Improved logistic chaotic sequence hybrid measurement matrix construction method

Abstract

The application discloses a kind of improved logistic chaotic sequence hybrid measurement matrix construction method.The element of measurement matrix in the application is by the sequence with independent and identically distributed statistical properties generated by a modified logistic chaotic system, after two-level processing, it meets Bernoulli statistical distribution, satisfies the RIP criterion required to be obeyed by measurement matrix of compressive sensing technology, finally, the measurement matrix obtained by sequentially filling according to the toprizi matrix construction rule, with the randomness characteristics of chaotic deterministic system, and the measurement matrix has the advantages of low data amount required to be generated, easy to realize hardware resources etc., and the reconstruction accuracy of signal is better than random Gaussian measurement matrix.

Description

TECHNICAL FIELD

[0001] The application belongs to the technical field of signal processing communication, and particularly relates to a construction method of an improved Logistic chaotic sequence hybrid measurement matrix. BACKGROUND

[0002] With the deep development of the third industrial revolution, in many practical applications, a large amount of sampling data is required for storage and processing to achieve a sampling rate, thereby resulting in high cost of sampling hardware, low acquisition efficiency, and even difficulty in implementation in some cases. In 2004, Candès et al. proposed compressed sensing (CS) technology, which breaks through the traditional Nyquist sampling theorem, is an information processing method based on signal sparsity and compressibility, thereby realizing sub-sampling of signals, saving hardware resources such as analog-to-digital converters, and greatly reducing data transmission costs.

[0003] Compressed sensing technology mainly includes three key technologies, namely, sparsification of the original signal, design of the measurement matrix, and optimization algorithm for reconstructing the observed signal. It has been verified that most signals processed in practice can be sparsely represented in a certain domain through corresponding mapping rules, such as Fourier transform, wavelet transform, and discrete cosine transform. Sparsification of the signal is a key prerequisite for compressed sensing, and the construction of a good observation matrix is the key to whether the original signal can be reconstructed with high quality by compressed sensing technology. A good observation matrix can accurately reconstruct the observed signal to the original signal with fewer observations, and is also a key part of determining the difficulty of hardware device implementation of compressed sensing technology.

[0004] At present, scholars have made a lot of research on the construction and optimization of the measurement matrix of compressed sensing technology. The measurement matrix is mainly divided into two categories. One is a random measurement matrix, represented by a Gaussian random measurement matrix, a sparse random measurement matrix, and a Bernoulli random measurement matrix. The random measurement matrix can measure and reconstruct the signal with good performance, but the generation of the random measurement matrix has great uncertainty, which means that a large number of repeated experiments need to be performed. On the other hand, the calculation complexity is high, and the hardware implementation cost is large. The other category is a deterministic measurement matrix that is more commonly used in practical applications, such as an encoding-based measurement matrix, a Toeplitz matrix, and a circulant matrix. The characteristic of this type of matrix is that the matrix elements are fixed according to the system and parameters. However, some of the above deterministic matrices do not perform as well as random matrices, and some deterministic matrices need to satisfy a large number of conditions to meet the RIP criterion, which greatly increases the complexity of the system, causes resource waste, and also violates the original intention of compressed sensing technology, thereby limiting the range of practical applications of compressed sensing technology in various fields. SUMMARY

[0005] The application aims at the shortcomings of the classical deterministic and random measurement matrix, such as poor signal performance, great uncertainty and high hardware implementation cost, and provides an improved Logistic chaotic sequence hybrid measurement matrix construction method with low memory consumption and easy hardware loading.

[0006] In the first aspect of the application, an improved Logistic chaotic sequence hybrid measurement matrix construction method is provided, comprising the following steps:

[0007] Step (1), determining the size of the measurement matrix Φ as M*N according to the dimension N of the actual signal to be sampled and the prior condition sparsity k of the signal to be sampled;

[0008] Step (2), generating a chaotic pseudo-random sequence x with a length of M+N-1 by using the Logistic chaotic system according to the size of the measurement matrix Φ n ;

[0009] Step (3), performing nonlinear transformation on the chaotic pseudo-random sequence x n to convert it into a chaotic sequence y n with sequence values obeying uniform distribution;

[0010] Step (4), mapping the uniform distribution chaotic sequence y n by a sign function to generate a two-level chaotic sequence z n with statistical independent and identically distributed characteristics for each sequence value;

[0011] Step (5), determining the first column vector and the first row vector of the measurement matrix Φ according to the two-level chaotic sequence z n ;

[0012] Step (6), according to the rule of the Toeplitz matrix, the remaining elements of the measurement matrix are filled by cyclically copying the boundary elements to construct the final measurement matrix Φ.

