CN112731306B - UWB-LFM signal parameter estimation method based on CS and simplified FrFT - Google Patents
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Abstract
The invention discloses a UWB-LFM signal parameter estimation method based on CS and simplified FrFT, which introduces a compressed sensing method, and firstly performs sparse representation and AIC sampling on an ultra-wideband linear frequency modulation signal; secondly, a discrete simplified fractional Fourier transform dictionary is established as a base dictionary matrix for observation, and algorithm complexity is reduced; and finally, an improved CoSaMP reconstruction algorithm is used for reconstructing the observation vector, regularization in the traditional ROMP reconstruction algorithm is combined with an adaptive idea in the SAMP reconstruction algorithm, the defect that prior signal information is lacked in practical application and the calculated amount is huge is overcome, so that the reconstruction precision and the anti-interference capability of the signal are improved, and the method has good accuracy and instantaneity.
Description
Technical Field
The invention relates to the fields of ultra-wideband linear frequency modulation signal technology, microwave photon technology and signal processing, in particular to a parameter estimation method of an ultra-wideband linear frequency modulation signal based on compressed sensing and simplified fractional Fourier transform.
Background
For ultra-wideband linear frequency modulation signals, in view of the ultra-high bandwidth and low signal-to-noise ratio detection characteristics, the traditional Nyquist sampling law is used for sampling, the sampling rate needs to be twice or more than the maximum bandwidth to achieve accurate reconstruction of the signals, but huge sampling data brings great pressure to a signal acquisition and processing mechanism. The development of compressed sensing theory (Compressed Sensing, CS) effectively alleviates the problem and becomes the subject of research in recent years.
The compressed sensing theory utilizes the sparse characteristic of signals, projects the signals to a given domain to obtain a group of compressed sampling data, processes the compressed sampling data by utilizing an optimization algorithm, and estimates important information of the original signals so as to relieve the limitation of the Nyquist sampling theorem. Whereas Ultra wideband chirp signals (Ultra-Wide Band Linear Frequency Modulation, UWB-LFM) just meet the characteristics of the impulse signal in the transform domain of the reduced fractional fourier transform (Concise Fractional Fourier Transform, CFrFT) making it possible to implement parameter estimation for UWB-LFM signals in a compressed sensing framework.
The following two defects are common in existing studies based on ultra wideband chirp parameter estimation: firstly, the defect of huge calculation amount of an algorithm under the condition of lacking priori LFM information is overcome, in a scene of unknown actual signal environment, the sparsity of a signal to be estimated in a CFrFT transformation domain is unknown, and blind optimization and peak search can bring considerable calculation complexity; secondly, the method has the defect of not having excellent detection efficiency and estimation accuracy under the low signal-to-noise ratio, and aims at avoiding the influence of noise interference under the low signal-to-noise ratio environment in the environment of microwave photon broadband radar electronic war, so that the method is an important problem to be solved.
Disclosure of Invention
Accordingly, the present invention is directed to a method for estimating parameters of ultra wideband chirp signals based on compressed sensing and simplified fractional fourier transform.
In order to achieve the above purpose, the present invention provides the following technical solutions:
the invention provides a parameter estimation method of ultra-wideband linear frequency modulation signals based on compressed sensing and simplified fractional Fourier transform, which comprises the following steps:
acquiring an ultra-wideband linear frequency modulation signal containing Gaussian white noise;
constructing a discrete simplified fractional Fourier transform dictionary matrix to perform sparse representation on the ultra-wideband linear frequency modulation signal;
performing compression sampling on the ultra-wideband linear frequency modulation signal;
reconstructing the coefficient vector using an improved compressed sample matching pursuit algorithm;
the initial frequency and the frequency modulation slope of the ultra-wideband linear frequency modulation signal are estimated according to the distribution characteristics of the signal in the discrete fractional Fourier transform domain.
Further, an ultra wideband chirp signal containing gaussian white noise is obtained, and a model x expression of the signal is as follows:
x＝s+n；
wherein s is an ultra-wideband linear frequency modulation signal, and n is additive Gaussian white noise.
