window function is:

[0045]

[0046] In Equation (11), W K (ω) is the frequency-domain expression of the Kaiser window function.

[0047] Step 3.3: Truncating the discrete broadband signal using the Nuttall-Kaiser combined window function, and the expression of the Nuttall-Kaiser combined window function is as follows:

[0048] W(ω) = c × W N (ω) + d × W K (ω)(12);

[0049] In Equation (12), W(ω) is the combined window function, and c and d are the proportionality coefficients of the Nuttall-Kaiser combined window function respectively; substituting the Nuttall window function and the Kaiser window function into the formula ω = 2πk / N, the specific expression of the Nuttall-Kaiser combined window is as follows:

[0050]

[0051] Step 4 includes the following steps:

[0052] Step 4.1: Weight and truncate the broadband signal with a length of 2N - 1 using the Nuttall-Kaiser combined window function with a length of 2N - 1; then, add the data points with an interval of N in pairs to obtain the processed broadband signal data with a length of N; finally, perform FFT analysis on the processed broadband signal with N points.

[0053] Step 4.2: Perform dual-window improved FFT spectrum analysis on the broadband signal. Flip the window function w(n) as the front window to obtain the back window w(-n), and convolve the two to obtain the convolution window w c (n):

[0054] w c (n) = w(n) * w(-n), -N + 1 ≤ n ≤ N - 1 (14);

[0055] Step 4.3: Perform Fourier transform on the front window w(n) to obtain:

[0056]

[0057] In Equation (15), F(jω) is the Fourier transform of the front window, f(n) is the discrete time-domain broadband signal, and F g (ω) is the amplitude spectrum of the front window; τ is the amplitude spectrum of the front window;

[0058] Step 4.4: Weight the 2N - 1 data before and after the central sample point x(0) with the convolution window ω c (n), and then perform overlapping addition on the data with an interval of N in pairs. The output result y(n) is:

[0059] y(n) = [w c (n)x(n) + w c (n - N)x(n - N)] (16);

[0060] In Equation (16), x(n) is the sampled broadband signal, w c (n - N) is the window function after delaying N sample points, and x(n - N) is the sampled broadband signal after delaying N sample points.

[0061] Step 4.5: Perform discrete Fourier transform on the output result y(n), without considering negative frequencies. The spectral output result of the traditional window function FFT is as follows:

[0062]

[0063] In Equation (17), X[k] is the broadband signal component at the k-th frequency, ω0 is the central angular frequency of the broadband signal, and △ω is the change in angular frequency.

[0064] The output result of the Fourier transform of the signal processed by the improved FFT is the square of the output result of the direct Fourier transform;

[0065] F h (ω) = |F g (ω)| 2 (18);

[0066] In Equation (18), F g (ω) is the amplitude spectrum of the front window, and F h (ω) is the amplitude spectrum processed by the improved FFT.

[0067] A broadband signal spectrum analysis method based on spectral leakage suppression and improved FFT algorithm of the present invention has the following technical effects: 1) In step 1 of the present invention, according to the characteristics of high bandwidth and frequency dispersion of the broadband signal, a composite signal of fundamental frequency offset and superposition of multiple broadband oscillation signals is designed.

[0068] 2) In step 2 of the present invention, based on the Fourier transform theory, the basic principle of spectral leakage of the broadband signal is analyzed, laying a solid foundation for further analyzing spectral leakage and exploring how to effectively reduce spectral leakage.

[0069] 3) In step 3 of the present invention, the Nuttall and Kaiser window functions are combined for the first time, which can more effectively suppress the spectral leakage of the broadband signal.

[0070] 4) In step 4 of the present invention, the FFT is improved by combining the Nuttall-Kaiser combined window, and the convolution window property is used to effectively suppress the sidelobe attenuation, thereby reducing the influence of spectral leakage. BRIEF DESCRIPTION OF THE DRAWINGS

[0071] The present invention will be further described below with reference to the drawings and embodiments:

[0072] Figure 1 It is a flowchart for performing FFT analysis after weighting and truncating a broadband signal using a Nuttall-Kaiser combined window function.

[0073] Figure 2Results of spectrum analysis of broadband signals by traditional FFT and the improved FFT of the present invention for comparison.

