CN109787637B - Integer finite field compressed sensing method - Google Patents

Abstract

The invention discloses a method for compressed sensing of integer finite fields, which comprises the following steps: performing grouping with fixed length, projection transformation, modular division operation and modulation on an integer domain on information to be transmitted to obtain a transmission signal, and transmitting the transmission signal; the receiving end receives the transmission signal and screens and demodulates the transmission signal; obtaining a compressed sensing model according to the demodulated signal and the integer mould compression observation value corresponding to the demodulated signal and based on the compression observation value; and solving an original signal by using a compressed sensing model and a reconstruction algorithm. The invention directly carries out compressed sensing processing on the signal in the digital domain, wherein the digitized signal source symbol is directly processed at the signal source end, thereby omitting the process of compressing information redundancy in signal source coding and reducing the complexity of the signal source equipment end. Because the processing is directly carried out in the digital domain, the algorithm is beneficial to being realized by a digital circuit, and simultaneously has the function similar to channel error correction coding, and the anti-interference capability of a communication system can be improved.

Description

An Integer Finite Field Compressive Sensing Method

technical field

The invention relates to the technical field of information and communication, in particular to an integer finite field compressed sensing method.

Background technique

Reliable transmission of wireless communication under interference or noise environment can largely be achieved by forward error correction coding technology. However, error correction coding needs to increase the information redundancy for resisting channel interference. When it is applied in an actual system, for efficient transmission, it is necessary to first perform source coding on the information source, and compress the information brought by the source itself. Information redundancy increases the complexity of the source device objectively.

On the other hand, compressed sensing technology directly utilizes the information redundancy of the source itself, which can reduce the sampling rate at the source end, thereby reducing the complexity of the source end equipment. At present, compressed sensing technology is not actually used for the transmission process of wireless communication, to a large extent because the current compressed sensing technology is mainly a calculation in the real or complex number domain, which is not convenient for signal processing in the digital domain. In digital signal processing, there are quantization errors, finite word length effects and other problems that reduce the accuracy.

In addition, there have been some similar researches on the compressed sensing technology of integer or finite field. The document "Deterministic construction of compressed sensing matrices", (Devore RA..Journal of Complexity, 2007, 23(4-6):918-925) proposed to use the polynomial matrix on the finite field as the observation matrix, because the polynomial finite field is the channel It is a common tool for coding. Since then, many literatures have proposed to use error-correcting codes or check matrices as observation matrices. , "Computer Applications", No. 11, 2014) mentioned that the check matrix of the LDPC code on the GF(2 ⁶ ) polynomial finite field is used as the observation matrix, but in these methods, the binary elements 0 and 1 are used to construct the observation matrix , where the finite field is GF(2 ^q ), and the linear programming decoding algorithm is used for the recovery algorithm of the compressed sensing system using the coding matrix. However, using this method makes the entire compressed sensing system suitable for relatively simple conventional compressed sensing systems, such as the most commonly used image sensing field, but cannot be applied to the communication field where the modulation mode is varied, and physical and complex numbers can be mixed, and in this way The method also has some restrictions on the coding or parity check matrix used, such as the code loop length limitation.

SUMMARY OF THE INVENTION

One of the objectives of the present invention is at least to provide an integer finite field compressive sensing method for how to overcome the above-mentioned problems in the prior art, which can directly perform compressive sensing processing on digital signals. The integer matrix required by the compressed sensing observation matrix is transformed, and then the modulo operation is performed on the obtained signal using the co-prime transformation basis, so that multiple digital signal sequences of finite size will be obtained, which is convenient for communication and transmission using digital modulation.

In order to achieve the above objects, the technical solutions adopted in the present invention include the following aspects.

