CN109995376B - Signal reconstruction method based on joint block sparse model - Google Patents
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Abstract
The invention provides a signal reconstruction method based on a joint block sparse model, and belongs to the technical field of distributed compressed sensing. The method comprises the steps of firstly establishing a combined block sparse model based on a mixed support set model, then reconstructing a common part of signals by using the structural characteristics of the combined block sparse model based on the mixed support set model, then reconstructing a unique part of each signal one by using a BOMP algorithm, and finally adding reconstruction results of the common part and the unique part of the original signals to complete reconstruction of the original signals. The invention solves the problem of how to accurately reconstruct the original signal by a receiving end with low measurement value and low signal-to-noise ratio under the condition of multi-antenna and signal sparse coefficient blocking distribution. The invention can be used for signal reconstruction of a receiving end in an actual communication scene.
Description
Technical Field
The invention relates to a signal reconstruction method, and belongs to the technical field of distributed compressed sensing.
Background
Compressed sensing is a novel signal sampling theory for finding sparse solution of an underdetermined linear system, under the premise of signal sparseness or sparse representation, discrete samples of a signal are obtained by sampling at a sampling rate far lower than the Nyquist theorem, and then the original signal is accurately reproduced by a nonlinear recovery algorithm. The distributed compressed sensing is based on the theory of compressed sensing, the compressed sensing of a single signal is extended to a multi-signal model, the correlation among a plurality of signals is fully utilized, and the condition of joint recovery is created, so that the number of measured values required for successful recovery is further reduced.
The model of distributed compressed sensing is shown in formula (1):
wherein X ∈ R JN ,Y∈R JM ,Φ∈R MNJ (ii) a X is a received signal, Y is measurement data, and phi is an observation matrix; r JN 、R JM 、R MNJ Respectively representing JN, JM and MNJ dimensional real number vector sets; j is the number of signals in the received signal, M is the number of measured values, and N is the length of the original signal.
The mixed support set model is one of distributed compressed sensing models, and a signal under the model comprises two parts, namely a common part with the same support set and an independent part specific to each signal, and can be described as follows:
wherein J belongs to {1, 2.., J },
and
common and unique independent portions of each signal, respectively; phi is an observation matrix;
and
there is a separate non-zero portion.
Is provided with
The non-zero values are randomly distributed in independent parts of the support set
C, removing;
with K (c) The number of non-zero values and all signals share the supporting set, i.e.: common part of supporting set
j∈{1,2,...J }. The supporting set of each signal is therefore:
definition of
The total sparsity is greater than the specific partial sparsity.
The multiband signal is a signal with a blocky distribution of frequency domains, which is common in the field of signal processing, and the mathematical model of the multiband signal is shown in equation (7).
Wherein sinc (x) = sin (π x)/π x, s (t) comprises L pairs of frequency bands, t is time, E i And τ i Energy and time offsets of the individual bands, i =1, \8230;, L; b is the maximum bandwidth, f i Is the carrier frequency of each frequency band and has a distribution interval of [ -f [ - ] nyq /2,f nyq /2]，f nyq Is the Nyquist sampling rate.
In an actual communication scene, signals received by us usually have a joint block sparsity characteristic, and under the condition that multiple antennas and signal sparsity coefficients are distributed in a block mode, how a receiving end accurately reconstructs an original signal with a low measurement value and a low signal-to-noise ratio is a problem to be solved by us.
The current reconstruction algorithm based on block sparse signals is mainly divided into three types:
1. a Mixed l 2/l 1optimization (Mixed l 2/l 1optimization program, L-OPT) algorithm;
2. a Block-sparse Matching Pursuit (BMP) algorithm;
3. block-sparse Orthogonal Matching Pursuit (BOMP) algorithm.
The mixed l 2/l 1optimization algorithm is adopted as the reconstruction algorithm of the block sparse signal, so that the problems of high optimization complexity and difficult practical application exist; a block sparse matching tracking algorithm or a block sparse orthogonal matching tracking algorithm is adopted as a reconstruction algorithm of the block sparse signal, matching atoms are not changed after being determined, an over-matching phenomenon is easy to cause, and the method is not suitable for an actual communication scene.
Disclosure of Invention
The invention provides a signal reconstruction method based on a joint block sparse model, which aims to solve the problem of how to accurately reconstruct an original signal by a receiving end with a low measurement value and a low signal-to-noise ratio under the condition that multiple antennas are arranged and signal sparse coefficients are distributed in blocks.
