CN109586728B - Signal blind reconstruction method under modulation broadband converter framework based on sparse Bayes - Google Patents
Signal blind reconstruction method under modulation broadband converter framework based on sparse Bayes Download PDFInfo
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Abstract
A signal blind reconstruction method under a modulation broadband converter frame based on sparse Bayes is used in the technical field of reconstruction of compressed sensing signals. The invention solves the problem of poor reconstruction performance of the reconstruction method under the framework of the existing modulation broadband converter when the signal contains noise. Firstly multiplying an input sparse signal by a pseudorandom sequence, then carrying out low-speed sampling and filtering operation on the multiplied signal, then constructing an observation matrix, expressing the signal into a compressed sensing expression form, estimating the signal by adopting a sparse Bayesian method during recovery, iteratively solving a variance gamma of the input sparse signal by an EM (effective magnetic resonance) algorithm, and finishing the reconstruction of the sparse signal; under the condition that the signal-to-noise ratio is equal to-15 dB, compared with the prior method, the reconstruction method can reduce the steady state mean square error value by over 75 percent, and effectively improves the reconstruction performance. The invention can be applied to the field of reconstruction of compressed sensing signals.
Description
Technical Field
The invention belongs to the technical field of reconstruction of compressed sensing signals, and particularly relates to a signal blind reconstruction method under a modulation broadband converter framework.
Background
The CS theorists have proposed Analog Information Conversion (AIC) based on compressed sensing, one of which is mainly the sub-nyquist sampler based on random demodulation, and this scheme mainly aims at multi-tone signals with multiple discrete components, and has the advantages of low power consumption, simple circuit structure, etc., but its disadvantages are also obvious, in practice, because the frequency spectrum of the bandwidth is wide, it can be regarded as infinite discrete components, if a random demodulation scheme is adopted, the frequency spectrum should be discretized first, if the selected frequency resolution is too low, serious spectrum distortion will be caused, and if the selected resolution is higher, the storage and operation will be too heavy. In addition, studies have shown that random demodulation schemes are sensitive to signal models, and large errors can occur when the true sparse basis and the estimated sparse basis deviate.
In 2008, israeli proposed a modulation broadband converter (MWC), and derived and realized related formulas and signal reconstruction, and realized the construction work of hardware. Compared with other analog information converters, MWCs have the following main advantages: the selected signals are all frequency domain continuous signals which are common in life, the requirement on the matching degree of a signal model is not very high, and the existing low-speed ADC can be completely realized. The method has the advantages that the method has disadvantages, and the reconstruction method under the existing modulation broadband converter framework has certain defects, for example, the reconstruction condition of the existing reconstruction method is harsh, so that the reconstruction rate is low, the signal to noise ratio requirement on the signal is high, and the signal reconstruction performance is poor when the signal contains noise.
Disclosure of Invention
The invention aims to solve the problem that the reconstruction performance of the reconstruction method under the framework of the existing modulation broadband converter is poor when a signal contains noise.
The technical scheme adopted by the invention for solving the technical problems is as follows:
a signal blind reconstruction method under a modulation broadband converter framework based on sparse Bayes comprises the following steps:
the method comprises the following steps: inputting a sparse signal x (t), and respectively connecting the input sparse signal x (t) with the pseudo random sequences of m channels of the modulation broadband converterColumn multiplication, the pseudo-random sequences of each channel being mutually orthogonal; then the multiplied result corresponding to each channel is passed through F 0 The sampling frequency of the' is sampled to obtain the sampling result of each channel, and the sampling result of each channel is filtered by a low-pass filter with the cut-off frequency fs/2 to obtain the sampling value y output by each channel i (n) a frequency domain DTFT;
step two: for the obtained sampling value y of each channel output i (n) performing windowing to obtain a windowed signal;
step three: adding Gaussian white noise to the signal subjected to windowing processing obtained in the step two to obtain a signal added with the Gaussian white noise;
step four: calculating an expression of the observation matrix A, and expressing the signal added with the Gaussian white noise obtained in the step three by using a compressed sensing model;
step five: and calculating the edge probability and the posterior probability of the signal added with the Gaussian white noise, applying a sparse Bayesian algorithm to the compressed sensing model obtained in the step four, and iteratively solving the variance gamma of the input sparse signal through an EM (effective-energy-efficient) algorithm to complete the reconstruction of the sparse signal.