Further, constructing a discrete CFrFT dictionary matrix ultra-wideband linear frequency modulation signal for sparse representation for compressed sampling, selecting an optimal transformation order, and detecting and parameter estimating subsequent signals, and constructing a discrete CFrFT matrix expression as follows:
wherein ψ represents a discrete reduced-fraction Fourier transform dictionary matrix, j is an imaginary part of an exponential function, n is a sampling point number of a discretized continuous signal x, m is a discrete point number of the discretized continuous signal x after CFrFT, alpha represents a coordinate axis rotation angle of reduced-fraction Fourier transform,ΔT＝N/f s n is the number of sampling points, f s Is the sampling frequency;
using the dictionary matrix as an observation matrix, performing sparse representation according to the following formula:
wherein x is the input signal, θ i As coefficient vector, ψ= { ψ 1 ，ψ 2 ，...，ψ i And (3) sparse basis.
Further, the ultra-wideband linear frequency modulation signal is compressed and sampled, and the method is realized according to the following steps:
by using an observation matrix Φ e R which is uncorrelated with the sparse basis ψ M×N To observe the input signal x to obtain M observations y, where M < N. For wideband signals, specific compressed sampling is typically accomplished by Analog-to-Information Converter (AIC) converters.
Further, the coefficient vector after compressed sampling is reconstructed by using an improved compressed sampling matching pursuit algorithm, and the method comprises the following steps of:
firstly, calculating a fuzzy threshold value of a selected index in the operation process of a reconstruction algorithm according to the following formula:
wherein a is pr And b pr To blur threshold parameter, t hr In order to select the condition parameters of the index, rand (1) is a random number within 0-1, F is a support set, r is a residual error, k is the iteration number, A is a sensing matrix, A T I is an index selected from the support set, j is a maximum index meeting the selection condition;
forming a measurement matrix by using atoms corresponding to the indexes in the support set;
calculating a correlation coefficient, and extracting indexes corresponding to maximum values of the size for secondary screening;
selecting a group of indexes corresponding to the maximum energy value to be merged into a supporting set by utilizing a regularization idea;
reconstructing the signal by using a least square method, and solving a new residual error;
the reconstruction mode of the signal is determined according to the relation between the energy of the residual norm and the clipping threshold.
Further, the initial frequency and the frequency modulation slope of the ultra-wideband linear frequency modulation signal are estimated according to the distribution characteristic of the ultra-wideband linear frequency modulation signal in a discrete fractional Fourier transform domain, and the method is realized according to the following steps:
first, a simplified fractional Fourier transform distribution of the ultra wideband chirp signal is calculated according to the following formula:
CF p (u) represents the frequency domain distribution after CFrFT transformation, t is time, j is the imaginary part of an exponential function, u is frequency, and alpha is the rotation angle of the coordinate axis of CFrFT;
secondly, through carrying out peak search on the simplified fractional Fourier transform distribution, the initial frequency and the frequency modulation slope of the signal are estimated according to the following formula:
wherein,for the estimation of the frequency modulation slope, +.>For the estimated value of the starting frequency +.>For the best rotation angle>For peak frequency position, S is a normalization factor, calculated according to the following formula:
wherein Δt=n/f s N is the number of sampling points, f s Is the sampling frequency.
The invention has the beneficial effects that:
according to the UWB-LFM parameter estimation method based on CS and CFrFT, firstly, UWB-LFM signals in Gaussian white noise are subjected to sparse representation and AIC compressed sampling to obtain compressed sampled coefficient vectors of the signals; performing rough search on a discrete CFrFT dictionary matrix within the range of the construction order [0,2 ] with the precision of 0.01; solving coefficient vectors in each order of discrete CFrFT matrix of the UWB-LFM signal in rough estimation by utilizing an improved CoSaMP algorithm to obtain a preliminary peak value position; performing fine search by using an energy gravity center principle to obtain the position of the maximum peak value; and finishing the final frequency modulation slope and the initial frequency according to the position of the maximum value and the relation between the rotation angle and the order.
The compression sensing framework is used for relieving the pressure of the Nyquist sampling theorem, discrete simplified fractional Fourier transform is used for establishing the dictionary matrix, the complexity of algorithm calculation is reduced, the traditional CoSaMP signal reconstruction algorithm is improved, the regularization thought of the traditional ROMP algorithm and the self-adaptive thought in the SAMP algorithm are introduced, and the defect that a large amount of data redundancy and huge calculation amount are caused due to the fact that the prior UWB-LFM signal information is lacking in practical application is overcome, so that the reconstruction accuracy and the anti-interference capability of the signal are finally improved.
Additional advantages, objects, and features of the invention will be set forth in part in the description which follows and in part will become apparent to those having ordinary skill in the art upon examination of the following or may be learned from practice of the invention. The objects and other advantages of the invention may be realized and obtained by means of the instrumentalities and combinations particularly pointed out in the specification.