[0074] Figure 3 It is a simulation diagram of the fan grid - connection system.

[0075] Figure 4 It is the time - domain waveform of the current signal i w to be analyzed.

[0076] Figure 5 It is the spectrum analysis diagram obtained by the method of the present invention.

[0077] Figure 6 It is a comparison diagram of the errors of signal reconstruction by different methods. Specific embodiments

[0078] A broadband signal spectrum analysis method based on spectrum leakage suppression and improved FFT algorithm includes the following steps:

[0079] Step 1: According to the characteristics of the broadband signal, a broadband signal model including a fundamental - frequency offset signal and multiple broadband oscillation signals is set up;

[0080] Step 2: Based on the Fourier transform theory, analyze the generation mechanism of spectrum leakage of the broadband signal;

[0081] Step 3: Design a Nuttall - Kaiser combined window function to suppress the spectrum leakage of the broadband signal and improve the measurement accuracy of the broadband signal;

[0082] Step 4: Improve the FFT algorithm and use its convolution window property to suppress the sidelobe attenuation to reduce the influence of spectrum leakage.

[0083] In the said Step 1, the mathematical expression of the broadband signal model is:

[0084]

[0085] In formula (1): x(t) is the broadband signal represented in the time domain, A is the amplitude of the fundamental - frequency offset signal, f is the frequency of the fundamental - frequency offset signal, △f is the frequency offset of the fundamental - frequency offset signal, t is the time, A i is the amplitude of the broadband oscillation signal, f i is the frequency of the broadband oscillation signal, is the phase of the broadband oscillation signal, I is the total number of broadband oscillation signals, Acos(2π(f ± △f)t) represents the fundamental - frequency offset signal, represents the broadband oscillation signal.

[0086] In the said Step 2, when the sampling frequency does not match the broadband signal frequency and the truncation effect on the broadband signal, both will cause spectrum leakage of the broadband signal.

[0087] Step 2 includes the following steps:

[0088] S2.1: Convert the continuous broadband signal into a discrete broadband signal model:

[0089] x[n] = Acos(2πf0nΔt + φ)(2);

[0090] In formula (2): x[n] is the discrete broadband signal, A is the signal amplitude, f0 is the signal frequency, φ is the initial signal phase, n = 1, 2, 3..., and Δt is the unit time.

[0091] S2.2: Sample the broadband signal within the time interval [0, T] at the sampling frequency f s with a total number of sampling points N. Perform a discrete Fourier transform on the discrete broadband signal represented in the time domain;

[0092]

[0093] In formula (3): X[k] is the frequency-domain representation of the broadband signal, and k is the frequency index.

[0094] S2.3: Apply a rectangular window function w[n] to the discrete broadband signal x[n], perform a discrete Fourier transform on the truncated broadband signal x w [n], and expand it through Euler's formula;

[0095]

[0096] In formula (4): and are both complex exponential factors in the Fourier transform.

[0097] If the sampling frequency matches the broadband signal frequency, i.e., f0 = m·f s / N, where m is an integer, then the exponential terms in the above two summations are integer multiples of N, and X[k] has a non-zero peak at k = m and is zero at other k values. At this time, there is no spectral leakage phenomenon; however, if the sampling frequency does not match the broadband signal frequency, the exponential terms in the above summation are not integer multiples of N, which will cause the energy to disperse over multiple k values, resulting in spectral leakage.

[0098] According to formula (4), the specific form of spectral leakage depends on the choice of the rectangular window function w[n]. The direct truncation of a finite-length signal is equivalent to adding a rectangular window function to the broadband signal, and the rectangular window function w[n] can be expressed as:

[0099]

[0100] The frequency-domain expression of the rectangular window function can be obtained through the discrete-time Fourier transform, and the frequency-domain expression is the Sinc Function:

[0101]

[0102] In Equation (6): W(f) is the frequency-domain representation of the Sinc function, and T s is the time interval for sampling the wideband signal.

[0103] The frequency-domain expression X w (f) of the truncated signal is the convolution of the original signal X(f) and the frequency-domain expression W(f) of the rectangular window function, as shown in Equation (7) below:

[0104]

[0105] It can be seen that the spectral energy of the signal is no longer concentrated at a single frequency point f0, but is spread around f0.