The beneficial effects of the invention are: the invention provides a signal blind reconstruction method under a modulation broadband converter frame based on sparse Bayes, and particularly discloses a method for combining sparse Bayes with a modulation broadband converter, wherein the method comprises the steps of multiplying an input sparse signal by a pseudorandom sequence, then carrying out low-speed sampling and filtering operation on the signal obtained by multiplication, then constructing an observation matrix, representing the signal into a compressed sensing representation form, estimating the signal by adopting a sparse Bayes method during recovery, and iteratively solving a variance gamma of the input sparse signal by an EM (effective electromagnetic) algorithm to complete the reconstruction of the sparse signal; the invention combines the sparse Bayes with the modulation broadband converter framework to realize the reconstruction under the condition that the signal sparsity is unknown, and compared with the prior art, the reconstruction method can reduce the steady state mean square error value by more than 75 percent under the condition that the signal to noise ratio of the signal is-15 dB, thereby effectively improving the reconstruction performance.
Drawings
FIG. 1 is a time domain waveform of an original signal in the presence of noise;
FIG. 2 is a time domain waveform of a reconstructed signal in the presence of noise;
the abscissa of fig. 1 and 2 is time and the ordinate is signal amplitude;
FIG. 3 is a frequency domain waveform of an original signal in the presence of noise;
FIG. 4 is a frequency domain waveform of a reconstructed signal in the presence of noise;
the abscissa of fig. 3 and 4 is frequency and the ordinate is signal amplitude;
fig. 5 is a graph comparing the reconstruction performance using the method of the present invention with that using the conventional method.
The abscissa of fig. 5 is the signal-to-noise ratio and the ordinate is the steady state mean square error.
Detailed Description
The technical solution of the present invention is further described below with reference to the accompanying drawings, but not limited thereto, and any modification or equivalent replacement of the technical solution of the present invention without departing from the spirit and scope of the technical solution of the present invention shall be covered by the protection scope of the present invention.
The first embodiment is as follows: the blind signal reconstruction method based on the sparse Bayesian modulation broadband converter framework in the embodiment comprises the following steps:
the method comprises the following steps: inputting sparse signals x (t), and multiplying the input sparse signals x (t) with pseudo-random sequences of m channels of a modulation broadband converter respectively, wherein the pseudo-random sequences of each channel are mutually orthogonal; then the multiplied result corresponding to each channel is passed through F 0 The sampling frequency of the' is sampled to obtain the sampling result of each channel, and the sampling result of each channel is filtered by a low-pass filter with the cut-off frequency fs/2 to obtain the sampling value y output by each channel i (n) a frequency domain DTFT;
step two: for the obtained sampling value y of each channel output i (n) performing windowing to obtain a windowed signal;
step three: adding Gaussian white noise to the signal subjected to windowing processing obtained in the step two to obtain a signal added with the Gaussian white noise;
step four: calculating an expression of the observation matrix A, and expressing the signal added with the Gaussian white noise obtained in the third step by using a compressed sensing model;
step five: and calculating the edge probability and the posterior probability of the signal added with the Gaussian white noise, applying a sparse Bayesian algorithm to the compressed sensing model obtained in the step four, and iteratively solving the variance gamma of the input sparse signal through an EM (effective-energy-efficient) algorithm to complete the reconstruction of the sparse signal.
The second embodiment is as follows: the embodiment further defines the signal blind reconstruction method based on the modulation broadband converter framework based on the sparse bayes in the first embodiment, and the specific process of the first step is as follows:
the pseudo-random sequence of the ith channel of the modulated wideband converter is p i (t) the pseudo-random sequence p is known from Fourier transform i The specific expression of (t) is as follows:
wherein: l is a Fourier series, c il Is a Fourier coefficient, j is a complex unit, T P Is the period of the pseudorandom sequence, t is time;
obtaining a Fourier coefficient c according to the inverse Fourier transform il The expression of (a) is:
inputting a sparse signal x (t), multiplying the input sparse signal x (t) with the pseudo-random sequences of m channels of the modulation broadband converter respectively to obtain a frequency domain Y of the ith channel of a multiplied result i (f) The expression of (a) is:
wherein: f is the frequency domain;
by sampling frequency F 0 Sampling the obtained frequency domain of each channel to obtain a sampling result;
the sampling result is passed through a low-pass filter whose cut-off frequency is fs/2, and the high-frequency component of the filtered signal is filtered out and only the frequency of 0 to fs/2 is retained to obtain the filtered signal, i.e. the sampling value y outputted by the ith channel is obtained by filtering i The expression of frequency domain DTFT of (n) is:
wherein:
sample value y representing output of ith channel i Frequency domain of (n), T s Is F 0 Reciprocal of `, f p Is the frequency of the pseudorandom sequence, X (-) is the frequency domain of X (t), L 0 Is the smallest positive integer, L, that causes all non-zero values of the sparse signal x (t) to be contained in the low-pass filter 0 The expression of (a) is:
wherein: f. of nyq At the nyquist sampling rate.