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In order to make the objects, technical solutions and advantageous effects of the present invention more clear, the present invention provides the following drawings for description:
fig. 1 is a flow chart of UWB-LFM signal parameter estimation based on CS and CFrFT.
Fig. 2 is a schematic diagram of a UWB-LFM time domain signal.
Fig. 3 is a schematic diagram of the distribution of UWB-LFM signals in the discrete CFrFT domain.
Fig. 4 is a schematic diagram of the structure of AIC compressed sampling.
Fig. 5 is a schematic diagram showing UWB-LFM signal comparison before and after reconstruction.
Fig. 6 is a diagram showing the optimal transform order distribution of UWB-LFM signals in the discrete CFrFT domain.
Detailed Description
The present invention will be further described with reference to the accompanying drawings and detailed description so that those skilled in the art may better understand the present invention and practice the same, but the examples are not meant to be limiting.
As shown in fig. 1, the present embodiment provides a UWB-LFM parameter estimation method based on CS, cfrtf and improved CoSaMP reconstruction algorithm, which first obtains an ultra wideband chirp signal containing white gaussian noise; constructing a discrete simplified fractional Fourier transform dictionary matrix ultra-wideband linear frequency modulation signal to perform sparse representation; performing compression sampling on the ultra-wideband linear frequency modulation signal; reconstructing the coefficient vector using an improved compressed sample matching pursuit algorithm; the initial frequency and the frequency modulation slope of the ultra-wideband linear frequency modulation signal are estimated according to the distribution characteristics of the signal in the discrete fractional Fourier transform domain.
As shown in fig. 1, the method specifically comprises the following steps:
the mathematical representation of a single-component LFM signal is:
if the signal contains Gaussian white noise, the signal model may be expressed as: x=s+n (2)
Wherein t is time, A is amplitude, f 0 Is the initial frequency, k 0 Is the frequency modulation slope, Δt is the duration of the signal, j is the imaginary part of the exponential function.
Research on ultra-wideband signals is commonly performed using the definition rules specified by the federal communications commission (Federal Communications Commission, FCC) in the united states: the signal with the working frequency band of 3.1-10.6GHz and the bandwidth of more than 500MHz is called UWB signal, and the time domain diagram is shown in figure 2.
The aim of the technique is to estimate the initial frequency f of the UWB-LFM signal 0 And frequency modulation slope k 0 。
The CS is introduced to process the single-component UWB-LFM signal, and the CS framework can be divided into three parts of sparse representation, compressed sampling and reconstruction algorithm, and the specific steps are as follows:
suppose for discrete signal x εR N×1 In other words, fullThe non-zero element has at most K (K < N) l 0 -norm conditions: |x| 0 K, i.e., K-sparse signals, are collectively expressed as: sigma (sigma) K ＝{x：||x|| 0 ≤K} (3)
Assume that there is a sparse basis ψ ε R N×N The sparse representation of the K-sparse signal which can be obtained under the sparse basis is as follows:
wherein x is the input signal, θ i As coefficient vector, ψ= { ψ 1 ，ψ 2 ，...，ψ i And (3) sparse basis.
Since the chirp signal exhibits characteristics of an impulse signal in the fractional fourier transform domain, as shown in fig. 3, the UWB-LFM signal satisfies the above-mentioned l of K-sparse signal in the domain 0 -norm conditions, thus sparse representation of UWB-LFM signal construction discrete reduced fractional fourier transform dictionary matrix ultra wideband chirp signals, comprising the specific steps of:
1. discrete sampling is carried out on the input continuous signal (2), and dimension normalization is carried out to obtain a resampling sequence expression of the signal, wherein the resampling sequence expression is as follows:
wherein f s For the sampling frequency, Δt is the duration of the signal, Δx is the sampling interval, and n is the number of x (T) samples of the discretized continuous signal.
2. Discrete Chirp multiplication is performed on the resampled sequence x (n/delta x) to obtain a new sequence expression:
wherein n is the number of sampling points, j is the imaginary part of an exponential function, and alpha is the coordinate axis rotation angle of CFrFT;
3. performing fast fourier transform on the sequence s (n) to obtain a discrete cfrtf expression:
wherein m is the discrete point number after fast Fourier transform,as discrete elements
Kernel function, X of cfrtt transform α (m) can be regarded asAt K α (m, n) projection onto the set of basis, thus the kernel function
The expression can be used as a base dictionary of discrete CFrFT, and is as follows:
in preparation for performing the rough search step at this stage, the DCFrFT matrix in the order pE [0,2 ] range is constructed with an accuracy of 0.01
Compressive sampling is performed by using an observation matrix Φ ε R that is uncorrelated with sparse basis ψ M×N To observe the input signal x to obtain M observed data y, wherein M < N, the matrix has the expression: y=Φx=Φψθ (9)
Where y is an Mx1-dimensional column vector, x is an Nx1-dimensional column vector, and θ is an observation vector.