The invention discloses a signal reconstruction method based on a joint block sparse model, which is realized by the following technical scheme:
step one, establishing a combined block sparse model based on a mixed support set model;
step two, initialization:
setting the iteration times l =1, and setting the vector selection index set omega to be null; let initialization residual r j,0 Is equal to the measured value y j Initializing the resulting sparse coefficient vector
Step three, sub-block selection:
selecting an atomic block with the largest average inner product value of all J residual errors in the observation matrix, and recording an index corresponding to the atomic block;
step four, sub-block reconstruction:
measured value y j Projecting in a vector space formed by the updated observation matrix D to obtain a measured value y j Coefficient beta linearly represented by each column vector in D j ；
Step five, residual error r is updated j,l (ii) a Updating sparse coefficient vectors
Step six, judging whether | | | r is satisfied j,l || 2 >ε||y j || 2 And l is less than K, if so, making l = l +1, and returning to the step three(ii) a If not, the Fourier transform matrix is used
As a result of reconstruction of the common portion of each original signal; epsilon is a residual error determination coefficient; k is the bulk sparsity;
step seven, respectively using BOMP algorithm to the residual errors, and reconstructing the specific part of each signal one by one;
and step eight, adding the common part of the original signal and the reconstruction result of the special part to complete the reconstruction of the original signal.
The most prominent characteristics and remarkable beneficial effects of the invention are as follows:
the invention relates to a signal reconstruction method based on a combined block sparse model, which takes the actual multi-antenna communication scene into consideration, combines a mixed support set model in distributed compressed sensing with a block sparse structure, establishes the combined block sparse model based on the mixed support set model closer to the actual communication scene, and then divides the reconstruction process into a public part and a special part to be respectively carried out; the method makes full use of the structural characteristics of the combined block sparse model based on the mixed support set model, and jointly utilizes the block sparse characteristic and the combined sparse information of the signals in the sub-block atom selection process, so that the sub-blocks containing non-zero points can be more accurately screened out, and the reconstruction accuracy is improved; the method can reconstruct the original signal with higher precision under the conditions of multi-antenna and signal sparse coefficient blocking distribution by utilizing the structural information of the model under the conditions of low measurement value and low signal to noise ratio. The reconstruction performance under the same measurement value number is obviously superior to the traditional BOMP algorithm, and the anti-noise performance is also superior to the traditional BOMP algorithm. In simulation experiments, the mean square error of reconstruction under the same measured value quantity is about 1.5dB lower than that of the traditional BOMP algorithm.
Drawings
FIG. 1 is a flow chart of the present invention;
FIG. 2 is a diagram of a spectrum of an original signal X1 in an embodiment;
FIG. 3 is a diagram of the spectrum of an original signal X2 in an example;
FIG. 4 is a diagram of the spectrum of an original signal X3 in an example;
FIG. 5 is a diagram illustrating the effect of reconstructing the original signal X1 by using the method of the present invention in the embodiment;
FIG. 6 is a diagram illustrating the effect of reconstructing the original signal X2 according to the method of the present invention in the embodiment;
fig. 7 is a diagram illustrating the effect of reconstructing the original signal X3 by using the method of the present invention in the embodiment;
FIG. 8 is a graph comparing the reconstructed Mean Square Error (MSE) of the BOMP and the number of measurements M in accordance with the present invention;
FIG. 9 is a graph comparing the noise immunity of the method of the present invention with BOMP.
Detailed Description
The first embodiment is as follows: the present embodiment is described with reference to fig. 1, and the method for reconstructing a signal based on a joint block sparse model provided in the present embodiment specifically includes the following steps:
step one, establishing a combined block sparse model based on a mixed support set model;
step two, initialization:
setting the iteration times l =1, and setting the vector selection index set omega to be null; let initialization residual r j,0 Is equal to the measured value y j Initializing the obtained sparse coefficient vector
Step three, sub-block selection:
for an observation matrix containing BNum atom blocks, selecting an atom block with the largest average inner product value with all J residual errors, and recording an index corresponding to the atom block;
step four, sub-block reconstruction:
measured value y j Projecting in a vector space formed by the updated observation matrix D to obtain a measured value y j Coefficient beta linearly represented by each column vector in D j ；
Step five, updating residual error r j,l (ii) a Updating sparse coefficient vectors
Step six, judging whether | | | r is satisfied j,l || 2 >ε||y j || 2 If l is less than K, making l = l +1, and returning to the step three; if not, the Fourier transform matrix is used
As a result of reconstruction of the common portion of each original signal; stopping iteration when the convergence condition is met; wherein epsilon is a residual error judgment coefficient; k is the bulk sparsity;
step seven, after the public part of each signal is obtained, the BOMP algorithm is respectively used for the residual errors, and the specific part of each signal is reconstructed one by one;
and step eight, adding the reconstruction results of the common part and the specific part of the original signal to finish the accurate reconstruction of the original signal.