The third concrete implementation mode: the embodiment further defines the signal blind reconstruction method under the framework of the modulation broadband converter based on the sparse bayes in the first embodiment, and the specific process of the fourth step is as follows:
and (4) expressing the signal added with the Gaussian white noise obtained in the step three by using a compressed sensing model as follows:
y(f)＝Az(f) (5)
wherein: intermediate variable z (f) = [ z = 1 (f),...,z L (f)] T And z is i′ (f)＝X(f+(i′-L 0 -1)f p ),1≤i′≤L，L＝2L 0 +1; y (f) is a vector of length m, and
a is an observation matrix;
after sampling and filtering, the Fourier coefficient is represented by c il Is changed into c il ′：
Wherein: a is ik Is the value of the pseudo-random sequence of the ith channel, k =0,1, \8230;, L 0 -1；
Defining the integral term of equation (6) as d l ：
Wherein: intermediate variables
Then
The observation matrix a is then expressed as:
A＝SFD (8)
wherein: the intermediate variable matrix F is L x L dimensional matrix, and the ith column of the matrix F is represented as F i″ ＝[θ 0 ,θ 1*i″ ,...,θ (L-1)*i″ ] T Is representative of multiplying, -L 0 ＜i″＜L 0 (ii) a D is a diagonal array of L rows and L columns, and the diagonal array D is in the form of
S is a symbol matrix with m rows and L columns, and the form of the symbol matrix S is
Therefore, the formula y (f) = Az (f) is converted into the form of formula (9):
the fourth concrete implementation mode: the embodiment further defines the signal blind reconstruction method under the framework of the modulation broadband converter based on the sparse bayes in the first embodiment, and the specific process of the fifth step is as follows:
step five, the edge probability of the signal after the Gaussian white noise is added is expressed as follows:
wherein: p (Y) ·s ；γ,σ 2 ) The representative parameters are gamma and sigma 2 Edge probability of p (Y) ·s |X ·s ；σ 2 ) Representative parameter is σ 2 Conditional probability of (A), p (X) ·s (ii) a γ) represents the prior probability of the parameter being γ;
Y ·s sampled value y for output of ith channel i At the s-th column of the frequency domain DTFT of (n), γ is the variance of the input sparse signal, σ 2 For the output sampled value y i (n) a variance; x ·s Is the s-th column of the input sparse signal;
is a matrix Y ·s The transpose of (a) is performed,
is a matrix sigma Y The inverse matrix of (d); edge probability covariance matrix sigma Y The expression of (a) is:
Σ Y ＝σ 2 I+AΓA T (11)
wherein: Γ = diag (γ), Γ being the diagonal matrix of the matrix γ; a is an observation matrix, I is an identity matrix,
posterior probability p (X) of signal after adding Gaussian white noise ·s |Y ·s ；γ,σ 2 ) Expressed as: the posterior probability is also Gaussian distribution
Wherein: p (X) ·s |Y ·s ；γ,σ 2 ) The representative parameters are gamma and sigma 2 A posterior probability of (d), mu ·s Is an average value, and
posterior probability covariance matrix
The expression of (a) is:
fifthly, randomly initializing the variance gamma of the input sparse signal and the output sampling value y i (n) variance σ 2 ；
For the first iteration, by formula
To calculate the variance gamma of the input sparse signal corresponding to the first iteration 1 (ii) a Wherein:
||·|| 2 represents a norm;
intermediate variables
Σ ii Is composed of
Any element on the diagonal of (1);
by passingMaximum a posteriori probability to update
And mu ·s ；
For the second iteration, update Σ by equation (14) ii Reuse of updated sigma ii And mu ·s To calculate the variance gamma of the input sparse signal corresponding to the second iteration 2 ：
Then according to the updated gamma 2 To update equations (14) and (15);
and so on until the variance gamma of the input sparse signal converges to a certain fixed point gamma * Stopping iteration, obtaining maximum posterior probability by using gamma value when stopping iteration, and obtaining mean value mu by using maximum posterior probability ·s Mean value of μ ·s Namely the most sparse data to be recovered, and the reconstruction of the sparse signal is completed.
