CN109586728A - Signal blind reconstructing method under modulation wide-band transducer frame based on sparse Bayesian - Google Patents
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Abstract
Signal blind reconstructing method under modulation wide-band transducer frame based on sparse Bayesian, it is used for the reconfiguration technique field of compressed sensing signal.The present invention solves the problems, such as that reconstruction property of the reconstructing method when signal contains noise under existing modulation wide-band transducer frame is poor.Input sparse signal is multiplied by the present invention with pseudo-random sequence first, then low speed sampling and filtering operation are carried out to the signal that multiplication obtains, then observing matrix is constructed, signal is expressed as to the representation of compressed sensing, when restoring, signal is estimated using sparse Bayesian method, the variance γ of input sparse signal is acquired by EM algorithm iteration, completes the reconstruct of sparse signal；In the case where Signal-to-Noise is all -15dB, compared with the conventional method, Steady State Square Error value can be reduced by 75% or more by reconstructing method of the invention, effectively improve reconstruction property.Present invention could apply to the reconstruction fields of compressed sensing signal.
Description
Technical field
The invention belongs to the reconfiguration technique fields of compressed sensing signal, and in particular under a kind of modulation wide-band transducer frame
Signal blind reconstructing method.
Background technique
CS theory scholar proposes analog information conversion (AIC) on the basis of compressed sensing, one of them is main just
It is the sub- nyquist sampling device based on random demodulation, the program is primarily directed to have the multitone of multiple discrete components to believe
Number, have many advantages, such as low in energy consumption, circuit structure is simple, but he the shortcomings that it is similarly obvious, in practice, due to bandwidth frequency spectrum compared with
Width can regard infinite multiple discrete components as, according to random demodulation scheme, first discretization should be carried out to frequency spectrum, if selection
Frequency resolution it is too low, it will cause serious distortion spectrum, if the resolution ratio of selection is higher, it will cause storage and operations
Burden it is too heavy.In addition, studies have shown that random demodulation scheme is more sensitive to signal model, when true sparse basis and estimation
Sparse basis when having deviation, it may appear that large error.
2008, Israel scholar proposed modulation wide-band transducer (MWC), and derivation realizes relevant formula and letter
Number reconstruct, and realizes the building of hardware.MWC mainly has the advantage that it compared with other analog information converters
The signal of selection is all relatively common frequency domain continuous signal in life, to the matching degree of signal model require be not it is very high,
Existing low speed ADC may be implemented completely.It is advantageous just to have disadvantage, the reconstructing method under existing modulation wide-band transducer frame
There is also certain deficiency, such as the reconstruction condition of existing reconstructing method are harsher, cause reconstruct rate lower, especially pair
The signal-to-noise ratio of signal is more demanding, and signal reconstruction performance when containing noise is poor.
Summary of the invention
The purpose of the present invention is contain noise in signal to solve the reconstructing method under existing modulation wide-band transducer frame
When reconstruction property difference problem.
The technical solution adopted by the present invention to solve the above technical problem is:
Signal blind reconstructing method under modulation wide-band transducer frame based on sparse Bayesian, this method include following step
It is rapid:
Step 1: input sparse signal x (t), the sparse signal x (t) of input is a with the m of modulation wide-band transducer respectively
The pseudo-random sequence in channel is multiplied, and the pseudo-random sequence in each channel is mutually orthogonal；Again by the corresponding multiplication in each channel
Result passes through F afterwards0' sample frequency sampled to obtain the sampled result in each channel, and by the sampled result in each channel
It is filtered by the low-pass filter that cutoff frequency is fs/2, obtains the sampled value y of each channel outputi(n) frequency domain
DTFT；
Step 2: to the sampled value y of each channel output of acquisitioni(n) after progress windowing process obtains windowing process
Signal；
Step 3: the signal after the windowing process obtained to step 2 adds white Gaussian noise, obtains addition Gauss white noise
Signal after sound；
Step 4: calculating the expression formula of observing matrix A, the signal benefit after the addition white Gaussian noise that step 3 is obtained
It is indicated with compressed sensing model；
Step 5: the marginal probability and posterior probability of the signal after calculating addition white Gaussian noise, and by sparse Bayesian
Algorithm is applied to the compressed sensing model that step 4 obtains, and the variance γ of input sparse signal is acquired by EM algorithm iteration, complete
At the reconstruct of sparse signal.
