CN112737636B - Compressed sensing and reconstruction method based on power signal transmission singularity measurement - Google Patents
Compressed sensing and reconstruction method based on power signal transmission singularity measurement Download PDFInfo
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Abstract
The invention belongs to the field of compression and denoising of power line signals, and relates to a compression sensing and reconstruction method based on power signal transmission singularity measurement. The method comprises the following steps: constructing a low-voltage three-phase power transmission line model; collecting voltage data of power lines of each phase of line; adding noise to the collected voltage data; selecting a proper wavelet base; decomposing the voltage data added with the noise by using binary wavelet transform to obtain a modulus maximum value of each layer of the voltage signal; judging the modulus maximum value of the useful signal and the modulus maximum value of the noise according to the Lee's index, removing the modulus maximum value of the noise and improving the sparsity of the modulus maximum value of the signal; compressing and sensing the modulus maximum value reserved after denoising by utilizing an OMP algorithm; and reconstructing the signal at a receiving end by utilizing the received signal modulus maximum value restored by the OMP algorithm and reserved. The accuracy of power signal transmission is effectively improved. The method overcomes the defect of unstable denoising effect of the common given threshold method and combines the advantage of small compressed sensing calculation amount.
Description
Technical Field
The invention belongs to the field of compression and denoising of power line signals, and relates to a compressed sensing and reconstruction method based on power signal transmission anisotropy measurement.
Background
With the continuous development of information technology and the accelerated popularization of broadband, the explosive growth of data traffic inevitably brings about the large-scale construction of communication infrastructure. The low-voltage power lines which are widely distributed at present save the trouble of re-laying lines, save communication cost, have firm structures and low price, and are focused by researchers, so that power line carrier communication (PLC) is developed. However, as the related art is continuously developed, the information transmission rate of the PLC is rapidly increased, and a large amount of noise accompanying the power line signal transmission seriously affects the accuracy of the information transmission. To address the problems associated with PLC, researchers have introduced compressed sensing techniques that have emerged in recent years into this area.
The current applications of compressed sensing in power signal processing are mainly as follows: the method comprises the steps of completing missing data of the power load, compressing a large amount of power quality data, positioning line faults, finding harmonic sources and the like. Therefore, in the power system, the compression sensing is mainly used for compressing massive power data, so that efficient transmission is realized, and power signal denoising is rarely involved. While denoising for power signals is mainly used at present: the method comprises the steps of variable step size adaptive filtering, combination of threshold and translation invariant wavelet and an improved threshold translation invariant method, wherein the denoising effect of the method depends on the selection of the threshold, but the selection of the threshold does not have a uniform standard, so the denoising effect is unstable, and the calculation amount is huge.
It is known that the current research mainly uses compressed sensing to reduce the data processing amount of the power line signal, but rarely involves simultaneously compressing and denoising the power line signal containing noise. The denoising method represented by a given threshold value method has large calculation amount and low efficiency.
Disclosure of Invention
The technical problem to be solved by the invention is to provide a compression sensing and reconstruction method based on power line signal transmission singularity measurement, which takes the Lipschitz index as a noise removal basis to remove the module maximum value of noise, and then compresses and senses the denoised module maximum value, thereby effectively improving the accuracy of power line signal transmission. The method overcomes the defect of unstable denoising effect of the common given threshold method and combines the advantage of small computational complexity of compressed sensing.
The invention is realized in such a way, and discloses a compression sensing and reconstruction method based on power signal transmission singularity measurement, which comprises the following steps:
1) Constructing a low-voltage three-phase power transmission line model;
2) Collecting voltage data of power lines of each phase of line;
3) Adding noise to the collected voltage data;
4) Selecting a proper wavelet base;
5) Decomposing the voltage data added with the noise in the step 3 by using binary wavelet transform to obtain a modulus maximum value of each layer of the voltage signal;
6) Judging the modulus maximum value of the useful signal and the modulus maximum value of the noise according to the Lee's index, removing the modulus maximum value of the noise and improving the sparsity of the modulus maximum value of the signal;
7) Compressing and sensing the modulus maximum value retained after denoising by utilizing an OMP algorithm;
8) And reconstructing the signal at a receiving end by utilizing the received signal modulus maximum value restored by the OMP algorithm and reserved.