The effects of the invention are explained with the attached drawings: as shown in fig. 1 and 2, fig. 1 and 2 are a time domain waveform diagram of an original signal and a time domain waveform diagram of a reconstructed signal in the presence of noise, respectively; as shown in fig. 3 and 4, fig. 3 and 4 are a frequency domain waveform diagram of an original signal and a frequency domain waveform diagram of a reconstructed signal, respectively, in the presence of noise;
the method can recover the sparse signal under the condition of unknown sparsity of the signal, as shown in fig. 5, compared with the traditional reconstruction method, under the condition that the signal-to-noise ratio is-15 dB, the reconstruction method can reduce the steady state mean square error value by over 75 percent, and the method can effectively reduce the sparsity of the signal and improve the reconstruction performance.
The fifth concrete implementation mode is as follows: the present embodiment further defines the signal blind reconstruction method under the framework of the modulation wideband converter based on the sparse bayes in the first or second embodiment, where F is 0 'the value range is [ fs/4,fs/2']。
The sixth specific implementation mode: in this embodiment, the signal blind reconstruction method under the framework of the modulation wideband converter based on the sparse bayes in the first embodiment is further defined, and in the second step, the sampling value y output by each channel is obtained i And (n) performing windowing, wherein a windowing function adopted in the windowing is a Hamming window. The fence effect of the signal is avoided.
The above description is only for the preferred embodiment of the present invention, but the scope of the present invention is not limited thereto, and any person skilled in the art should be considered to be within the technical scope of the present invention, and the technical solutions and the inventive concepts thereof according to the present invention should be equivalent or changed within the scope of the present invention.
Claims (3)
1. A signal blind reconstruction method under a modulation broadband converter framework based on sparse Bayes is characterized by comprising the following steps:
the method comprises the following steps: inputting sparse signals x (t), and multiplying the input sparse signals x (t) with pseudo-random sequences of m channels of a modulation broadband converter respectively, wherein the pseudo-random sequences of each channel are mutually orthogonal; then the multiplied result corresponding to each channel is passed through F 0 The sampling frequency of the sampling frequency is sampled to obtain the sampling result of each channel, and the sampling result of each channel is filtered by a low-pass filter with the cut-off frequency of fs/2 to obtain the sampling value y output by each channel i (n) a frequency domain DTFT;
the specific process of the step one is as follows:
the pseudo-random sequence of the ith channel of the modulated wideband converter is p i (t) obtaining a pseudorandom sequence p by Fourier transform i The specific expression of (t) is as follows:
wherein: l is a Fourier series, c il Is a Fourier coefficient, j is a complex unit, T P Is the period of the pseudorandom sequence, t is time;
obtaining a Fourier coefficient c according to the inverse Fourier transform il The expression of (a) is:
inputting a sparse signal x (t), multiplying the input sparse signal x (t) with the pseudo-random sequences of m channels of the modulation broadband converter respectively to obtain a frequency domain Y of the ith channel of a multiplied result i (f) The expression of (a) is:
wherein: f is the frequency domain;
sampling the obtained frequency domain of each channel through a sampling frequency F to obtain a sampling result;
and the sampling result is passed through low-pass filter whose cut-off frequency is fs/2 to obtain filtered signal, i.e. filtering to obtain sampling value y outputted by ith channel i The expression of frequency domain DTFT of (n) is:
wherein:
sample value y representing output of ith channel i Frequency domain of (n), T s Is the reciprocal of F, F p Is the frequency of the pseudorandom sequence, X (-) is the frequency domain of X (t), L 0 All non-zero values of the sparse signal x (t) are included in the low-pass filteringMinimum positive integer of the device, L 0 The expression of (c) is:
wherein: f. of nyq At the nyquist sampling rate;
step two: for the obtained sampling value y of each channel output i (n) performing windowing to obtain a windowed signal;
step three: adding Gaussian white noise to the signal subjected to windowing processing obtained in the step two to obtain a signal added with the Gaussian white noise;