[0013] In the second aspect of the application, the application of the measurement matrix constructed by the above method in compressive sensing is provided.

[0014] In the third aspect of the application, an improved Logistic chaotic sequence hybrid measurement matrix construction device is provided, comprising a memory, a processor and a computer program stored in the memory and executable on the processor, wherein the processor implements the above construction method when executing the program.

[0015] In the fourth aspect of the application, a computer readable storage medium is provided, characterized in that the storage medium stores a computer program for executing the above construction method.

[0016] Compared with the prior art, the present application has the following advantages:

[0017] 1、The measurement matrix constructed by the method has complex dynamic chaotic pseudo-random characteristics. Since each random variable in the measurement matrix is generated by a chaotic pseudo-random code sequence, the appearance of each element in the measurement matrix is random, thereby ensuring the accurate recovery of the original signal.

[0018] 2、The matrix multiplication performed when the measurement matrix constructed by the method is applied can use fast Fourier transform (FFT), and the calculation complexity of the method is greatly reduced compared with traditional multiplication.

[0019] 3、The measurement matrix in the method has a certain determinacy, and only needs to input a certain input parameter under the full chaotic mapping of the chaotic system to obtain the required measurement matrix.

[0020] 4、The random variables of the measurement matrix in the method are bipolar ±1 sequences generated by a chaotic system, which is easy to directly realize in a hardware circuit and is conducive to saving the storage space of hardware.

[0021] 5、Each random variable in the measurement matrix constructed by the method has independent and identically distributed characteristics and obeys Bernoulli distribution, and greatly satisfies the RIP limit and other capacity limits, and the reconstructed signal has high precision. BRIEF DESCRIPTION OF DRAWINGS

[0022] Figure 1 is a flow chart of the measurement matrix construction method proposed by the present application;

[0023] Figure 2 is a schematic diagram of a signal waveform to be sampled;

[0024] Figure 3 is a distribution diagram of an improved Logistic chaotic mapping;

[0025] Figure 4 is a histogram of an improved Logistic chaotic mapping;

[0026] Figure 5 is a schematic diagram of a signal waveform recovered using the measurement matrix proposed by the present application;

[0027] Figure 6 is a comparison diagram of a reconstructed signal using the measurement matrix proposed by the present application and an original signal to be sampled;

[0028] Figure 7 is a detailed diagram of recovery residuals of signals respectively observed using the measurement matrix proposed by the present application and a Gaussian random matrix;

[0029] Figure 8The recovery residual value of the signal observed by using the measurement matrix of the application and using a Gaussian random matrix is respectively observed. DETAILED DESCRIPTION

[0030] A modified Logistic chaotic sequence hybrid measurement matrix construction method according to an embodiment of the application is described below in combination with the drawings,

[0031] As shown in the figure, the specific steps of the embodiment are as follows: Figure 1

[0032] Step 1: Determine the size of the measurement matrix Φ as M×N (where M is much smaller than N) according to the dimension N of the actual to-be-sampled signal in the communication field that needs to be processed and the prior condition sparsity k of the to-be-sampled signal.

[0033] Further, the general natural signal is not a sparse signal in the time domain, but can become a sparse signal after sparse transformation. Common sparse transformations include Fourier transform (FFT), discrete cosine transform (DCT), discrete wavelet transform (DWT), etc.

[0034] If the actual to-be-sampled signal x of length N can be represented as a linear combination of k sparse bases in a set of sparse bases Ψ = [ψ1, ψ2, ψ3…ψ N ] of size N, then the actual to-be-sampled signal x is a k-sparse signal in the sparse bases Ψ, where the coefficients a ≤ x,

[0035] Ψ = [ψ1, ψ2, ψ3…ψ N ] of size N, then the actual to-be-sampled signal x is a k-sparse signal in the sparse bases Ψ, where the coefficients a ≤ x,

[0036]

[0037] The actual to-be-sampled signal x is a k-sparse signal in the sparse bases Ψ, where the coefficients a ≤ x, After sparse transformation, the actual to-be-sampled signal x ∈ R N Generally, the dimension is too large, and another set of bases that are not related to the sparse bases where M is much smaller than N, so that the actual to-be-sampled signal x can be compressed and observed, that is,

[0038] y = Φx = ΦΨa

[0039] where y is the reduced dimension signal after the compressed observation of the actual to-be-sampled signal x, that is, the measurement matrix to be constructed by the application, with a size of M×N.

