CN107483057A - Sparse multi-band signals reconstructing method based on conjugate gradient tracking - Google Patents
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Abstract
The invention discloses a kind of sparse multi-band signals reconstructing method based on conjugate gradient tracking, first, signal reconstruction model is established；Then, structure reconstruct framework；Finally, use MMV CGP algorithm reconstruction signals, OMPMMV Algorithm for Solving problems are converted into Unconstrained Optimization Problem by the algorithm, using the greedy system of selection similar to OMPMMV algorithms, and optimized by conjugate gradient and replace pseudo-inverse operation, reach the purpose for reducing signal reconstruction process computational complexity and amount of calculation.
Description
Technical Field
The invention belongs to the technical field of optimized digital signal processing, and particularly relates to a sparse multi-band signal reconstruction method based on conjugate gradient tracking, which reduces the calculation complexity and the calculation amount and is easy for engineering realization.
Background
In nyquist sampling, the sampling rate must be greater than or equal to twice the highest frequency or bandwidth of the signal. In recent years, an emerging Compressed Sensing (CS) theory has brought a breakthrough to data acquisition technology. In this theoretical framework, the sampling rate is no longer determined by the highest frequency or bandwidth of the signal, but by the signal structure and the useful information, and as long as the signal is sparse in a certain transform domain, the useful information of the signal can be acquired at a rate far below the nyquist sampling rate, the analog signal digitization is completed, and the signal is reconstructed by an optimization algorithm. The CS theory avoids high-speed sampling, which means that the acquisition and processing of signals can be performed at a lower rate, and the data storage and transmission cost, and the signal processing time and calculation cost can be significantly reduced. The successful application of the theory can change the existing signal processing framework and has wide application prospect.
In the compressive sensing theory, the OMPMMV algorithm for multi-band signal reconstruction needs pseudo-inverse operation in each iteration process, the calculation complexity and the calculation amount are large, and the engineering realization has great difficulty.
Disclosure of Invention
Aiming at the problems existing in the OMPMMV Algorithm, the problem solved by the OMPMMV Algorithm is converted into an unconstrained optimization problem, and a sparse multiband signal reconstruction method based on Conjugate gradient tracking is provided based on an MMV-Conjugate gradient tracking Algorithm (MMV-Conjugate Pursuit Algorithm, MMV-CGP).
In order to achieve the purpose, the technical scheme of the invention is realized as follows: a sparse multi-band signal reconstruction method based on conjugate gradient tracking comprises the following steps:
firstly, establishing a signal reconstruction model;
then, constructing a reconstruction frame;
finally, the signal is reconstructed using the MMV-CGP algorithm.
Further, the establishing of the signal reconstruction model specifically comprises: establishing a signal acquisition model aiming at compressed sensing acquisition and reconstruction of multiband signals; the analog multi-band signal x (t) enters N channels simultaneously and is respectively mixed with the mixing function p of each channel i (t) multiplication, each channel p i (t) is an independent and identically distributed random function; the width of the pass band of the mixed signal is 1/T s Then passes through a low pass filter with a sampling frequency of 1/T s The AD realizes low-speed sampling to obtain compressed sensing acquisition data y of an analog signal x (t) i [n]And finally, by y i [n]The original signal is reconstructed.
Further, constructing a reconstruction framework specifically includes: reconstruction of sparse spectrum Z (f) by y (f) = AZ (f)) Then reconstructing an original signal spectrum X (f) by Z (f); wherein y (f) is compressed sensing acquisition data y i [n]A is the ith channel p i Fourier coefficient c of (t) il And forming an M multiplied by L dimensional observation matrix, wherein Z (f) is a matrix formed by the original frequency spectrum X (f) of the signal.
Further, to solve Z (f), it is first necessary to construct a framework V, and then solve the rarest solution for V = AUFor estimating the support set S of the signal Z (n), Z (n) being the inverse fourier transform of Z (f), whereinIs sparse.
Further, in order to estimate the support set S, a frame matrix needs to be constructed, and the frame matrix is obtained by the following formula decomposition; order to
y[n]＝{y 1 [n],y 2 [n],…,y M [n]}
y (f) is the Fourier transform of y [ n ];
decomposing Q to obtain
Q＝VV H V＝AU
Wherein (C) H To find the conjugate transpose of the matrix, () T Transpose for matrix;
the rarest solution is then found for V = AUThe supporting set S; pass type reconstruction Z (f)
Z(f)＝{z 1 (f)，z 2 (f)，…,z i (f)，…,z M (f)}i＝1,…,M
Wherein the content of the first and second substances,to find the inverse of the matrix, A S Is the set of elements in matrix a that are made up of the support set.