[0106] Step 3 includes the following steps:

[0107] Step 3.1:

[0108] 1): The Nuttall window is a weighted window function, and the rapid attenuation of the sidelobe can be achieved by adjusting the coefficient:

[0109]

[0110] In Equation (8), w N (n) is the Nuttall window function, N is the total number of samples of the window function, M is the number of terms of the window function, b m is the window coefficient, and m is the m-th term of the window function.

[0111] Performing the discrete Fourier transform on the wideband signal, the spectrum of the Nuttall window function can be expressed as:

[0112]

[0113] In Equation (9), W R (f) = (sinNπf / sinπf)e -j(N-1)πf , and ω is the angular frequency.

[0114] The sidelobe attenuation rate of the Nuttall window is relatively fast, superior to many other window functions, and performs well in the spectral analysis of wide dynamic ranges, and is suitable for wideband signals.

[0115] 2): The Kaiser window is a parameterized window function, and the balance between the main lobe width and sidelobe attenuation can be controlled through parameter optimization;

[0116]

[0117] In Equation (10), w k (n) is the Kaiser window function, I0 is the zero-order modified Bessel function, and the parameter β controls the characteristics of the window function. A smaller β value results in a narrower main lobe and higher side lobes, while a larger β value results in a wider main lobe and lower side lobes. Step 3.2: Perform a discrete Fourier transform on the broadband signal. The frequency-domain expression of the Kaiser window function is:

[0118]

[0119] In Equation (11), W K (ω) is the frequency-domain expression of the Kaiser window function.

[0120] The Kaiser window function introduces the parameter β, and an appropriate balance can be found between the frequency resolution (main lobe width) and leakage (side lobe amplitude) by adjusting β. The main lobe width of the Kaiser window is relatively narrow, and its frequency resolution is better than that of the Nuttall window. However, the side lobe attenuation rate of the Kaiser window is slower than that of the Nuttall window, which is prone to causing leakage at relatively far frequencies.

[0121] Step 3.3: Truncate the discrete broadband signal using the Nuttall-Kaiser combined window function. The expression of the Nuttall-Kaiser combined window function is as follows:

[0122] W(ω) = c × W N (ω) + d × W K (ω) (12);

[0123] In Equation (12), W(ω) is the combined window function, and c and d are the proportionality coefficients of the Nuttall-Kaiser combined window function, and the values of c and d are 0.5

[0124] Substitute the Nuttall window function and the Kaiser window function into the formula ω = 2πk / N, and the specific expression of the Nuttall-Kaiser combined window is:

[0125]

[0126] Compared with the single Kaiser window function and Nuttall window function, the Kaiser-Nuttall combined window has a narrower main lobe width and a higher side lobe attenuation rate, so it has advantages in suppressing spectral leakage and improving the signal measurement accuracy.

[0127] The said Step 4 includes the following steps:

[0128] Step 4.1: Weight and truncate the broadband signal with a length of 2N - 1 using a Nuttall-Kaiser combined window function with a length of 2N - 1; then, add the data points with an interval of N in pairs to obtain the processed broadband signal data with a length of N; finally, perform FFT analysis on the processed broadband signal with N points, and the processing process is as Figure 1 shown;

[0129] Step 4.2: Perform double-window improved FFT spectrum analysis on the broadband signal. Fold the window function w(n) to obtain the rear window w(-n), and convolve the two to obtain the convolution window w c (n):

[0130] w c (n) = w(n) * w(-n), -N + 1 ≤ n ≤ N - 1 (14);

[0131] Step 4.3: Perform Fourier transform on the front window w(n) to obtain:

[0132]

[0133] In formula (15), F(jω) is the Fourier transform of the front window, f(n) is the discrete time-domain broadband signal, and F g (ω) is the amplitude spectrum of the front window; τ is the amplitude spectrum of the front window;

[0134] Step 4.4: Weight the 2N - 1 data before and after the central sample point x(0) with the convolution window ω c (n), and then perform overlapping addition on the data with an interval of N in pairs. The output result y(n) is:

[0135] y(n) = [w c (n)x(n) + w c (n - N)x(n - N)] (16);

[0136] In formula (16), x(n) is the sampled broadband signal, w c (n - N) is the window function after delaying N sample points, and x(n - N) is the sampled broadband signal after delaying N sample points.

