CN106656202A - Robustness compressed sensing method based on Bayes - Google Patents

Abstract

The invention belongs to the technical field of signal detection and estimation and communication and is applied to a scene that a DOA sensor is partially damaged or is abnormal. A realization method is robustness compressed sensing based on Bayes. The invention aims at providing robustness compressed sensing method based on Bayes. According to the method, the problem that abnormities occur in partial observation signals is taken into consideration, thereby carrying out effective recovery on sparse original signals. According to the method, the core thought is automatically detecting positions of abnormal values based on a Bayes frame through utilization of a pointer complying with bernoulli distribution and removing the positions of the abnormal values, thereby carrying out the effective recovery.

Description

Based on Bayesian robustness compression sensing method

Technical field

The invention belongs to signal detection and estimation (signal detection and estimation) is led with the communication technology Domain, is used in the inductor partial destruction of DOA or the scene of exception, and implementation method is based on the robustness pressure of Bayesian Estimation Contracting is perceived.

Background technology

DOA (array signal angle of arrival) estimates the key issue for being Array Signal Processing field.Correlation is expanded spatial spectrum Estimate, CS can overcome the problem for needing big amount measurement data.Compressive sensing theory is applied into DOA estimation problems, needs to set up Suitable angular estimation rarefaction representation, i.e. space rarefaction.Below will be from Ordinary Compression scene to DOA scenes, difference is, general The calculation matrix A of logical compression scene is Gaussian distributed, and the calculation matrix A of DOA scenes is to obey FFT matrixes, therefrom It is random to take out M rows (also having equidistantly selection).For problems, it is contemplated that a more general problem, that is, measure Value is abnormal with the presence of part or lacks.In such cases, very high requirement is proposed to the robustness of algorithm, so depositing at present Thinking be to compensate to abnormal signal, this festival-gathering propose a new thinking, abnormal signal is done self adaptation rejecting, effect Have than larger lifting in fruit and robustness.

The content of the invention

It is an object of the invention to provide a kind of be based on Bayesian robustness compression sensing method.The present invention considers part There is abnormal, to carry out effectively recovering to sparse primary signal problem in observation signal.The core concept of the present invention is using one The pointer of Bernoulli Jacob's distribution is obeyed, based on Bayesian frame come the position of automatic detection exceptional value, and it is rejected is carried out effectively Recover.

Understand for convenience, the model that the present invention is used is introduced first：

For the abnormal sparse signal of recovered part observation, basic model is y=Ax+s+e to the present invention, wherein, observation Signal y ∈ R^M, obey the calculation matrix A ∈ R of standard gaussian distribution (or FFT matrixes)^M×N, sparse signal x ∈ R^NDegree of rarefication be K, sparse exceptional value s ∈ R^M, white Gaussian noise e ∈ R^M。

One kind is based on Bayesian robustness compression sensing method, comprises the following steps that：

S1, the perception matrix A with stochastical sampling property is constructed, sampling is carried out to signal and obtains y, step-up error preset value ε；

Priori, the Posterior distrbutionp of S2, construction parameters：

S3, target update function, relevant variableMeanwhile,

S4, each parameter priori：

S5, each parameter is updated using Variational-EM algorithms, comprised the following steps that：

S51, renewal q_x(x)：Due to

Wherein, D_s=diag (s) and D_α= Diag (α),

Due to x Gaussian distributeds, then

S52, renewal q_α(α)：Due to

Due to

S53, renewal q_γ(γ)：Due to

P (γ | c, d)=Gamma (γ；c,d)

Then,

S54, renewal q_s(s)：Due to

Wherein,Simultaneously

S55, renewal q_λ(λ)：Due to

ThenTherefore

If S6, S5 iterative process meets end conditionStop iteration, otherwise return S5 carries out next Secondary iteration.

The invention has the beneficial effects as follows：

The all parameters of the present invention can be automatically updated, and when exceptional value quantity is more, advantage is particularly evident, and the time Complexity is lower.

