CN109787637B - Integer finite field compressed sensing method - Google Patents
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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
Technical Field
The invention relates to the technical field of information and communication, in particular to an integer finite field compressed sensing method.
Background
Reliable transmission of wireless communications in an interfering or noisy environment can be achieved to a large extent by means of forward error correction coding techniques. However, error correction coding needs to increase information redundancy for resisting channel interference, and when the error correction coding is applied in an actual system, in order to achieve efficient transmission, firstly, a source coding needs to be performed on a source, information redundancy brought by the source itself is compressed, and complexity of source equipment is objectively increased.
On the other hand, the compressed sensing technology directly utilizes the information redundancy of the information source, and can reduce the sampling rate at the information source end, thereby reducing the complexity of the information source end equipment. Currently, the compressive sensing technology is not actually used in the transmission process of wireless communication, and the current compressive sensing technology is mainly a real number or complex number domain calculation, so that the current compressive sensing technology is not convenient for signal processing in a digital domain, and when the current compressive sensing technology is directly used for digital signal processing, the problems of accuracy reduction such as quantization error and finite word length effect exist.
In addition, there are some similar researches in the aspect of compressed sensing technology of integer or finite fields. The literature "determination of complex sensing matrices" (Deore RA.. Journal of Complexity,2007,23(4-6):918-6) LDPC codes over polynomial finite fieldsThe check matrix of (2) is used as an observation matrix, but in the methods, the observation matrix is constructed by adopting binary elements 0 and 1, and the finite field in which the observation matrix is positioned is GF (2)q) And a linear programming decoding algorithm is adopted for a recovery algorithm of the compressed sensing system adopting the coding matrix. However, the whole compressed sensing system is suitable for a simpler conventional compressed sensing system, such as the most commonly used image sensing field, but cannot be applied to the communication field with variable modulation modes and capable of mixing and using real objects and complex numbers, and the method also has some limitations on the adopted coding or check matrix, such as the code loop length limitation.
Disclosure of Invention
At least one of the objectives of the present invention is to overcome the above problems in the prior art, and to provide an integer finite field compressive sensing method, which can directly perform compressive sensing processing on digital signals, including first transforming digitized signal source signals by using an integer matrix meeting the requirement of a compressive sensing observation matrix, and then performing a modular operation on the obtained signals by using a transform basis of co-prime, so as to obtain a plurality of digital signal sequences of finite size, which is convenient for communication transmission by using digital modulation.
In order to achieve the above object, the present invention adopts the following aspects.
A method of compressed sensing based on integer finite fields, the method comprising:
grouping information to be transmitted in a fixed length to obtain a first transmission signal; carrying out projection transformation on the integer domain on the first transmission signal to obtain a second transmission signal; performing modular division operation on a plurality of pairwise coprime integer modular values on the second transmission signal to obtain a plurality of remainder sequences, wherein the plurality of remainder sequences are the third transmission signal; modulating the third transmission signal to obtain a fourth transmission signal, and transmitting the fourth transmission signal to a receiving end through a carrier;
the receiving end receives the fourth transmission signal, screens the fourth transmission signal based on the requirement of a demodulation threshold, selects a modulation symbol with the carrier-to-noise ratio higher than the demodulation threshold, and demodulates the screened signal to obtain a fifth transmission signal; obtaining an observation matrix according to the integer domain projection transformation matrix and the modulation symbol selection matrix; solving a compression observed value of the second transmission signal according to the fifth transmission signal and the corresponding integer modulus value; sparse representation is carried out on the first transmission signal, and a 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; and solving an estimation value of the first transmission signal by using a compressed sensing model and a reconstruction algorithm.
Preferably, the projective transformation is a multiplication of the first transmission signal with a random number matrix in the integer domain.
Preferably, in the integer finite field-based compressed sensing method, the modulus division operation is to divide the second transmission signal by a plurality of pairwise relatively prime integers to obtain the plurality of remainder sequences.
Preferably, in the integer finite field-based compressed sensing method, the number of columns of the modulation symbol selection matrix is equal to the dimension of the first transmission signal, the number of rows is equal to the number of modulation symbols in the fourth transmission signal with the carrier-to-noise ratio higher than the demodulation threshold, each row of the symbol selection matrix is a vector with only 1 element being 1 and the rest elements being 0, and the position of the element 1 corresponds to the position of the modulation symbol in the fourth transmission signal with the carrier-to-noise ratio higher than the demodulation threshold.
