Frequency super-resolution time-frequency analysis method based on eigenvalue decomposition

Abstract

The invention belongs to the field of digital signal time-frequency analysis, and provides a frequency super-resolution time-frequency analysis method based on eigenvalue decomposition, which comprises the steps of intercepting a signal sequence along a time axis, constructing a virtual matrix, carrying out eigendecomposition on a covariance matrix, constructing a noise matrix, calculating pseudo-spectrum functions and combining the pseudo-spectrum functions along the time axis to form a time-frequency graph. The method does not depend on the number of time domain sampling points, geometric structure information of signals is extracted through subspace decomposition insensitive to noise, limitation of an STFT time window is broken through, high-frequency resolution can still be achieved under the condition of a short window, the time-frequency contradiction problem of STFT is relieved, and the method has the advantages of being simple in structure and convenient to operate. And a new technical approach is provided for accurate detection and parameter extraction of non-cooperative signals in an optical down-conversion electric field measurement system.

Description

Technical Field

[0001] The present invention belongs to the field of digital signal time-frequency analysis, and in particular relates to a frequency super-resolution time-frequency analysis method based on eigenvalue decomposition. Background Art

[0002] With advances in electronic communications technology, the electromagnetic spectrum is flooded with countless signals, some of which are characterized by wide bandwidth, large number, and high speed, placing higher demands on electromagnetic environment perception. Optical down-conversion electric field measurement systems can down-convert signals to a narrowband, shortening measurement time and enhancing signal reception. However, narrowband compression often leads to spectral aliasing, hindering effective parameter extraction because traditional separation methods are unable to process aliased signals. Time-frequency image processing and separation techniques can separate time-frequency aliased signals. Compared with traditional signal separation techniques, this method does not require prior conditions and has a stronger ability to process complex signals.

[0003] The frequency recovery algorithm of the optical down-converted electric field measurement system requires a very high frequency measurement accuracy of the down-converted signal. The traditional short-time Fourier transform (STFT) uses a fixed-length window function, such as a rectangular window or a Hamming window, to intercept the signal and use the fast Fourier transform (FFT) to analyze the signal segment by segment. If the sampling rate of the signal is f s , the number of sampling points is N, then the frequency resolution of FFT is f s / N, where the resolution is limited by the number of sampling points. Therefore, the STFT window length determines both time and frequency resolution performance: a short window improves time resolution, but due to the small number of FFT points, it reduces frequency resolution. A long window improves frequency resolution but sacrifices time resolution to accurately capture rapid signal changes. Partial cycle truncation introduces spectral leakage, resulting in peak blurring, further limiting improvements in frequency resolution.

[0004] Furthermore, the commonly used time-frequency analysis method, the wavelet transform, exhibits good time resolution in high-frequency regions, but low frequency resolution. Conversely, in low-frequency regions, it provides high-frequency resolution but poor time resolution. The performance of the wavelet transform depends largely on the choice of wavelet basis function, as different basis functions have different adaptability to signals. For non-stationary or multi-component signals, a single wavelet basis function may not be able to adapt to all signal components simultaneously. Furthermore, the wavelet basis function and window length in the wavelet transform are fixed and cannot be adaptively adjusted according to signal characteristics, which can lead to a reduction in the resolution of the time-frequency representation and an inability to effectively extract signal parameters under multi-signal conditions. Summary of the Invention

[0005] In response to the shortcomings of the existing technology, the present invention proposes a frequency super-resolution time-frequency analysis method based on eigenvalue decomposition. This method expands the spectrum of the aliased signal along time and uses the orthogonality of the signal and noise subspaces to achieve high-precision readings of the down-conversion frequency, providing higher estimation accuracy and spectral resolution, thereby achieving more effective aliased signal separation and parameter extraction in the optical down-conversion electric field measurement system.

[0006] The technical solutions of the present invention are as follows:

[0007] A frequency super-resolution time-frequency analysis method based on eigenvalue decomposition, comprising intercepting a signal sequence along a time axis, constructing a virtual matrix, performing eigendecomposition on a covariance matrix, constructing a noise matrix, calculating a pseudo-spectral function, and combining the functions along the time axis to form a time-frequency diagram;

[0008] The virtual matrix X is determined by the number of interception points K, the number of virtual sensor array elements M, and the intercepted signal sequence x[k], as shown in the following formula:

[0009]

[0010] Preferably, the number M of virtual sensor array elements should be 1 / 3 to 1 / 2 of the number K of interception points.

