A uniform missing linear frequency modulation signal fast iterative threshold deconvolution reconstruction method

Abstract

The application discloses a kind of uniform missing linear frequency modulation signal fast iterative threshold deconvolution reconstruction method, solve the matrix singularity problem that sparse reconstruction algorithm appears in pseudo-inverse process in the reconstruction of uniform missing linear frequency modulation signal, the present application is based on fast iterative threshold algorithm, according to the pattern of uniform missing, deduce the point spread function due to signal time domain missing causes spectrum aliasing, then utilize point spread function for the spectrum of zero padding and dechirp processing after missing linear frequency modulation signal by fast iterative threshold algorithm carries out complex deconvolution, to recover the spectrum of signal in frequency domain, in turn reconstruct complete signal.The present application realizes complex deconvolution by fast iterative threshold algorithm, need not carry out matrix pseudo-inverse operation, to avoid matrix singularity problem.

Description

Technical Field

[0001] This invention belongs to the field of radar signal processing, and particularly relates to the reconstruction of uniformly missing radar linear frequency modulated signals. Background Technology

[0002] Due to specific application scenarios, interference, and the unique structure of sensors, the signals received by sensors may exhibit uniform gaps. Traditional signal processing methods for uniformly missing signals typically fail to yield satisfactory results. Therefore, it is necessary to research and develop signal processing algorithms that can eliminate or suppress the negative impact of uniform signal gaps. Deconvolution, as a solution to the inverse problem, is an important method for reconstructing uniformly missing signals.

[0003] Deconvolution, as a method for reconstructing uniformly missing signals, obtains the point spread function based on the pattern of uniform signal loss and then reconstructs the missing signal based on the point spread function. In the paper "Amplitude Spectrum Estimation for Two-Dimensional Gapped Data," IEEE Transactions on Signal Processing, Vol. 50, No. 6, June, 2002, pp. 1343-1354, Erik G. Larsson et al. proposed a spectral estimation algorithm capable of handling missing data. This algorithm reconstructs the missing data and completes spectral estimation by interpolating the data at the missing locations based on the available data in the missing data through continuous iteration using the least squares criterion. The paper "Source Localization and Sensing: A Nonparametric Iterative Adaptive Approach Based on Weighted Least Squares", IEEE Transactions on Aerospace and Electronic Systems, Vol. 46, No. 1, January, 2010, pp. 425-443, proposes an algorithm for estimating the spectrum of missing data. This algorithm is based on weighted least squares and a nonparametric iterative adaptive algorithm was developed to reconstruct missing data and perform spectrum estimation. Common algorithms for reconstructing missing or sparse data typically utilize least squares algorithms for sparse reconstruction or missing data completion. However, least squares algorithms involve finding the pseudo-inverse of matrices, which can lead to matrix singularity issues and cause problems in the algorithm's solution. To avoid this problem, it is necessary to research and develop a uniform missing signal reconstruction algorithm that does not require finding the pseudo-inverse of matrices. Summary of the Invention

[0004] Algorithms for reconstructing missing or sparse data typically employ least squares algorithms to reconstruct or complete missing or sparse signals. However, the pseudo-inverse of matrices in least squares algorithms can lead to matrix singularity problems, causing issues in the solution process. This invention addresses this problem by proposing a fast iterative threshold deconvolution reconstruction method for uniformly missing linear frequency modulated (LFM) signals, based on the Fast Iterative Thresholding Algorithm (FISTA). The method derives the point spread function (PSF) for spectral aliasing caused by time-domain missing data based on the uniform missing pattern. Then, using the PSF, the spectrum of the missing LFM signal after zero-padding and de-modulation is deconvolved using the FSF algorithm to recover the signal's spectrum in the frequency domain, thereby reconstructing the complete signal.

[0005] The technical solution of this invention is as follows:

[0006] Step 1: Based on the requirements and application background of the radar system, obtain the system parameters, the uniform missing pattern parameters of the linear frequency modulated (LFM) signal, and the uniform missing LFM signal. This mainly includes the signal bandwidth, sampling rate, pulse width, signal uniform missing pattern parameters, and the uniform missing LFM signal to be processed.

[0007] Step 2: Define complete signal data as having N sampling points, and uniformly missing LFM signals as having N sampling points. D 100 sampling points. For a uniformly missing LFM time-domain signal, compared to the complete signal, in (NN) D Zero-filling is performed at the locations where ) sampling points are missing.