Further, in the step 3), adding narrow-band noise and Middleton Class a noise respectively, where the narrow-band noise is a result of superposition of independent sinusoidal signals:
wherein, A i Is the amplitude of the signal, which is uniformly normalized and the amplitude is randomly generated in the background noise range, f i Is the frequency of the radio wave to be transmitted,
is the phase;
the probability density function for Middleton Class a noise is as follows:
where z is the impulse noise sample,
is the noise variance, a determines the impulse degree of the noise, and Γ is the gaussian pulse power ratio.
Further, in the step 5), the collected voltage data is decomposed by using binary wavelet transform, so as to obtain a modulus maximum value of each layer of the voltage signal, which is specifically as follows:
given a basis function:
where a is the scale and b is the displacement, discretizing a
a 0 Greater than 0, j ∈ Z then a 0 ＝2，
The wavelet transform at this scale is called binary wavelet transform:
and decomposing the noisy power signal by using binary wavelet transform to obtain the modulus maximum value of each layer.
Further, the step 6) of distinguishing the mode maximum value of the useful signal and the mode maximum value of the noise according to the lees index, removing the mode maximum value of the noise, and improving the sparsity of the mode maximum value of the signal includes: calculating the maximum value of the voltage signal wavelet mode to calculate the Lee index;
if the signal is disconnected at a certain point or the derivative of a certain order is discontinuous, the point of the signal is singular, and in the singularity analysis of the signal, a Lipschitz index is introduced to express singularity characteristics;
given a function f (x) epsilon R, where 0 ≦ α ≦ 1, k is constant, where x 0 To x 0 Such that:
|f(x)-f(x 0 )|≤k|x-x 0 | α (9)
then f (x) is said to be in x 0 Is Lipschitz α if the wavelet transform of f (x) is at point(s) 0 ，x 0 ) Satisfies the following conditions:
|W f (s 0 ，x)|≤|W f (s 0 ，x 0 )| (10)
when the binary wavelet transform is performed on f (x) in the interval [ a, b ], equation (10) satisfies:
|W f (2 j ，x)|≤k(2 j ) α (11)
wherein k is a constant and α is a Lipschitz index;
when W is f (2 j X) is a modulus maximum, on both sides of equation (11) at two different scales s j ，s j+1 The modulus maximum with base 2 is taken up:
subtracting the two ends of the formula (12) and simplifying to obtain the formula (13):
as can be seen from equation (13), when α > 0, the modulus maximum of the wavelet transform is proportional to the scale j, whereas inversely proportional, and when α =0, the modulus maximum is unchanged;
according to the standard that the Lee index of the mode maximum value of the noise is smaller than zero, removing the point that the Lee index of the mode maximum value of the signal is smaller than zero, namely, removing the mode maximum value of the noise and reserving the mode maximum value of the useful signal, and improving the sparsity of the mode maximum value.
Further, in the step 7), the modulus maximum value retained after denoising is compressed and sensed by using an OMP algorithm, which is specifically as follows:
assuming that the original signal is a one-dimensional signal with length N and the sparsity k is unknown, the measurement matrix phi is a matrix of M x N, the sparse matrix psi is a matrix of N x N, theta is a sparse decomposition coefficient, x = psi theta, and the observation vector y is a matrix of M x 1:
y＝φψθ (14)
let the perceptual matrix a = phi ψ, then equation (14) at this time is:
y＝Aθ (15)
the compressed sensing is mainly to solve the underdetermined linear equation y = A theta, and in the case that y and A are unknown, to solve theta,
solving theta by using an OMP algorithm, and recording the solved theta as
The reconstructed signal is then noted as:
further, the flow of the OMP algorithm is as follows:
inputting:
a 1. Mxn sensing matrix a = Φ ψ;
an observation vector y of 2. Nx 1;
3. sparsity K;
and (3) outputting:
1. signal sparse representation coefficient obtained through algorithm calculation
2.Nx 1 residual error
In-process algorithm meterIn the calculation process: r is a radical of hydrogen t Representing the residual, t is the number of iterations,
represents the empty set, Λ t Set of indices, λ, representing t iterations t Representing the index found in the t-th iteration, a j Represents the jth column of matrix A, A t Representative by index Λ t Selected set of columns, θ, of matrix A t Is a column vector of t x 1, the symbol @ denotes a set merging operation,<■，■>the inner product is calculated as follows:
(1) Initialization r 0 ＝y，
t＝1；
(2) Find index lambda t So that λ t ＝arg max j＝1，2...N |＜r t-1 ，a j ＞|；
(3) Let Λ t ＝Λ t-1 ∪{λ t }，A t ＝A t-1 ∪a λ ；
(4) Ask y = A t θ t Least squares solution of (c):
(5) And (3) residual error updating:
(6) t = t +1, if t is less than or equal to K, returning to the step (2), otherwise, stopping iteration and entering the step (7);
(7) Reconstructing the resultant
At Λ t There is a non-zero term whose value is respectively based on the result of the last iteration>
Further, in step 8), the receiving end reconstructs the signal by using the retained signal modulus maximum recovered by the received OMP algorithm, which is specifically as follows:
receiving a reserved modulus maximum value recovered by the OMP algorithm;
reconstructing a signal according to the abscissa of the modulus maximum value, the modulus maximum value and the profile of the last stage; wherein:
in binary wavelet transform, the minimum value of j is 1, the assumed maximum scale j is large enough, and information with the scale larger than j is concentrated in low-frequency a j (t) at which time a j (t) is a j-th level profile; let t j，n Transforming WT for binary wavelets x The abscissa when taking the modulus maximum, then | WT x (j，t j，n ) I is the modulo maximum value of the matrix,
at an arbitrary point t for the signal 0 The method comprises the following steps:
|WT x (j，t 0 )|＝＜x(t)，ψ j (t 0 -t)＞ (17)
and satisfies the following conditions:
＜h(t)，ψ j (t j，n -t)＞＝<x(t)，ψ j (t j，n -t)＞ (18)
where x (t) is the original signal, h (t) is the set of assumed signals, i.e. the approximate signal of x (t), t j，n Is the abscissa of the maximum of the x (t) wavelet transform mode, denoted by t j，n Modulo maximum and last level profile a j (t) to reconstruct the signal x (t).
Compared with the prior art, the invention has the beneficial effects that:
the method combines the advantage of small compressed sensing calculation amount and overcomes the defect of unstable denoising effect of the commonly used given threshold value method, achieves good denoising effect while realizing power line signal compression, is simple to realize, and is easy to realize in practical engineering application.
The invention improves the noise removal effect of the power line, takes the Leersian exponent as the basis for removing the noise, and overcomes the instability of the noise removal effect caused by the uncertainty of threshold selection in the traditional given threshold method. The data processing amount is reduced, the transmission efficiency of power line signals is improved, the compression sensing is utilized to compress and transmit the modulus maximum value which has sparsity after denoising, and the data processing efficiency is greatly improved. Theoretical analysis shows that the method achieves good denoising effect while achieving high-efficiency voltage of the power signal, and can reconstruct an original signal without noise well when the compressibility of a modulus maximum value reserved after signal denoising reaches 40%. The invention combines compressed sensing and opposite sex detection, thereby improving the data compression efficiency of the transmission signal while ensuring the quality and accuracy of the signal. Moreover, the method can greatly save the transmission cost, is easy to implement and has high engineering application value.
Drawings
FIG. 1 is a general flow diagram of a method of compressed sensing and reconstruction based on power signal transmission singularity metrics;
FIG. 2 is a flow chart for removing noise modulus maxima from the Lee index;
FIG. 3 is a flow chart of the OMP algorithm;
fig. 4 is a flow of compressing, sensing, and reconstructing the original signal using the modulus maxima restored by the OMP algorithm.
Detailed Description
In order to make the objects, technical solutions and advantages of the present invention more apparent, the present invention is further described in detail with reference to the following embodiments. It should be understood that the specific embodiments described herein are merely illustrative of the invention and do not delimit the invention.
The invention combines the compressed sensing with the singularity detection of the power line signal, overcomes the problem that the denoising effect is unstable in the currently commonly applied method taking a threshold value as a core, and simultaneously combines the compressed sensing to realize the high-efficiency compression of the power line signal while denoising, thereby realizing the accuracy of the power line signal transmission and saving the storage space.