[0040] In a certain example, the application directly sets the actual to-be-sampled signal as a signal of length 256. First, create x with a size of 256×1 and initialize all elements of x to 0. Then, insert 10 non-zero values with random Gaussian distribution at 10 indexes, and perform normalization processing on x. At this time, x has become a sparse signal, as shown in the figure.Figure 2 x can be expressed as a linear combination of 10 sparse bases in the set of sparse bases Ψ = [ψ1, ψ2, ψ3... ψN] as follows: 256 ] = ΦΨa

[0041]

[0042]

[0043] The signal to be sampled x is a k-sparse signal in the sparse bases Ψ, since the signal to be sampled x ∈ R N Generally, the dimension is too large, by selecting another set of bases which are not correlated with the sparse bases where M is much smaller than N, the actual signal to be sampled x can be compressed and observed as follows:

[0044] y = Φx = ΦΨa

[0045] Here x = a, Φ = ΦΨ, y is the compressed and observed signal of the actual signal to be sampled x, where The measurement matrix Φ to be constructed in the present application is 50 x 256.

[0046] Step two, according to the size of the measurement matrix Φ, a good performance chaotic pseudo-random sequence x of length M + N - 1 is generated by using the Logistic chaotic system n .

[0047] The Logistic chaotic system has the feature of internal randomness of the system, and the form is:

[0048] x k+1 = ux k (1 - x k ), 0 < x k < 1

[0049] When the system parameter u ∈ (3.5699456, 4], the Logistic mapping can generate complex dynamic behavior, and the system is in a chaotic state, here the system parameter u = 4, the system is in a full chaotic state, then

[0050] x k+1 = 4x k (1 - x k ), 0 < x k < 1

[0051] The generated chaotic sequence has good performance of pseudo-randomness.

[0052] For different system initial values x0 of the Logistic chaotic system, completely different pseudo-random sequences will be generated, here the initial value x0 = 0.1 is set. According to the required measurement matrix chaotic pseudo-random sequence x of length 305 elements is generated n At this time, the chaotic value range of the sequence is [0, 1], and the chaotic value frequency is highest near 0 and 1. The probability density function of the Logistic mapping at the system parameter u = 4 is:

[0053]

[0054] It can be concluded that x n does not satisfy the uniform distribution.

[0055] Step three, the chaotic pseudo-random sequence x n of step two is subjected to a nonlinear transformation to convert it into a chaotic sequence y n that satisfies the uniform distribution.

[0056] Based on the probability principle, a transformation method is used to perform a nonlinear transformation on the function y that does not satisfy the uniform distribution to obtain x that satisfies the uniform distribution after equalization, that is,

[0057]

[0058] Then the chaotic sequence x n can be converted into a uniformly distributed sequence y1 n according to the following formula:

[0059]

[0060] And its chaotic value range is converted to [-1, 1] by the following formula to obtain a sequence y n that is uniformly distributed and ranges from [-1, 1]:

[0061] y n = 2y1 n -1

[0062] The distribution diagram of the improved chaotic sequence is shown in Figure 3 , which is in a chaotic state and uniformly distributed. The chaotic mapping histogram is shown in Figure 4 , which is uniformly distributed in the range [-1, 1].

[0063] Step four, the uniformly distributed chaotic sequence y n of step three is mapped through a sign function to generate a two-level chaotic sequence z n that has statistical independent and identically distributed characteristics for each sequence value.

[0064] According to the properties of uniform distribution, the chaotic sequence y n has equal occurrence probabilities of 1 / 2 in [-1, 0] and [0, 1], so the chaotic sequence y n is mapped to the sequence z nOnly contains random variable 1 and -1, where sgn(·) represents the sign function, which can be obtained by the following formula:

[0065]

[0066] Then the sequence z n Has the property of statistical independent and identically distributed, and it can be proved that it satisfies the Bernoulli distribution. The sequence z n Has only two values 1 and -1, so it can be called a two-level chaotic sequence. The advantage of this sequence is that it is easy to implement in hardware circuit. The specific value of z n ={-1,1,1,1,1,…,-1}.