[0008] Step 3: Generate the corresponding time-domain matched function based on the parameters of the uniformly missing LFM signal. Multiply the time-domain matched function with the zero-padded uniformly missing LFM signal to complete the de-modulation process. After de-modulation, only linear phase remains in the signal phase, and the signal is sparsified in the frequency domain.

[0009] Step 4: For the signal after frequency modulation, perform a Fast Fourier Transform (FFT) to transform the signal to the frequency domain. Because the missing positions in the time domain signal are padded with zeros, the frequency domain signal exhibits aliasing.

[0010] Step 5: Based on the discrete form expression of the uniform signal loss pattern, derive the point spread function for spectral aliasing caused by the time-domain loss of the signal according to the definition of the Discrete Fourier Transform. Then, using the FISTA algorithm based on the point spread function, perform deconvolution processing on the aliased signal spectrum to suppress or eliminate spectral aliasing, thereby obtaining the recovered signal spectrum.

[0011] Step 6: Then, the signal is transformed to the time domain by inverse fast Fourier transform (IFFT) and multiplied by the complex conjugate of the time domain matching function in step 3 to obtain the final recovered time domain signal.

[0012] Compared to existing technologies, this invention achieves deconvolution through FISTA, avoiding the matrix singularity problem that may exist in the pseudo-inverse process in traditional sparse reconstruction algorithms, thereby improving the robustness of missing data reconstruction. Attached Figure Description

[0013] Figure 1 This is a schematic diagram of the processing flow of an embodiment of the present invention.

[0014] Figure 2 This is a schematic diagram of the zero-padding operation in an embodiment of the present invention. Detailed Implementation

[0015] This invention proposes a fast iterative threshold deconvolution reconstruction method for uniformly missing linear frequency modulated signals. This method eliminates the need for matrix pseudo-inverse operations, thus avoiding the problem of matrix singularities affecting algorithm solutions. A schematic diagram of the processing flow of this invention is shown below. Figure 1 As shown, the implementation process is as follows:

[0016] Step 1: Based on the requirements and application background of the radar system, obtain the system parameters, the uniform missing pattern parameters of the linear frequency modulated (LFM) signal, and the uniform missing LFM signal. This mainly includes the signal bandwidth, sampling rate, pulse width, signal uniform missing pattern parameters, and the uniform missing LFM signal to be processed.

[0017] Define the signal bandwidth as B r The pulse width is T r The sampling rate is F r Frequency modulation slope K r =B r / T r A complete dataset is defined as having N sampling points. A complete linear frequency modulated signal is:

[0018]

[0019]

[0020] In formula (1), rect represents a rectangular function.

[0021] For a uniform signal loss pattern, in the signal sequence, define M per sample A Each sampling point is missing M M There are N sampling points. Define the number of missing data points as N. D There are N sampling points. In a uniformly missing signal, there are N... C Each sampling period contains M samples. A Available sampling points and M M There are discontinuous missing sampling points, and M is defined. P Equal to MA With M M The sum of. Therefore, N = (M A +M M )N C =M P N C And N D =M A N C .

[0022] Based on the complete linear frequency modulated signal and the uniform missing pattern of the signal as shown in formula (1), the uniform missing LFM time domain signal to be processed can be obtained.

[0023] Step 2: Define complete signal data as having N sampling points, and uniformly missing LFM signals as having N sampling points. D 100 sampling points. For a uniformly missing LFM time-domain signal, compared to the complete signal, in (NN) D Zero-padding is performed at the locations where ) sampling points are missing. A schematic diagram of the zero-padding process is shown in the attached figure of the instruction manual. Figure 2 As shown.

[0024] Step 3: Generate the corresponding time-domain matched function based on the parameters of the uniformly missing LFM signal. Multiply the time-domain matched function with the zero-padded uniformly missing LFM signal to complete the frequency modulation removal process. After frequency modulation removal, only linear phase remains in the signal phase, and the signal is sparsified in the frequency domain. The expression of the time-domain matched function is as follows:

[0025]

[0026] Step 4: For the signal after frequency modulation, perform a Fast Fourier Transform (FFT) to transform the signal to the frequency domain. Because the missing positions in the time domain signal are padded with zeros, the frequency domain signal exhibits aliasing.

[0027] Step 5: After zero-padding, the signal can be viewed as the product of the complete data and the rectangular pulse sequence. The expression for the rectangular pulse sequence is as follows:

[0028]

[0029] In expression (3), mod(n,M) P The remainder operation is represented by ), where n is the dividend and M is the variable. P It is the divisor. The modulo operation means obtaining n divided by M. P The remainder obtained afterwards.

