CN103841583B - A kind of radio network optimization magnanimity signaling data acquisition method based on compressed sensing - Google Patents
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Abstract
The invention discloses a kind of radio network optimization magnanimity signaling data acquisition method based on compressed sensing, its step includes：Set up signaling acquisition system；Set up initial data pretreated model；Signaling data categorizing selection device is set up, categorizing selection carries out rarefaction to each index；Calculation matrix is set up, the data after rarefaction are compressed with the method for compressed sensing；Data recovery algorithm is set up, accelerates to complete recovery task using parallel computation.This method can realize the vast scale compression to signaling data well in the case of signaling data magnanimity, it is possible to close to sig data information is precisely or accurately recovered, especially for two-value situation；The collection of signaling data can be realized with a kind of divisible, discardable transmission, the load of transmission is reduced, can be successfully applied in the middle of follow-up same or analogous task, save energy in the transmission environment of wireless network.
Description
Technical field
The present invention relates to the technical field of radio network optimization magnanimity signaling data collection, refer in particular to a kind of based on compression
The radio network optimization magnanimity signaling data acquisition method of perception.
Background technology
How developed rapidly with the granting and domestic mobile Internet of 3G license, the exponential growth of terminal data amount
On existing ripe 2G networks, developing mobile Internet, development Various types of data business rapidly and healthily turns into the country three at present
Big operator focus of attention.By taking China Mobile as an example, one of wireless carriers of Largest In China, the base of its user are used as
Plinth business datum and value-added service data are skyrocketed through in recent years, and the arrival in big data epoch is both to choose for operator
War is again opportunity.Operator thinks the status of solid base communication service, wins the admission ticket of internet, can combine the excellent of itself
Gesture, connects the behavioural habits of miscellaneous service and user, more preferable service user.And what is more important, in data industry
It is engaged under the epoch of great development, how efficiently stores the progress network optimization of wireless user's mass data urgent as what is confronted
Problem.
The research of data compression is very active always, but substantially all more strict to the integrity demands of data.As long as number
Damaged according to the data for going out active or reception during transmission, data are difficult to be recovered.In recent years, study very
Popular " compressed sensing " algorithm brings new approaches to solve the problem.The original intention of compressed sensing is when gathering signal
Wait and reduce sample rate, and accurately initial data is recovered by the way that restoration algorithm is close afterwards, the single pixel of rice university
Camera is namely based on principle realization.But advantage of the compressed sensing in terms of compression is also what is taken charge of a department, particularly works as net
Network speed is undesirable, it is difficult to which when receiving a complete data, Ordinary Compression algorithm is difficult stable work, and compressed sensing
Can preferably it work.Compressed sensing algorithm is used when being acquired to radio network optimization mass data, be yet there are no
Document report.Its compression not only effectively to data, saves transmission cost, and data when can also allow transmission after compression
Lose, the data after compressing in general are lost on a small quantity to be impacted to the recovery of former data.
In recent years, compressed sensing achieves gratifying achievements in numerous areas such as medical examination, Internet of Things.Compression sense
Know a kind of innovation for also becoming collection signal method, existing strict mathematical theory proves its correctness and validity.Due to
The peculiar effect of compressed sensing, and compressed sensing in non-linear optimal solution theory with having developed into a series of ripe solution sides
The fusion faculty of method so that the data acquisition algorithm based on compressed sensing is ground extensively in fields such as image, sound, mass datas
Study carefully and apply.
Existing radio network optimization data acquisition is mostly simply to carry out copying and saving using signal collecting card, not
Have and it is compressed using data compression algorithm, and it is all sparse or can be sparse to have many fields per data, is caused
The waste in data storage space, and the load of transmission is larger.If be compressed using compressed sensing to it, it can not only subtract
The space of few data storage and the load of transmission, can also strengthen the security of data.Particularly network signal it is weaker and warp
In the case of often there is bust this, the strategy of discarding can be simply taken, the reception without influenceing data.
Two most crucial technologies of compressed sensing are exactly to build calculation matrix and selection restoration algorithm, and calculation matrix needs full
Sufficient RIP properties, restoration algorithm is related to efficiency and accuracy rate when recovering data；Summary demand, the survey that the present invention is used
Moment matrix is got by the graceful matrix of hada, and restoration algorithm is using ripe orthogonal matching pursuit algorithm.