Description of the drawings

Fig. 1 respectively measures number M and recovers the relation of accuracy, exceptional value number and the relation for recovering accuracy.

Fig. 2 respectively measures the relation of the relation of number M and NMSE, exceptional value number and NMSE.

Specific embodiment

With reference to specific embodiment, the present invention is described in further detail.

A kind of bit compression cognitive method of strong robustness 1, comprises the following steps that：

S1, the perception matrix A with stochastical sampling property is constructed, sampling is carried out to signal and obtains y, step-up error preset value ε；

Priori, the Posterior distrbutionp of S2, construction parameters：

S3, target update function, relevant variableMeanwhile,

S4, each parameter priori：

S5, each parameter is updated using Variational-EM algorithms, comprised the following steps that：

S51, renewal q_x(x)：Due to

Wherein, D_s=diag (s) and D_α= Diag (α),

Due to x Gaussian distributeds, then

S52, renewal q_α(α)：Due to

Due to

S53, renewal q_γ(γ)：Due to

P (γ | c, d)=Gamma (γ；c,d)

Then,

S54, renewal q_s(s)：Due to

Wherein,Simultaneously

S55, renewal q_λ(λ)：Due to

ThenTherefore

If S6, S5 iterative process meets end conditionStop iteration, otherwise return S5 carries out next Secondary iteration.

Below other related algorithms are verified into the present invention's with the algorithm performance comparative analysis of the inventive method with further Performance.

Using two kinds of measurement indexs come the performance of metric algorithm.One is to recover accuracy (success rate), in nothing Test under noise conditions；One is recovery accuracy for weighing sparse signal, is called normalized mean squared error (Normalized Mean Squared Error, abbreviation NMSE), tests under conditions of Gaussian noise is mingled with.NMSE's determines Justice is

N=64 in Fig. 1 (a), K=3, T=7, N=64 in Fig. 1 (b), K=3, M=25, T represents the number of exceptional value.From The figure can be seen that the algorithm (C-RBCS) compared those and compensate to exceptional value, and the present invention has bigger performance advantage； N=64 in Fig. 2 (a), K=3, T=7, N=64, K=3, M=25 in Fig. 2 (b), noise variance is 0.01, and we can have found this Patent proposes that algorithm (BP-RBCS) robustness is more preferable.

To sum up told, this patent proposes a new bayes method and is directed to robustness compressed sensing.By obeying Whether the variable of Bernoulli Jacob's distribution is abnormal to represent observation, is effectively automatically updated by bayes method, and we can be with Sparse signal is effectively recovered after detection exceptional value and rejecting.Experimental result shows, algorithm performance proposed by the present invention More excellent and robustness is more preferable.

Claims (1)

1. it is a kind of to be based on Bayesian robustness compression sensing method, comprise the following steps that：

S1, the perception matrix A with stochastical sampling property is constructed, sampling is carried out to signal and obtains y, step-up error preset value ε；

Priori, the Posterior distrbutionp of S2, construction parameters：

S3, target update function, relevant variableMeanwhile,

${\mathrm{lnq}}_x\left(x\right)=<\ln p(y,\mathrm{\&theta;}){>}_{q_{\mathrm{\&alpha;}}\left(\mathrm{\&alpha;}\right)q_{\mathrm{\&gamma;}}\left(\mathrm{\&gamma;}\right)q_s\left(s\right)q_{\mathrm{\&lambda;}}\left(\mathrm{\&lambda;}\right)}+cons\tan t$

${\mathrm{lnq}}_{\mathrm{\&alpha;}}\left(\mathrm{\&alpha;}\right)=<\ln p(y,\mathrm{\&theta;}){>}_{q_x\left(x\right)q_{\mathrm{\&gamma;}}\left(\mathrm{\&gamma;}\right)q_s\left(s\right)q_{\mathrm{\&lambda;}}\left(\mathrm{\&lambda;}\right)}+cons\tan t$

