Frequency-Hopping Transmitter Fingerprint Feature

Research ArticleFrequency-Hopping Transmitter FingerprintFeature Classification Based on Kernel CollaborativeRepresentation Classifier

Ping Sui Ying Guo Kun-feng Zhang and Honguang Li

Institute of Information and Navigation Air Force Engineering University Xirsquoan Shaanxi 710077 China

Correspondence should be addressed to Ping Sui ziwuningxin163com

Received 15 June 2017 Accepted 27 August 2017 Published 9 October 2017

Academic Editor Donatella Darsena

Copyright copy 2017 Ping Sui et alThis is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Noncooperation frequency-hopping (FH) transmitter fingerprint feature classification is a significant but challenging issue forFH transmitter recognition since not only is it sensitive to noise but also it has the nonlinear non-Gaussian and nonstabilitycharacteristics which make it difficult to guarantee the classification in the original signal space To address these problems amethod of frequency-hopping transmitter fingerprint feature classification based on kernel collaborative representation classifieris proposed in this paper First the noise suppression pretreatment of the FH transmitter signal is carried out by using the waveatoms frame method Then the nuances of the FH transmitters in the feature space are characterized by the surrounding-lineintegral bispectra features And finally incorporating the kernel function a classifier which can generalize a linear algorithm tononlinear counterpart is constructed for the final transmitter fingerprint feature classification Extensive experiments on real-worldFH transmitter ldquoturn-onrdquo transient signals demonstrate the robust classification of our method

1 Introduction

Frequency-hopping (FH) signals are generated by varying thecarrier frequencies according to a certain hopping patternwhich is typically pseudo-random Due to their inherentcapability of low interception good confidentiality andstrong anti-interference FH signals have become an impor-tant tactical means of antireconnaissance and anti-jammingin military communication

FH transmitter identification is a traditional significantbut challenging issue in the electromagnetic war domainespecially under serious noise and noncooperation condi-tions Due to the individual nuances of FH transmitters thereexist the inherent features which can be used to identifythe transmitter individuals This inherent feature based onthe individual nuances can be called the fingerprint of thetransmitters

Benefiting from the nuances among the transmitters anumber of studies using transient response features wereemphatically researched to achieve the identification Xu[1] combined the maximum correlation processing with

time-dependent statistical method to realize the FH signalrecognition Eric et al [2] proposed the MUSIC methodbased on signal direction information to sort the FH signalsLuo et al [3] used the transient response feature of thetransmitting power amplifiers to obtain the recognitionresults However when it comes to the condition of seri-ous noise and complicated electromagnetic environmentespecially when the transmitters are noncooperative thepractical classification of such methods is not ideal Inrecent years with the advantage of insensitivity to noiseand outperform of classification some methods based onsparse representation have been rapidly developed in variousfields of signal classification In the early studies the sparserepresentation of the training data can be found via the 1198970-minimization problem However this problem is proved tobe an NP-hard problem [4] Fortunately theoretical resultsshow that if the sparsely code solution of the training data issparse enough then the solution of 1198971 and 1198970-minimizationproblems is equivalent [5 6] Therefore Wright et al [7]proposed a sparse representation-based classifier (SRC) inwhich sparse representation problem can be addressed by

HindawiWireless Communications and Mobile ComputingVolume 2017 Article ID 9403590 9 pageshttpsdoiorg10115520179403590

2 Wireless Communications and Mobile Computing

the 1198971-minimization optimization Although there are lots offast numerical algorithms that have been proposed for the1198971-minimization optimization it is still very computationallydemanding In contrast Zhang et al [8] argued both the-oretically and empirically that collaborative representationclassifier (CRC) based on the 1198972-minimization is significantlymore computationally efficient and can result in similarperformance compared with the SRC And also Yang et al[9] proposed a relaxed collaborative representation modelwhich is simple but very competitive with the state-of-the-artclassificationmethodThusCRChas beenwidely used for theresearch of signal classification and a recent study has shownthat CRC is more efficient than the SRC without sacrificingthe classification accuracy However CRC is conducted inthe original signal space rather than the nonlinear highdimensional feature space thus the effectiveness of CR forthe classification is difficult to be guaranteed in the originalsignal space when it is used to describing the nonlinearnon-Gaussian and nonstability feature of the FH transmitterfingerprints

Above all there are still some issues that have not yetbeen properly addressed in particular (1) the influence ofnoise Fingerprints of the FH transmitter are so subtle andseriously sensitive to noise (2) effectiveness of classification arecent study has shown that CR is a promising regularizationframework for signal classification However it is conductedin the original signal space rather than the nonlinear highdimensional feature space so that the effectiveness of theclassification is difficult to be guaranteed

To address these issues in this paper an effective FHtransmitter fingerprint feature classification method basedon kernel collaborative representation classifier is proposedFirstly we denoise the signal data by wave atoms frameinstead of the traditional wavelet-based method Secondlyutilizing the surrounding-line integral bispectra analysis weextract the fingerprint features of the FH transmitters by theirreal-world ldquoturn-onrdquo transient signals Finally the kernel col-laborative representation classifier by introducing the kernelfunction was used to realize the classification results Themain advantages of our method include the following (1) thedata noises are especially inevitable in electric environmentOur noise suppression method based wave atoms frameoutperforms the traditionalwavelet-based denoisingmethod(2) instead of the CRC which is conducted in the originalsignal space our method generalizes the linear CRC to itsnonlinear counterpart by using the kernel function and inthis way the fingerprint features belonging to the same classcan be easily separated Experiments on real-world FH ldquoturn-onrdquo transient signals demonstrate the effectiveness and highclassification of our method

The rest of this paper is organized as follows Thetheories of wave atoms frame and CRC are expounded inSection 2 Then Section 3 describes the procedures of noisesuppression and the details of the proposed KCRC methodSection 4 demonstrates the experimental results and analyzesthe classification performance Finally Section 5 concludesthe paper

2 Preliminaries

This section starts with a succinct description of the waveatoms and offers insight into its implementation And thensome previous works in classification are introduced In thispart we present the traditional sparse representationmethodfirst and then give the collaborative representation-basedclassifier and its deficiencies

21 Wave Atoms Frame In a general signal processing prob-lem the wave atoms are represented as 120601120583(119909) with subscript120583 = (119895mn) = (119895 1198981 1198982 1198991 1198992) where 1198951198981 1198982 1198991 1198992 isin 119885and index is a point (119909120583 119908119906) in phase-space as

119909120583 = 2minus119895n119908119906 = 1205872119895m

11986212minus119895 le max119894=12

10038161003816100381610038161198981198941003816100381610038161003816 le 11986222119895(1)

where 1198621 1198622 gt 0 are two constants the position vector 119909120583 isthe center of 120601120583(119909) and the wave vector 119908119906 determines thecenters of both bumps of 120601120583(119909) as plusmn119908119906 in frequency plane