The graceful matrix of hada, with very strong incoherence, vector is all pairwise orthogonal, and each number all for 1 or-
1, greatly reduce amount of calculation.The major defect of current this method is the data compression for being difficult in adapt to random length, and it will
Ask initial data length to be necessary for 2 n powers (n is positive integer), and be often possible in reality undesirable.Due to this hair
The bright radio network optimization data length to be gathered is controllable, as long as so data length is set to 2 n powers.
Data recovery algorithm is used to the reduction to compressed data, that is, seeks a globally optimal solution, be largely divided into greedy calculation
Method, L1 convex optimized algorithms, TVL3 restoration algorithms etc..Orthogonal matching pursuit algorithm, abbreviation OMP algorithms, circulates all look for each time
Go out the column vector minimum with one and residual error inner product, but a step quadrature operation more than MP algorithm, accelerate the process of iteration with
Improve the accuracy rate of selection atom.Performance in terms of efficiency is also outstanding than base tracing algorithm, and can be using parallel meter
Calculate and accelerate solving speed.Therefore, when former data length is larger, using OMP algorithms are simple and good performance.
Many Practical Project systems, because the data of system have openness or for can be sparse in itself, can be used
A kind of method based on compressed sensing is acquired and stored to it, and this is for terminal small machines, and performance is to meet to require
's.And amount of calculation is larger during recovery due to compressed sensing, the later stage is using the stronger computer of computing capability to data
Restored, the problem can be solved, so can be close to effectively compressed data in the case of accurate.
The content of the invention
It is an object of the invention to the deficiencies in the prior art, there is provided a kind of wireless network based on compressed sensing is excellent with defect
Change magnanimity signaling data acquisition method, this method can be well realized to signaling data in the case of signaling data magnanimity
Vast scale compresses, it is possible to close to sig data information is precisely or accurately recovered, especially for two-value situation；Can be
In the transmission environment of wireless network, the collection of signaling data is realized with a kind of divisible, discardable transmission, transmission is reduced
Load, can be successfully applied in the middle of follow-up same or analogous task, save energy.
To achieve the above object, technical scheme provided by the present invention is：A kind of wireless network based on compressed sensing is excellent
Change magnanimity signaling data acquisition method, comprise the following steps：
1）The acquisition system using mobile communication signaling as target is set up, data during for mobile phone terminal call and online
It is acquired, can be set up in each base station or upper strata terminal, Gb mouth of the particular location such as between BSC and SGSN,
Or the IuPS mouths between RNC and SGSN etc.；
2）Initial data pretreated model is set up, useless or incomplete original signaling data is deleted, and it is right
Core data is extracted and pre-processed；
3）Signaling data categorizing selection device is set up, according to step 1）The signaling acquisition system and step 2 of foundation）The original of foundation
Data after pretreatment are classified by beginning data prediction model, the openness of data are classified based on, for the dilute of signal
Property is dredged, if wherein only a small number of elements are non-zeros, the signal is sparse.Natural sign in usual time domain, such as language
Sound, image etc. is all non-sparse, but is probably sparse, such as wavelet field in some transform domains；Wherein, classifying rules is：
If data meet rarefaction feature in itself, rarefaction operation need not be carried out；If it is not satisfied, then according to wavelet transformation or Fourier
The rarefaction method of conversion, pair can the data of rarefaction enter line translation,, can if still not enough openness for the data after conversion
Further processing, such as, can be by setting a threshold values for image（Statistical analysis acquirement is carried out according to image pixel value）,
And the data below the threshold values are all cleared to 0, make data fit openness；
4）Setting up has non coherent calculation matrix, and the data after rarefaction are pressed with the method for compressed sensing
Contracting, the method for setting up calculation matrix is relatively more, and conventional has using graceful matrix of random matrix, hada of Gaussian Profile etc., these
Linear incoherent property can be met；Wherein the graceful matrix of hada each measurement vector between it is orthogonal each other, each measure to
Amount is made up of 1 or -1 value, and the length requirement of each vector is 2 m powers, and m is positive integer；
5）Set up data recovery algorithm and complete data recovery task, according to step 4）Described calculation matrix is to step 3）Enter
Measurement result obtained by row measurement, the data recovery algorithm based on orthogonal matching pursuit OMP algorithms is completed to sparse data
Restore, this is actually one and asks the deficient of optimum solution to determine process, is finally converted into initial data by inverse transformation again.