For wideband signals, the compressive sampling process is accomplished by an analog information converter, a typical AIC architecture is shown in FIG. 4, where the continuous signal x (t) passes through the perceptual function Θ i I=1, M is carried outModulation, providing the necessary randomness to the signal, followed by a window function of T W Is filtered to ensure that the signal is undistorted, and is finally sampled by an Analog-to-Digital Converter (ADC) to obtain compressed sampled data
The UWB-LFM signal after coefficient representation and compressed sampling is reconstructed through a modified CoSaMP algorithm, and the specific flow steps of the reconstruction algorithm are as follows:
1. input sensing matrix A, observation vector y, fuzzy threshold parameter a pt And b pr Large step clipping threshold and stop threshold epsilon 1
And epsilon 2 ；
2. Initializing sparsity K=1, sparsity estimation step length, supporting set F, residual error t=y, observation vector y, stage=1 in initial stage, iteration number k=1, supporting set length size=k, and index setCandidate set->An empty set C;
3. atomic pre-selection operation: and selecting an index by using a fuzzy threshold value, and storing the index into a support set F, wherein the expression of the fuzzy threshold value is as follows:
wherein the blur threshold parameter a pr And b pr Selecting too small reduces reconstruction accuracy and increases computation time, but large step clipping and stopping thresholds ε 1 And epsilon 2 But the selection principle of (2) is exactly the opposite, which must be followed by ε 1 ＞＞ε 2 The actual parameter selection depends on the case-specific deviation.
4. Using the atomic composition corresponding to the index in the support set to satisfy the measurement matrix of constraint equidistance (Restricted Isometry Property, RIP), if not, making k=k+1, and returning to the step 3;
5. calculating the atomic correlation coefficient |A T r k-1 Extracting the indexes corresponding to the maximum values of the size and storing the indexes into a set C for secondary screening;
6. dividing the size correlation coefficients into a plurality of groups by using a regularization idea in the ROMP according to half of the maximum value sparseness as a threshold value, and selecting a group of indexes corresponding to the group with the maximum energy value to be combined into a support set F;
7. the signal is reconstructed using the least square method and a new residual is found, expressed as follows:
wherein,to reconstruct the signal r new As a new residual error, A F To support the integrated sensing matrix, A F + The inverse matrix of the sensing matrix is represented by y, which is the observation vector;
8. when the energy of the residual norm is less than the clipping threshold, i.e.Step 9 is performed when +.> When the step is executed, the step 10 is executed;
9. requiring the stage=stage+1, the step length step is unchanged, the size of the support set=size+step, the iteration number k=k+1, and executing the step 5 until the reconstruction is completed;
10. the stage stage=stage+1, step=step/2, support set size=size+step, number of iterations k=k+1, until reconstruction is completed.
The steps (1) - (2) finish the estimation of the initial value of the sparsity, the steps (3) - (4) introduce the regularization idea to select matching atoms, the steps (6) - (8) control the sizes of the support set and the iteration step length by introducing two thresholds to improve the recovery precision, and the UWB-LFM signal comparison diagrams before and after reconstruction are shown in fig. 5.
Solving for the DCFrFT matrix ψ of each order in the rough estimation of the UWB-LFM signal using the above-described improved CoSaMP reconstruction algorithm DCFrFT-p Coefficient vector in (a)Obtaining a preliminary peak position->
Performing fine search according to the energy gravity principle: constructing order p E [ p ] with precision of 0.001 coarse -0.015，p coarse +0.015) range DCFrFT matrix
Solving the DCFrFT matrix ψ of each order of UWB-LFM signals in fine search by using the improved CoSaMP reconstruction algorithm DCFrFT-p Coefficient vector in (a)Obtain the position of maximum peak +.>Frequency modulation slope and starting frequency +.>The expression is as follows:
α＝pπ/2 (14)
wherein,for the optimal rotation angle, p is the order corresponding to the optimal rotation angle, S is a dimension normalization factor, and is used for reducing the frequency modulation slope of signals and reducing the angle search range of CFrFT during signal detection so as to improve the efficiency and reduce the complexity in the CFrFT calculation process, and the expression is as follows:
where ΔT is the duration of the signal, f s Is the sampling frequency.