A compressed sensing method based on an integer finite field, comprising:

The information to be transmitted is grouped with a fixed length to obtain the first transmission signal; the projection transformation on the integer domain is performed on the first transmission signal to obtain the second transmission signal; the second transmission signal is subjected to a plurality of pairwise relatively prime integers The modular division operation of the modulo value obtains a plurality of remainder sequences, and the plurality of remainder sequences is the third transmission signal; after modulating the third transmission signal, a fourth transmission signal is obtained, and the fourth transmission signal is sent to the receiving end through the carrier wave ;

The receiving end receives the fourth transmission signal, screens the fourth transmission signal based on the demodulation threshold requirements, selects a modulation symbol whose carrier-to-noise ratio is higher than the demodulation threshold, and performs demodulation processing on the filtered signal to obtain the fifth transmission signal; obtain the observation matrix according to the integer domain projection transformation matrix and the modulation symbol selection matrix; solve the compressed observation value of the second transmission signal according to the fifth transmission signal and its corresponding integer modulus value; perform sparse representation on the first transmission signal , obtain the compressed sensing model according to the compressed observation value of the second transmission signal, the observation matrix, and the sparse representation of the first transmission signal; use the compressed sensing model and the reconstruction algorithm to obtain the estimated value of the first transmission signal.

Preferably, in a compressed sensing method based on an integer finite field, the projection transformation is to multiply the first transmission signal by a random number matrix in the integer field.

Preferably, in a compressive sensing method based on an integer finite field, the modular division operation is to divide the second transmission signal by a plurality of integers that are relatively prime in pairs to obtain the plurality of remainder sequences.

Preferably, a compressive sensing method based on an integer finite field, the number of columns of the modulation symbol selection matrix is equal to the dimension of the first transmission signal, and the number of rows is equal to the number of rows where the carrier-to-noise ratio in the fourth transmission signal is higher than the demodulation threshold The number of modulation symbols, and each row of the symbol selection matrix is a vector with only one element of 1 and the remaining elements of 0. The position of element 1 corresponds to the modulation symbol in the fourth transmission signal whose carrier-to-noise ratio is higher than the demodulation threshold s position.

Preferably, a compressed sensing method based on an integer finite field, the observation matrix is obtained by multiplying a modulation symbol selection matrix and a projection transformation matrix, and the observation matrix is composed of a modulation symbol selection moment and a partial row vector of the projection transformation matrix of.

Preferably, in a compressive sensing method based on an integer finite field, the compressed observation value of the second transmission signal is obtained by solving a system of congruence equations according to the fifth transmission signal and constructing the integer modulus of the third transmission signal.

Preferably, a compressed sensing method based on an integer finite field, the compressed sensing model is obtained according to the compressed observation value of the second transmission signal, the observation matrix, and the sparse representation of the first transmission signal; using the compressed sensing model and the reconstruction algorithm , and finding the estimated value of the first transmission signal includes:

Using the compressed observation value, observation matrix, and sparse transformation matrix of the second transmission signal as the input parameters of the sparse reconstruction algorithm, through the orthogonal matching pursuit algorithm, and taking 1 as the optimization target value of the algorithm, the sparse representation estimate of the signal is obtained. coefficients; the estimated coefficients are estimated according to the sparse transformation matrix and the signal sparse representation, and the estimated value of the first signal is obtained through signal reconstruction.

To sum up, due to the adoption of the above technical solutions, the present invention has at least the following beneficial effects:

1. The whole process of compressive sensing technology is placed in the integer digital domain for processing, which overcomes the difficulty that the current compressed sensing processing based on floating-point numbers is inconvenient for digital circuit implementation, and is convenient for application in the field of wireless communication.

2. The information redundancy of the source signal itself is directly used for channel error correction, which simplifies the process of first performing source coding to reduce information redundancy in traditional communication systems, and then performing channel coding to increase information redundancy. The system processing steps are simplified, a new form of source-channel joint coding is realized, and the robustness of the transmission system in the noise and interference environment is increased.

Description of drawings

FIG. 1 is a flowchart of an integer finite field compressed sensing method according to an exemplary embodiment of the present invention.

FIG. 2 is a schematic structural diagram of an integer finite field compressed sensing system according to an exemplary embodiment of the present invention.

Detailed ways

The present invention will be further described in detail below with reference to the accompanying drawings and embodiments, so as to make the objectives, technical solutions and advantages of the present invention more clear. It should be understood that the specific embodiments described herein are only used to explain the present invention, but not to limit the present invention.