[0030] Based on the discrete form expression of the uniform signal loss pattern, and according to the definition of the discrete Fourier transform, the point spread function V(k) for spectral aliasing caused by signal time-domain loss is derived:

[0031]

[0032] W in expression (4) N q is obtained through expressions (5) and (6).

[0033]

[0034]

[0035] By using the formula for the summation of a geometric series, the point spread function V(k) can be expressed as:

[0036]

[0037] Let Z be the spectrum of the uniformly missing signal after zero-padding and frequency modulation removal, and X be the spectrum of the complete signal after frequency modulation removal. The following relationship exists between Z and X:

[0038] Z=V⊙X (8)

[0039] In expression (8), ⊙ represents a circular convolution operation on two vectors. To more easily solve the deconvolution problem, the convolution expression in expression (8) can be transformed into the following discrete linear system:

[0040] Z = ΨX (9)

[0041]

[0042] In expression (9), matrix Ψ is the circular convolution operator. The matrix multiplication of matrix Ψ with vector X is equal to the result of the circular convolution operation of vector V with vector X. Matrix Ψ is a cyclic matrix, obtained based on vector V according to the definition of circular convolution. The composition of matrix Ψ is shown in expression (10).

[0043] Subsequently, the deconvolution problem is transformed into estimating the unknown X based on the known Z and Ψ. Solving this problem can be viewed as a linear inversion problem, and the method proposed in this patent uses the FISTA algorithm to solve this problem. The main steps of FISTA are shown in expressions (11)-(14):

[0044] X k =Γ βt [y k -2tΨ T (Ψy k -Z)] (11)

[0045]

[0046]

[0047] y k+1 =X k +(p k -1)(X k -X k-1 ) / p k+1 (14)

[0048] In expressions (11)-(14), t is the appropriate step size, k represents the number of iterations, and Γ βt It is a convergence operator. The initial value of p is 1, and the initial value of y is X0.

[0049] Based on the point spread function, the FISTA algorithm is used to deconvolve the aliased signal spectrum to suppress or eliminate the spectral mixing phenomenon and thus obtain the recovered signal spectrum.

[0050] Step 6: Then, the signal is transformed to the time domain by inverse fast Fourier transform (IFFT) and multiplied by the complex conjugate of the time domain matching function in step 3 to obtain the final recovered time domain signal.

Claims

1. A fast iterative threshold deconvolution reconstruction method for uniformly missing linear frequency modulated signals, characterized in that: Step 1: Based on the requirements and application background of the radar system, obtain the system parameters, parameters of the uniform missing pattern of the linear frequency modulated signal, and the uniform missing linear frequency modulated signal, mainly including the signal bandwidth, sampling rate, pulse width, parameters of the uniform missing pattern of the signal, and the uniform missing linear frequency modulated signal to be processed. Step 2: For the time-domain signal of the uniformly missing linear frequency modulated signal, perform zero-padding at the missing sampling points; Step 3: Generate the corresponding time-domain matching function based on the parameters of the uniformly missing linear frequency modulated signal; multiply the time-domain matching function with the uniformly missing linear frequency modulated signal after zero-padding to complete the frequency modulation process; Step 4: Perform a Fast Fourier Transform on the signal after frequency modulation to transform the signal to the frequency domain; Step 5: Derive the point spread function for spectral aliasing caused by signal time-domain missing patterns based on the uniform missing pattern of the signal, and perform deconvolution processing on the signal spectrum obtained in Step 4 using a fast iterative thresholding algorithm based on the point spread function to obtain the recovered signal spectrum. Step 5 includes: After zero-padding, the signal is considered as the product of complete data and a rectangular pulse sequence; the rectangular pulse sequence is the discrete form of the uniform signal loss pattern. Based on the discrete form expression of the uniform signal loss pattern, the point spread function caused by spectral aliasing due to time-domain signal loss is derived according to the definition of the discrete Fourier transform; the discrete form expression of the uniform signal loss pattern is: Complete signal data has N sampling points. In a uniformly missing signal, there are N C Each sampling period contains M samples. A Available sampling points and M M M discontinuously missing sampling points P Equal to M A With M M sum; Step 6: Transform the recovered signal spectrum to the time domain using inverse fast Fourier transform, and multiply it by the complex conjugate of the time-domain matching function in step 3 to obtain the final recovered time-domain signal.

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