The beneficial effects of the present invention are: the invention proposes a kind of modulation wide-band transducer frame based on sparse Bayesian
Lower signal blind reconstructing method specifically discloses a kind of method for combining sparse Bayesian with modulation wide-band transducer, the present invention
Input sparse signal is multiplied with pseudo-random sequence first, low speed sampling then is carried out to the signal that multiplication obtains and filtering is grasped
Make, then construct observing matrix, signal is expressed as to the representation of compressed sensing, when restoring, using sparse Bayesian side
Method estimates signal, and the variance γ of input sparse signal is acquired by EM algorithm iteration, completes the reconstruct of sparse signal；This
Sparse Bayesian in conjunction with modulation wide-band transducer frame, is realized the reconstruct in the case where signal degree of rarefication is unknown by invention,
In the case where Signal-to-Noise is all -15dB, compared with the conventional method, reconstructing method of the invention can be by stable state mean square error
Difference reduces by 75% or more, effectively improves reconstruction property.
Detailed description of the invention
Fig. 1 is original signal time domain waveform in the presence of noise；
Fig. 2 is reconstruction signal time domain waveform in the presence of noise；
The abscissa of Fig. 1 and Fig. 2 is the time, and ordinate is signal amplitude；
Fig. 3 is original signal frequency-domain waveform figure in the presence of noise；
Fig. 4 is reconstruction signal frequency-domain waveform figure in the presence of noise；
The abscissa of Fig. 3 and Fig. 4 is frequency, and ordinate is signal amplitude；
Fig. 5 is the comparison diagram using method and the reconstruction property using conventional method of the invention.
The abscissa of Fig. 5 is signal-to-noise ratio, and ordinate is Steady State Square Error.
Specific embodiment
Further description of the technical solution of the present invention with reference to the accompanying drawing, and however, it is not limited to this, all to this
Inventive technique scheme is modified or replaced equivalently, and without departing from the spirit and scope of the technical solution of the present invention, should all be covered
Within the protection scope of the present invention.
Specific embodiment 1: believing under modulation wide-band transducer frame described in present embodiment based on sparse Bayesian
Number blind reconstructing method, method includes the following steps:
Step 1: input sparse signal x (t), the sparse signal x (t) of input is a with the m of modulation wide-band transducer respectively
The pseudo-random sequence in channel is multiplied, and the pseudo-random sequence in each channel is mutually orthogonal；Again by the corresponding multiplication in each channel
Result passes through F afterwards0' sample frequency sampled to obtain the sampled result in each channel, and by the sampled result in each channel
It is filtered by the low-pass filter that cutoff frequency is fs/2, obtains the sampled value y of each channel outputi(n) frequency domain
DTFT；
Step 2: to the sampled value y of each channel output of acquisitioni(n) after progress windowing process obtains windowing process
Signal；
Step 3: the signal after the windowing process obtained to step 2 adds white Gaussian noise, obtains addition Gauss white noise
Signal after sound；
Step 4: calculating the expression formula of observing matrix A, the signal benefit after the addition white Gaussian noise that step 3 is obtained
It is indicated with compressed sensing model；
Step 5: the marginal probability and posterior probability of the signal after calculating addition white Gaussian noise, and by sparse Bayesian
Algorithm is applied to the compressed sensing model that step 4 obtains, and the variance γ of input sparse signal is acquired by EM algorithm iteration, complete
At the reconstruct of sparse signal.
Specific embodiment 2: present embodiment turns the modulation broadband described in embodiment one based on sparse Bayesian
Signal blind reconstructing method is further limited under parallel operation frame, the detailed process of the step 1 are as follows:
The pseudo-random sequence for modulating i-th channel of wide-band transducer is pi(t), according to Fourier transformation it is found that it is pseudo- with
Machine sequence pi(t) expression are as follows:
Wherein: l is Fourier space, cilFor Fourier coefficient, j is complex unit, TPIt is the period of pseudo-random sequence, t
For the time；
According to Fourier inversion, Fourier coefficient c is obtainedilExpression formula are as follows:
It inputs sparse signal x (t), by the m channel with modulation wide-band transducer respectively the sparse signal x (t) of input
Pseudo-random sequence is multiplied, the frequency domain Y in i-th channel of result after being multipliedi(f) expression formula are as follows:
Wherein: f is frequency domain；
Pass through sample frequency F0' frequency domain in every obtained channel is sampled to obtain sampled result；
And sampled result is passed through into the low-pass filter that cutoff frequency is fs/2, by filtered signal, high fdrequency component
It is filtered out and only retains 0 frequency for arriving fs/2, obtain filtered signal, i.e., the sampling of i-th channel output is obtained by filtering
Value yi(n) expression formula of frequency domain DTFT are as follows:
Wherein:Represent the sampled value y of i-th channel outputi(n) frequency domain, TsFor F0' inverse, fpFor puppet
The frequency of random sequence, X () are the frequency domain of x (t), L0It is that whole nonzero values of sparse signal x (t) is made to be included in low-pass filtering
The minimum positive integer of device, L0Expression formula are as follows:
Wherein: fnyqFor Nyquist sampling rate.