In step 1）In, the acquisition system, all signaling datas sent to mobile phone terminal are preserved, the signaling
Data include useful family identity, session start time, conversation end time, surfing flow, network data.
In step 2）In, described initial data pretreated model, for each signaling data, if signaling data is not
The complete, discarding, then directly remove the Data Entry；For each signaling data, it is understood that there may be multiple indexs, need
Analysis or combined analysis are separated, the network data of such as user is a pictures or one section of voice, video, then needs to merge one
Rise, signaling data is pre-processed according to demand.
In step 3）In, described signaling data categorizing selection device is as follows：
xi=f (si)
Wherein, for step 2）Obtained data change into polytype tables of data si, may be word, image, language
Sound；By taking image data as an example, then xiFor image data siTo carrying out the value after small echo sparse transformation base is multiplied, become by small echo
The rarefaction method such as change so that data xiMeet sparse characteristic；After wavelet transformation, sampling is compressed and by restoring
Algorithm restore obtained xi, also simultaneously should be by wavelet inverse transformation, to obtain si；If the index siItself meet dilute
Thinning property, then xi=si, then without inverse transformation of progress after restoring.
In step 4）In, described calculation matrix, with step 3）Resulting sparse data has incoherence；Hada is graceful
Orthogonal each other between each measurement vector of matrix, each measurement vector is made up of 1 or -1 value, the length requirement of each vector
For 2 m powers, m is positive integer, and using the graceful matrix of hada as calculation matrix, its perfect matrix is as follows：
H0=1
Wherein, it is desirable to which the m powers that the length of each vector is 2, m is positive integer；When length n is taken as 65536, k is
16, H16For 65536 matrixes for being multiplied by 65536, the numerical value of the matrix is unification, meets compression and the equity requirement restored；
If compression ratio is 0.1, preceding 65536 row of calculation matrix is chosen, 65536*65536 calculation matrix A is formed；Measurement
Matrix carries out the y=Ax that is multiplied with sparse initial data, obtains measured value y, then y length is the 1/10 of x；
In step 5）In, described data recovery algorithm is when needing to restore data, according to step 4）Gained
The measured value y arrived, and the calculation matrix arranged according to y length solve x process；And use orthogonal matching pursuit
OMP algorithms, initial data x can be recovered exactly, at the beginning of orthogonal matching pursuit OMP algorithms are a successive ignition process, residual error
Beginning turns to initial data y, Increment Matrix ΦiSky is initialized as, subscript i represents the variable in ith iteration, each interior circulation
Process it is as follows：
1）The column vector atom maximum with residual error inner product is found out from calculation matrix, the column vector is then added to increment
In matrix, and the column vector is deleted from calculation matrix；
2）Solve an optimal problem
In this solution procedure, matrix decomposition can be used to make calling program can be with parallel processing.It is defined as C,
Because C is Nonsingular symmetrical matt, LDL can be used when invertingTDecomposed, and L and D are updated according to newest atom；
Then the inverse of C is：
3）Update residual error
After successive ignition, above-mentioned algorithm will obtain an optimal solutionIf primary signal has by rarefaction
Conversion, then it is rightInitial data x can just be obtained by carrying out once corresponding inverse transformation；If do not converted by rarefaction,
The present invention compared with prior art, has the following advantages that and beneficial effect：
1st, magnanimity signaling data acquisition method of the invention takes new compressed sensing to wireless network optimization data
Algorithm, can realize adopting for signaling data in the transmission environment of wireless network with a kind of divisible, discardable transmission
Collection, reduces the load of transmission, can be successfully applied in the middle of follow-up same or analogous task, save transmission belt
Width, reduces the memory space of data；
2nd, magnanimity signaling data acquisition method of the invention can not only be realized ensure that in the case of unstable networks
The integrality of data, it is to avoid the consumption that data are retransmitted, and a certain degree of encryption can be carried out to data, it is ensured that data
Security.