[0011] The covariance matrix R xx is the covariance matrix of the dummy matrix X.

[0012] The noise matrix is ​​assumed to contain D incoherent signals in the sequence, and the covariance matrix R xx After eigendecomposition, the eigenvalues ​​are arranged from large to small, and the noise matrix E is constructed by taking the MD eigenvectors corresponding to the smallest MD eigenvalues. n .

[0013] The pseudo-spectral function is shown below:

[0014]

[0015] Where H represents the conjugate transpose of the matrix and a(f) is the phase vector, which can be expressed as follows:

[0016]

[0017] Where j is the imaginary unit, f i is the i-th frequency, f s is the sampling frequency.

[0018] The time-frequency diagram is formed by calculating the pseudo-spectral function of each intercepted signal sequence and then arranging and combining them along the time axis.

[0019] Compared with the prior art, the present invention has the following beneficial effects:

[0020] The present invention does not rely on the number of time-domain sampling points. Instead, it extracts the geometric structure of the signal through noise-insensitive subspace decomposition, thus breaking through the limitations of the STFT time window and achieving high-frequency resolution under short window conditions. This frequency super-resolution time-frequency analysis method based on eigenvalue decomposition not only alleviates the time-frequency contradiction problem of the STFT, but also provides a new technical approach for the accurate detection and parameter extraction of non-cooperative signals in optical down-conversion electric field measurement systems. BRIEF DESCRIPTION OF THE DRAWINGS

[0021] In order to more clearly illustrate the embodiments of the present invention or the technical solutions in the prior art, the following briefly introduces the drawings required for use in the embodiments. By referring to the drawings, the features and advantages of the present invention can be more clearly understood. The drawings are schematic and should not be understood as limiting the present invention in any way. For ordinary technicians in this field, other drawings can be obtained based on these drawings without any creative work.

[0022] Figure 1 It is a schematic diagram of the key process of the time-frequency analysis method of the present invention.

[0023] Figure 2 It is the time-frequency analysis effect diagram of the present invention. DETAILED DESCRIPTION

[0024] In order to more clearly understand the above-mentioned objects, features and advantages of the present invention, the present invention is further described in detail below with reference to the accompanying drawings and specific embodiments. It should be noted that the embodiments of the present invention and the features therein can be combined with each other without conflict.

[0025] In the following description, many specific details are set forth to facilitate a full understanding of the present invention. However, the present invention may also be implemented in other ways different from those described herein. Therefore, the scope of protection of the present invention is not limited to the specific embodiments disclosed below.

[0026] The working principle of the frequency super-resolution time-frequency analysis method based on eigenvalue decomposition of the present invention is as follows:

[0027] For the received signal r(t), a short signal x(t) is intercepted and M sensor arrays (sampled M times) are placed on the time axis to form the signal's time streamline matrix X. Assuming that x(t) contains D incoherent signals, the vector of M samples can be represented as a superposition of signal and noise.

[0028] X=AS+N (4)

[0029] Where A is the phase difference between the signal and the first reference sensor element, S is the signal matrix, and N is the noise matrix.

[0030] A=[a(f1)a(f2)…a(f M )] T (5)

[0031] Then the covariance matrix R of X xx It can be expressed as:

[0032] R xx =E[XX H ]=E[(AS+N)(AS+N) H ]=AE[SS H ]A H +E[NN H ]=AR s A H +σ 2 I(6)

[0033] Among them, E represents the mean, R s is the covariance matrix of the signal matrix S, σ 2 is the noise power, and I is the identity matrix.

[0034] Due to σ 2 >0, R xx There are M positive real eigenvalues ​​λ1,λ2...λ M , corresponding to M eigenvectors, and since R xx It is a Hermite matrix, so the eigenvectors are orthogonal. The eigenvalue represents the energy of the signal at different frequencies, and the eigenvector represents the directionality of the signal. Sorted by the eigenvalue size, there are only the largest D eigenvalues ​​related to the signal, which are equal to the matrix AR s A H The eigenvalues ​​of σ 2 The sum corresponds to D eigenvectors belonging to the signal subspace; the remaining MD eigenvalues ​​are σ 2 , corresponding to MD eigenvectors belonging to the noise subspace, such as Figure 1 (b) shown.