Specific embodiment 3: present embodiment turns the modulation broadband described in embodiment one based on sparse Bayesian
Signal blind reconstructing method is further limited under parallel operation frame, the detailed process of the step 4 are as follows:
Signal after addition white Gaussian noise that step 3 obtains is expressed as using compressed sensing model:
Y (f)=Az (f) (5)
Wherein: intermediate variable z (f)=[z1(f),...,zL(f)]T, and zi′(f)=X (f+ (i '-L0-1)fp),1≤i′
≤ L, L=2L0+1；Y (f) is the vector that length is m, andA is observing matrix；
After over-sampling and filtering, Fourier coefficient is by cilBecome cil':
Wherein: aikFor the value of the pseudo-random sequence in i-th channel, k=0,1 ..., L0-1；
The integral term of defined formula (6) is dl:
Wherein: intermediate variableThen
Then observing matrix A is indicated are as follows:
A=SFD (8)
Wherein: intermediate variable matrix F is that L*L ties up matrix, matrix F i-th " list is shown as Fi″=[θ0,θ1*i″,...,
θ(L-1)*i″]T, * representative, which is done, to be multiplied ,-L0< i " < L0；D is the diagonal matrix of L row L column, and the form of diagonal matrix D is
S is the sign matrix of m row L column, and the form of sign matrix S is
So formula y (f)=Az (f) to be converted to the form of formula (9):
Specific embodiment 4: present embodiment turns the modulation broadband described in embodiment one based on sparse Bayesian
Signal blind reconstructing method is further limited under parallel operation frame, the detailed process of the step 5 are as follows:
The marginal probability of signal after step 5 one, addition white Gaussian noise indicates are as follows:
Wherein: p (Y·s；γ,σ2) representation parameter be γ and σ2Marginal probability, p (Y·s|X·s；σ2) representation parameter be σ2's
Conditional probability, p (X·s；γ) representation parameter is the prior probability of γ；
Y·sFor the sampled value y of the output in i-th channeli(n) s of frequency domain DTFT is arranged, and γ is input sparse signal
Variance, σ2For the sampled value y of outputi(n) variance；X·sFor the s column for inputting sparse signal；For matrix Y·sTransposition,For matrix ΣYInverse matrix；Marginal probability covariance matrix ΣYExpression formula are as follows:
ΣY=σ2I+AΓAT (11)
Wherein: Γ=diag (γ), Γ are to seek diagonal matrix to matrix γ；A is observing matrix, and I is unit matrix,
Posterior probability p (the X of signal after adding white Gaussian noise·s|Y·s；γ,σ2) indicate are as follows: posterior probability is similarly
Gaussian Profile
Wherein: p (X·s|Y·s；γ,σ2) representation parameter be γ and σ2Posterior probability, μ·sFor mean value, andPosterior probability covariance matrixExpression formula are as follows:
The sampled value y of step 5 two, the variance γ of random initializtion input sparse signal and outputi(n) variances sigma2；
For first time iteration, pass through formulaIt is dilute to calculate the corresponding input of first time iteration
Dredge the variance γ of signal1；Wherein:||·||2Represent norm；
Intermediate variableΣiiForDiagonal line on either element；
It is updated by maximum a posteriori probabilityAnd μ·s；
For second of iteration, Σ is updated by formula (14)ii, recycle updated ΣiiAnd μ·sTo calculate second
The variance γ of the corresponding input sparse signal of iteration2:Further according to updated γ2To update public affairs
Formula (14) and (15)；
And so on, until the variance γ of input sparse signal converges to some fixed point γ*When stop iteration, utilize stopping
γ value when iteration acquires maximum a posteriori probability, recycles the maximum a posteriori probability to acquire mean μ·s, mean μ·sIt is as to be restored
Most sparse data, complete the reconstruct of sparse signal.