3rd, magnanimity signaling data acquisition method of the invention being capable of the realization pair well in the case of signaling data magnanimity
The vast scale compression of signaling data, it is possible to close to sig data information is precisely or accurately recovered, especially for two-value
Situation.
Brief description of the drawings
Fig. 1 is signaling acquisition system overall framework figure.
Fig. 2 is the packet produced when cellphone subscriber surfs the Net.
Fig. 3 is the part field of China Mobile user Internet data bag.
Fig. 4 is 256*256 picture example.
Fig. 5 is data of the image data after rarefaction in Fig. 4.
Fig. 6 is the schematic diagram after data recovery.
Embodiment
With reference to specific embodiment, the invention will be further described.
It is shown in Figure 1, the collection of the radio network optimization magnanimity signaling data based on compressed sensing described in the present embodiment
Method, its concrete condition is as follows：
1）The acquisition system using mobile communication signaling as target is set up, data during for mobile phone terminal call and online
It is acquired, can be set up in each base station or upper strata terminal.
2）Initial data pretreated model is set up, useless or incomplete original signaling data is deleted, and it is right
Core data is extracted and pre-processed.
3）Signaling data categorizing selection device is set up, according to step 1）The signaling acquisition system and step 2 of foundation）The original of foundation
Data after pretreatment are classified by beginning data prediction model；If data meet rarefaction feature in itself, it need not enter
Row rarefaction is operated；Otherwise, according to wavelet transformation or the rarefaction method of Fourier transformation, pair can the data of rarefaction become
Change, set a threshold values, the data below threshold values are all cleared to 0, make the sparse degree of data fit.
4）Setting up has non coherent calculation matrix, according to compressive sensing theory, except step 3）It is required that measurement number
According to the feature with rarefaction is needed, another requirement is that calculation matrix has very strong incoherence；Set up the side of calculation matrix
Method is relatively more, just can accurate release signal as long as meeting linear incoherent feature；Conventional has using Gaussian Profile
Graceful matrix of random matrix, hada etc.；The present invention is used to be had between the graceful matrix of hada, each measurement vector of the graceful matrix of hada
There is very strong incoherence, and calculate relatively simple, but require the m powers that the length of each vector is 2 (m is positive integer).
5）Set up data recovery algorithm and complete data recovery task, according to step 4）Described calculation matrix is to step 3）Enter
Measurement result obtained by row measurement, the data recovery algorithm based on OMP completes the recovery to sparse data, and this is actually one
It is individual to ask the deficient of optimum solution to determine process, initial data is finally converted into by inverse transformation again.
In step 1）In, the acquisition system, all signaling datas sent to mobile phone terminal are preserved, and it gathers system
System can be realized that each signaling data all has hundreds of by the existing signal collecting card in self-designed or market
Field, it is topmost including user identity, session start time, conversation end time, surfing flow, network data etc..
In step 2）In, described initial data pretreated model, for each data signaling, if data signaling is
It is incomplete, the discarding, then directly remove the Data Entry；It is that we are unconcerned for a data, such as control instruction,
Can also directly it delete；For each data, it is understood that there may be multiple indexs, it is necessary to separately analyze or combined analysis, such as
The network data of user is a pictures or one section of voice, video, then needs to merge, signaling data is entered according to demand
Row pretreatment.
In step 3）In, described signaling data categorizing selection device is as follows：
xi=f (si)
Wherein, for step 2）Obtained data change into polytype tables of data si, may be word, image, language
Sound；By taking image data as an example, then xiFor image data siTo carrying out the value after small echo sparse transformation base is multiplied, become by small echo
The rarefaction method such as change so that data xiMeet sparse characteristic；After wavelet transformation, sampling is compressed and by restoring
Algorithm restore obtained xi, also simultaneously should be by wavelet inverse transformation, to obtain si；If the index siItself meet dilute
Thinning property, then xi=si, then without inverse transformation of progress after restoring.