Let 119909119906 and 119908119906 be as in (1) for some 1198621 1198622 gt 0 Theelements of a frame of wave packets 120601120583 are called waveatoms for all 119872 gt 0 when

10038161003816100381610038161003816120601120583 (119909)10038161003816100381610038161003816 le 119862119872 sdot 2119895 (1 + 2119895 10038161003816100381610038161003816119909 minus 11990912058310038161003816100381610038161003816)minus11987210038161003816100381610038161003816120601120583 (119908)10038161003816100381610038161003816 le 119862119872 sdot 2minus119895 (1 + 2119895 10038161003816100381610038161003816119908 minus 11990812058310038161003816100381610038161003816)minus119872 + 119862119872

sdot 2minus119895 (1 + 2minus119895 10038161003816100381610038161003816119908 + 11990812058310038161003816100381610038161003816)minus119872 (2)

Consider the localization condition one-dimensionalwave atoms can be obtained by constructing the frequencydomain tightly support symmetric 0119898(119908) and deal with it bytwo-dimensional scaling and translation as follows [10]

0119898 (119908) = 119890minus1198941199082 [119890119894120572119898119892(120576119898 (119908 minus 120587(119898 + 12)))

+ 119890minus119894120572119898119892(120576119898+1 (119908 + 120587(119898 + 12)))]

(3)

where 120576119898 = (minus1)119898 120572119898 = (1205872)(119898 + 12) The function 119892 isan appropriate real-valued119862infin bump functionwith a supportinterval length of 2120587 and it satisfiedsum119898 |0119898(119908)|2 = 1 And bycombining dyadic dilates and translating 0119898 on the frequencyaxis one-dimensional wave atoms can be noted as

120595119895119898119899 (119909) = 120595119895119898 (119909 minus 2minus119895119899) = 211989521205950119898 (2119895119909 minus 119899) (4)

To preserve the orthonormality of the 120595119895119898119899(119909) the profile 119892needs to be asymmetric in addition to all the other propertiesin the sense that 119892(minus2119908 minus 1205872) = 119892(1205872 + 119908) for 119908 isin[minus1205873 1205873] with 119892 itself being supported on [minus71205876 51205876]

Wireless Communications and Mobile Computing 3

Considering 119867 is a Hilbert transform the orthogonalbasis and its dual orthogonal basis can be defined as

120601+120583 (1199091 1199092) = 1205951198951198981 (1199091 minus 2minus1198951198991) 1205951198951198981 (1199092 minus 2minus1198951198992)120601minus120583 (1199091 1199092) = 1198671205951198951198981 (1199091 minus 2minus1198951198991)1198671205951198951198981 (1199092 minus 2minus1198951198992) (5)

It is easy to see that the recombination

120601(1)120583 = 120601+120583 + 120601minus1205832

120601(2)120583 = 120601+120583 minus 120601minus1205832

(6)

provides basic functions with two bumps in the frequencyplane symmetric with respect to the origin and hence thedirectionalwave packets Together120601(1)120583 and120601(2)120583 form thewaveatom frame and may be denoted jointly as 120601120583 If the frame istight there are

sum120583

10038161003816100381610038161003816⟨120601(1)120583 120583⟩100381610038161003816100381610038162 + sum120583

10038161003816100381610038161003816⟨120601(2)120583 120583⟩100381610038161003816100381610038162 = 1199062 (7)

Then we have the two-dimensional wave atoms transformcoefficient as [10]

119862120583 = ⟨120601(1)120583 120583⟩ + ⟨120601(2)120583 120583⟩ (8)

22 Collaborative Representation-Based Classifier Assumethat the training data set is represented as A = [A1A2 A119862] isin R119889times119873 where 119862 is the number of classes A119894 =[1198861198941 1198861198942 119886119894119899119894] isin R119889times119899119894 is the collection of the data points forclass 119894 and 119873 = sum119862119894=1 119899119894 is the total number of training dataThen the sparse representation of a new test data y isin R119889 canbe found via the 1198970-minimization problem as

= argmin120572

1205720st y = A120572 (9)

where sdot 0 denotes the 1198970-norm which is defined to be thenumber of nonzero elements in a vector Generally a sparsesolution via the 1198970-minimization ismore robust and facilitatesthe consequent classification of the test data y isin R119889 Howeverthis problem (9) is proved to be an NP-hard problem [4]Fortunately theoretical results show that if the solution obtained is sparse enough then the solution of the 1198970 and1198971-minimization problems is equivalent [5 6] Therefore inSRC [7] the test data y isin R119889 can be sparsely coded by the1198971-minimization problem as

= argmin120572

1205721st y = A120572 (10)

Although the 1198971-minimization problem is extensively studiedand lots of numerical algorithms have been proposed itis still a computationally demanded problem In contrast

Zhang et al [8] verified both empirically and theoreticallythat 1198972-minimization based classifier relying on collaborativerepresentation can result in in a similar performance Thecollaborative representation of y isin R119889 can be written as

= argmin120572

12057222st y = A120572 (11)

Then the classifier based on collaborative representation isruled as

class (119910) = argmin119894

1003817100381710038171003817y minus A119894100381710038171003817100381722 119894 = 1 2 119862 (12)

Compared with the 1198971-minimization based sparse repre-sentation extensive experimental results in [8] demonstratethat the 1198972-minimization base collaborative representation ismore computationally efficient

3 Proposed Method

As for FH transmitters there will be many actual and signifi-cant transient states during its work Some of these transitionsare from the system itself such as ldquoturn-onoffrdquo instantaneouschanges andmode switch of the transmitters others are fromthe outside interference which is accidental and does notexist for every transmitter And these transitionswhich reflectthe characteristics of the transient states contain a wealth ofindividual nuances information of the FH transmitters Basedon the analysis above in order to fully characterize individualnuances of the FH transmitters we choose the instantaneousldquoturn-onrdquo response of the FH signals to calculate the finefeatures for the final transmitter classification

31 Noise Suppression The FH signals collected in the actualenvironment often have noise and clutter interference Inorder to effectively reduce the influence of external distur-bances on the feature extraction of transient signals ourmethod performs the noise reduction pretreatment on thecollected transient signals firstly

At the present time one-dimensional signal noise sup-pression is mainly based on the wavelet analysis Due to thenonstationarity of the transient signals of FH transmitter thetraditional wavelet-based denoising algorithms can reducesome noise but they blur the signal information at the sametime In contrast experimental results in [11] demonstratethat wave atoms frame is more significantly denoising effi-cient without sacrificing more signal accuracy Therefore inthis paper we use the wave atoms frame to carry out thedenoising result of the ldquoturn-onrdquo transient FH signal