[0137] Step 4.5: Perform discrete Fourier transform on the output result y(n). Without considering negative frequencies, the spectrum output result of the traditional window function FFT is:

[0138]

[0139] In formula (17), X[k] is the broadband signal component at the k-th frequency, ω0 is the central angular frequency of the broadband signal, and △ω is the change in angular frequency.

[0140] The output result of the Fourier transform of the signal processed by the improved FFT is the square of the output result of the direct Fourier transform;

[0141] F h (ω) = |F g (ω)| 2 (18);

[0142] In Equation (18), F g (ω) is the amplitude spectrum of the front window, and F h (ω) is the amplitude spectrum processed by the improved FFT.

[0143] This means that the main lobe of the improved FFT is more prominent, the attenuation of the side lobes is faster, and the spectral leakage is smaller.

[0144] Example 1:

[0145] To verify the measurement effect of the method proposed by the present invention on the broadband signals of the power system, a signal model with known parameters is used to simulate the actual broadband oscillation signal for verification research and algorithm characteristic analysis.

[0146] According to the characteristics of the broadband signal, a broadband signal model including a fundamental frequency offset signal and multiple broadband oscillation signals is set up, and its mathematical expression is:

[0147]

[0148] In Equation (1), Acos(2π(f ± Δf)t) is the fundamental frequency offset signal, is the broadband oscillation signal. The specific parameters of the signal are given in Table 1.

[0149] Table 1 Given parameters of the broadband signal

[0150] Parameter Frequency / Hz Amplitude / A Phase / deg Fundamental wave 50.1 22.0 90 Oscillation 1 145.3 24.7 20 Oscillation 2 270.5 11.4 150 Oscillation 3 420.6 5.9 210

[0151] Set the sampling frequency to 2560 Hz and the number of sampling points N to 500. The traditional FFT and the improved FFT of the present invention are used to perform spectral analysis on the broadband signal, and the comparison results are as Figure 2 shown. It can be analyzed from Figure 2 that when the traditional FFT performs spectral analysis on the broadband signal with non-integer period truncation, there is serious spectral leakage, and the frequency and amplitude measurement errors are large, which seriously affects the subsequent further analysis of the broadband signal. However, the method of the present invention using the hybrid window function and the improved FFT significantly reduces the influence of spectral leakage.

[0152] Example 2:

[0153] To verify that the method of the present invention can also accurately measure the broadband signals of the actual power system, a simulation example of a wind turbine grid-connected system is constructed, asFigure 3 As shown, the parameter of some components of the circuit of the fan grid-connected inverter is: L w = 0.0003H, L v = 0.0001H, L g = 0.0002H, u g = 220V. Select Figure 4 The current i w in it as the signal to be analyzed. The time-domain waveform of the signal and the spectral analysis obtained by using the method of the present invention are respectively as Figure 4 and Figure 5 shown. It can be obtained from the analysis of the frequency-domain diagram of Figure 5 that the signal amplitude of the 50Hz component accounts for the main part, and there are oscillation components near 200Hz, 400Hz and 600Hz at the same time.

[0154] To further evaluate the measurement accuracy of the method proposed by the present invention for broadband signals, the following quantization indexes about frequency, amplitude and phase error are set.

[0155]

[0156] In formula (19), FE is the total frequency error, which is composed of the frequency estimation error FE n and the frequency missing penalty term θ p , f i is the actual frequency, and f ig is the estimated frequency. The quantization methods of amplitude and phase error are the same as those of frequency error, and the deviation between the estimated value and the actual value is evaluated by using the corresponding indexes.

[0157] The signals are reconstructed by using different methods respectively, and the error comparison is as Figure 6 shown. Analyzing Figure 6 of the test results, it can be seen that the FFT with Nuttall window has a certain improvement in measurement accuracy compared with the traditional FFT, and the measurement accuracies of the time-domain methods Prony and wavelet transform are also higher than those of the traditional FFT. Compared with other methods, the method proposed by the present invention has significant advantages in measurement accuracy, and its root mean square error is below 0.3.