In step 4）In, described calculation matrix, with step 3）Resulting sparse data has very strong incoherence；
Orthogonal each other between each measurement vector of the graceful matrix of hada, each measurement vector is made up of 1 or -1 value, the length of each vector
Degree requires the m powers (m is positive integer) for 2, and using the graceful matrix of hada as calculation matrix, its perfect matrix is as follows：
H0=1
Wherein, it is desirable to which the m powers that the length of each vector is 2, m is positive integer；When length n is taken as 65536, k is
16, H16For 65536 matrixes for being multiplied by 65536, the numerical value of the matrix is unification, meets compression and the equity requirement restored；
If compression ratio is 0.1, preceding 65536 row of calculation matrix is chosen, 65536*65536 calculation matrix A is formed；Measurement
Matrix carries out the y=Ax that is multiplied with sparse initial data, obtains measured value y, then y length is the 1/10 of x；Using the graceful square of hada
Battle array is as calculation matrix, in addition to orthogonal property, and the benefit of calculating speed can be improved when measuring compression by also having, and be kept away
Exempt from the inefficiency of gaussian random matrix floating number multiplication, and only carry out plus and minus calculation.
In step 5）In, described data recovery algorithm is when needing to restore data, according to step 4）Gained
The measured value y arrived, and the calculation matrix arranged according to y length solve x process；And use orthogonal matching pursuit
OMP algorithms, initial data x can be recovered exactly, at the beginning of orthogonal matching pursuit OMP algorithms are a successive ignition process, residual error
Beginning turns to initial data y, Increment Matrix ΦiSky is initialized as, subscript i represents the variable in ith iteration, each interior circulation
Process it is as follows：
1）The column vector atom maximum with residual error inner product is found out from calculation matrix, the column vector is then added to increment
In matrix, and the column vector is deleted from calculation matrix；
2）Solve an optimal problem
In this solution procedure, matrix decomposition can be used to make calling program can be with parallel processing.It is defined as C,
Because C is Nonsingular symmetrical matt, LDL can be used when invertingTDecomposed, and L and D are updated according to newest atom；
Then the inverse of C is：
4）Update residual error
After successive ignition, above-mentioned algorithm will obtain an optimal solutionIf primary signal has by rarefaction
Conversion, then it is rightInitial data x can just be obtained by carrying out once corresponding inverse transformation；If do not converted by rarefaction,
With reference to Fig. 2 to Fig. 6, with the compression acquisition problems of China Mobile user surfing flow data, to present invention side
Method is specifically described, and its situation is as follows：
1）China Mobile user Internet data model
China Mobile user Internet data bag such as Fig. 2 such as shows, the packet of user's online from terminal be dealt into base station when, transformation
Into one or more signaling, including personally identifiable information and miscellaneous service information, each Data Entry has up to a hundred fields
(Fig. 3 illustrate only least a portion of exemplary field), due to the difference of business, it is understood that there may be it is empty many fields.With one
Exemplified by image data, in order to quickly gather this pictures, each column data of picture can be used separately compressed sensing calculation
Method is compressed storage：
The Internet data for belonging to same user is extracted from packet, and identifies a pictures, due in picture
Pixel value transition than shallower, small echo sparse transformation can be passed through so that it has sparse characteristic.Cromogram is to be divided into three
Individual RGB conjunction figure, can also regard the superposition of three primary colours different gray-scale map as, in order to simpler and intuitively represent that its is dilute
Characteristic is dredged, will be showed using gray-scale map, original image data is as shown in Figure 4.
2）Rarefaction is carried out using wavelet transform matrix
Wavelet transform matrix w=DWT (n) is generated, wherein n is the length of initial data X each columns, it is assumed that X is 256*256's
Matrix, then n is 256.
Then the data after rarefaction are：XX=w*sparse (X) * w', the data on flows after wavelet transformation is such as
Shown in Fig. 5, ater expression value is 0；If the wavelet transformation of the step, then need to carry out inverse transformation one during restored data
It is secondary.