Preferably, the compressed sensing method based on the integer finite field multiplies the modulation symbol selection matrix and the projective transformation matrix to obtain the observation matrix, and the observation matrix is composed of the modulation symbol selection moment and a part of row vectors of the projective transformation matrix.
Preferably, the compressed sensing method based on the integer finite field is obtained by solving a congruence equation set according to a fifth transmission signal and an integer modulus constructed for the third transmission signal.
Preferably, a compressed sensing method based on an integer finite field obtains a compressed sensing model according to a compressed observation value of a second transmission signal, an observation matrix and a sparse representation of a first transmission signal; the method for calculating the estimation value of the first transmission signal by using the compressed sensing model and the reconstruction algorithm comprises the following steps:
taking the compressed observation value, the observation matrix and the sparse transformation matrix of the second transmission signal as input parameters of a sparse reconstruction algorithm, and obtaining a signal sparse representation estimation coefficient by using an orthogonal matching pursuit algorithm and 1 as a sparsity degree as an optimized target value of the algorithm; and obtaining an estimated value of the first signal through signal reconstruction according to the sparse transform matrix and the signal sparse representation estimation coefficient.
In summary, due to the adoption of the technical scheme, the invention at least has the following beneficial effects:
1. the whole process of the compressed sensing technology is put in an integer digital domain for processing, the difficulty that the current compressed sensing processing based on floating point number is not convenient for realizing a digital circuit is overcome, and the compressed sensing technology is convenient to apply to the field of wireless communication.
2. The information redundancy of the information source signal is directly used for channel error correction, the process that information source coding is firstly carried out to reduce the information redundancy and then channel coding is carried out to increase the information redundancy in the traditional communication system is simplified, the system processing steps are simplified, a novel information source and channel combined coding mode is realized, and the robustness of the transmission system in noise and interference environments is increased.
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 diagram of an integer finite field compressed sensing system according to an exemplary embodiment of the present invention.
Detailed Description
The present invention will be described in further detail with reference to the accompanying drawings and embodiments, so that the objects, technical solutions and advantages of the present invention will be more clearly understood. It should be understood that the specific embodiments described herein are merely illustrative of the invention and are not intended to limit the invention.
Example 1
FIG. 1 illustrates 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, a sending end carries out grouping with fixed length on information to be transmitted to obtain a first transmission signal;
specifically, it is assumed that the digital signal to be transmitted is obtained by directly digitizing from the source, and is not subjected to the information compression process in the source coding. A terminal (sending end) groups information code streams in an integer form to be transmitted according to a certain code element number to obtain a first transmission signal x;
102, performing projection transformation on an integer domain on the first transmission signal to obtain a second transmission signal;
specifically, the first transmission signal x is precoded by using a projective transformation matrix Σ in an integer domain, so that each element in the coded packet signal y can carry information of all symbols in x, which is called projective transformation. The projective transformation process can be written in matrix form:
in the above formula, Σ is a projective transformation matrix (which can also be written as Σ)N×NRepresenting Σ as a square matrix with dimension N), all elements σ in the matrixijThe matrixes meet the requirement on integer observation matrixes in compressed sensing; y is the signal packet after the sigma-delta conversion, and is called as a second transmission signal.
103, performing a plurality of pairwise coprime integer modulus division operations on the second transmission signal to obtain a plurality of remainder sequences, so as to obtain a third transmission signal;
specifically, the second transmission signal y is modulo-divided by p (p > ═ 2) integers which are relatively prime in pairs, so as to obtain p transmission packet signals z1, z2, … and zp, which are called z1, z2 and …, and zp is a third transmission signal, i.e., the plurality of remainder sequences are the third transmission signal.
And each group zi (i ═ 1,2, …, p) is the remainder sequence obtained by modulo division of y by the ith of p pairwise relatively prime integers.
Step 104, modulating the third transmission signal to obtain a fourth transmission signal, and transmitting the fourth transmission signal to a receiving end through a carrier;
specifically, the third transmission signal is modulated to obtain a fourth transmission signal, and the fourth transmission signal is sent to the channel through the radio frequency sending end. In the channel, the modulated signal will be subject to natural or man-made interference.
105, receiving the fourth transmission signal by the receiving end, screening the fourth transmission signal based on the requirement of a demodulation threshold, selecting a modulation symbol with a carrier-to-noise ratio higher than the demodulation threshold, and demodulating the screened signal 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 signal with sufficient carrier to noise ratio is demodulated to obtain the fifth transmission signal s1, s2, …, sp, i.e. the packet zi is considered to be transmitted and demodulated to obtain the packet si (i stands for 1,2, …, p). The position of those discarded modulation symbols in the transmission packet zi is recorded in the demodulation.