Furthermore, the MMV-CGP algorithm reconstructed signal is specifically: the MMV-CGP algorithm firstly initializes residual errors to V, and residual errors R are obtained in each iteration t And correlating with each column of A to obtain a group of correlation vectors, calculating the norm of each correlation vector, and putting the column of the matrix A corresponding to the maximum t-th iteration norm into a reconstruction vector setCalculating gradientsAnd iterative search directionUpdating residual error R t And repeating the iteration process until an iteration stop condition is met, and outputting a signal support set S.
Furthermore, in the MMV-CGP algorithm, the subscript t in the following formula represents the number of iterations:
let ε 1 A threshold for controlling the iteration stop;
inputting: observing a matrix A, and constructing a frame V;
and (3) outputting: t iteration support set S t ；
Initialization: residual error R 0 = V, supporting setIteration counter t =1;
method for calculating correlation coefficientsNorm mu t ，μ t ＝{μ i |μ i ＝||<R t-1 ,A i >|| 2 ,i＝1,2,…,L}；
Selecting a kth column of observation matrix A, the column satisfying k = argmax k |μ t |；
Support set S for updating t ＝[S t-1 ,k]Recording a set of reconstruction vectors
Step four: calculating gradients
If t =1
Computing iterative search direction
Calculating intermediate variables(turn (b))
Otherwise
Calculating intermediate variables
Calculating iterative search direction correction coefficient C t (:,j)＝-B' t-1 (:,j)W t (:,j)/η t-1 (j,j)
Computing iterative search directions
Calculating intermediate variables
b: calculating the intermediate variable eta t ＝<B t ,B t
c: calculating the step factor alpha t ＝tr[<R t-1 ,B t >]/tr(η t ) Tr (-) is the trace of the matrix
Fifthly, updating residual error R t ＝R t-1 -α t B t ；
Sixthly, if the R is satisfied t || 2 /||V|| 2 ≤ε 1 Stopping iteration and reconstructing a signal by formula; if not, let t = t +1 and enter the next iteration.
Due to the adoption of the technical scheme, the method can achieve the following technical effects:
1) The multiband signal can be accurately reconstructed, the reconstruction precision is similar to an OMPMMV algorithm, and the relative reconstruction errors are less than 2%;
2) The CPU running time of the MMV-CGP algorithm is less than that of the OMPMMV algorithm;
3) The MMV-CGP algorithm and the OMPMMV algorithm have good noise stability, and when the signal-to-noise ratio is reduced to-10 dB, the reconstruction probability is still higher than that of the algorithm of 90 percent, and the noise stability is good.
In summary, compared to the OMPMMV algorithm, the MMV-CGP algorithm can reduce the CPU running time of signal reconstruction while ensuring reconstruction accuracy and good noise stability.
Drawings
The invention has the following figures 5:
FIG. 1 Signal acquisition model;
FIG. 2 a multi-band signal model;
fig. 3 a sparse multi-band LFM signal;
FIG. 4 multi-band signal reconstruction spectrum of MMV-CGP algorithm;
FIG. 5 shows the probability of algorithm reconstruction under different signal-to-noise ratios.
Detailed Description
In order to make the objects, technical solutions and advantages of the present invention more apparent, the present invention will be described in detail with reference to the accompanying drawings and specific embodiments.
A sparse multi-band signal reconstruction method based on conjugate gradient tracking comprises the following steps:
step 1: establishing a signal reconstruction model;
establishing a signal acquisition model aiming at compressed sensing acquisition and reconstruction of multiband signals; the analog multi-band signal x (t) enters N channels simultaneously and is respectively mixed with the mixing function p of each channel i (t) multiplication, each channel p i (t) is a random function which is independent and identically distributed; the width of the pass band of the mixed signal is 1/T s Then passes through a low pass filter with a sampling frequency of 1/T s The AD realizes low-speed sampling to obtain compressed sensing acquisition data y of an analog signal x (t) i [n]And finally, by y i [n]The original signal is reconstructed.