[0035] Since the signal and noise subspaces are orthogonal, the frequency of the signal can be obtained by searching for the peak value using equation (2). The denominator in the equation represents the inner product of the phase vector of the signal and the noise vector. When a(f) is equal to E n When the columns of are orthogonal, the denominator is approximately zero, so a very sharp spectrum peak can be obtained at this frequency, and other frequencies are flat minimum values. By searching the frequency, the frequency information of all signals can be obtained, thus achieving high-resolution frequency analysis. The frequency analysis results of each signal are combined along the time axis to form a time-frequency analysis with frequency super-resolution, such as Figure 1 (c) shown.

[0036] According to the above analysis, by replacing the FFT in STFT with a frequency analysis method based on eigenvalue decomposition, higher frequency resolution can be achieved when the sampling points are limited, and the frequency resolution capability of the time-frequency representation can be significantly enhanced.

[0037] The above principle is verified through a specific example below.

[0038] Example 1

[0039] The present invention provides a frequency super-resolution time-frequency analysis method based on eigenvalue decomposition, such as Figure 1 The parameters of the method in this example are as follows:

[0040] The signal sampling rate is 400 MHz, the number of signal sampling points is 327680, the number of interception points K is 2048, the number of virtual sensor array elements M is 1000, and the number of irrelevant signals D is 4.

[0041] Figure 2 The time-frequency display effect of the frequency super-resolution time-frequency analysis method based on eigenvalue decomposition of the present invention is demonstrated when there are two sinusoidal continuous signals with frequencies of 15MHz and 25MHz. This shows that the present invention realizes the time-frequency analysis function.

[0042] In order to evaluate the frequency analysis accuracy of the present invention, the frequency of the simulation signal was linearly increased from 0.5 MHz to 100 MHz in increments of 0.5 MHz. Figure 3 The relationship between the frequency error and signal frequency using the STFT and the present invention is shown. When using the present invention, the average frequency error is reduced from 48.75kHz using the STFT to 0.9375kHz, equivalent to a 98.08% reduction in error, demonstrating superior frequency estimation accuracy compared to traditional time-frequency analysis.

[0043] The foregoing description is merely a preferred embodiment of the present invention and is not intended to limit the present invention. Those skilled in the art will readily appreciate that various modifications and variations of the present invention are possible. Any modifications, equivalent substitutions, or improvements made within the spirit and principles of the present invention are intended to be within the scope of protection of the present invention.

Claims

1. A frequency super-resolution time-frequency analysis method based on eigenvalue decomposition, characterized in that: The method includes intercepting the signal sequence along the time axis, constructing a virtual matrix, performing eigendecomposition on the covariance matrix, constructing a noise matrix, calculating the pseudo-spectral function and combining them along the time axis to form a time-frequency diagram.

2. The frequency super-resolution time-frequency analysis method according to claim 1, characterized in that: The number of rows of the virtual matrix is ​​the number of virtual sensor array elements M, and the number of columns is the number of points K for intercepting signals minus the number of virtual sensor array elements plus 1.

3. The frequency super-resolution time-frequency analysis method according to claim 2, characterized in that: The number M of virtual sensor array elements is 1 / 3 to 1 / 2 of the number K of intercepted signal points.

4. The frequency super-resolution time-frequency analysis method according to claim 1, characterized in that: The covariance matrix R xx is the covariance matrix of the dummy matrix.

5. The frequency super-resolution time-frequency analysis method according to claim 1, characterized in that: The noise matrix is ​​assumed to contain D incoherent signals in the sequence, and the covariance matrix R xx After eigendecomposition, the eigenvalues ​​are arranged from large to small, and the noise matrix E is constructed by taking the MD eigenvectors corresponding to the smallest MD eigenvalues. n .

6. The frequency super-resolution time-frequency analysis method according to claim 1, characterized in that: The pseudo-spectral function is shown below: Where H represents the conjugate transpose of the matrix and a(f) is the phase vector, which can be expressed as follows: Where j is the imaginary unit, f i is the i-th frequency, f s is the sampling frequency.

7. The frequency super-resolution time-frequency analysis method according to claim 1, characterized in that: The time-frequency diagram is formed by calculating the pseudo-spectral function of each intercepted signal sequence and then arranging and combining them along the time axis.

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