step four: calculating an expression of the observation matrix A, and expressing the signal added with the Gaussian white noise obtained in the third step by using a compressed sensing model;
the specific process of the step four is as follows:
and (3) expressing the signal added with the Gaussian white noise obtained in the step three by using a compressed sensing model as follows:
y(f)＝Az(f) (5)
wherein: intermediate variable z (f) = [ z = 1 (f),...,z L (f)] T And z is i′ (f)＝X(f+(i′-L 0 -1)f p ),1≤i′≤L，L＝2L 0 +1; y (f) is a vector of length m, and
a is an observation matrix;
after sampling and filtering, the Fourier coefficient is represented by c il Is changed into c il ′：
Wherein: a is ik Is the value of the pseudo-random sequence of the ith channel, k =0,1, \8230;, L 0 -1；
Defining the integral term of equation (6) as d l ：
Wherein: intermediate variables
Then
The observation matrix a is then expressed as:
A＝SFD (8)
wherein: the intermediate variable matrix F is a L-dimension matrix, and the ith' column of the matrix F is represented as F i″ ＝[θ 0 ,θ 1*i″ ,...,θ (L-1)*i″ ] T ，-L 0 ＜i″＜L 0 (ii) a D is a diagonal array of L rows and L columns, and the diagonal array D is in the form of
S is a symbol matrix with m rows and L columns, and the form of the symbol matrix S is
Therefore, the formula y (f) = Az (f) is converted into the form of formula (9):
step five: calculating the edge probability and the posterior probability of the signal added with the Gaussian white noise, applying a sparse Bayesian algorithm to the compressed sensing model obtained in the step four, iteratively solving the variance gamma of the input sparse signal through an EM (effective electromagnetic) algorithm, and completing the reconstruction of the sparse signal;
the concrete process of the step five is as follows:
step five, the edge probability of the signal after the Gaussian white noise is added is expressed as follows:
wherein: p (Y) ·s ；γ,σ 2 ) Representative parameters are γ and σ 2 Edge probability of p (Y) ·s |X ·s ；σ 2 ) The representative parameter is σ 2 Conditional probability of (A), p (X) ·s (ii) a γ) represents the prior probability of the parameter being γ;
Y ·s sample value y for output of ith channel i At the s-th column of the frequency domain DTFT of (n), γ is the variance of the input sparse signal, σ 2 For the output sampled value y i (n) variance; x ·s Is the s-th column of the input sparse signal;
is a matrix Y ·s The method (2) is implemented by the following steps,
is a matrix sigma Y The inverse matrix of (d); edge probability covariance matrix sigma Y The expression of (a) is:
Σ Y ＝σ 2 I+AΓA T (11)
wherein: Γ = diag (γ), Γ being the diagonal matrix of the matrix γ; a is an observation matrix, I is an identity matrix,
posterior probability p (X) of signal after adding Gaussian white noise ·s |Y ·s ；γ,σ 2 ) Expressed as:
wherein: p (X) ·s |Y ·s ；γ,σ 2 ) Representative parameters are γ and σ 2 A posterior probability of (d), mu ·s Is an average value, and
posterior probability covariance matrix
The expression of (c) is:
fifthly, randomly initializing the variance gamma of the input sparse signal and the output sampling value y i (n) variance σ 2 ；
For the first iteration, by formula
To calculate the variance gamma of the input sparse signal corresponding to the first iteration 1 (ii) a Wherein:
||·|| 2 represents a norm;
intermediate variables
Σ ii Is composed of
Any element on the diagonal of (1);
updating by maximum a posteriori probability
And mu ·s ；
For the second iteration, update Σ by equation (14) ii Reuse of updated sigma ii And mu ·s To calculate the variance gamma of the input sparse signal corresponding to the second iteration 2 ：
Then according to the updated gamma 2 To update equations (14) and (15);
and so on until the variance gamma of the input sparse signal converges to a certain fixed point gamma * Stopping iteration, obtaining maximum posterior probability by using gamma value when iteration is stopped, and obtaining mean value mu by using maximum posterior probability ·s Mean value of μ ·s Namely the most sparse data to be recovered, and the reconstruction of the sparse signal is completed.
2. The sparse Bayesian-based blind reconstruction method for signals under the framework of modulated wideband converters as claimed in claim 1, wherein F is 0 The value range of (b) is [ fs/4,fs/2]。
3. The method for blind reconstruction of signals under the framework of the sparse Bayesian-based modulation wideband converter as claimed in claim 1, wherein the sampling value y of each channel output obtained in the second step i And (n) performing windowing, wherein a windowing function adopted in the windowing is a Hamming window.
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