Let 119904(119905) be the source ldquoturn-onrdquo signal and 119899(119905) theadditive noise which follows the Gaussian distribution andalso they are uncorrelated then we have the observed signalas

119909 (119905) = 119904 (119905) + 119899 (119905) (13)

Wave atoms frame is a special two-dimensional wavepocket deformation and is usually used for image and other


two-dimensional signal denoising Thus we construct avirtual observation matrix X for the 119909(119905) by adding whitenoise

X =[[[[[[[

1199091 (119905)1199092 (119905)

119909119875 (119905)

]]]]]]]

=[[[[[[[

119909 (119905)119909 (119905) + 1198992 (119905)

119909119875 (119905) + 119899119875 (119905)

]]]]]]]

=[[[[[[[

119904 (119905) + 119899 (119905)119904 (119905) + 119899 (119905) + 1198992 (119905)

119904 (119905) + 119899 (119905) + 119899119875 (119905)

]]]]]]]

(14)

where 119899(119905) 1198992(119905) 119899119875(119905) obey the Gaussian distributionIn matrix X the signal between the lines is determinedby the nuances characteristics of the FH transmitter theinformation correlation is strong at all times When 119899(119905) israndom noise the correlation is weak

Sampling the virtual observation matrix X at [0 119905] thenumber of sampling points is 119871 obtain its discrete form X =[X1times119871 X2times119871 X119875times119871] as

X =[[[[[[[[[[[[

1199091 ( 119905119871) 1199091 (2119905

119871 ) sdot sdot sdot 1199091 (119905)1199092 ( 119905

119871) d

119909119875 ( 119905119871) sdot sdot sdot 119909119875 (119905)

]]]]]]]]]]]]

(15)

In this way the matrix X can be treated as a two-dimensional signal matrix for noise reduction The effectiveinformation in X is equivalent to the vertical texture featureof the two-dimensional image And then we can use thesuperiority of wave atoms frame for two-dimensional textureinformation expression to obtain the denoising result Xlowast thefinal denoising one-dimensional signal can be written as

x = 1119875119875sum119894=1

Xlowast119894times119871 (16)

32 Feature Extraction Compared with the traditional first-order and second-order spectrums high-order spectrumcan extract more significant features of nonstationary non-Gaussian and nonlinear signals In this paper we extract thesurrounding-line integral bispectra features to characterizethe fingerprints of the FH transmitters in the feature spaceThe bispectrum of noise suppression data x is defined as

119861x (1205961 1205962)= int+infinminusinfin

int+infinminusinfin

1198883x (1205911 1205912) 119890minus119895(12059611205911+12059621205912)11988912059111198891205912(17)

1

2

Figure 1 The integral paths of integral bispectra

where 1198883x(1205911 1205912) is the three-order cumulant of x Afterobtaining the bispectrum we use the surrounding-line inte-gral bispectra analysis method to process the bispectralestimation result As shown in Figure 1 the integral path is asquare centered around the origin and each point representsa bispectral estimate

This integral path does not miss out or reuse any bispec-trum values which ensure the integrity of the target infor-mation Furthermore this calculation transforms the resultsfrom two dimensions into one dimension reducing thecomputationally complexity [11] And finally datasets of fin-gerprint characteristics are established by the surrounding-line integral bispectra features 119861x(1205961 1205962)33 Kernel Collaborative Representation Classifier In themachine learning field the kernel function is a well-knowntechnique which can generalize a linear algorithm to itsnonlinear counterpart in which features belonging to thesame class are better grouped together and thus are easilyclassified Assume there is a nonlinear feature mappingfunctionΦ(sdot) 119877119889 rarr 119877119863 (119889 ≪ 119863) It transforms the test datay isin R119889 and the training dictionary A = [A1A2 A119862] isinR119889times119873 to their high dimensional feature space y rarr Φ(y)A =[A1A2 A119862] rarr Φ(A) = [Φ(A1) Φ(A2) Φ(A119862)]Then we get the kernel collaborative representation classifier(KCRC)

= argmin120572

12057222st Φ(y) = Φ (A)120572 (18)

where Φ(sdot) is typically unknown and can only accessed bykernel function 119896(x y) = ⟨Φ(x) Φ(y)⟩ = Φ119879(x)Φ(y) Inthis paper we take advantage of the Gaussian kernel PCAalgorithm to solve the optimization problem in (18) due toits excellent performance reported in the paper [12ndash15] TheGaussian kernel can be written as

119896 (x y) = exp (minus120574 1003817100381710038171003817x minus y10038171003817100381710038172) (19)

Suppose there is a transformation matrix P which projectsthe high dimensional data points into a low dimensionalsubspace with dimensionality 119898

P = Φ (A)B (20)


ldquoTurn-onrdquo transient signal Wave atoms denoising

Training stage

Testing stage

New test signal Denoising result Bispectra feature

[[[[[[[[[[[[[

16724

04649

69583

middot middot middot



]]]]]]]]]]]]]

Bispectra feature

[[[[[[[[[[[[[

[[[[[[[[[[[[[

[[[[[[[[[[[[[

15391

05647

66186




]]]]]]]]]]]]]

Transformation matrix

Q = (I + KBBTK)minus1 KBBT

Kernel matrixK = ΦT(A)Φ(A)

Classication result

Class (y) = ＬＡＧＣＨi

=y minus AI2

2

= Qk(A y)

Figure 2 The transmitter fingerprint feature classification process of our method

whereB = [12057311205732 120573119898] isin R119873times119898120573119895 119895 = 1 2 119898 (119898 lt119873) is a normalized eigenvector associated with the 119895thlargest eigenvalues corresponding to the kernel matrix K =Φ119879(A)Φ(A)

By introducing the transformation matrix P to Eq (14)we can get

= argmin120572

12057222st P119879Φ(y) = P119879Φ (A)120572

(21)

Let 120582 gt 0 be a Lagrange parameter and introduce theLagrangian to (21) the object function of KCRC can berewritten as follows

119871 (120572) = 10038171003817100381710038171003817P119879Φ(y) minus P119879Φ (A)1205721003817100381710038171003817100381722 + 120582 12057222= Φ119879 (y)PP119879Φ(y) minus Φ119879 (y)PP119879Φ (A)120572

minus 120572119879Φ119879 (A)PP119879Φ(y)+ 120572119879Φ119879 (A)PP119879Φ (A)120572 + 120582120572119879120572

(22)

The optimal solution to (22) requires

120597119871120597120572 = 120582120572 + Φ119879 (A)PP119879Φ (A)120572 minus Φ119879 (A)PP119879Φ(y)

= 0(23)

Substituting (20) into (23) we have

= Q119896 (A y) (24)

where 119896(A y) = Φ119879(A)Φ(y) and Q = (120582I + KBB119879K)minus1 sdotKBB119879 Clearly Q is independent of y such that it could be

precalculated which reduces the computational complexityand thus should be more efficient