Example 1

FIG. 1 shows an integer finite field compressed sensing method according to an exemplary embodiment of the present invention. The method of this embodiment mainly includes:

Step 101, the transmitting end groups the information to be transmitted into fixed-length groups to obtain a first transmission signal;

Specifically, it is assumed that the digital signal to be transmitted is obtained after being directly digitized by the source, without going through the information compression process in the source encoding. The terminal (sending end) groups the information code stream in integer form to be transmitted according to a certain number of symbols to obtain the first transmission signal x;

Step 102, performing a projection transformation on the first transmission signal in the integer domain to obtain a second transmission signal;

Specifically, the first transmission signal x is pre-coded by using the projection transformation matrix Σ in the integer domain, so that each element in the encoded packet signal y can carry the information of all symbols in x, which is called this transformation process. is the projection transformation. The projective transformation process can be written in matrix form:

In the above formula, Σ is the projection transformation matrix (it can also be written as Σ _N×N , indicating that Σ is a square matrix with a dimension of N), all elements σ _ij in the matrix are integers, and this matrix conforms to the observation matrix for integers in compressed sensing requirements; y is the signal grouping after Σ transformation, and y is called the second transmission signal.

Step 103, performing a modular division operation on a plurality of pairwise relatively prime integer modulus values on the second transmission signal to obtain a plurality of remainder sequences, thereby obtaining the third transmission signal;

Specifically, the second transmission signal y is modulo p (p>=2) integers that are relatively prime in pairs, and modulo division operation is performed to obtain p transmission packet signals z1, z2, . . . , zp, which are called z1, z2 , . . . , zp are the third transmission signals, that is, the multiple residue sequences are the third transmission signals.

And each group zi (i=1, 2, . . . , p) is a remainder sequence obtained by modulo dividing y by the i-th of p pairwise coprime integers.

Step 104, modulate the third transmission signal to obtain a fourth transmission signal, and send it to the receiving end through a carrier wave;

Specifically, the third transmission signal is modulated to obtain the fourth transmission signal, which is sent into the channel through the radio frequency transmitting end. In the channel, the modulated signal will be subject to natural or man-made interference.

Step 105, the receiving end receives the fourth transmission signal, and screens the fourth transmission signal based on the demodulation threshold requirements, selects modulation symbols whose carrier-to-noise ratio is higher than the demodulation threshold, and demodulates the filtered signal. modulation and processing to obtain a fifth transmission signal;

Specifically, the radio frequency receiving end demodulates the received signal, determines the carrier-to-noise ratio of the received signal before demodulation, and discards the modulated signal if the carrier-to-noise ratio is lower than a certain threshold. The fifth transmission signal s1, s2, . The positions of those discarded modulation symbols in the transport packet zi are recorded in the demodulation.

Step 106, obtaining the observation matrix according to the integer domain projection transformation matrix and the modulation symbol selection matrix;

Specifically, each received packet si (i=1, 2, ..., p) in the fifth transmission signal is expanded into a vector form si=[si ₁ si ₂ ... si _M ] ^T . Due to the partial discarding in the previous step, the elements in each packet si are less than the elements in the packet zi at the time of transmission. Assuming that the number of elements in each si is M, without loss of generality, assuming M<<N, then si can be written in the following form:

Where [] _pi means to perform a modular division operation on all elements in parentheses modulo pi.

pi (i=1,2,...p) is the integer modulus of the pairwise coprime selected in step 3; the matrix B _M×N is the modulation symbol selection matrix, which is obtained by the following methods: first design a unit matrix, the matrix The number of rows is equal to the number N of elements in the first transmission signal x; then delete the row corresponding to the position of the discarded modulation symbol in the vector of the identity matrix whose row number is equal to the fifth transmission signal, and the final number of rows for M. Therefore, the final matrix Φ _M×N is composed of partial row vectors of the projection matrix Σ, which is called the observation matrix.

Step 107, obtain the sparse representation of the first transmission signal;

Specifically, since the first transmission signal x is obtained directly from the source without the information compression process, it can be sparsely represented. Therefore, equation (3) can be further written as the following equation:

si=[Φ _M×N ·x] _pi =[Φ _M×N ·Ψ _N×P ·β _P×1 ] _pi (4)

If the dimensions of the above matrices and vectors are ignored, the above formula can also be written as:

si=[ΦΨβ] _pi (5)

In the above formula, the signal vector si demodulated by the receiver from the threshold higher than the carrier-to-noise ratio is the observation vector, Φ is the observation matrix, Ψ is the sparse transformation matrix, and its dimension is N×P. Since the signal x is obtained directly from the source, it can be sparsely represented as x=Ψβ, or it can be written as x=Ψ _N×P ·β _P×1 , where Ψ _N×P indicates that the matrix Ψ has dimension N×P , β _P×1 means that the vector β has dimension P×1 and is the sparse transform coefficient vector of x on Ψ.