The above-described embodiments are merely preferred embodiments for fully explaining the present invention, and the scope of the present invention is not limited thereto. Equivalent substitutions and changes made by those skilled in the art on the basis of the present invention are all within the scope of the present invention. The protection scope of the invention is subject to the claims.
Claims (3)
1. The UWB-LFM signal parameter estimation method based on CS and simplified FrFT is characterized by comprising the following steps of: the method comprises the following steps:
acquiring an ultra-wideband linear frequency modulation signal containing Gaussian white noise;
constructing a discrete simplified fractional Fourier transform dictionary matrix ultra-wideband linear frequency modulation signal to perform sparse representation;
performing compression sampling on the ultra-wideband linear frequency modulation signal;
reconstructing the coefficient vector using an improved compressed sample matching pursuit algorithm;
estimating the initial frequency and the frequency modulation slope of the ultra-wideband linear frequency modulation signal according to the distribution characteristic of the signal in a discrete fractional Fourier transform domain;
the ultra-wideband linear frequency modulation signal containing Gaussian white noise is obtained, and the expression of a model x of the signal is as follows:
x＝s+n；
wherein s is an ultra-wideband linear frequency modulation signal, and n is additive Gaussian white noise;
constructing a discrete simplified fractional Fourier transform dictionary matrix to perform sparse representation on the sparse-represented ultra-wideband linear frequency modulation signal so as to perform compressive sampling, select the optimal transformation order, and detect and estimate subsequent parameters of subsequent signals;
calculating a discrete reduced fractional order fourier transform dictionary matrix according to the following formula:
wherein ψ represents a discrete reduced-fraction Fourier transform dictionary matrix;
alpha represents the rotation angle of the coordinate axis of the reduced fractional Fourier transform;
n is the number of sampling points of the discretized continuous signal x, and m is the number of discrete points of the discretized continuous signal x after CFrFT;
ΔT＝N/f s n is the number of sampling points, f s For the sampling frequency to be the same,
using the dictionary matrix as an observation matrix, performing sparse representation according to the following formula:
wherein x is the input signal, θ i As coefficient vector, ψ= { ψ 1 ,ψ 2 ,…,ψ i And (3) sparse basis;
the ultra-wideband linear frequency modulation signal is compressed and sampled, and the method is realized according to the following steps:
compressive samplingThe essence is by using an observation matrix phi epsilon R which is uncorrelated with the sparse basis psi M×N To observe the input signal x to obtain M observations y, where M < N; for wideband signals, specific compressed sampling is typically done by an analog information converter AIC.
2. The CS-based and simplified FrFT-based UWB-LFM signal parameter estimation method of claim 1, wherein: in the improved compressed sampling matching pursuit algorithm, the fuzzy threshold of the selection index in the algorithm operation process is calculated according to the following formula:
wherein a is pr And b pr F is a support set, r is a residual error, and A is a sensing matrix; k is the iteration number; i is an index selected to be stored in the supporting set, and j is a maximum index meeting the selection condition; t is t hr Selecting condition parameters of the index;
forming a measurement matrix by using atoms corresponding to the indexes in the support set;
calculating a correlation coefficient, and extracting indexes corresponding to maximum values of the size for secondary screening;
selecting a group of indexes corresponding to the maximum energy value to be merged into a supporting set by utilizing a regularization idea;
reconstructing the signal by using a least square method, and solving a new residual error;
the reconstruction mode of the signal is determined according to the relation between the energy of the residual norm and the clipping threshold.
3. The CS-based and simplified FrFT-based UWB-LFM signal parameter estimation method of claim 1, wherein: the parameter estimation of the ultra-wideband linear frequency modulation signal in the discrete fractional Fourier transform domain is carried out according to the following steps:
the reduced fractional fourier transform distribution of the ultra wideband chirp signal is calculated according to the following formula:
alpha represents the rotation angle of the coordinate axis of the reduced fractional Fourier transform; CF (compact flash) p (u) represents the frequency domain distribution after CFrFT transformation; t is time; u is the frequency;
by performing peak search on the reduced fractional Fourier transform distribution, the initial frequency and the frequency modulation slope of the signal are estimated according to the following formula:
wherein,for the estimation of the frequency modulation slope, +.>For the estimated value of the initial frequency +.>For peak frequency position, +.>For the optimal rotation angle, S is a normalization factor, calculated according to the following formula:
wherein Δt=n/f s N isSampling point number f s Is the sampling frequency.
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