Effect of the invention is described with reference to the drawings: as depicted in figs. 1 and 2, Fig. 1 and Fig. 2 are respectively in noisy situation
Under, the time domain waveform of original signal and the time domain waveform of reconstruction signal；As shown in Figure 3 and Figure 4, Fig. 3 and Fig. 4 respectively exists
In noisy situation, the frequency-domain waveform figure of original signal and the frequency-domain waveform figure of reconstruction signal；
The present invention can carry out sparse signal recovery in the case where the degree of rarefication of unknown signaling, as shown in figure 5, with tradition
Reconstructing method compare, in the case where signal-to-noise ratio is all -15dB, reconstructing method of the invention can be by Steady State Square Error value
75% or more is reduced, method of the invention can effectively reduce signal degree of rarefication, promote reconstruction property.
Specific embodiment 5: present embodiment is wide to the modulation described in embodiment one or two based on sparse Bayesian
Signal blind reconstructing method is further limited under tape switching unit frame, the F0' value range be [fs/4, fs/2].
Specific embodiment 6: present embodiment turns the modulation broadband described in embodiment one based on sparse Bayesian
Signal blind reconstructing method is further limited under parallel operation frame, adopting to each channel output of acquisition in the step 2
Sample value yi(n) windowing process is carried out, the windowed function that windowing process uses is Hamming window.Avoid the fence effect of signal.
The foregoing is only a preferred embodiment of the present invention, but scope of protection of the present invention is not limited thereto,
Anyone skilled in the art in the technical scope disclosed by the present invention, according to the technique and scheme of the present invention and its
Inventive concept is subject to equivalent substitution or change, should be covered by the protection scope of the present invention.
Claims (6)
1. signal blind reconstructing method under the modulation wide-band transducer frame based on sparse Bayesian, which is characterized in that this method packet
Include following steps:
Step 1: input sparse signal x (t), by the sparse signal x (t) of input respectively with modulation wide-band transducer m channel
Pseudo-random sequence be multiplied, the pseudo-random sequence in each channel is mutually orthogonal；It will be tied after the corresponding multiplication in each channel again
Fruit passes through F0' sample frequency sampled to obtain the sampled result in each channel, and the sampled result in each channel is passed through
Cutoff frequency is that the low-pass filter of fs/2 is filtered, and obtains the sampled value y of each channel outputi(n) frequency domain DTFT；
Step 2: to the sampled value y of each channel output of acquisitioni(n) it carries out windowing process and obtains the signal after windowing process；
Step 3: the signal after the windowing process obtained to step 2 adds white Gaussian noise, after obtaining addition white Gaussian noise
Signal；
Step 4: calculating the expression formula of observing matrix A, the signal utilization pressure after the addition white Gaussian noise that step 3 is obtained
Contracting sensor model indicates；
Step 5: the marginal probability and posterior probability of the signal after calculating addition white Gaussian noise, and by sparse Bayesian algorithm
It is applied to the compressed sensing model that step 4 obtains, the variance γ of input sparse signal is acquired by EM algorithm iteration, is completed dilute
Dredge the reconstruct of signal.
2. signal blind reconstructing method under the modulation wide-band transducer frame according to claim 1 based on sparse Bayesian,
It is characterized in that, the detailed process of the step 1 are as follows:
The pseudo-random sequence for modulating i-th channel of wide-band transducer is pi(t), according to Fourier transformation, pseudo-random sequence is obtained
pi(t) expression are as follows:
Wherein: l is Fourier space, cilFor Fourier coefficient, j is complex unit, TPIt is the period of pseudo-random sequence, when t is
Between；
According to Fourier inversion, Fourier coefficient c is obtainedilExpression formula are as follows:
Input sparse signal x (t), by the sparse signal x (t) of input respectively with modulation wide-band transducer m channel puppet with
Machine sequence is multiplied, the frequency domain Y in i-th channel of result after being multipliedi(f) expression formula are as follows:
Wherein: f is frequency domain；
Pass through sample frequency F0' frequency domain in every obtained channel is sampled to obtain sampled result；
And sampled result is passed through into the low-pass filter that cutoff frequency is fs/2, filtered signal is obtained, i.e., by filtering
The sampled value y exported to i-th channeli(n) expression formula of frequency domain DTFT are as follows:
Wherein:Represent the sampled value y of i-th channel outputi(n) frequency domain, TsFor F0' inverse, fpFor pseudorandom
The frequency of sequence, X () are the frequency domain of x (t), L0It is that whole nonzero values of sparse signal x (t) is made to be included in low-pass filter
Minimum positive integer, L0Expression formula are as follows:
Wherein: fnyqFor Nyquist sampling rate.