In order to fit actual communication scenes better, a combined block sparse model based on a mixed support set model is constructed, and a reconstruction method under the model is provided. The method fully utilizes the structural characteristics of the model, and jointly utilizes the block sparse characteristic and the joint sparse information of the signals in the sub-block atom selection process, so that the sub-blocks containing non-zero points can be screened out more accurately, and the reconstruction accuracy is improved.
The second embodiment is as follows: this embodiment is different from the first embodiment in that,
the combined block sparse model based on the mixed support set model in the first step specifically comprises:
wherein, X = [ X = 1 x 2 ... x J ] T J belongs to {1, 2.., J } for the received signal, and J is the number of signals in the received signal;
and
common and unique independent portions of each signal, respectively; phi is an observation matrix;
is provided with
A plurality of non-zero blocks;
with K (c) A non-zero block.
Other steps and parameters are the same as those in the first embodiment.
The third concrete implementation mode: the difference between this embodiment and the first or second embodiment is that the third step is specifically:
the solution model for distributed compressive sensing can be expressed as:
wherein the content of the first and second substances,
denotes x j K is the block sparsity;
according to the traditional BOMP algorithm based on the block sparse model, when sub-block atoms are selected, the principle that the inner product of original dictionary sub-blocks and signal residual errors is the largest is utilized, and when signals have the combined block sparse characteristic, the idea of synchronous orthogonal matching tracking algorithm in distributed compressed sensing can be referred to, and then the original sub-block with the largest average inner product value of all J residual errors in an observation matrix is selected:
wherein n is l Representing the index corresponding to the most matched atomic block selected by the iteration;<·>expressing the inner product, | | | · | | expresses a norm; phi j,n To observe the nth column in matrix Φ, N =1, \ 8230;, N; n is the length of the original signal (also the total number of columns of the observation matrix Φ); it can be seen from equation (9) that, in the sub-block atom selection process, the principle that the sum of the residual products of the original sub-blocks of the dictionary and all the J signals is the maximum is utilized, and as the J signals have the same common support set, the method is easier to find the most relevant atomic block than the traditional BOMP algorithm, and even if noise exists, the noise influence can be reduced by taking the sum of the residual products and the residual products of the J signals to be the maximum due to the random distribution of the noise, so that the anti-noise performance is obviously enhanced, the joint sparse information of the signals is fully utilized, and the probability of finding the optimal sub-block is improved.
Then record the index corresponding to the block atom:
Ω＝[Ω,n l ] (10)
other steps and parameters are the same as those in the first or second embodiment.
The fourth concrete implementation mode: the present embodiment is different from the third embodiment in that β in the fourth step j The least squares method yields:
wherein D is an updated observation matrix composed of columns taken out of the original observation matrix.
Other steps and parameters are the same as those in the third embodiment.
The fifth concrete implementation mode: the difference between this embodiment and the fourth embodiment is that the specific process of updating the residual error in the fifth step is as follows:
other steps and parameters are the same as those in the fourth embodiment.
The sixth specific implementation mode: the difference between this embodiment and the fifth embodiment is that the specific process of updating the sparse coefficient vector in the fifth step is as follows:
the other steps and parameters are the same as those in the fifth embodiment.
The seventh embodiment: in this embodiment, the difference from the first embodiment is that the residual determination coefficient ∈ =0.01 in step six.
Other steps and parameters are the same as those in the first to sixth embodiments.
Examples
The following examples were employed to demonstrate the beneficial effects of the present invention:
the signal reconstruction method based on the joint block sparse model in the embodiment is carried out according to the following steps:
step one, establishing a combined block sparse model based on a mixed support set model:
wherein the content of the first and second substances,
is provided with
A plurality of non-zero blocks;
with K (c) A plurality of non-zero blocks;
step two, inputting: measured data Y = [ Y = [ ] 1 y 2 ... y J ] T Observation matrix phi, signal number J, block sparsity K and block division number BNum;
setting the iteration times l =1, and setting the vector selection index set omega to be null; let initialization residual r j,0 Is equal to the measured value y j Initializing the resulting sparse coefficient vector
Step three, sub-block selection:
selecting an atomic block with the maximum average inner product value of all J residual errors in the observation matrix, and recording an index corresponding to the atomic block;
step four, sub-block reconstruction:
measured value y j Projecting in a vector space formed by the updated observation matrix D to obtain a measured value y j Coefficient beta linearly represented by each column vector in D j (ii) a Obtained by the least squares method:
step five, residual error is updated
Updating sparse coefficient vectors
Step six, judging whether | | | r is satisfied j,l || 2 >ε||y j || 2 If l is less than K, making l = l +1, and returning to the step three; if not, the Fourier transform matrix is used
As a result of reconstruction of the common portion of each original signal; since the sparsity of the common part of the given signal set is K1 and the sparsity of the specific part is K2, K1 iterations are needed to find all the common parts.