3）Compress acquisition phase
In order to faster gathered data, the data length of each row is only taken as 256, generates the Kazakhstan of 256*256 sizes
Dammam calculation matrix H7：
H0=1
Compression ratio r (0≤r≤1) is set, it is assumed that r is 0.5, then takes preceding 256*0.5=128 row measurements of calculation matrix vectorial
A calculation matrix M is constituted, data are compressed with measurement using M.
Y=M*XX
Wherein, M is calculation matrix, and XX is the matrix of initial data rarefaction.
4）The data recovery stage
According to above resulting measured value y, and according to the process for above consistent calculation matrix solve x, this
In use orthogonal matching pursuit OMP algorithms, can accurately recover initial data x, due to restore be one very
Time-consuming process, so taking the thought of parallel computation to OMP algorithms, improves recovering efficiency.
Wherein, Φ is the atom set constantly extended, and element is the most imperial palace chosen during each is circulated to calculation matrix
Long-pending column vector.
One optimal solution is obtained by OMP algorithmsAfterwards, because primary signal has by sparse transformation, so rightCarry out
Once corresponding inverse transformationInitial data x is obtained, as shown in Figure 6；Can be with from the image of recovery
Find out, can be under very high precision with larger compression ratio storage user data.
Examples of implementation described above are only the preferred embodiments of the invention, and the implementation model of the present invention is not limited with this
Enclose, therefore the change that all shape, principles according to the present invention are made, it all should cover within the scope of the present invention.
Claims (4)
1. a kind of radio network optimization magnanimity signaling data acquisition method based on compressed sensing, it is characterised in that including following
Step：
1) acquisition system using mobile communication signaling as target is set up, data during for mobile phone terminal call and online are carried out
Collection, can set up in each base station or upper strata terminal；
2) initial data pretreated model is set up, useless or incomplete original signaling data is deleted, and to core
Data are extracted and pre-processed；
3) signaling data categorizing selection device is set up, according to step 1) set up signaling acquisition system and step 2) set up original number
Data after pretreatment are classified by Data preprocess model, and classifying rules is：If data meet rarefaction feature in itself,
Rarefaction operation need not then be carried out；, pair can be sparse if it is not satisfied, then according to wavelet transformation or the rarefaction method of Fourier transformation
The data of change enter line translation, for the data after conversion, if still not enough openness, can further handle, make data fit sparse
Property；Wherein, described signaling data categorizing selection device is as follows：
xi=f (si)
Wherein, for step 2) obtained data change into polytype tables of data si；xiFor siWith small echo sparse transformation base phase
Value after multiplying, passes through the rarefaction method of wavelet transformation so that data xiMeet sparse characteristic；After wavelet transformation,
It is compressed sampling and carries out restoring obtained x by restoration algorithmi, also simultaneously by wavelet inverse transformation, to obtain siIf, should
Index siItself sparse property is met, then xi=si, then without inverse transformation of progress after restoring；
4) setting up has non coherent calculation matrix, and the data after rarefaction are compressed with the method for compressed sensing；
5) set up data recovery algorithm complete data recovery task, according to step 4) described in calculation matrix to step 3) survey
Measurement result obtained by amount, the data recovery algorithm based on orthogonal matching pursuit OMP algorithms completes the recovery to sparse data,
This is actually one and asks the deficient of optimum solution to determine process, is finally converted into initial data by inverse transformation again.
2. a kind of radio network optimization magnanimity signaling data acquisition method based on compressed sensing according to claim 1,
It is characterized in that：In step 1) in, the acquisition system, all signaling datas sent to mobile phone terminal are preserved, described
Signaling data includes useful family identity, session start time, conversation end time, surfing flow, network data.
3. a kind of radio network optimization magnanimity signaling data acquisition method based on compressed sensing according to claim 1,
It is characterized in that：In step 2) in, described initial data pretreated model, for each signaling data, if signaling data
It is the incomplete, discarding, then directly removes the Data Entry；For each signaling data, it is understood that there may be multiple refer to
Mark is, it is necessary to separately analyze or combined analysis, and the network data of such as user is a pictures or one section of voice, video, then needs to close
And together, signaling data is pre-processed according to demand.