106, obtaining an observation matrix according to the integer domain projection transformation matrix and the modulation symbol selection matrix;
specifically, each reception packet si (i ═ 1,2, …, p) in the fifth transmission signal is expanded into a vector form si ═ si [ si ═1si2… siM]T. Since the previous step was partially discarded, the number of elements in each packet si is small relative to the number of 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, assume that M is<<N, then si can be written as follows:
wherein]piIndicating that all elements in parentheses are modulo pi.
pi (i-1, 2, … p) is the selected two-by-two coprime in step threeModulo an integer of (d); matrix BM×NSelecting a matrix for the modulation symbols by: firstly, designing an identity matrix, wherein the row number of the matrix is equal to the number N of elements in a first transmission signal x; and deleting the row corresponding to the position where the discarded modulation symbol is located in the vector of the fifth transmission signal, wherein the row number of the unit matrix is equal to M. Thus the final matrix phiM×NThe matrix is formed by partial row vectors of a projection matrix Σ and is referred to as an observation matrix.
Step 107, obtaining a sparse representation of the first transmission signal;
specifically, since the first transmission signal x is directly obtained by the source, and is not subjected to the information compression process, sparse representation can be performed, and therefore equation (3) can be further written as follows:
si＝[ΦM×N·x]pi＝[ΦM×N·ΨN×P·βP×1]pi(4)
if the dimensions of the above matrices and vectors are ignored, the above equation can also be written as:
si＝[ΦΨβ]pi(5)
in the above formula, a signal vector si demodulated by a receiving end from a value higher than a carrier-to-noise ratio threshold is an observation vector, Φ is an observation matrix, Ψ is a sparse transformation matrix, and its dimension is N × pN×P·βP×1In which ΨN×PThe representation matrix Ψ has the dimensions N × P, βP×1The representation vector β has a dimension P × 1 and is a sparse transform coefficient vector with x at Ψ.
Step 108, solving a compression observed value of the second transmission signal according to the fifth transmission signal and the integer modulus value corresponding to the fifth transmission signal;
specifically, let θ ═ Φ Ψ β, a congruence equation set can be listed according to (5):
si＝[θ]pi，(i＝1,2，…,p) (6)
the value of θ, which is actually a compressed observation value of the second transmission signal, is obtained by the chinese remainder theorem for the above equation.
Step 109, acquiring 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 the results obtained in step 107 and step 108, a compressed sensing model is obtained:
θ＝ΦΨβ
where θ is the compressed observation value of the second transmission signal solved in step eight, Φ is the observation matrix, Ψ is the sparse transformation matrix of the first transmission signal, and both are known values.
Step 110, an estimation value of the first transmission signal is solved by using a compressed sensing model and a reconstruction algorithm.
Specifically, a sparse coefficient vector β is finally obtained by using a common compressed sensing reconstruction algorithm such as an Orthogonal Matching Pursuit (OMP) algorithm, and finally, the transmitted original signal (i.e., the first transmission signal) x is obtained according to x ═ Ψ β.
Example 2
FIG. 2 is a diagram illustrating an integer finite field compressed sensing system according to an exemplary embodiment of the present invention. The integer finite field compressed sensing system specifically comprises: a first communication device, the first communication device comprising: the system comprises a quantization grouping module, a projection transformation module, a modulation module and a radio frequency transmitter; the second communication apparatus (transmitting end) includes: the device comprises a radio frequency receiver, a carrier-to-noise ratio judgment module, a symbol selection module, a demodulation module, an equation solving module and a compressed sensing reconstruction module.
Example 3
The integer domain based compressed sensing algorithm provided by the invention is described in detail below with reference to fig. 1 and 2. Based on the foregoing, we can design a specific embodiment.