3. signal blind reconstructing method under the modulation wide-band transducer frame according to claim 1 based on sparse Bayesian,
It is characterized in that, the detailed process of the step 4 are as follows:
Signal after addition white Gaussian noise that step 3 obtains is expressed as using compressed sensing model:
Y (f)=Az (f) (5)
Wherein: intermediate variable z (f)=[z1(f),...,zL(f)]T, and zi′(f)=X (f+ (i '-L0-1)fp), 1≤i '≤L, L
=2L0+1；Y (f) is the vector that length is m, andA is observing matrix；
After over-sampling and filtering, Fourier coefficient is by cilBecome cil':
Wherein: aikFor the value of the pseudo-random sequence in i-th channel, k=0,1 ..., L0-1；
The integral term of defined formula (6) is dl:
Wherein: intermediate variableThen
Then observing matrix A is indicated are as follows:
A=SFD (8)
Wherein: intermediate variable matrix F is that L*L ties up matrix, matrix F i-th " list is shown as Fi″=[θ0,θ1*i″,...,θ(L-1)*i″]T,-
L0< i " < L0；D is the diagonal matrix of L row L column, and the form of diagonal matrix D isS is the sign matrix of m row L column,
The form of sign matrix S is
So formula y (f)=Az (f) to be converted to the form of formula (9):
4. signal blind reconstructing method under the modulation wide-band transducer frame according to claim 1 based on sparse Bayesian,
It is characterized in that, the detailed process of the step 5 are as follows:
The marginal probability of signal after step 5 one, addition white Gaussian noise indicates are as follows:
Wherein: p (Y·s；γ,σ2) representation parameter be γ and σ2Marginal probability, p (Y·s|X·s；σ2) representation parameter be σ2Condition
Probability, p (X·s；γ) representation parameter is the prior probability of γ；
Y·sFor the sampled value y of the output in i-th channeli(n) s of frequency domain DTFT is arranged, and γ is the variance for inputting sparse signal,
σ2For the sampled value y of outputi(n) variance；X·sFor the s column for inputting sparse signal；For matrix Y·sTransposition,For square
Battle array ΣYInverse matrix；Marginal probability covariance matrix ΣYExpression formula are as follows:
ΣY=σ2I+AΓAT (11)
Wherein: Γ=diag (γ), Γ are to seek diagonal matrix to matrix γ；A is observing matrix, and I is unit matrix,
Posterior probability p (the X of signal after adding white Gaussian noise·s|Y·s；γ,σ2) indicate are as follows:
Wherein: p (X·s|Y·s；γ,σ2) representation parameter be γ and σ2Posterior probability, μ·sFor mean value, andPosterior probability covariance matrixExpression formula are as follows:
The sampled value y of step 5 two, the variance γ of random initializtion input sparse signal and outputi(n) variances sigma2；
For first time iteration, pass through formulaTo calculate the corresponding sparse letter of input of first time iteration
Number variance γ1；Wherein:||·||2Represent norm；
Intermediate variableΣiiForDiagonal line on either element；
It is updated by maximum a posteriori probabilityAnd μ·s；
For second of iteration, Σ is updated by formula (14)ii, recycle updated ΣiiAnd μ·sTo calculate second of iteration
The variance γ of corresponding input sparse signal2:Further according to updated γ2Carry out more new formula
(14) and (15)；
And so on, until the variance γ of input sparse signal converges to some fixed point γ*When stop iteration, utilize stop iteration
When γ value acquire maximum a posteriori probability, recycle the maximum a posteriori probability to acquire mean μ·s, mean μ·sAs it is to be restored most
Sparse data completes the reconstruct of sparse signal.
5. the blind reconstruct side of signal under the modulation wide-band transducer frame according to claim 1 or 2 based on sparse Bayesian
Method, which is characterized in that the F0' value range be [fs/4, fs/2].
6. signal blind reconstructing method under the modulation wide-band transducer frame according to claim 1 based on sparse Bayesian,
It is characterized in that, to the sampled value y of each channel output of acquisition in the step 2i(n) windowing process, windowing process are carried out
The windowed function used is Hamming window.
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