Step 2, constructing a reconstruction frame;
multi-band signal reconstruction process: reconstructing a sparse spectrum Z (f) by y (f) = AZ (f), and reconstructing an original signal spectrum X (f) by Z (f); wherein y (f) is compressed sensing acquisition data y i [n]A is the frequency spectrum of the ith channel p i Fourier coefficient c of (t) il And forming an M multiplied by L dimensional observation matrix, wherein Z (f) is a matrix formed by the original frequency spectrum X (f) of the signal.
Because f e [ -f s /2,+f s /2](f s For system sampling rate), over a continuum, sparse solutions of numerous underdetermined equations are required. The problem belongs to an infinite observed value vector problem, which can not be solved in practice and needs to be converted into a multi-observed value vector problem. It is necessary to first construct the framework V and then solve the rarest of V = AUIs used to estimate the support set S of the signal Z (n), which is the inverse fourier transform of Z (f).
To estimate the support set S, a frame matrix is constructed, which is obtained by the following decomposition; order to
y[n]＝{y 1 [n],y 2 [n],…,y M [n]}
y (f) is the Fourier transform of y [ n ];
decomposing Q to obtain
Q＝VV H V＝AU
Wherein (C) H To find the conjugate transpose of the matrix, () T Transpose for matrix;
the rarest solution is then found for V = AUThe supporting set S; pass type reconstruction Z (f)
Z(f)＝{z 1 (f)，z 2 (f)，…,z i (f)，…,z M (f)} i＝1,…,M
Wherein the content of the first and second substances,to find the inverse of the matrix, A S Is the set of elements in matrix a that are made up of the support set.
And 3, reconstructing the signal by using an MMV-CGP algorithm.
The MMV-CGP algorithm initializes the residual to V first, and for each iteration, the residual R t Making correlation with each column of A to obtain a group of correlation vectors, calculating the norm of each correlation vector, and placing the column of matrix A corresponding to the maximum t-th iteration norm into a reconstruction vector setCalculating gradientsAnd iterative search directionUpdating residual error R t And repeating the iteration process until an iteration stop condition is met, and outputting a signal support set S.
The MMV-CGP algorithm process specifically comprises the following steps: the subscript t in the following equations each represents the number of iterations:
let ε 1 A threshold for controlling the iteration stop;
inputting: observing the matrix A, and constructing a frame V;
and (3) outputting: t-th iteration support set S t ；
Initialization: residual error R 0 = V, supporting setIteration counter t =1;
method for calculating correlation coefficientsNorm mu t ，μ t ＝{μ i |μ i ＝||<R t-1 ,A i >|| 2 ,i＝1,2,…,L}；
Selecting the kth column of the observation matrix A, which column satisfies k = argmax k |μ t |；
Support set S for updating t ＝[S t-1 ,k]Recording a set of reconstruction vectors
Step four: calculating gradients
If t =1
Computing iterative search directions
Calculating intermediate variables(turn (b))
Otherwise
Calculating intermediate variables
Calculating iterative search direction correction coefficient C t (:,j)＝-B' t-1 (:,j)W t (:,j)/η t-1 (j,j)
Computing iterative search direction
Calculating intermediate variables
b: calculating the intermediate variable eta t ＝<B t ,B t
c: calculating the step factor alpha t ＝tr[<R t-1 ,B t >]/tr(η t ) Tr (·) is the trace of matrix solving
Updating residual error R t ＝R t-1 -α t B t ；
Sixthly, if the R is satisfied t || 2 /||V|| 2 ≤ε 1 Stopping iteration and reconstructing a signal by formula; if not, let t = t +1 and enter the next iteration.
Example 1
The embodiments of the present invention are implemented on the premise of the technical solution of the present invention, and detailed embodiments and specific operation procedures are given, but the scope of the present invention is not limited to the following embodiments.
Step 1: a multi-band signal model is established, which comprises a plurality of Linear Frequency Modulation (LFM) signal components, the supporting set of the fourier transform is N disjoint frequency bands, and the bandwidth of each frequency band does not exceed B, as shown in fig. 2.
The LFM signal is sparse in the frequency domain, and multiple signals can constitute a sparse multiband signal, which can be reconstructed using the CS algorithm. The method comprises the steps of selecting 6 narrow-band LFM signals, wherein the signal composition is shown in table 1, the pulse width is 6us, the frequency spectrum is shown in figure 3 (the lower sideband of each frequency band in the figure is the initial frequency of a signal, the upper sideband is the initial frequency plus the bandwidth of the signal), and a plurality of narrow-band LFMs can form sparse multiband signals.