(1) Discrete binary wavelet transform is carried out on the acquired power line signals, the modulus maximum value of each layer is solved, and the modulus maximum value of noise is removed by taking the Lee index as an index.
The wavelet transform is a continuous wavelet transform, and a binary discrete wavelet is formed after the binary discrete is performed on the scale of the continuous wavelet transform. And decomposing the noisy power line signal by using binary wavelet transformation to obtain a modulus maximum value of each layer, and removing the modulus maximum value of the noise by using the Lee index as a judgment standard.
(1) Performing binary wavelet transform on the signals to obtain decomposition coefficients of each layer, and calculating a modulus maximum value of each layer;
(2) and removing the modulus maximum of the noise according to the difference between the noise and the Lee index of the useful information.
(2) And compressing and sensing the reserved module maximum value by using an Orthogonal Matching Pursuit (OMP) algorithm, and reconstructing an original signal by using the module maximum value restored by the OMP.
The OMP algorithm is a typical representation of the greedy algorithm in the compressive perceptual reconstruction algorithm. The premise of compressed sensing is that the signal has certain sparsity, and the modulus maximum has obvious sparsity, so that the requirement is well met.
(1) Compressing the reserved modulus maximum values of the layers by using an OMP algorithm;
(2) and reconstructing the original signal at the receiving end by using the modulus maximum value recovered by the OMP.
Referring to fig. 1 to 4, in the embodiment, a method for compressed sensing and reconstruction based on power signal transmission singularity measurement includes the following specific steps:
1) And constructing a power transmission model according to the parameters of the actual low-voltage power line. The mathematical model of the real three-phase transmission line is as follows:
wherein u is a 、u b 、u c The phase voltage is respectively the phase A, the phase B and the phase C of the three-phase circuit. w is the angular frequency and α is the phase angle.
2) Collecting voltage data of power lines of each phase of line;
3) Adding noise to the acquired voltage data;
(1) narrow-band noise, the spectrum of which was found to consist of a series of narrow-band signals with multiple impulses, and thus can be considered as the result of the superposition of several independent sinusoidal signals:
wherein A is i Is the amplitude of the signal, which is uniformly normalized and the amplitude is randomly generated within the background noise range. f. of i Is the frequency of the radio,
is the phase.
(2) Middleton Class a noise, a typical model of power line impulse noise. This is an impulse noise modeling method based on physical statistical analysis, since its model derivation is based entirely on the statistical analysis of the sequence in which the power line noise is actually generated. The probability density function is as follows:
where z is the impulse noise sample,
is the variance of the noise, block ATo determine the pulse degree of the noise, Γ is the gaussian pulse power ratio.
4) A suitable wavelet basis is selected. According to the characteristics of the power signal and the waveform characteristics of the dbN wavelet, the db4 wavelet is selected as a wavelet basis, which has no explicit expression, but the square of the modulus of the transfer function is determined:
where the coefficients of the binomial equation are:
5) Decomposing the acquired voltage data by using binary wavelet transform to obtain the modulus maximum value of each layer of the voltage signal, which is as follows:
given a basis function:
where a is the dimension and b is the displacement. Discretizing a, making
a 0 Greater than 0, j ∈ Z then a 0 ＝2，
The wavelet transform at this scale is called a binary wavelet transform, as follows:
and decomposing the noisy power signal by using binary wavelet transform to obtain the modulus maximum value of each layer.
6) And judging the modulus maximum value of the useful signal and the modulus maximum value of the noise according to the Lee index, removing the modulus maximum value of the noise and improving the sparsity of the modulus maximum value of the signal.
(1) Constructing a low-voltage three-phase power transmission line model according to the step 1);
(2) collecting voltage data of power lines with noise in each phase line according to the step 2);
(3) adding noise to the collected voltage data according to step 3)
(4) Selecting a proper wavelet base according to the step 4);
(5) according to the step 5), decomposing the acquired voltage data by utilizing binary wavelet transform to obtain a modulus maximum value of each layer of the voltage signal;
(5) calculating the Lee index according to the maximum value of the wavelet mode of the calculated voltage signal;
the signal is singular if the signal is broken at a certain point or the derivative of a certain order is discontinuous, and in the singularity analysis of the signal, a Lipschitz index is introduced to express the singularity feature.