Step 108: Calculate the compressed observation value of the second transmission signal according to the fifth transmission signal and its corresponding integer modulus value;

Specifically, set θ=ΦΨβ, then according to (5), the system of congruence equations can be listed:

si=[θ] _pi , (i=1,2,...,p) (6)

Using the Chinese remainder theorem to find the value of θ for the above formula, the value of θ is actually the compressed observation value of the second transmission signal.

Step 109: Obtain a compressed sensing model according to the compressed observation value of the second transmission signal, the observation matrix, and the sparse representation of the first transmission signal;

Specifically, according to steps 107 and 108, a compressed sensing model is obtained:

θ=ΦΨβ

Wherein, θ is the compressed observation value of the second transmission signal solved in step 8, Φ is the observation matrix, and Ψ is the sparse transformation matrix of the first transmission signal, all of which are known values.

In step 110, the estimated value of the first transmission signal is obtained by using the compressed sensing model and the reconstruction algorithm.

Specifically, the sparse coefficient vector β is finally obtained by using a commonly used compressed sensing reconstruction algorithm such as the Orthogonal Matching Pursuit (OMP) algorithm, and finally the transmitted original signal (ie the first transmission signal) x is obtained according to x=Ψβ .

Example 2

FIG. 2 shows a schematic structural diagram of an integer finite field compressed sensing system according to an exemplary embodiment of the present invention. The integer finite field compressed sensing system of the present invention specifically includes: a first communication device, the first communication device includes: a quantization grouping module, a projection transformation module, a modulation module, and a radio frequency transmitter; the second communication device (transmitter) includes: RF receiver, carrier-to-noise ratio judgment module, symbol selection module, demodulation module, equation solving module, compressive sensing reconstruction module.

Example 3

The compressive sensing algorithm based on the integer domain provided by the present invention will be described in detail below with reference to FIG. 1 and FIG. 2 . Based on the foregoing ideas, we can design a specific implementation.

再根据稀疏变换关系求出编码码字x的估计

具体实施过程如下：It is assumed that the original data to be transmitted can be grouped as x=[x ₁ x ₂ ... x _N ] ^T , where N=15. Since the signal x is directly digitized from the source, it can be sparsely represented. A redundancy dictionary Ψ can be constructed, and the signal x can be represented as a sparse coefficient vector β of sparsity K in the Ψ domain. An integer matrix Σ (15×15 dimension) is used to perform projective transformation on the codeword x, and after modular division by two relatively prime integers such as (2, 3), two 15-dimensional grouped signals y1 and y2 are obtained, and then After modulation, it is sent into the channel. The receiver first passes the carrier-to-noise ratio judgment. Assuming that there are 7 modulation symbols at the same position on each packet signal whose carrier-to-noise ratio is less than the threshold, then these modulation symbols whose carrier-to-noise ratio is less than the threshold will be discarded, and then 2 packets will be selected. The remaining 8 modulation symbols are demodulated, and then the 16 demodulated data on the 2 groups are used to solve the congruence equation using the Chinese remainder theorem, and the 8 integer data actually sent are obtained. According to this, the OMP (Orthogonal Matching Pursuit) algorithm is used to restore and reconstruct the obtained M=8 integer data pairs, and the estimated value of the coefficient vector β is obtained.

Then according to the sparse transformation relationship, the estimation of the encoded codeword x is obtained

The specific implementation process is as follows:

Step 1: Group the original data x according to a certain number to obtain the first transmission signal;

The user data information sent by the terminal can be represented as an original data vector x=[x ₁ x ₂ ··· x _N ] ^T , wherein each element is an integer. For example, the original data is an integer sequence including integers 0 to 7, and the number of packet lengths N=15 is selected. Since these data are directly quantized from the source, it can be considered that there is information redundancy between the data symbols, which can be sparse. Representation, let x=Ψβ, Ψ is the sparse transformation matrix, and β is the transformation coefficient.