The procedures of FH transmitter classification based onkernel collaborative representation classifier are illustratedin Figure 2 In the training stage we first extract thesurrounding-line integral bispectra features A of the noisesuppressed ldquoturn-onrdquo transient FH signals and then calculatethe transformation matrix Q In the testing stage givena new test signal y the corresponding operation of noisesuppression and feature extraction is applied and then thetarget is classified as one of the known FH transmitters basedon the KCRC

4 Experimental Results

In this section experiments are conducted on 100 recordsof training signal for each of 5 FH transmitters And allexperiments are implemented in Matlab 2014a and run on aPC with Intel Core i7 293GHz CPU and 4GBRAM

41 Noise Robustness In order to verify the noise robustnessof our method we compared the recognition rate withwhether or not to produce denoising the recognition resultsof the classifier are shown in Figure 2 the red line is theexperimental result using the training data samples whichhave been preproduced bywave atoms frame based noise sup-pression and the other line is the experimental result withoutnoise suppression From Figure 3 we can see that our waveatoms frame based noise suppression method overcomes theinfluences of noise and performs high recognition rate in theexperiment Figure 4 shows the signal denoising results byourmethod andwavelet-basedmethod from the red circle inthis figure we can clearly find that our method outperforms


50 100 150 200 250 300 350 400 45055

60

65

70

75

80

85

90

95

100

Number of train data sample

Only KCRCDenoising + KCRC

Clas

sifica

tion

rate

()

Figure 3 Classification rate with whether or not to produce denoising

0 200 400 600 800 1000 1200minus02

0

02

04

06

08

1

12Wave atom frame denoising result

Sample point

Sign

al am

plitu

de

0 200 400 600 800 1000 1200minus02

0

02

04

06

08

1

12Wavelet denoising denoising result

Sample point

Sign

al am

plitu

de

0 200 400 600 800 1000 1200minus02

0

02

04

06

08

1

12Noisy signal

Sample point

Sign

al am

plitu

de

0 200 400 600 800 1000 1200minus02

0

02

04

06

08

1

Input signal

Sample point

Sign

al am

plitu

de

Figure 4 Denoising result with different algorithms


Input image

200 400 600 800 1000

200

400

600

800

1000 0

02

04

06

08

1Noisy image

200 400 600 800 1000

200

400

600

800

1000

0

05

1

Wave atom frame psnr = 3936

200 400 600 800 1000

200

400

600

800

1000 0

02

04

06

08

1Wavelet db5 psnr = 3773

200 400 600 800 1000

200

400

600

800

1000

0

02

04

06

08

1

Figure 5 Two-dimensional displays of signals in Figure 3

the state-of-the-art wavelet-based denoising algorithm andFigure 5 shows the two-dimensional display correspondingto the signals in Figure 4

42 Effectiveness of Classification Classification efficiencyis important for real-world FH transmitter classificationapplication As illustrated in Figure 2 the proposed methodmainly requires a kernel function to generalize a linear algo-rithm to its nonlinear counterpart in which the accuracy ofclassification could be ensuredWe compare the classificationefficiency of our method with KNN SVM SRC and CRCthe results are illustrated in Table 1 and Figure 6 Fromthis experiment we could conclude that our method alwaysshows a powerful classification ability compared with otherclassifiers

Furthermore for evaluating the proposed algorithmcomprehensively we also compared the methods of KNNSVM SRC and CRC with signals of the original wavelettreated and wave atoms treated respectively As seen fromTable 2 and Figure 7 the wave atoms frame denoised result

50 100 150 200 250 300 350 400 45055

60

65

70

75

80

85

90

95

100


KNNSVMSRC

CRCKCRC

Clas

sifica

tion

rate

()

Figure 6 The classification rate with different method


50 100 150 200 250 300 350 400 45055

60

65

70

75

80

85

90

95

100


Clas

sifica

tion

rate

() o

f KN

N

OriginalWavelet treatedWave atoms treated

(a) Classification result of KNN

50 100 150 200 250 300 350 400 45060

65

70

75

80

85

90

95

100


Clas

sifica

tion

rate

() o

f SV

M


(b) Classification result of SVM

50 100 150 200 250 300 350 400 45065

70

75

80

85

90

95

100


Clas

sifica

tion

rate

() o

f SRC


(c) Classification result of SRC

50 100 150 200 250 300 350 400 45070

75

80

85

90

95

100


Clas

sifica

tion

rate

() o

f CRC


(d) Classification result of CRC

Figure 7 The classification rate of different treated signals with the methods of KNN SVM SRC and CRC

Table 1 The classification rate with different methods ()

Number of train sample KNN SVM SRC CRC KCRC50 576 630 670 704 880100 638 734 760 786 906150 678 794 834 846 940200 704 812 868 874 950250 744 864 912 904 956300 780 890 926 922 974350 834 918 945 940 982400 894 954 964 966 996450 944 980 982 984 998

has the best performance in all experiments It demonstratesthat our wave atoms frame noise reduction pretreatment hassignificant contribution to the final identification result

5 Conclusion

In this paper a novel method based on kernel collaborativerepresentation is proposed for FH transmitter fingerprintfeature classification Firstly our method takes advantage ofthe wave atoms based noise suppression algorithm to removethe influence of noise thus the accuracy and efficiency ofthe fingerprint feature extraction are improved And thenour method utilizes the surrounding-line integral bispectra


Table 2 The classification rate of different treated signals ()

Number of train sample 50 100 150 200 250 300 350 400 450KNN

Original 576 638 678 704 744 780 834 894 944Wavelet 584 643 681 726 761 804 859 906 952Wave atoms 615 665 696 738 775 826 881 927 969

SVMOriginal 630 734 794 812 864 890 918 954 980Wavelet 658 761 806 839 873 903 924 964 984Wave atoms 694 783 818 852 887 915 946 972 987

SRCOriginal 670 760 834 868 912 926 945 964 982Wavelet 705 784 852 887 921 934 956 969 984Wave atoms 732 817 870 906 929 947 961 970 985

CRCOriginal 704 786 846 874 904 922 940 966 984Wavelet 728 816 858 883 915 937 949 969 986Wave atoms 755 837 867 897 924 941 955 971 989

features of ldquoturn-onrdquo transient signals more effectively byintroducing a kernel function to generalize the linear algo-rithm to its nonlinear counterpart Experimental results onreal-world 5 FH transmitters show that our method achievesobviously better performance than CRC and several state-of-the-art methods in terms of accuracy and efficiency

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

This research was supported by the National Natural ScienceFoundation of China (no 61601500)

References

[1] Q H Xu Recognition Method for Frequency-Hopping in SignalReconnaissance Xidian University Xian China 2005