TABLE 1 Signal composition
And 2, step: in the experiment, the OMPMMV algorithm and the MMV-CGP algorithm are respectively used for reconstructing one-dimensional multiband LFM signals (the sampling rate is 51.282 MHz), and the algorithms are compared and analyzed in performance.
Experiment one: in a noise-free environment
1) Verifying the feasibility of reconstructing the multi-band signal by the MMV-CGP algorithm, wherein a reconstructed frequency spectrum is depicted in the attached figure 4;
2) The signal was reconstructed by 12 iterations and the average CPU run time results for each iteration of the algorithm are shown in table 2. Relative reconstruction errors are 1.08% and 1.19% respectively;
experiment two: in an environment containing additive white noise
1) The signal-to-noise ratio was varied, 1000 monte carlo simulations were run for both algorithms, the reconstruction probabilities of the algorithms at different signal-to-noise ratios were counted, and the result curves are depicted in fig. 5.
TABLE 2 average CPU calculation time for each iteration of the algorithm
2.3 simulation analysis
The comprehensive simulation results can draw the following conclusions:
1) Comparing fig. 4 with fig. 3, it can be seen that the MMV-CGP algorithm proposed herein can accurately reconstruct a multi-band signal;
2) As can be seen from Table 2, the reconstruction accuracy of the MMV-CGP algorithm is similar to that of the OMPMMV algorithm, and the relative reconstruction errors are less than 2%. As known from the algorithm flow, the OMPMMV algorithm and the MMV-CGP algorithm are support sets for reconstructing signals firstly, and then original signals are reconstructed by the formula. When the support set is accurately reconstructed, the signal reconstruction error comes from the calculation error, so that the reconstruction precision of the two algorithms is approximate;
3) As can be seen from Table 2, the CPU runtime of the MMV-CGP algorithm is 76.7% of that of the OMPMMV algorithm. In both algorithms, as the number of iterations increases,the dimension of (a) increases. The complexity and the operation time of the solving type pseudo-inverse operation are obviously increased in the iterative process of the OMPMMV algorithm. The MMV-CGP algorithm does not need pseudo-inverse operation,the increase in dimensionality does not result in a significant increase in computational complexity. Therefore, the CPU running time of the MMV-CGP algorithm is smaller than that of the OMPMMV algorithm;
4) As can be seen from FIG. 5, both the MMV-CGP algorithm and the OMPMMV algorithm have good noise stability, and when the signal-to-noise ratio is reduced to-10 dB, the reconstruction probability is still higher than 90%. In both algorithms, the decrease in signal-to-noise ratio does not have much impact on the reconstruction of the signal support set. Thus, both algorithms have good noise stability for reconstructing the original signal.
In summary, compared to the OMPMMV algorithm, the MMV-CGP algorithm can reduce the CPU running time of signal reconstruction while ensuring reconstruction accuracy and good noise stability, and the time reduction is more significant when the number of signals increases.
The invention relates to the field of compressed sensing and error correcting code optimization theories, and designs a compressed sensing signal reconstruction method based on gradient tracking. The method firstly utilizes a compressed sensing theory to carry out low-rate acquisition on an analog signal, and completes the digitization of the analog signal. And then according to the spectral characteristics of the acquired low-speed digital signals, combining an optimization theory and improving the traditional OMPMMV algorithm by using a conjugate gradient tracking method to obtain an MMV-CGP algorithm and reconstruct the spectrum of the original signal. From the simulation results and analysis, it can be concluded that the method has good noise stability and can realize signal spectrum reconstruction, as shown in fig. 4. The problem to be solved by the invention is how to improve the OMPMMV algorithm by utilizing an optimization theory and reduce the CPU running time of signal reconstruction.
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 as the technical solutions and the inventive concepts of the present invention within the technical scope of the present invention.
Claims (7)
1. The sparse multiband signal reconstruction method based on conjugate gradient tracking is characterized by comprising the following steps of:
firstly, establishing a signal reconstruction model;
then, constructing a reconstruction frame;
finally, the signal is reconstructed using the MMV-CGP algorithm.