Step seven, respectively using a BOMP algorithm to the residual errors, and reconstructing the specific parts of each signal one by one, wherein the iteration times of the process is 2X K2 due to the existence of noise, which is also the case of considering the poor anti-noise performance of the traditional BOMP algorithm;
and step eight, adding the reconstruction results of the common part and the specific part of the original signal to finish the accurate reconstruction of the original signal.
As shown in fig. 5, 6 and 7, which are diagrams of the effect of reconstructing the original signal in fig. 2, 3 and 4 by using the method of the present invention, it can be seen that the method of the present invention has a better reconstruction effect; FIG. 8 is a relationship between the reconstructed Mean Square Error (MSE) of the method (B-SOMP) of the present invention and the number of measured values M, and it can be seen that the performance of the method of the present invention gradually becomes better as the number of measured values increases (the mean square error gradually decreases as the number of measured values increases), and the average reconstructed mean square error at the same number of measured values is about 1.5dB lower than that of the conventional BOMP algorithm; fig. 9 is a comparison of the noise immunity of the method of the present invention (B-SOMP) and the conventional block orthogonal matching pursuit algorithm (BOMP), and it can be seen that the noise immunity of the proposed method is significantly better than the BOMP and the advantage is more significant at low snr.
The present invention may be embodied in other specific forms without departing from the spirit or essential attributes thereof, and it is therefore intended that all such changes and modifications be considered as within the spirit and scope of the appended claims.
Claims (2)
1. The signal reconstruction method based on the joint block sparse model is characterized by comprising the following steps:
step one, establishing a combined block sparse model based on a mixed support set model;
the joint block sparse model based on the mixed support set model specifically comprises the following steps:
wherein, X = [ X ] 1 x 2 ...x J ] T J belongs to {1, 2.., J } for receiving signals, and J is the number of signals in the received signals;
and
common and unique independent portions of each signal, respectively; phi is an observation matrix;
is provided with
A plurality of non-zero blocks;
with K (c) A plurality of non-zero blocks;
step two, initialization:
setting the iteration times l =1, and enabling a vector selection index set omega to be empty; let initialization residual r j,0 Is equal to the measured value y j Initializing the resulting sparse coefficient vector
Step three, sub-block selection:
selecting an atomic block with the largest average inner product value of all J residual errors in the observation matrix, and recording an index corresponding to the atomic block, wherein the specific steps are as follows:
selecting the atomic block with the largest average inner product value with all J residual errors in the observation matrix:
wherein n is l Representing the index corresponding to the original block with the maximum average inner product value of all J residual errors in the observation matrix; phi (phi) of j,n Is the nth column in the observation matrix phi; n is the length of the original signal,<·>expressing the inner product, | | | · | | expresses a norm;
then recording the index corresponding to the atomic block:
Ω＝[Ω,n l ] (10)；
step four, sub-block reconstruction:
measured value y j Projecting in a vector space formed by the updated observation matrix D to obtain a measured value y j Coefficient beta linearly represented by each column vector in D j ；
Beta is said j And is obtained by the least square method:
wherein D is an updated observation matrix composed of columns taken out of the original observation matrix;
step five, residual error r is updated j,l (ii) a Updating sparse coefficient vectors
The specific process of updating the residual error is as follows:
the specific process for updating the sparse coefficient vector comprises the following steps:
step six, judging whether | | | r is satisfied j,l || 2 ＞ε||y j || 2 If l is less than K, making l = l +1, and returning to the step three; if not, the Fourier transform matrix is used
As a result of reconstruction of the common portion of each original signal; epsilon is a residual error determination coefficient; k is the bulk sparsity;
step seven, respectively using BOMP algorithm to the residual errors, and reconstructing the specific part of each signal one by one;
and step eight, adding the common part of the original signal and the reconstruction result of the special part to complete the reconstruction of the original signal.
2. The method for reconstructing a signal based on a joint block sparse model as claimed in claim 1, wherein the residual error decision coefficient e =0.01 in step six.
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| CN112202452A (en) * | 2020-09-02 | 2021-01-08 | 北京电子工程总体研究所 | Compressed sensing signal reconstruction method and system based on block sparsity and binary tree search |
| CN112653472A (en) * | 2020-12-15 | 2021-04-13 | 哈尔滨工程大学 | Dolphin whistle call signal reconstruction method based on block sparse compressed sensing |
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