[2] M Eric M L Dukic and M Obradovic ldquoFrequency-hoppingsignal separation by spatio-frequency analysis based on Musicmethodrdquo in Proceedings of The Sixth International Symposiumon Spread Spectrum Techniques and Applications vol 66 of 8278 pages 2000

[3] Z X Luo Z J Zhao and J Y Xu ldquoIndividual Frequency-hopping transmitter identification based on transient responseof power amplifierrdquo Journal of Hangzhou Dianzi University vol29 no 1 pp 29ndash32 2009

[4] E Amaldi and V Kann ldquoOn the approximability of minimizingnonzero variables or unsatisfied relations in linear systemsrdquoTheoretical Computer Science vol 209 no 1-2 pp 237ndash2601998

[5] D L Donoho ldquoFor most large underdetermined systems of lin-ear equations the minimal 1198971-norm solution is also the sparsestsolutionrdquo Communications on Pure and Applied Mathematicsvol 59 no 6 pp 797ndash829 2006

[6] E J Candes and T Tao ldquoNear-optimal signal recovery fromrandom projections universal encoding strategiesrdquo Institute ofElectrical and Electronics Engineers Transactions on InformationTheory vol 52 no 12 pp 5406ndash5425 2006

[7] JWright A Y Yang A Ganesh S S Sastry and YMa ldquoRobustface recognition via sparse representationrdquo IEEE Transactionson Pattern Analysis and Machine Intelligence vol 31 no 2 pp210ndash227 2009

[8] L Zhang M Yang and X Feng ldquoSparse representation orcollaborative representation Which helps face recognitionrdquo inProceedings of the IEEE International Conference on ComputerVision (ICCV rsquo11) pp 471ndash478 Barcelona Spain November2011

[9] M Yang L Zhang D Zhang and S Wang ldquoRelaxed collab-orative representation for pattern classificationrdquo in Proceedingsof the IEEE Conference on Computer Vision and Pattern Recog-nition (CVPR rsquo12) pp 2224ndash2231 IEEE Providence RI USAJune 2012

[10] L Demanet and L Ying ldquoWave atoms and sparsity of oscilla-tory patternsrdquo Applied and Computational Harmonic AnalysisTime-Frequency and Time-Scale Analysis Wavelets NumericalAlgorithms and Applications vol 23 no 3 pp 368ndash387 2007

[11] Z Tang and Y K Lei ldquoRadio transmitter identification basedon collaborative representationrdquoWireless Personal Communica-tions pp 1ndash15 2017

[12] M Yang ldquoKernel eigenfaces vs kernel fisherfaces face recogni-tion using kernel methodrdquo in proceedings of the IEEE Interna-tional Conference on Automatic Face and Gesture Recognitionpp 215ndash220 2008

[13] V N Vapnik Statistical Learning Theory Wiley-InterscienceNew York NY USA 1998

[14] B Scholkopf A Smola and K-R Muller ldquoNonlinear compo-nent analysis as a kernel eigenvalue problemrdquoNeural Computa-tion vol 10 no 5 pp 1299ndash1319 1998

[15] B Wang W Li N Poh and Q Liao ldquoKernel collaborativerepresentation-based classifier for face recognitionrdquo in Proceed-ings of the 2013 38th IEEE International Conference on AcousticsSpeech and Signal Processing ICASSP 2013 pp 2877ndash2881Vancouver Cannada May 2013

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014


Active and Passive Electronic Components

Control Scienceand Engineering

Journal of


International Journal of

RotatingMachinery


Hindawi Publishing Corporation httpwwwhindawicom

Journal of

Volume 201

Submit your manuscripts athttpswwwhindawicom

VLSI Design



Shock and Vibration


Civil EngineeringAdvances in

Acoustics and VibrationAdvances in



Electrical and Computer Engineering

Journal of

Advances inOptoElectronics


Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of


Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Chemical EngineeringInternational Journal of Antennas and

Propagation




Navigation and Observation



DistributedSensor Networks



the 1198971-minimization optimization Although there are lots offast numerical algorithms that have been proposed for the1198971-minimization optimization it is still very computationallydemanding In contrast Zhang et al [8] argued both the-oretically and empirically that collaborative representationclassifier (CRC) based on the 1198972-minimization is significantlymore computationally efficient and can result in similarperformance compared with the SRC And also Yang et al[9] proposed a relaxed collaborative representation modelwhich is simple but very competitive with the state-of-the-artclassificationmethodThusCRChas beenwidely used for theresearch of signal classification and a recent study has shownthat CRC is more efficient than the SRC without sacrificingthe classification accuracy However CRC is conducted inthe original signal space rather than the nonlinear highdimensional feature space thus the effectiveness of CR forthe classification is difficult to be guaranteed in the originalsignal space when it is used to describing the nonlinearnon-Gaussian and nonstability feature of the FH transmitterfingerprints

Above all there are still some issues that have not yetbeen properly addressed in particular (1) the influence ofnoise Fingerprints of the FH transmitter are so subtle andseriously sensitive to noise (2) effectiveness of classification arecent study has shown that CR is a promising regularizationframework for signal classification However it is conductedin the original signal space rather than the nonlinear highdimensional feature space so that the effectiveness of theclassification is difficult to be guaranteed

To address these issues in this paper an effective FHtransmitter fingerprint feature classification method basedon kernel collaborative representation classifier is proposedFirstly we denoise the signal data by wave atoms frameinstead of the traditional wavelet-based method Secondlyutilizing the surrounding-line integral bispectra analysis weextract the fingerprint features of the FH transmitters by theirreal-world ldquoturn-onrdquo transient signals Finally the kernel col-laborative representation classifier by introducing the kernelfunction was used to realize the classification results Themain advantages of our method include the following (1) thedata noises are especially inevitable in electric environmentOur noise suppression method based wave atoms frameoutperforms the traditionalwavelet-based denoisingmethod(2) instead of the CRC which is conducted in the originalsignal space our method generalizes the linear CRC to itsnonlinear counterpart by using the kernel function and inthis way the fingerprint features belonging to the same classcan be easily separated Experiments on real-world FH ldquoturn-onrdquo transient signals demonstrate the effectiveness and highclassification of our method

The rest of this paper is organized as follows Thetheories of wave atoms frame and CRC are expounded inSection 2 Then Section 3 describes the procedures of noisesuppression and the details of the proposed KCRC methodSection 4 demonstrates the experimental results and analyzesthe classification performance Finally Section 5 concludesthe paper

2 Preliminaries

This section starts with a succinct description of the waveatoms and offers insight into its implementation And thensome previous works in classification are introduced In thispart we present the traditional sparse representationmethodfirst and then give the collaborative representation-basedclassifier and its deficiencies

21 Wave Atoms Frame In a general signal processing prob-lem the wave atoms are represented as 120601120583(119909) with subscript120583 = (119895mn) = (119895 1198981 1198982 1198991 1198992) where 1198951198981 1198982 1198991 1198992 isin 119885and index is a point (119909120583 119908119906) in phase-space as