It is assumed that the original data to be transmitted can be grouped into x ═ x1x2…… xN]TA redundant 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, and integer matrix Σ (15 × 15 dimension) is used to pair codewordsx is processed by projection transformation, and then is subjected to modular division by two relatively prime integers such as (2, 3), so as to obtain two 15-dimensional packet signals y1 and y2, and then the packet signals are modulated and sent to a channel.A receiving end firstly judges an overload noise ratio, and if the carrier-to-noise ratio of 7 modulation symbols at the same position on each packet signal is smaller than a threshold, the modulation symbols with the carrier-to-noise ratio smaller than the threshold are discarded, then 8 modulation symbols respectively remaining on 2 packets are selected for demodulation, then congruence equation solution is carried out on 16 demodulation data on the 2 packets by adopting Chinese remainder theorem, so as to obtain 8 actually sent integer data, and then an OMP (orthogonal matching pursuit) algorithm is adopted to restore and reconstruct according to the obtained M-8 integer data pairs, so as to obtain an estimated value of a coefficient vector β
Then, the estimation of the code word x is solved according to the sparse transformation relation
The specific implementation process is as follows:
the method comprises the following steps: grouping original data x according to a certain number to obtain a first transmission signal;
the user data information transmitted by the terminal can be expressed as an original data vector x ═ x1x2…… xN]TFor example, the original data is an integer sequence including integers 0 to 7, the packet length number N is 15, and since these data are quantized directly from the source, it can be considered that there is information redundancy between data symbols, and sparse representation can be performed, where x is Ψ β, Ψ is a sparse transform matrix, and β is a transform coefficient.
Step two: the first transmission signal x is precoded by using a projective transformation matrix Σ in the integer domain, so that each element in the coded packet signal y can carry information of all symbols in x, which is called projective transformation. The projective transformation process can be written in matrix form:
in the above formula, Σ is a projective transformation matrix (which can also be written as Σ)N×NRepresenting Σ as a square matrix with dimension N), all elements σ in the matrixijAre all integers and can be chosen as a partial Hadamard matrix of order 15. y is the signal packet after the sigma-delta conversion, and is called as a second transmission signal.
Step three: the integers 2 and 3 which are relatively prime are selected as modulus, and the remainder is calculated for y, so that two remainder sequences z1 and z2 with the size of 15 are obtained, and the third transmission signal is called.
Step four: and then, the third transmission signal is subjected to QPSK modulation to obtain a fourth transmission signal, and the fourth transmission signal is sent into a channel. In the channel, the modulated signal will be subject to natural or man-made interference.
Step five: the receiving end demodulates the received signal, judges the carrier-to-noise ratio of the received signal before demodulation, if the carrier-to-noise ratio is lower than the QPSK demodulation threshold, discards the modulated signal. And demodulating the signal with the carrier-to-noise ratio higher than the QPSK demodulation threshold to obtain a fifth transmission signal s1, s2, namely considering that the packets z1 and z2 are transmitted and demodulated to obtain packets s1, s 2. The positions of those discarded modulation symbols in the transmission packets z1, z2, respectively, are recorded in the demodulation.
Step six: each received packet si (i ═ 1,2) in the fifth transmission signal is expanded into a vector form si ═ si1si2… siM]T. As a result of the loss, there are fewer elements in each packet si relative to the elements in the packet zi at the time of transmission. Assuming that the number of elements in each si is M-7, it is clear that 7<<N15, si can be written as follows:
wherein]piDenotes a remainder obtained by performing a modulo division operation on all elements in parentheses with modulo pi (i ═ 1,2), and p1 ═ 2, p2 ═ 3; matrix B7×15Selecting a matrix for the modulation symbols by: firstly, a unit moment is designedA matrix, the number of rows of which is equal to the number 15 of elements in the first transmission signal x; and then deleting the row corresponding to the position where the discarded modulation symbol in the vector of the fifth transmission signal is positioned, wherein the row number of the unit matrix is equal to that of the fifth transmission signal. Thus the final matrix phi7×15The matrix is formed by partial row vectors of a projection matrix Σ and is referred to as an observation matrix.
Step seven: since the first transmission signal x is directly obtained by the source and thus can be sparsely represented, equation (3) can be further written as follows:
si＝[Φ7×15·x]pi＝[Φ7×15·Ψ15×128·β128×1]pi(4)
if the dimensions of the above matrices and vectors are ignored, the above equation can also be written as:
si＝[ΦΨβ]pi(5)
in the above equation, the signal vector si demodulated by the receiving end from above the carrier-to-noise ratio threshold is an observation vector, Φ is an observation matrix, Ψ is a sparse transform matrix, and the dimension is 15 × 128, since the signal x is directly obtained from the source, it can be sparsely represented as x ═ Ψ β is a sparse transform coefficient vector of x on Ψ, and the dimension is 128 × 1.
In step eight, if θ ═ Φ Ψ β, a congruence equation set can be listed according to (5):
si＝[θ]pi，(i＝1,2) (6)
the value of θ, which is actually a compressed observation value of the second transmission signal, is obtained by the chinese remainder theorem for the above equation.