${\mathrm{lnq}}_{\mathrm{\&gamma;}}\left(\mathrm{\&gamma;}\right)=<\ln p(y,\mathrm{\&theta;}){>}_{q_x\left(x\right)q_{\mathrm{\&alpha;}}\left(\mathrm{\&alpha;}\right)q_s\left(s\right)q_{\mathrm{\&lambda;}}\left(\mathrm{\&lambda;}\right)}+cons\tan t$

${\mathrm{lnq}}_s\left(s\right)=<\ln p(y,\mathrm{\&theta;}){>}_{q_x\left(X\right)q_{\mathrm{\&alpha;}}\left(\mathrm{\&alpha;}\right)q_{\mathrm{\&gamma;}}\left(\mathrm{\&gamma;}\right)q_{\mathrm{\&lambda;}}\left(\mathrm{\&lambda;}\right)}+cons\tan t$

${\mathrm{lnq}}_{\mathrm{\&lambda;}}\left(\mathrm{\&lambda;}\right)=<\ln p(y,\mathrm{\&theta;}){>}_{q_x\left(x\right)q_{\mathrm{\&alpha;}}\left(\mathrm{\&alpha;}\right)q_{\mathrm{\&gamma;}}\left(\mathrm{\&gamma;}\right)q_s\left(s\right)}+cons\tan t$

S4, each parameter priori：

$\begin{array}{c}p\left(y\right|x,s,\mathrm{\&gamma;})=\underset{i=1}{\overset{M}{\&Pi;}}N{(y_i;A_{i}x,{\mathrm{\&gamma;}}^{-1})}^{s_i}\\ p\left(x\right|\mathrm{\&alpha;})=\underset{i=1}{\overset{N}{\&Pi;}}N(x_i;0,{\mathrm{\&alpha;}}_i^{-1})\\ p\left(\mathrm{\&alpha;}\right|a,b)=\underset{i=1}{\overset{N}{\&Pi;}}Gamma({\mathrm{\&alpha;}}_i;a,b)\\ p\left(\mathrm{\&gamma;}\right|c,d)=Gamma(\mathrm{\&gamma;};c,d)\\ p\left(s\right|\mathrm{\&lambda;})={{\mathrm{\&lambda;}}_i}^{s_i}{(1-{\mathrm{\&lambda;}}_i)}^{1-s_i}\\ p\left(\mathrm{\&lambda;}\right|\mathrm{\&epsiv;})=\underset{i=1}{\overset{M}{\&Pi;}}Beta(\mathrm{\&epsiv;},1-\mathrm{\&epsiv;})=\underset{i=1}{\overset{M}{\&Pi;}}\frac{\mathrm{\&Gamma;}\left(1\right)}{\mathrm{\&Gamma;}\left(\mathrm{\&epsiv;}\right)\mathrm{\&Gamma;}(1-\mathrm{\&epsiv;})}{\mathrm{\&lambda;}}_i^{\mathrm{\&epsiv;}-1}{(1-{\mathrm{\&lambda;}}_i)}^{1-\mathrm{\&epsiv;}-1}\end{array};$

S5, each parameter is updated using Variational-EM algorithms, comprised the following steps that：