119909120583 = 2minus119895n119908119906 = 1205872119895m

11986212minus119895 le max119894=12

10038161003816100381610038161198981198941003816100381610038161003816 le 11986222119895(1)

where 1198621 1198622 gt 0 are two constants the position vector 119909120583 isthe center of 120601120583(119909) and the wave vector 119908119906 determines thecenters of both bumps of 120601120583(119909) as plusmn119908119906 in frequency plane

Let 119909119906 and 119908119906 be as in (1) for some 1198621 1198622 gt 0 Theelements of a frame of wave packets 120601120583 are called waveatoms for all 119872 gt 0 when

10038161003816100381610038161003816120601120583 (119909)10038161003816100381610038161003816 le 119862119872 sdot 2119895 (1 + 2119895 10038161003816100381610038161003816119909 minus 11990912058310038161003816100381610038161003816)minus11987210038161003816100381610038161003816120601120583 (119908)10038161003816100381610038161003816 le 119862119872 sdot 2minus119895 (1 + 2119895 10038161003816100381610038161003816119908 minus 11990812058310038161003816100381610038161003816)minus119872 + 119862119872

sdot 2minus119895 (1 + 2minus119895 10038161003816100381610038161003816119908 + 11990812058310038161003816100381610038161003816)minus119872 (2)

Consider the localization condition one-dimensionalwave atoms can be obtained by constructing the frequencydomain tightly support symmetric 0119898(119908) and deal with it bytwo-dimensional scaling and translation as follows [10]

0119898 (119908) = 119890minus1198941199082 [119890119894120572119898119892(120576119898 (119908 minus 120587(119898 + 12)))

+ 119890minus119894120572119898119892(120576119898+1 (119908 + 120587(119898 + 12)))]

(3)

where 120576119898 = (minus1)119898 120572119898 = (1205872)(119898 + 12) The function 119892 isan appropriate real-valued119862infin bump functionwith a supportinterval length of 2120587 and it satisfiedsum119898 |0119898(119908)|2 = 1 And bycombining dyadic dilates and translating 0119898 on the frequencyaxis one-dimensional wave atoms can be noted as

120595119895119898119899 (119909) = 120595119895119898 (119909 minus 2minus119895119899) = 211989521205950119898 (2119895119909 minus 119899) (4)

To preserve the orthonormality of the 120595119895119898119899(119909) the profile 119892needs to be asymmetric in addition to all the other propertiesin the sense that 119892(minus2119908 minus 1205872) = 119892(1205872 + 119908) for 119908 isin[minus1205873 1205873] with 119892 itself being supported on [minus71205876 51205876]





120601(1)120583 = 120601+120583 + 120601minus1205832

120601(2)120583 = 120601+120583 minus 120601minus1205832

(6)


sum120583

10038161003816100381610038161003816⟨120601(1)120583 120583⟩100381610038161003816100381610038162 + sum120583

10038161003816100381610038161003816⟨120601(2)120583 120583⟩100381610038161003816100381610038162 = 1199062 (7)


119862120583 = ⟨120601(1)120583 120583⟩ + ⟨120601(2)120583 120583⟩ (8)


= argmin120572

1205720st y = A120572 (9)


= argmin120572

1205721st y = A120572 (10)



= argmin120572

12057222st y = A120572 (11)


class (119910) = argmin119894

1003817100381710038171003817y minus A119894100381710038171003817100381722 119894 = 1 2 119862 (12)


3 Proposed Method





119909 (119905) = 119904 (119905) + 119899 (119905) (13)




X =[[[[[[[

1199091 (119905)1199092 (119905)

119909119875 (119905)

]]]]]]]

=[[[[[[[

119909 (119905)119909 (119905) + 1198992 (119905)

119909119875 (119905) + 119899119875 (119905)

]]]]]]]

=[[[[[[[

119904 (119905) + 119899 (119905)119904 (119905) + 119899 (119905) + 1198992 (119905)

119904 (119905) + 119899 (119905) + 119899119875 (119905)

]]]]]]]

(14)



X =[[[[[[[[[[[[

1199091 ( 119905119871) 1199091 (2119905

119871 ) sdot sdot sdot 1199091 (119905)1199092 ( 119905

119871) d

119909119875 ( 119905119871) sdot sdot sdot 119909119875 (119905)

]]]]]]]]]]]]

(15)


x = 1119875119875sum119894=1




int+infinminusinfin

1198883x (1205911 1205912) 119890minus119895(12059611205911+12059621205912)11988912059111198891205912(17)

1

2




= argmin120572

12057222st Φ(y) = Φ (A)120572 (18)




P = Φ (A)B (20)



Training stage

Testing stage


[[[[[[[[[[[[[

16724

04649

69583




]]]]]]]]]]]]]

Bispectra feature

[[[[[[[[[[[[[

[[[[[[[[[[[[[

[[[[[[[[[[[[[

15391

05647

66186




]]]]]]]]]]]]]




Classication result


=y minus AI2

2

= Qk(A y)




= argmin120572

12057222st P119879Φ(y) = P119879Φ (A)120572

(21)




(22)



= 0(23)


= Q119896 (A y) (24)








50 100 150 200 250 300 350 400 45055

60

65

70

75

80

85

90

95

100



Clas

sifica

tion

rate

()


0 200 400 600 800 1000 1200minus02

0

02

04

06

08

1


Sample point

Sign

al am

plitu

de

0 200 400 600 800 1000 1200minus02

0

02

04

06

08

1


Sample point

Sign

al am

plitu

de

0 200 400 600 800 1000 1200minus02

0

02

04

06

08

1

12Noisy signal

Sample point

Sign

al am

plitu

de

0 200 400 600 800 1000 1200minus02

0

02

04

06

08

1

Input signal

Sample point

Sign

al am

plitu

de



Input image

200 400 600 800 1000

200

400

600

800

1000 0

02

04

06

08

1Noisy image

200 400 600 800 1000

200

400

600

800

1000

0

05

1


200 400 600 800 1000

200

400

600

800

1000 0

02

04

06

08


200 400 600 800 1000

200

400

600

800

1000

0

02

04

06

08

1





50 100 150 200 250 300 350 400 45055

60

65

70

75

80

85

90

95

100


KNNSVMSRC

CRCKCRC

Clas

sifica

tion

rate

()