[0067] Step five, the two-level chaotic sequence z n ={-1,1,1,1,1,…,-1} obtained in step 4 completely determines the first column vector and the first row vector of the measurement matrix Φ, where the first column element of the measurement matrix Φ is Φ 1,1 =-1, Φ 2,1 =-1, Φ 3,1 =1,…, Φ 50,1 =-1, the first row element of the measurement matrix Φ is Φ 1,1 =-1, Φ 1,2 =1, Φ 1,3 =-1,…, Φ 1,256 =-1, so all elements in the two-level chaotic sequence z n are used to completely determine the measurement matrix Φ, which greatly reduces the waste of data resources, that is:

[0068]

[0069] Step six, according to the rule of Toeplitz matrix, the remaining elements of the measurement matrix are filled by cyclic copying of the boundary elements, and the final measurement matrix Φ is constructed.

[0070] As long as the measurement times, that is, the number of rows of the measurement matrix M≥C δ K 2 log(N / K), the high probability satisfies the RIP limit of the Toeplitz matrix, and it is proved that the reconstruction effect of the signal is equivalent to that of the Gaussian random measurement matrix. While the Gaussian random matrix needs 50×256 independent random variables each time, the Toeplitz matrix only needs 305 independent random variables, which greatly reduces the data waste. And the matrix multiplication of the Toeplitz matrix can be effectively realized by using fast Fourier transform (FFT), which only requires o(8) steps, which is much lower than the o(12800) steps required by the multiplication of the independent and identically distributed conventional measurement matrix, greatly improving the calculation efficiency. The general form of the Toeplitz matrix is as follows:

[0071]

[0072] In this embodiment, the secondary chaotic sequence z obtained in step four is filled into the matrix according to the rule of the Toeplitz matrix n After the random variables of the first column vector and the random variables of the first row vector are determined in step five, the remaining part of the matrix is filled in turn based on the rule of the Toeplitz matrix, and multiplied by a scalar In order to ensure that the energy of the signal to be sampled x and the reduced dimension signal y after compressed observation is consistent during observation, the matrix Φ is constructed as follows:

[0073]

[0074] Finally, in order to test the performance of the improved Logistic chaotic sequence hybrid measurement matrix generated in this embodiment, it is compared with a Gaussian random matrix, and the signal to be sampled x is observed respectively, according to the situation that may be encountered in practical application, a certain noise is added to the reduced dimension signal y after compressed observation, and the OMP reconstruction algorithm is used to reconstruct the signal y containing noise to finally obtain the reconstructed signal, such as Figure 5 is a signal waveform diagram recovered using the measurement matrix proposed in this application, Figure 6 is a comparison diagram of the reconstructed signal and the original signal to be sampled using the measurement matrix proposed in this application, Figure 7 is a recovery residual detail diagram of observing the signal using the measurement matrix proposed in this application and using a Gaussian random matrix respectively, Figure 8 is the recovery residual value of observing the signal using the measurement matrix proposed in this application and using a Gaussian random matrix respectively, the recovery error (defined as ) corresponding to the measurement matrix proposed in this application and using a Gaussian random matrix is 6.0630e-12 and 1.9241e-11 respectively, and Figure 5 and Figure 6 It can be seen that the signal to be sampled x can be reconstructed with very small error (the order of magnitude of the recovery error is 10 -12 ), which also shows that the measurement matrix construction method proposed in this application has excellent measurement efficiency.

[0075] The application embodiment further provides an application of the measurement matrix constructed by the improved Logistic chaotic sequence hybrid measurement matrix construction method in compressed sensing.

[0076] The application embodiment further provides an improved Logistic chaotic sequence hybrid measurement matrix construction device, comprising a memory, a processor and a computer program stored in the memory and executable on the processor, wherein the processor implements the improved Logistic chaotic sequence hybrid measurement matrix construction method when executing the program.

[0077] The embodiment of the application further provides a computer readable storage medium, which stores a computer program for executing the improved Logistic chaotic sequence hybrid measurement matrix construction method.

[0078] Embodiments of the device of the application can be applied on network equipment. The device embodiments can be implemented by software, or by hardware or a combination of software and hardware. Taking software implementation as an example, as a logically meaningful device, the device is formed by reading corresponding computer program instructions in a non-volatile memory into a memory for execution by a processor of the device, wherein the computer program is used for executing the improved Logistic chaotic sequence hybrid measurement matrix construction method. From the hardware level, the device can also include other hardware to extend the hardware level.