Given a function f (x) epsilon R, where 0 ≦ α ≦ 1, k is constant, where x 0 To x 0 The neighborhood x of (c) is such that:
|f(x)-f(x 0 )|≤k|x-x 0 | α (9)
then f (x) is said to be in x 0 Is Lipschitz α. If the wavelet transform of f (x) is at point(s) 0 ，x 0 ) Satisfies the following conditions:
|W f (s 0 ，x)|≤|W f (s 0 ，x 0 )| (10)
when the binary wavelet transform is performed on f (x) in the interval [ a, b ], equation (10) satisfies:
|W f (2 j ，x)|≤k(2 j ) α (11)
where k is a constant and α is the Lipschitz index.
When W is f (2 j X) is a modulus maximum, the equation (15) is bounded on both sides by two different scales s j ，s j+1 The modulus maximum with base 2 is taken up:
subtracting and simplifying the two ends of the formula (12) to obtain a formula (13):
as can be seen from equation (13), when α > 0, the modulus maximum of the wavelet transform is proportional to the scale j, whereas inversely proportional, and when α =0, the modulus maximum is unchanged.
(6) According to the standard that the Lee's index of the mode maximum value of the noise is smaller than zero, the mode maximum value of the useful signal is reserved after the point that the Lee's index of the mode maximum value of the signal is smaller than zero is removed, and the sparsity of the mode maximum value is improved.
7) And compressing and sensing the modulus maximum value reserved after denoising by utilizing an OMP algorithm.
The assumed signal is a one-dimensional signal with the length of N, and the sparsity k is unknown. The measurement matrix phi is a matrix of M x N, the sparse matrix psi is a matrix of N x N, theta is a sparse decomposition coefficient, x = psi theta, the observation vector y is a matrix of M x 1:
y＝φψθ (14)
let the perceptual matrix Λ = Φ ψ, then equation (14) at this time can be:
y＝Aθ (15)
the compressed sensing mainly solves the underdetermined linear equation y = a θ, i.e. in case y and a are unknown, θ is solved. The invention uses OMP algorithm to solve theta, and records the solved theta as
The reconstructed signal is then noted as:
the flow of the OMP algorithm is as follows:
inputting:
a 1. Mxn sensing matrix a = Φ ψ;
an observation vector y of 2. Nx 1;
3. sparsity K;
and (3) outputting:
1. signal sparse representation coefficient obtained through algorithm calculation
/>
2.Nx 1 residual error
Referring to fig. 3, in the following algorithm calculation flow: r is a radical of hydrogen t Representing the residual, t is the number of iterations,
represents the empty set, Λ t Set of indices, λ, representing t iterations t Representing the index (column index), a, found in the t-th iteration j Represents the j column of matrix A, Λ t Representative by index Λ t The selected column set of matrix A (M x t matrix), θ t Is a column vector of t × 1, and the symbol @ represents a union operation.<■，■>The inner product is calculated as follows:
(1) Initialization r 0 ＝y，
t＝1；
(2) Find index lambda t So that λ t ＝arg max j＝1，2...N |<r t-1 ，a j >|；
(3) Let Λ t ＝Λ t-1 ∪{λ t }，A t ＝A t-1 ∪a λ ；
(4) Ask y = A t θ t Least squares solution of (c):
(5) And (3) residual error updating:
(6) t = t +1, if t is less than or equal to K, returning to the step (2), otherwise, stopping iteration and entering the step (7);
(7) Reconstructing the resultant
At Λ t There is a non-zero term whose value is respectively based on the result of the last iteration>
According to the characteristic that the modulus maximum value has certain sparsity, the unit matrix is selected as a sparse base, and the Gaussian matrix is selected as a measurement matrix. And then compressing and sensing the denoised modulus maximum value by using an OMP algorithm.
8) And reconstructing the signal at a receiving end by utilizing the received retained signal modulus maximum value recovered by the OMP algorithm.