50 100 150 200 250 300 350 400 45055

60

65

70

75

80

85

90

95

100


Clas

sifica

tion

rate

() o

f KN

N



50 100 150 200 250 300 350 400 45060

65

70

75

80

85

90

95

100


Clas

sifica

tion

rate

() o

f SV

M



50 100 150 200 250 300 350 400 45065

70

75

80

85

90

95

100


Clas

sifica

tion

rate

() o

f SRC



50 100 150 200 250 300 350 400 45070

75

80

85

90

95

100


Clas

sifica

tion

rate

() o

f CRC







5 Conclusion












Acknowledgments


References
















RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation













120601(1)120583 = 120601+120583 + 120601minus1205832

120601(2)120583 = 120601+120583 minus 120601minus1205832

(6)


sum120583

10038161003816100381610038161003816⟨120601(1)120583 120583⟩100381610038161003816100381610038162 + sum120583

10038161003816100381610038161003816⟨120601(2)120583 120583⟩100381610038161003816100381610038162 = 1199062 (7)


119862120583 = ⟨120601(1)120583 120583⟩ + ⟨120601(2)120583 120583⟩ (8)


= argmin120572

1205720st y = A120572 (9)


= argmin120572

1205721st y = A120572 (10)



= argmin120572

12057222st y = A120572 (11)


class (119910) = argmin119894

1003817100381710038171003817y minus A119894100381710038171003817100381722 119894 = 1 2 119862 (12)


3 Proposed Method





119909 (119905) = 119904 (119905) + 119899 (119905) (13)




X =[[[[[[[

1199091 (119905)1199092 (119905)

119909119875 (119905)

]]]]]]]

=[[[[[[[

119909 (119905)119909 (119905) + 1198992 (119905)

119909119875 (119905) + 119899119875 (119905)

]]]]]]]

=[[[[[[[

119904 (119905) + 119899 (119905)119904 (119905) + 119899 (119905) + 1198992 (119905)

119904 (119905) + 119899 (119905) + 119899119875 (119905)

]]]]]]]

(14)



X =[[[[[[[[[[[[

1199091 ( 119905119871) 1199091 (2119905

119871 ) sdot sdot sdot 1199091 (119905)1199092 ( 119905

119871) d

119909119875 ( 119905119871) sdot sdot sdot 119909119875 (119905)

]]]]]]]]]]]]

(15)


x = 1119875119875sum119894=1




int+infinminusinfin

1198883x (1205911 1205912) 119890minus119895(12059611205911+12059621205912)11988912059111198891205912(17)

1

2




= argmin120572

12057222st Φ(y) = Φ (A)120572 (18)




P = Φ (A)B (20)



Training stage

Testing stage


[[[[[[[[[[[[[

16724

04649

69583




]]]]]]]]]]]]]

Bispectra feature

[[[[[[[[[[[[[

[[[[[[[[[[[[[

[[[[[[[[[[[[[

15391

05647

66186




]]]]]]]]]]]]]




Classication result


=y minus AI2

2

= Qk(A y)




= argmin120572

12057222st P119879Φ(y) = P119879Φ (A)120572

(21)




(22)



= 0(23)


= Q119896 (A y) (24)








50 100 150 200 250 300 350 400 45055

60

65

70

75

80

85

90

95

100



Clas

sifica

tion

rate

()


0 200 400 600 800 1000 1200minus02

0

02

04

06

08

1


Sample point

Sign

al am

plitu

de

0 200 400 600 800 1000 1200minus02

0

02

04

06

08

1


Sample point

Sign

al am

plitu

de

0 200 400 600 800 1000 1200minus02

0

02

04

06

08

1

12Noisy signal

Sample point

Sign

al am

plitu

de

0 200 400 600 800 1000 1200minus02

0

02

04

06

08

1

Input signal

Sample point

Sign

al am

plitu

de



Input image

200 400 600 800 1000

200

400

600

800

1000 0

02

04

06

08

1Noisy image

200 400 600 800 1000

200

400

600

800

1000

0

05

1


200 400 600 800 1000

200

400

600

800

1000 0

02

04

06

08


200 400 600 800 1000

200

400

600

800

1000

0

02

04

06

08

1





50 100 150 200 250 300 350 400 45055

60

65

70

75

80

85

90

95

100


KNNSVMSRC

CRCKCRC

Clas

sifica

tion

rate

()



50 100 150 200 250 300 350 400 45055

60

65

70

75

80

85

90

95

100


Clas

sifica

tion

rate

() o

f KN

N



50 100 150 200 250 300 350 400 45060

65

70

75

80

85

90

95

100


Clas

sifica

tion

rate

() o

f SV

M



50 100 150 200 250 300 350 400 45065

70

75

80

85

90

95

100


Clas

sifica

tion

rate

() o

f SRC



50 100 150 200 250 300 350 400 45070

75

80

85

90

95

100


Clas

sifica

tion

rate

() o

f CRC







5 Conclusion












Acknowledgments


References
















RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











X =[[[[[[[

1199091 (119905)1199092 (119905)

119909119875 (119905)

]]]]]]]

=[[[[[[[

119909 (119905)119909 (119905) + 1198992 (119905)

119909119875 (119905) + 119899119875 (119905)

]]]]]]]

=[[[[[[[

119904 (119905) + 119899 (119905)119904 (119905) + 119899 (119905) + 1198992 (119905)

119904 (119905) + 119899 (119905) + 119899119875 (119905)

]]]]]]]

(14)



X =[[[[[[[[[[[[

1199091 ( 119905119871) 1199091 (2119905

119871 ) sdot sdot sdot 1199091 (119905)1199092 ( 119905

119871) d

119909119875 ( 119905119871) sdot sdot sdot 119909119875 (119905)

]]]]]]]]]]]]

(15)


x = 1119875119875sum119894=1




int+infinminusinfin

1198883x (1205911 1205912) 119890minus119895(12059611205911+12059621205912)11988912059111198891205912(17)

1

2




= argmin120572

12057222st Φ(y) = Φ (A)120572 (18)




P = Φ (A)B (20)



Training stage

Testing stage


[[[[[[[[[[[[[

16724

04649

69583




]]]]]]]]]]]]]

Bispectra feature

[[[[[[[[[[[[[

[[[[[[[[[[[[[

[[[[[[[[[[[[[

15391

05647

66186




]]]]]]]]]]]]]




Classication result


=y minus AI2

2

= Qk(A y)




= argmin120572

12057222st P119879Φ(y) = P119879Φ (A)120572

(21)




(22)



= 0(23)


= Q119896 (A y) (24)








50 100 150 200 250 300 350 400 45055

60

65

70

75

80

85

90

95

100



Clas

sifica

tion

rate

()