Claims

1. A wideband signal spectrum analysis method based on spectrum leakage suppression and improved FFT algorithm, characterized in that It includes the following steps: Step 1: According to the characteristics of broadband signals, a broadband signal model including a fundamental frequency offset signal and multiple broadband oscillation signals is set up; Step 2: Based on the Fourier transform theory, analyze the generation mechanism of spectral leakage of broadband signals; Step 3: Design a Nuttall-Kaiser combined window function to suppress the spectral leakage of broadband signals; Step 4: Improve the FFT algorithm and utilize its convolution window property to suppress sidelobe attenuation to reduce the influence of spectral leakage.

2. The broadband signal spectrum analysis method based on spectrum leakage suppression and improved FFT algorithm according to claim 1, wherein: In the said Step 1, the mathematical expression of the broadband signal model is: In Equation (1): x(t) is a broadband signal represented in the time domain, A is the amplitude of the fundamental frequency offset signal, f is the frequency of the fundamental frequency offset signal, △f is the frequency offset of the fundamental frequency offset signal, t is time, A i is the amplitude of the broadband oscillation signal, f i is the frequency of the broadband oscillation signal, is the phase of the broadband oscillation signal, I is the total number of broadband oscillation signals, A cos(2π(f±△f)t) represents the fundamental frequency offset signal, represents the broadband oscillation signal.

3. The broadband signal spectrum analysis method based on spectrum leakage suppression and improved FFT algorithm according to claim 1, characterized in that: In the said Step 2, when the sampling frequency does not match the frequency of the broadband signal and the truncation effect on the broadband signal will both cause spectral leakage of the broadband signal.

4. The broadband signal spectrum analysis method based on spectrum leakage suppression and improved FFT algorithm according to claim 3, characterized in that: Step 2 It includes the following steps: S2.1: Convert the continuous broadband signal into a discrete broadband signal: x[n] = A cos(2πf0nΔt + φ) (2); In formula (2): x[n] is the discrete broadband signal, A is the signal amplitude, f0 is the signal frequency, φ is the signal initial phase, n = 1, 2, 3..., Δt is the unit time; S2.2: The broadband signal is sampled within the time interval [0, T] at a sampling frequency f s to obtain a total of N sampling points; perform a discrete Fourier transform on the discrete broadband signal represented in the time domain; In formula (3): X[k] is the frequency-domain representation of the broadband signal, k is the frequency index; S2.3: Apply a rectangular window function w[n] to the discrete wideband signal x[n], perform a discrete Fourier transform on the truncated wideband signal x w [n], and expand it using Euler's formula; In formula (4): and are both complex exponential factors in the Fourier transform.

5. The broadband signal spectrum analysis method based on spectrum leakage suppression and improved FFT algorithm according to claim 4, characterized in that: If the sampling frequency matches the broadband signal frequency, i.e., f0 = m·f s / N, where m is an integer, then the exponential terms in the above two summations are integer multiples of N, X[k] has a non-zero peak at k = m, and is zero at other k values. At this time, no spectral leakage phenomenon occurs; however, if the sampling frequency does not match the broadband signal frequency, the exponential terms in the above summation are not integer multiples of N, which will cause the energy to disperse over multiple k values, thereby generating spectral leakage.

6. The broadband signal spectrum analysis method based on spectrum leakage suppression and improved FFT algorithm according to claim 5, characterized in that: According to formula (4), the specific form of spectral leakage depends on the selection of the rectangular window function w[n]; the direct truncation of a finite-length signal is equivalent to adding a rectangular window function to the broadband signal, and the rectangular window function w[n] can be expressed as: The frequency-domain expression of the rectangular window function can be obtained through the discrete-time Fourier transform, and the frequency-domain expression is the sinc function: In Equation (6): W(f) is the frequency-domain representation of the sinc function, and T s is the time interval of the wideband signal sampling.