[0079] For the device embodiments, since they basically correspond to the method embodiments, the related parts are described with reference to the part of the method embodiments. The above-described device embodiments are only illustrative, and those skilled in the art can understand and implement them without creative effort.

[0080] In summary, the application proposes an improved Logistic chaotic sequence hybrid measurement matrix construction scheme to solve the problems of poor signal performance, large uncertainty and high hardware implementation cost of the classical deterministic and random measurement matrix. In the scheme, the elements of the measurement matrix are generated by a modified Logistic chaotic system with independent and identically distributed statistical properties. After two-level processing, the measurement matrix meets the RIP criterion required by the measurement matrix of the compressive sensing technology, and finally the measurement matrix is filled in accordance with the construction rules of the Toeplitz matrix. The measurement matrix has the characteristics of random chaos deterministic system, and has the advantages of low data volume required for generation, easy hardware resource implementation and the like. The reconstruction accuracy of the signal is better than that of the random Gaussian measurement matrix.

[0081] Other embodiments of the application will be apparent to those skilled in the art from consideration of the specification and practice of the application disclosed herein. The application is intended to cover any variations, uses or adaptations of the application following, in general, the principles of the application and including such departures from the present disclosure as come within known or customary practice in the art to which the application pertains. The specification and examples are to be regarded as illustrative only.

[0082] It is also to be noted that the term "comprising" or "including" or any other variation thereof is intended to cover a non-exclusive inclusion, such that a process, method, article, or apparatus that comprises a list of elements does not include only those elements but can also include other elements not expressly listed or inherent to such process, method, article, or apparatus.

[0083] The above description is merely illustrative of the application, and not in limitation thereof. It is the object of the appended claims to particularly point out the application. Any modifications, equivalent substitutions, improvements, etc. made within the spirit and principle of the application should be included in the scope of the application.

Claims

1. An improved method for constructing a hybrid measurement matrix of Logistic chaotic sequence, characterized in that The method comprises the following steps: Step (1): Based on the dimension of the actual signal to be sampled And the sparsity of the prior conditions of the signal to be sampled. Determine the measurement matrix The size is ; Step (2): Based on the measurement matrix The size, using a Logistic chaotic system to generate a length of... Chaotic pseudo-random sequence ; Step (3), performing a nonlinear transformation on the chaotic pseudo-random sequence to convert it into a chaotic sequence whose sequence values obey a uniform distribution ; Step (4), uniform distribution chaotic sequence By mapping through the sign function, a two-level chaotic sequence with statistically independent and identically distributed characteristics of each sequence value is generated ; Step (5), determining the first column vector and the first row vector of the measurement matrix determining a measurement matrix of the first column vector and the first row vector; Step (6), according to the rule of Toeplitz matrix, the rest of the elements of the measurement matrix are filled by cyclically copying the boundary elements, and the final measurement matrix is constructed ; In the step (3), the chaotic pseudo-random sequence converted into a uniformly distributed sequence according to : The chaotic value range is converted to a uniformly distributed sequence with a range of 0 to 1 by the following formula , obtaining a uniformly distributed sequence with a range of 0 to 1 : : ; In the step (5), the secondary chaotic sequence is expressed as: ； the first column elements of the measurement matrix ;​ measurement matrix the first row elements of ; In the step (5), according to the rule of the Toeplitz matrix, the secondary chaotic sequence obtained in the step (4) is filled in the matrix After the random variables of the first column vector and the first row vector are determined in the step (5), the remaining part of the matrix is filled in turn based on the rule of the Toeplitz matrix and multiplied by a scalar , so as to keep the energy of the signal to be sampled consistent with that of the reduced dimension signal after the compressed observation in the measurement process.

2. The improved method for constructing a Logistic chaotic sequence hybrid measurement matrix according to claim 1, characterized in that: In step (1), the signal to be sampled is expressed as a linear combination of a set of sparse bases; another set of bases, which is incoherent with the sparse bases, is selected to perform compressed observation on the signal to be sampled, and the determined measurement matrix .

3. Application of the measurement matrix constructed by the improved logistic chaotic sequence hybrid measurement matrix construction method of claim 1 or 2 in compressive sensing.

4. An improved logistic chaotic sequence hybrid measurement matrix construction device, characterized by, Comprise: A memory, a processor and a computer program stored in the memory and executable on the processor, wherein the processor implements the construction method of claim 1 or 2 when executing the program.

5. A computer readable storage medium, characterized in that, The storage medium stores a computer program for executing the construction method of claim 1 or 2.

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