4. a kind of radio network optimization magnanimity signaling data acquisition method based on compressed sensing according to claim 1,
It is characterized in that：
In step 4) in, described calculation matrix, with step 3) obtained by sparse data there is incoherence；The graceful matrix of hada
Each measurement vector between it is orthogonal each other, each measurement vector is made up of 1 or -1 value, and the length requirement of each vector is 2
M powers, m is positive integer, and using the graceful matrix of hada as calculation matrix, its perfect matrix is as follows：
H0=1
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<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mrow>
<mo>-</mo>
<mn>1</mn>
</mrow>
</mtd>
</mtr>
</mtable>
</mfenced>
</mrow>
<mrow>
<msub>
<mi>H</mi>
<mi>k</mi>
</msub>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<msqrt>
<mn>2</mn>
</msqrt>
</mfrac>
<mfenced open = "[" close = "]">
<mtable>
<mtr>
<mtd>
<msub>
<mi>H</mi>
<mrow>
<mi>k</mi>
<mo>-</mo>
<mn>1</mn>
</mrow>
</msub>
</mtd>
<mtd>
<msub>
<mi>H</mi>
<mrow>
<mi>k</mi>
<mo>-</mo>
<mn>1</mn>
</mrow>
</msub>
</mtd>
</mtr>
<mtr>
<mtd>
<msub>
<mi>H</mi>
<mrow>
<mi>k</mi>
<mo>-</mo>
<mn>1</mn>
</mrow>
</msub>
</mtd>
<mtd>
<mrow>
<mo>-</mo>
<msub>
<mi>H</mi>
<mrow>
<mi>k</mi>
<mo>-</mo>
<mn>1</mn>
</mrow>
</msub>
</mrow>
</mtd>
</mtr>
</mtable>
</mfenced>
</mrow>
1
Wherein, it is desirable to which the m powers that the length of each vector is 2, m is positive integer；When length n is taken as 65536, k is 16, H16
For 65536 matrixes for being multiplied by 65536, the numerical value of the matrix is unification, meets compression and the equity requirement restored；If compression
Than for 0.1, then choosing preceding 65536 row of calculation matrix, 65536*65536 calculation matrix A is formed；Calculation matrix with
Sparse initial data carries out multiplication y=Ax, obtains measured value y, then y length is the 1/10 of x；
In step 5) in, described data recovery algorithm is when needing to restore data, according to step 4) obtained by
Measured value y, and the calculation matrix arranged according to y length solve x process；And use orthogonal matching pursuit OMP
Algorithm, initial data x can be recovered exactly, and orthogonal matching pursuit OMP algorithms are a successive ignition process, residual error initialization
For initial data y, Increment Matrix ΦiSky is initialized as, subscript i represents the variable in ith iteration, the mistake of each interior circulation
Journey is as follows：
1) the column vector atom maximum with residual error inner product is found out from calculation matrix, the column vector is then added to Increment Matrix
In, and the column vector is deleted from calculation matrix；
2) optimal problem is solved
<mrow>
<msub>
<mover>
<mi>x</mi>
<mo>^</mo>
</mover>
<mi>i</mi>
</msub>
<mo>=</mo>
<mi>arg</mi>
<mi> </mi>
<mi>m</mi>