(1) Receiving a reserved modulus maximum value recovered by the OMP algorithm;
(2) reconstructing a signal according to the abscissa of the modulus maximum value, the modulus maximum value and the profile of the last stage;
in binary wavelet transform, the minimum value of j is 1. Assuming that the maximum dimension j is large enough, information with a dimension larger than j is concentrated in a low frequency j (t) at which time a j And (t) is the j-th level of the profile. Let (t) j，n ) Transforming WT for binary wavelets x The abscissa when taking the modulus maximum, then | WT x (j，t j，n ) I is the modulo maximum.
At an arbitrary point for the signalt 0 The method comprises the following steps:
|WT x (j，t 0 )|＝＜x(t)，ψ j (t 0 -t)＞ (17)
and satisfies the following conditions:
＜h(t)，ψ j (t j，n -t)＞＝<x(t)，ψ j (t j，n -t)＞ (18)
where x (t) is the original signal, and h (t) is the assumed signal set, i.e., the approximate signal of x (h). t is t j，n Is the abscissa of the x (t) wavelet transform mode maximum. From t j，n Modulo maximum and last level profile a j (t) to reconstruct the signal x (t).
The above description is only for the purpose of illustrating the preferred embodiments of the present invention and is not to be construed as limiting the invention, and any modifications, equivalents and improvements made within the spirit and principle of the present invention are intended to be included within the scope of the present invention.
Claims (2)
1. A compressed sensing and reconstruction method based on power signal transmission singularity measurement is characterized by comprising the following steps:
1) Constructing a low-voltage three-phase power transmission line model;
2) Collecting voltage data of power lines of each phase of line;
3) Adding noise to the collected voltage data;
4) Selecting a proper wavelet base;
5) Decomposing the voltage data added with the noise in the step 3 by using binary wavelet transform to obtain a modulus maximum value of each layer of the voltage signal;
6) Judging the modulus maximum value of the useful signal and the modulus maximum value of the noise according to the Lee's index, removing the modulus maximum value of the noise and improving the sparsity of the modulus maximum value of the signal;
7) Compressing and sensing the modulus maximum value reserved after denoising by utilizing an OMP algorithm;
8) Reconstructing a signal at a receiving end by utilizing a received modulus maximum value of the retained useful signal recovered by the OMP algorithm;
in the step 5), the collected voltage data is decomposed by using binary wavelet transform to obtain a modulus maximum value of each layer of the voltage signal, which is as follows:
given a basis function:
where a is the scale and b is the displacement, discretizing a
a 0 >0,j ∈ Z then a 0 ＝2，
The wavelet transform at this scale is called binary wavelet transform:
decomposing the noisy voltage signal by using binary wavelet transform to obtain a modulus maximum value of each layer;
said step 6), judge the module maximum value of the useful signal and the module maximum value of the noise according to the lees' index, remove the module maximum value of the noise, promote the sparsity of the module maximum value of the signal, including: calculating the maximum value of the voltage signal wavelet mode to calculate the Lee index;
if the signal is disconnected at a certain point or the derivative of a certain order is discontinuous, the point of the signal is singular, and in the singularity analysis of the signal, a Lipschitz index is introduced to express singularity characteristics;
given a function f (x) epsilon R, where 0 ≦ α ≦ 1, k is constant, where x 0 To x 0 Such that:
|f(x)-f(x 0 )|≤k|x-x 0 | α (9)
then f (x) is said to be in x 0 Is Lipschitz α if the wavelet transform of f (x) is at point(s) 0 ，x 0 ) Satisfies the following conditions:
|W f (s 0 ,x)|≤|W f (s 0 ,x 0 )| (10)
when the binary wavelet transform is performed on f (x) in the interval [ a, b ], equation (10) satisfies:
|W f (2 j ,x)|≤k(2 j ) α (11)
wherein k is a constant and α is a Lipschitz index;
when W is f (2 j X) is a modulus maximum, on both sides of equation (11) at two different scales s j ，s j+1 The modulus maximum with base 2 is taken up:
subtracting the two ends of the formula (12) and simplifying to obtain the formula (13):
as can be seen from equation (13), when α > 0, the modulus maximum of the wavelet transform is proportional to the scale j, whereas inversely proportional, and when α =0, the modulus maximum is unchanged;