Step 2: Use the projection transformation matrix Σ on the integer domain to precode the first transmission signal x, so that each element in the encoded packet signal y can carry the information of all symbols in x, which is called this transformation process. is the projection transformation. The projective transformation process can be written in matrix form:

In the above formula, Σ is the projection transformation matrix (it can also be written as Σ _N×N , indicating that Σ is a square matrix with a dimension of N), and all elements σ _ij in the matrix are an integer, which can be selected as a partial Hadamard matrix of order 15 . y is the signal grouping after Σ transformation, and y is called the second transmission signal.

Step 3: Select coprime integers 2 and 3 as modulo, calculate the remainder for y, and obtain two remainder sequences z1 and z2 of size 15, which are called the third transmission signal.

Step 4: The third transmission signal is then modulated by QPSK to obtain a fourth transmission signal, which is sent into the channel. In the channel, the modulated signal will be subject to natural or man-made interference.

Step 5: The receiving end demodulates the received signal, judges the carrier-to-noise ratio of the received signal before demodulation, and discards the modulated signal if the carrier-to-noise ratio is lower than the demodulation threshold of QPSK. The fifth transmission signals s1 and s2 are obtained after demodulating the signal with the carrier-to-noise ratio higher than the QPSK demodulation threshold, that is, it is considered that the packets z1 and z2 are transmitted and demodulated to obtain the packets s1 and s2. The positions of those discarded modulation symbols in the transport packets z1, z2, respectively, are recorded in the demodulation.

Step 6: Expand each received packet si (i=1, 2) in the fifth transmission signal into a vector form si=[si ₁ si ₂ ··· si _M ] ^T . Due to the loss, there are fewer elements in each packet si relative to the elements in packet zi at the time of transmission. Assuming that the number of elements in each si is M=7, obviously 7<<N=15, then si can be written in the following form:

Wherein [] _pi represents the remainder obtained by modulo division of all elements in parentheses by modulo pi (i=1, 2), and p1=2, p2=3; matrix B _7×15 is the modulation symbol selection matrix, through The following method is obtained: first design a unit matrix, the number of rows of the matrix is equal to the number of elements in the first transmission signal x 15; then the row number of the unit matrix is equal to the position of the discarded modulation symbol in the vector of the fifth transmission signal The corresponding line is deleted. Therefore, the final matrix Φ _7×15 is composed of partial row vectors of the projection matrix Σ, which is called the observation matrix.

Step 7: Since the first transmission signal x is obtained directly from the source, it can be sparsely represented, so the formula (3) can be further written as the following formula:

si=[Φ _7×15 ·x] _pi =[Φ _7×15 ·Ψ _15×128 ·β _128×1 ] _pi (4)

If the dimensions of the above matrices and vectors are ignored, the above formula can also be written as:

si=[ΦΨβ] _pi (5)

In the above formula, the signal vector si demodulated by the receiver from the threshold higher than the carrier-to-noise ratio is the observation vector, Φ is the observation matrix, Ψ is the sparse transformation matrix, and its dimension is 15×128. Since the signal x is obtained directly from the source, it can be sparsely represented as x=Ψβ, where β is the sparse transform coefficient vector of x on Ψ, and its dimension is 128×1.

Step 8: Set θ=ΦΨβ, then according to (5), the system of congruence equations can be listed:

si=[θ] _pi , (i=1,2) (6)

Using the Chinese remainder theorem to find the value of θ for the above formula, the value of θ is actually the compressed observation value of the second transmission signal.

Step 9: After Step 8, the compressed sensing model can be written:

θ=ΦΨβ

Among them, when θ, Φ, and Ψ become known values, the sparse coefficient vector β is obtained by using a common compressed sensing reconstruction algorithm such as the Orthogonal Matching Pursuit (OMP) algorithm, and finally, according to x=Ψβ, the original transmitted original value is obtained. signal x.

Further, the process of processing the signal in the present invention is actually similar to the congruential transformation of the vector signal. When the congruent transform signals of different modes are transmitted through a channel with noise or interference, part of the signal will be lost, and then the present invention uses the Chinese remainder theorem and the compressed sensing reconstruction algorithm to recover the signal at the receiving end. This process ensures that the entire compression sampling process, observation process and recovery process are processed in the integer digital domain, thus avoiding the effect of finite word length and quantization errors in digital-to-analog conversion. Through the scheme proposed by the present invention, the compressed sensing method in the digital signal domain can be more easily realized by using digital circuits, thereby reducing the complexity of the transmitting end device and realizing an anti-interference communication method different from the traditional channel coding technology.