<mi>i</mi>
<mi>n</mi>
<mo>|</mo>
<mo>|</mo>
<mi>&gamma;</mi>
<mo>-</mo>
<msub>
<mover>
<mi>&Phi;</mi>
<mo>~</mo>
</mover>
<mi>i</mi>
</msub>
<mi>x</mi>
<mo>|</mo>
<mo>|</mo>
</mrow>
<mrow>
<mover>
<mi>x</mi>
<mo>^</mo>
</mover>
<mo>=</mo>
<msup>
<mrow>
<mo>(</mo>
<msup>
<mover>
<mi>&Phi;</mi>
<mo>~</mo>
</mover>
<mi>T</mi>
</msup>
<mover>
<mi>&Phi;</mi>
<mo>~</mo>
</mover>
<mo>)</mo>
</mrow>
<mrow>
<mo>-</mo>
<mn>1</mn>
</mrow>
</msup>
<msup>
<mover>
<mi>&Phi;</mi>
<mo>~</mo>
</mover>
<mi>T</mi>
</msup>
<mi>y</mi>
</mrow>
C is defined as, because C is Nonsingular symmetrical matt, LDL can be used when invertingTDecomposed, and according to newest
Atom update L and D；
<mrow>
<msub>
<mi>L</mi>
<mrow>
<mi>i</mi>
<mo>,</mo>
<mi>j</mi>
</mrow>
</msub>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<msub>
<mi>D</mi>
<mrow>
<mi>j</mi>
<mo>,</mo>
<mi>i</mi>
</mrow>
</msub>
</mfrac>
<mrow>
<mo>(</mo>
<mrow>
<msub>
<mi>C</mi>
<mrow>
<mi>i</mi>
<mo>,</mo>
<mi>j</mi>
</mrow>
</msub>
<mo>-</mo>
<munderover>
<mo>&Sigma;</mo>
<mrow>
<mi>k</mi>
<mo>=</mo>
<mn>1</mn>
</mrow>
<mrow>
<mi>j</mi>
<mo>-</mo>
<mn>1</mn>
</mrow>
</munderover>
<msub>
<mi>L</mi>
<mrow>
<mi>i</mi>
<mo>,</mo>
<mi>k</mi>
</mrow>
</msub>
<msub>
<mi>L</mi>
<mrow>
<mi>j</mi>
<mo>,</mo>
<mi>k</mi>
</mrow>
</msub>
<msub>
<mi>D</mi>
<mrow>
<mi>k</mi>
<mo>,</mo>
<mi>k</mi>
</mrow>
</msub>
</mrow>
<mo>)</mo>
</mrow>
<mo>,</mo>
<mrow>
<mo>(</mo>
<mrow>
<mi>i</mi>
<mo><</mo>
<mi>j</mi>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mrow>
<msub>
<mi>D</mi>
<mrow>
<mi>j</mi>
<mo>,</mo>
<mi>j</mi>
</mrow>
</msub>
<mo>=</mo>
<msub>
<mi>C</mi>
<mrow>
<mi>j</mi>
<mo>,</mo>
<mi>j</mi>
</mrow>
</msub>
<mo>-</mo>
<munderover>
<mo>&Sigma;</mo>
<mrow>
<mi>k</mi>
<mo>=</mo>
<mn>1</mn>
</mrow>
<mrow>
<mi>j</mi>
<mo>-</mo>
<mn>1</mn>
</mrow>
</munderover>
<msub>
<mi>L</mi>
<mrow>
<mi>j</mi>
<mo>,</mo>
<mi>k</mi>
</mrow>
</msub>
<msub>
<mi>L</mi>
<mrow>
<mi>j</mi>
<mo>,</mo>
<mi>k</mi>
</mrow>
</msub>
<msub>
<mi>D</mi>
<mrow>
<mi>k</mi>
<mo>,</mo>
<mi>k</mi>
</mrow>
</msub>
</mrow>
Then the inverse of C is：
<mrow>
<msup>
<mi>C</mi>
<mrow>
<mo>-</mo>
<mn>1</mn>
</mrow>
</msup>
<mo>=</mo>
<msup>
<mi>L</mi>
<mrow>
<mo>-</mo>
<msup>
<mn>1</mn>
<mi>T</mi>
</msup>
</mrow>
</msup>
<msup>
<mi>D</mi>
<mrow>
<mo>-</mo>
<mn>1</mn>
</mrow>
</msup>
<msup>
<mi>L</mi>
<mrow>
<mo>-</mo>
<mn>1</mn>
</mrow>
</msup>
</mrow>
3) residual error is updated
<mrow>
<mi>y</mi>
<mo>=</mo>
<mi>y</mi>
<mo>-</mo>
<msub>
<mi>&Phi;</mi>
<mi>i</mi>
</msub>
<msub>
<mover>
<mi>x</mi>
<mo>^</mo>
</mover>
<mi>i</mi>
</msub>
</mrow>
After successive ignition, above-mentioned algorithm will obtain an optimal solutionConverted if primary signal has by rarefaction,
It is then rightInitial data x can just be obtained by carrying out once corresponding inverse transformation；If do not converted by rarefaction,
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