according to the standard that the Lee's index of the mode maximum value of the noise is smaller than zero, removing the point that the Lee's index of the mode maximum value of the signal is smaller than zero, namely, removing the mode maximum value of the noise and reserving the mode maximum value of the useful signal, and improving the sparsity of the mode maximum value;
in the step 7), the OMP algorithm is used for compressing and sensing the modulus maximum value reserved after denoising, and the method specifically comprises the following steps:
assuming that the original signal is a one-dimensional signal with length N and the sparsity k is unknown, the measurement matrix phi is a matrix of M x N, the sparse matrix psi is a matrix of N x N, theta is a sparse decomposition coefficient, x = psi theta, and the observation vector y is a matrix of M x 1:
y＝φψθ (14)
let the perceptual matrix a = phi ψ, then equation (14) at this time is:
y＝Aθ (15)
the compressed sensing mainly solves the underdetermined linear equation y = A theta, and under the condition that y and A are not known, theta is solved,
solving theta by using an OMP algorithm, and recording the solved theta as
The reconstructed signal is then noted as:
the flow of the OMP algorithm is as follows:
inputting:
a 1. Mxn sensing matrix a = Φ ψ;
an observation vector y of 2. Nx 1;
3. sparsity K;
and (3) outputting:
1. signal sparse representation coefficient obtained through algorithm calculation
2.Nx 1 residuals
In the algorithm calculation flow: r is t Representing the residual, t is the number of iterations,
represents the empty set, Λ t Set of indices, λ, representing t iterations t Representing the index found in the t-th iteration, a j Represents the jth column of matrix A, A t Representative by index Λ t Selected set of columns, θ, of matrix A t Is a column vector of t × 1, the symbol @ represents a union operation,<■，■>the inner product is calculated as follows:
(1) Initialization r 0 ＝y，
t＝1；
(2) Find the index λ t So that λ t ＝argmax j＝1，2…N |<r t-1 ，a j >|；
(3) Let Λ t ＝Λ t-1 ∪{λ t }，A t ＝A t-1 ∪a j ；
(4) Ask y = A t θ t Least squares solution of (c):
(5) And (3) residual error updating:
(6) t = t +1, if t is less than or equal to K, returning to the step (2), otherwise, stopping iteration and entering the step (7);
(7) Reconstructing the resultant
At Λ t With non-zero terms having values obtained in the last iteration
In step 8), reconstructing a signal at the receiving end by using the retained signal modulus maximum recovered by the received OMP algorithm, which is specifically as follows:
receiving a reserved modulus maximum value recovered by the OMP algorithm;
reconstructing a signal according to the abscissa of the modulus maximum value, the modulus maximum value and the profile of the last stage; wherein:
in binary wavelet transform, the minimum value of j is 1, the assumed maximum scale j is large enough, and information with the scale larger than j is concentrated in low-frequency a j (t) at which time a j (t) is a j-th level profile; let t j，n Transforming WT for binary wavelets x The abscissa when taking the modulus maximum, then | WT x (j，t j，n ) I is the modulo maximum value of the matrix,
at an arbitrary point t for the signal 0 The method comprises the following steps:
|WT x (j，t 0 )|＝<x(t)，ψ j (t 0 -t)> (17)
and satisfies the following conditions:
<h(t)，ψ j (t j，n -t)>＝<x(t)，ψ j (t j，n -t)> (18)
where x (t) is the original signal, h (t) is the set of assumed signals, i.e. the approximate signal of x (t), t j，n Is the abscissa of the maximum of the x (t) wavelet transform modulus, denoted by t j，n Modulo maximum and last level profile a j (t) to reconstruct the signal x (t).
2. The method as claimed in claim 1, wherein the step 3) adds narrowband noise and Middleton Class a noise separately, wherein the narrowband noise is a result of superposition of independent sinusoidal signals:
wherein A is i Is the amplitude of the signal, which is uniformly normalized and the amplitude is randomly generated in the background noise range, f i Is the frequency of the radio wave to be transmitted,
is the phase;
the probability density function for Middleton Class a noise is as follows:
where z is the impulse noise sample,
is the noise variance, a determines the impulse degree of the noise, and Γ is the gaussian pulse power ratio.
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