7. The broadband signal spectrum analysis method based on spectrum leakage suppression and improved FFT algorithm according to claim 6, characterized in that: The frequency-domain expression X of the truncated signal w (f) is the convolution of the frequency-domain expression X(f) of the original signal and the frequency-domain expression W(f) of the rectangular window function, as shown in Equation (7) below:

8. The broadband signal spectrum analysis method based on spectrum leakage suppression and improved FFT algorithm according to claim 1, characterized in that: The said Step 3 includes the following steps: Step 3.1: 1): The Nuttall window can achieve rapid attenuation of sidelobes by adjusting coefficients: In Equation (8), w N (n) is the Nuttall window function, N is the total number of samples of the window function, M is the number of terms of the window function, b m is the window coefficient, and m is the m-th term of the window function; Perform a discrete Fourier transform on the broadband signal, and the spectrum of the Nuttall window function can be expressed as: In formula (9), W R (f) = (sinNπf / sinπf)e -j(N-1)πf , where ω is the angular frequency; 2): The Kaiser window controls the balance between the main lobe width and sidelobe attenuation through parameter optimization; In Equation (10), w k (n) is the Kaiser window function, I0 is the zero-order modified Bessel function, and the parameter β controls the characteristics of the window function. Step 3.2: Perform a discrete Fourier transform on the broadband signal. The frequency-domain expression of the Kaiser window function is: In Equation (11), W K (ω) is the frequency domain expression of the Kaiser window function; Step 3.3: Truncate the discrete broadband signal using the Nuttall-Kaiser combined window function, and the expression of the Nuttall-Kaiser combined window function is as follows: W(ω) = c × W N (ω) + d × W K (ω) (12); In formula (12), W(ω) is the combined window function, c and d are respectively the proportional coefficients of the Nuttall-Kaiser combined window function; substitute the Nuttall window function and the Kaiser window function into the formula ω = 2πk / N, and the specific expression of the Nuttall-Kaiser combined window is:

9. The broadband signal spectrum analysis method based on spectrum leakage suppression and improved FFT algorithm according to claim 1, characterized in that: The said Step 4 includes the following steps: Step 4.1: Weight and truncate the broadband signal with a length of 2N - 1 using the Nuttall-Kaiser combined window function with a length of 2N - 1; then, add the data points at intervals of N in pairs to obtain the processed broadband signal data with a length of N; finally, perform FFT analysis on the processed broadband signal with N points; Step 4.2: Perform double-window improved FFT spectrum analysis on the broadband signal. The window function w(n) is flipped as the front window to obtain the rear window w(-n), and the convolution of the two gives the convolution window w c (n): w c r(n) = w(n) * w(-n), -N + 1 ≤ n ≤ N - 1 (14); Step 4.3: Perform a Fourier transform on the front window w(n) to obtain: In Equation (15), F(jω) is the Fourier transform of the front window, f(n) is the discrete time-domain wideband signal, and F g (ω) is the amplitude spectrum of the front window; τ is the amplitude spectrum of the front window; Step 4.4: Weight the 2N - 1 data before and after the central sample point x(0) with the convolution window ω c (n), and then perform overlapping addition on the data with a pairwise interval of N. The output result y(n) is as follows: y(n) = [w c (n)x(n) + w c (n - N)x(n - N)] (16); In Equation (16), x(n) is the sampled broadband signal, and w c (n - N) is the window function after delaying N sample points, and x(n - N) is the sampled broadband signal after delaying N sample points; Step 4.5: Perform a discrete Fourier transform on the output result y(n), without considering negative frequencies, and the spectral output result of the traditional window function FFT is: In Equation (17), X[k] is the broadband signal component at the k-th frequency, ω0 is the center angular frequency of the broadband signal, and △ω is the change in angular frequency; The output result of the Fourier transform of the signal processed by the improved FFT is the square of the output result of the direct Fourier transform; F h F(ω)=|F g (ω)| 2 (18); In Equation (18), F g (ω) is the amplitude spectrum of the front window, and F h (ω) is the amplitude spectrum processed by the improved FFT.

10. The broadband signal spectrum analysis method based on spectrum leakage suppression and improved FFT algorithm according to claim 9, characterized in that: To further evaluate the measurement accuracy of the broadband signal, the following quantization indexes regarding frequency, amplitude, and phase error are set: In Equation (19), FE is the total frequency error, which consists of the frequency estimation error FE n and the frequency missing penalty term θ p . f i is the actual frequency, and f ig is the estimated frequency.

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