In the above-mentioned embodiment, by directly performing compressed sensing processing on the signal in the digital domain, the digitized signal source symbol (also including the digital signal before modulation) is directly processed at the source end, and the redundant information in the compressed information in the source coding is omitted. The redundancy process reduces the complexity of the source device. Due to the direct processing in the digital domain, the algorithm is beneficial to be implemented by digital circuits, and has a function similar to channel error correction coding, which can improve the anti-interference ability of the communication system.

The above description is only a detailed description of the specific embodiments of the present invention, rather than a limitation of the present invention. Various substitutions, modifications and improvements made by those skilled in the relevant technical field without departing from the principle and scope of the present invention should be included within the protection scope of the present invention.

Claims (7)

1. a compressive sensing method based on integer finite field, is characterized in that, comprises: The information to be transmitted is grouped with a fixed length to obtain the first transmission signal; the projection transformation on the integer domain is performed on the first transmission signal to obtain the second transmission signal; the second transmission signal is subjected to a plurality of pairwise relatively prime integers The modular division operation of the modulo value obtains a plurality of remainder sequences, and the plurality of remainder sequences is the third transmission signal; after modulating the third transmission signal, a fourth transmission signal is obtained, and the fourth transmission signal is sent to the receiving end through the carrier wave ; The receiving end receives the fourth transmission signal, screens the fourth transmission signal based on the demodulation threshold requirements, selects a modulation symbol whose carrier-to-noise ratio is higher than the demodulation threshold, and performs demodulation processing on the filtered signal to obtain the fifth transmission signal; multiply the projection transformation matrix and the modulation symbol selection matrix to obtain the observation matrix; solve the compressed observation value of the second transmission signal according to the fifth transmission signal and its corresponding integer modulus value; perform sparse representation on the first transmission signal , obtain the compressed sensing model according to the compressed observation value of the second transmission signal, the observation matrix, and the sparse representation of the first transmission signal; use the compressed sensing model and the reconstruction algorithm to obtain the estimated value of the first transmission signal; The number of columns of the modulation symbol selection matrix is equal to the dimension of the first transmission signal, and the number of rows is equal to the number of modulation symbols whose carrier-to-noise ratio is higher than the demodulation threshold in the fourth transmission signal. 2 . The method according to claim 1 , wherein the projective transformation is to multiply the first transmission signal by a random number matrix in the integer domain. 3 . 3 . The method according to claim 1 , wherein the modular division operation is to divide the second transmission signal by a plurality of integers that are relatively prime in pairs to obtain the plurality of remainder sequences. 4 . 4. The method according to claim 1, wherein each row of the modulation symbol selection matrix is a vector with only 1 element being 1 and the remaining elements being 0, and the position of the element 1 corresponds to the fourth transmission signal. The position of the modulation symbol whose carrier-to-noise ratio is above the demodulation threshold. 5 . The method according to claim 1 , wherein the observation matrix is composed of a modulation symbol selection matrix and a partial row vector of a projection transformation matrix. 6 . 6 . The method according to claim 1 , wherein the compressed observation value of the second transmission signal is obtained by solving a system of congruence equations according to the fifth transmission signal and constructing the integer modulus of the third transmission signal. 7 . 7. The method according to any one of claims 1 to 6, wherein the compressed sensing model is obtained according to the compressed observation value of the second transmission signal, the observation matrix, and the sparse representation of the first transmission signal; using The compressed sensing model and the reconstruction algorithm to obtain the estimated value of the first transmission signal include: Using the compressed observation value, observation matrix, and sparse transformation matrix of the second transmission signal as the input parameters of the sparse reconstruction algorithm, through the orthogonal matching pursuit algorithm, and taking 1 as the optimization target value of the algorithm, the sparse representation estimate of the signal is obtained. coefficients; the estimated coefficients are estimated according to the sparse transformation matrix and the signal sparse representation, and the estimated value of the first signal is obtained through signal reconstruction.

Images