Step nine: after step eight, a compressed sensing model can be written:
θ＝ΦΨβ
if θ, Φ, and Ψ are known values, a sparse coefficient vector β is obtained by using a common compressed sensing reconstruction algorithm such as an Orthogonal Matching Pursuit (OMP) algorithm, and finally, the original signal x to be transmitted is obtained according to x — Ψ β.
Further, the processing of the signals according to the invention is substantially similar to the congruential transformation of vector signals. When partial signals of the congruence transformation signals of different modes are lost after being transmitted through a channel with noise or interference, the method utilizes Chinese remainder theorem and a compressed sensing reconstruction algorithm to recover the signals at a receiving end. The process ensures that the whole compression sampling process, observation process and recovery process are processed in an integer digital domain, thereby avoiding the influence caused by the finite word length effect and the quantization error in the digital-to-analog conversion. The scheme provided by the invention can make the compressed sensing method of the digital signal domain easier to realize by using a digital circuit, thereby reducing the complexity of the equipment at the transmitting end and realizing an anti-interference communication method different from the traditional channel coding technology.
In the above embodiment, the compressed sensing processing is directly performed on the signal in the digital domain, wherein the digitized signal source symbol (including the digital signal before modulation) is directly processed at the signal source end, so that the process of compressing information redundancy in the signal source coding is omitted, and the complexity of the signal source equipment end is reduced. 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.
The foregoing is merely a detailed description of specific embodiments of the invention and is not intended to limit the invention. Various alterations, modifications and improvements will occur to those skilled in the art without departing from the spirit and scope of the invention.
Claims (7)
1. A compressed sensing method based on integer finite field is characterized by comprising the following steps:
grouping information to be transmitted in a fixed length to obtain a first transmission signal; carrying out projection transformation on the integer domain on the first transmission signal to obtain a second transmission signal; performing modular division operation on a plurality of pairwise coprime integer modular values on the second transmission signal to obtain a plurality of remainder sequences, wherein the plurality of remainder sequences are the third transmission signal; modulating the third transmission signal to obtain a fourth transmission signal, and transmitting the fourth transmission signal to a receiving end through a carrier;
the receiving end receives the fourth transmission signal, screens the fourth transmission signal based on the requirement of a demodulation threshold, selects a modulation symbol with the carrier-to-noise ratio higher than the demodulation threshold, and demodulates the screened signal to obtain a fifth transmission signal; multiplying the projection transformation matrix and the modulation symbol selection matrix to obtain an observation matrix; solving a compression observed value of the second transmission signal according to the fifth transmission signal and the corresponding integer modulus value; sparse representation is carried out on the first transmission signal, and a 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; calculating an estimated value of the first transmission signal by using a compressed sensing model and a reconstruction algorithm;
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 of which the carrier-to-noise ratio in the fourth transmission signal is higher than the demodulation threshold.
2. The method of claim 1, wherein the projective transformation multiplies the first transmission signal by a matrix of random numbers in the integer domain.
3. The method of claim 1, wherein the modulo division operation is dividing the second transmitted signal by a plurality of pairwise relatively prime integers to obtain the plurality of remainder sequences.
4. The method of claim 1, wherein each row of the modulation symbol selection matrix is a vector having only 1 element of 1 and the remaining elements of 0, and wherein the position of element 1 corresponds to the position of the modulation symbol in the fourth transmission signal having the carrier-to-noise ratio higher than the demodulation threshold.
5. The method of claim 1, wherein the observation matrix is formed by a modulation symbol selection matrix and a partial row vector of a projective transformation matrix.
6. The method of claim 1, wherein the compressed observation of the second transmitted signal is obtained by solving a congruence equation set based on a fifth transmitted signal and an integer modulus constructed for the third transmitted signal.
7. The method according to any one of claims 1 to 6, wherein the compressed sensing model is obtained from compressed observations of the second transmission signal, an observation matrix, and a sparse representation of the first transmission signal; the method for calculating the estimation value of the first transmission signal by using the compressed sensing model and the reconstruction algorithm comprises the following steps:
taking the compressed observation value, the observation matrix and the sparse transformation matrix of the second transmission signal as input parameters of a sparse reconstruction algorithm, and obtaining a signal sparse representation estimation coefficient by using an orthogonal matching pursuit algorithm and 1 as a sparsity degree as an optimized target value of the algorithm; and obtaining an estimated value of the first signal through signal reconstruction according to the sparse transform matrix and the signal sparse representation estimation coefficient.
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