2. The conjugate gradient pursuit-based sparse multiband signal reconstruction method according to claim 1, wherein the establishing of the signal reconstruction model specifically comprises: establishing a signal acquisition model aiming at compressed sensing acquisition and reconstruction of multiband signals; the analog multi-band signal x (t) enters N channels simultaneously and is respectively mixed with the mixing function p of each channel i (t) multiplication, each channel p i (t) is a random function which is independent and identically distributed; the width of the mixed signal passing through the passband is 1/T s Then passes through a low pass filter with a sampling frequency of 1/T s The AD realizes low-speed sampling to obtain compressed sensing acquisition data y of an analog signal x (t) i [n]And finally, by y i [n]The original signal is reconstructed.
3. Conjugate gradient tracking based sparse multi-band signal reconstruction as claimed in claim 2The method is characterized in that the construction of the reconstruction frame specifically comprises the following steps of: reconstructing a sparse spectrum Z (f) by y (f) = AZ (f), and reconstructing an original signal spectrum X (f) by Z (f); wherein y (f) is compressed sensing acquisition data y i [n]A is the ith channel p i Fourier coefficient c of (t) il And forming an M multiplied by L dimensional observation matrix, wherein Z (f) is a matrix formed by the original frequency spectrum X (f) of the signal.
4. The conjugate gradient pursuit-based sparse multiband signal reconstruction method of claim 3, wherein solving for Z (f) entails first constructing a frame V and then solving for V = AU for its sparsest solutionIs used to estimate the support set S of the signal Z (n), which is the inverse fourier transform of Z (f), where U is the sparse solution.
5. The conjugate gradient pursuit-based sparse multiband signal reconstruction method of claim 4, wherein for estimating the support set S, a frame matrix is constructed, and the frame matrix is obtained by the following formula decomposition; order to
y[n]＝{y 1 [n],y 2 [n],…,y M [n]}
y (f) is the Fourier transform of y [ n ];
decomposing Q to obtain
Q＝VV H V＝AU
Wherein, the (A) H To find the conjugate transpose of the matrix, () T Transpose for matrix;
the rarest solution is then found for V = AUThe supporting set S; pass through reconstruction Z (f)
Z(f)＝{z 1 (f)，z 2 (f)，…,z i (f)，…,z M (f)} i＝1,…,M
Wherein the content of the first and second substances,to find the inverse of the matrix, A S Is the set of elements in matrix a that are made up of the support set.
6. The conjugate gradient pursuit-based sparse multiband signal reconstruction method according to claim 5, wherein the MMV-CGP algorithm reconstructed signal is specifically: the MMV-CGP algorithm initializes the residual to V first, and for each iteration, the residual R t Making correlation with each column of A to obtain a group of correlation vectors, calculating the norm of each correlation vector, and placing the column of matrix A corresponding to the maximum t-th iteration norm into a reconstruction vector setCalculating gradientsAnd iterative search directionUpdating residual error R t And repeating the iteration process until an iteration stop condition is met, and outputting a signal support set S.
7. The conjugate gradient pursuit-based sparse multiband signal reconstruction method of claim 6, wherein the MMV-CGP algorithm is specifically that a subscript t in the following formula represents an iteration number:
let ε 1 A threshold for controlling the iteration stop;
inputting: observing a matrix A, and constructing a frame V;
and (3) outputting: t-th iteration support set S t ；
Initialization: residual error R 0 = V, supporting setIteration counter t =1;
method for calculating correlation coefficientNorm mu t ，μ t ＝{μ i |μ i ＝||<R t-1 ,A i >|| 2 ,i＝1,2,…,L}；
Selecting the kth column of the observation matrix A, which satisfies k = argmax k |μ t |；
Support set S for updating t ＝[S t-1 ,k]Recording a set of reconstruction vectors
Step four: calculating gradients
If t =1
Computing iterative search direction
Calculating intermediate variables(turn (b))
Otherwise
Calculating intermediate variables
Calculating iterative search direction correction coefficient C t (:,j)＝-B' t-1 (:,j)W t (:,j)/η t-1 (j,j)
Computing iterative search direction
Calculating intermediate variables
b: calculating the intermediate variable eta t ＝〈B t ,B t 〉
c: calculating the step factor alpha t ＝tr[〈R t-1 ,B t >]/tr(η t ) Tr (·) is the trace of matrix solving
Updating residual error R t ＝R t-1 -α t B t ；
Sixthly, if the R is satisfied t || 2 /||V|| 2 ≤ε 1 Stopping iteration and reconstructing a signal by formula; if not, let t = t +1 and enter the next iteration.
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Application publication date: 20171215 |