S51, renewal q_x(x)：Due to

Wherein, D_s=diag (s) and D_α=diag (α),

Due to x Gaussian distributeds, then

S52, renewal q_α(α)：Due to

$\begin{array}{c}p\left(x\right|\mathrm{\&alpha;})=\underset{i=1}{\overset{N}{\&Pi;}}N(x_i;0,{\mathrm{\&alpha;}}_i^{-1}),p\left(\mathrm{\&alpha;}\right|a,b)=\underset{i=1}{\overset{N}{\&Pi;}}Gamma({\mathrm{\&alpha;}}_i;a,b)\\ \&DoubleRightArrow;{\mathrm{lnq}}_{\mathrm{\&alpha;}}\left(\mathrm{\&alpha;}\right)\&Proportional;<\ln p\left(x\right|\mathrm{\&alpha;})+\ln p\left(\mathrm{\&alpha;}\right|a,b){>}_{q_x\left(x\right)}\\ =\underset{i=1}{\overset{N}{\&Sigma;}}<0.5{\mathrm{ln\&alpha;}}_i-0.5x_i^2{\mathrm{\&alpha;}}_i+{\mathrm{aln\&alpha;}}_i-{\mathrm{b\&alpha;}}_i>\\ =\underset{i=1}{\overset{N}{\&Sigma;}}<(0.5+a){\mathrm{ln\&alpha;}}_i-(0.5x_i^2+b){\mathrm{\&alpha;}}_i>\end{array},$

Due to

S53, renewal q_γ(γ)：Due to

Then,

$\mathrm{\&gamma;}=\frac{(c+0.5M)}{d+0.5({(y-\mathrm{Ax})}^T{D}_s(y-\mathrm{Ax})+\mathrm{trace}\left(A^{T}A{\mathrm{\&Phi;}}_x\right))};$

S54, renewal q_s(s)：Due to

$\begin{array}{c}p\left(y\right|x,s,\mathrm{\&gamma;})=N{(y_i;A_{i}x,{\mathrm{\&gamma;}}^{-1})}^{s_i}\\ p\left(s\right|\mathrm{\&lambda;})={{\mathrm{\&lambda;}}_i}^{s_i}{(1-{\mathrm{\&lambda;}}_i)}^{1-s_i}\end{array},$

Wherein,Simultaneously

S55, renewal q_λ(λ)：Due to

$\begin{array}{c}p\left(s\right|\mathrm{\&lambda;})=\underset{i=1}{\overset{M}{\mathrm{\&Pi;}}}{{\mathrm{\&lambda;}}_i}^{s_i}{(1-{\mathrm{\&lambda;}}_i)}^{1-s_i}\\ p\left(\mathrm{\&lambda;}\right|\mathrm{\&epsiv;})=\underset{i=1}{\overset{M}{\mathrm{\&Pi;}}}\mathrm{Beta}(\mathrm{\&epsiv;},1-\mathrm{\&epsiv;})=\underset{i=1}{\overset{M}{\mathrm{\&Pi;}}}\frac{\mathrm{\&Gamma;}\left(1\right)}{\mathrm{\&Gamma;}\left(\mathrm{\&epsiv;}\right)\mathrm{\&Gamma;}(1-\mathrm{\&epsiv;})}{{\mathrm{\&lambda;}}_i}^{\mathrm{\&epsiv;}-1}{(1-{\mathrm{\&lambda;}}_i)}^{1-\mathrm{\&epsiv;}-1}\\ \&DoubleRightArrow;\ln q_{\mathrm{\&lambda;}}\left(\mathrm{\&lambda;}\right)\&Proportional;<\ln p\left(s\right|\mathrm{\&lambda;})+\ln p\left(\mathrm{\&lambda;}\right|\mathrm{\&epsiv;}){>}_{q_s\left(s\right)}\\ \&Proportional;\underset{i=1}{\overset{M}{\mathrm{\&Sigma;}}}<s_i\ln {\mathrm{\&lambda;}}_i+(1-s_i)\ln (1-{\mathrm{\&lambda;}}_i)+(\mathrm{\&epsiv;}-1)\ln {\mathrm{\&lambda;}}_i-\mathrm{\&epsiv;}\ln (1-{\mathrm{\&lambda;}}_i)>,\\ =\underset{i=1}{\overset{M}{\mathrm{\&Sigma;}}}<(s_i+\mathrm{\&epsiv;}-1)\ln {\mathrm{\&lambda;}}_i+(1-s_i-\mathrm{\&epsiv;})\ln (1-{\mathrm{\&lambda;}}_i)>\end{array},$

ThenTherefore

If S6, S5 iterative process meets end conditionStop iteration, otherwise return S5 and changed next time Generation.