0 200 400 600 800 1000 1200minus02

0

02

04

06

08

1


Sample point

Sign

al am

plitu

de

0 200 400 600 800 1000 1200minus02

0

02

04

06

08

1


Sample point

Sign

al am

plitu

de

0 200 400 600 800 1000 1200minus02

0

02

04

06

08

1

12Noisy signal

Sample point

Sign

al am

plitu

de

0 200 400 600 800 1000 1200minus02

0

02

04

06

08

1

Input signal

Sample point

Sign

al am

plitu

de



Input image

200 400 600 800 1000

200

400

600

800

1000 0

02

04

06

08

1Noisy image

200 400 600 800 1000

200

400

600

800

1000

0

05

1


200 400 600 800 1000

200

400

600

800

1000 0

02

04

06

08


200 400 600 800 1000

200

400

600

800

1000

0

02

04

06

08

1





50 100 150 200 250 300 350 400 45055

60

65

70

75

80

85

90

95

100


KNNSVMSRC

CRCKCRC

Clas

sifica

tion

rate

()



50 100 150 200 250 300 350 400 45055

60

65

70

75

80

85

90

95

100


Clas

sifica

tion

rate

() o

f KN

N



50 100 150 200 250 300 350 400 45060

65

70

75

80

85

90

95

100


Clas

sifica

tion

rate

() o

f SV

M



50 100 150 200 250 300 350 400 45065

70

75

80

85

90

95

100


Clas

sifica

tion

rate

() o

f SRC



50 100 150 200 250 300 350 400 45070

75

80

85

90

95

100


Clas

sifica

tion

rate

() o

f CRC







5 Conclusion












Acknowledgments


References
















RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











Training stage

Testing stage


[[[[[[[[[[[[[

16724

04649

69583




]]]]]]]]]]]]]

Bispectra feature

[[[[[[[[[[[[[

[[[[[[[[[[[[[

[[[[[[[[[[[[[

15391

05647

66186




]]]]]]]]]]]]]




Classication result


=y minus AI2

2

= Qk(A y)




= argmin120572

12057222st P119879Φ(y) = P119879Φ (A)120572

(21)




(22)



= 0(23)


= Q119896 (A y) (24)








50 100 150 200 250 300 350 400 45055

60

65

70

75

80

85

90

95

100



Clas

sifica

tion

rate

()


0 200 400 600 800 1000 1200minus02

0

02

04

06

08

1


Sample point

Sign

al am

plitu

de

0 200 400 600 800 1000 1200minus02

0

02

04

06

08

1


Sample point

Sign

al am

plitu

de

0 200 400 600 800 1000 1200minus02

0

02

04

06

08

1

12Noisy signal

Sample point

Sign

al am

plitu

de

0 200 400 600 800 1000 1200minus02

0

02

04

06

08

1

Input signal

Sample point

Sign

al am

plitu

de



Input image

200 400 600 800 1000

200

400

600

800

1000 0

02

04

06

08

1Noisy image

200 400 600 800 1000

200

400

600

800

1000

0

05

1


200 400 600 800 1000

200

400

600

800

1000 0

02

04

06

08


200 400 600 800 1000

200

400

600

800

1000

0

02

04

06

08

1





50 100 150 200 250 300 350 400 45055

60

65

70

75

80

85

90

95

100


KNNSVMSRC

CRCKCRC

Clas

sifica

tion

rate

()



50 100 150 200 250 300 350 400 45055

60

65

70

75

80

85

90

95

100


Clas

sifica

tion

rate

() o

f KN

N



50 100 150 200 250 300 350 400 45060

65

70

75

80

85

90

95

100


Clas

sifica

tion

rate

() o

f SV

M



50 100 150 200 250 300 350 400 45065

70

75

80

85

90

95

100


Clas

sifica

tion

rate

() o

f SRC



50 100 150 200 250 300 350 400 45070

75

80

85

90

95

100


Clas

sifica

tion

rate

() o

f CRC







5 Conclusion












Acknowledgments


References
















RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










50 100 150 200 250 300 350 400 45055

60

65

70

75

80

85

90

95

100



Clas

sifica

tion

rate

()


0 200 400 600 800 1000 1200minus02

0

02

04

06

08

1


Sample point

Sign

al am

plitu

de

0 200 400 600 800 1000 1200minus02

0

02

04

06

08

1


Sample point

Sign

al am

plitu

de

0 200 400 600 800 1000 1200minus02

0

02

04

06

08

1

12Noisy signal

Sample point

Sign

al am

plitu

de

0 200 400 600 800 1000 1200minus02

0

02

04

06

08

1

Input signal

Sample point

Sign

al am

plitu

de



Input image

200 400 600 800 1000

200

400

600

800

1000 0

02

04

06

08

1Noisy image

200 400 600 800 1000

200

400

600

800

1000

0

05

1


200 400 600 800 1000

200

400

600

800

1000 0

02

04

06

08


200 400 600 800 1000

200

400

600

800

1000

0

02

04

06

08

1





50 100 150 200 250 300 350 400 45055

60

65

70

75

80

85

90

95

100


KNNSVMSRC

CRCKCRC

Clas

sifica

tion

rate

()



50 100 150 200 250 300 350 400 45055

60

65

70

75

80

85

90

95

100


Clas

sifica

tion

rate

() o

f KN

N



50 100 150 200 250 300 350 400 45060

65

70

75

80

85

90

95

100


Clas

sifica

tion

rate

() o

f SV

M



50 100 150 200 250 300 350 400 45065

70

75

80

85

90

95

100


Clas

sifica

tion

rate

() o

f SRC



50 100 150 200 250 300 350 400 45070

75

80

85

90

95

100


Clas

sifica

tion

rate

() o

f CRC







5 Conclusion












Acknowledgments


References
















RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Input image

200 400 600 800 1000

200

400

600

800

1000 0

02

04

06

08

1Noisy image

200 400 600 800 1000

200

400

600

800

1000

0

05

1


200 400 600 800 1000

200

400

600

800

1000 0

02

04

06

08


200 400 600 800 1000

200

400

600

800

1000

0

02

04

06

08

1





50 100 150 200 250 300 350 400 45055

60

65

70

75

80

85

90

95

100


KNNSVMSRC

CRCKCRC

Clas

sifica

tion

rate

()



50 100 150 200 250 300 350 400 45055

60

65

70

75

80

85

90

95

100


Clas

sifica

tion

rate

() o

f KN

N



50 100 150 200 250 300 350 400 45060

65

70

75

80

85

90

95

100


Clas

sifica

tion

rate

() o

f SV

M



50 100 150 200 250 300 350 400 45065

70

75

80

85

90

95

100


Clas

sifica

tion

rate

() o

f SRC



50 100 150 200 250 300 350 400 45070

75

80

85

90

95

100


Clas

sifica

tion

rate

() o

f CRC







5 Conclusion












Acknowledgments


References
















RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










50 100 150 200 250 300 350 400 45055

60

65

70

75

80

85

90

95

100


Clas

sifica

tion

rate

() o

f KN

N



50 100 150 200 250 300 350 400 45060

65

70

75

80

85

90

95

100


Clas

sifica

tion

rate

() o

f SV

M



50 100 150 200 250 300 350 400 45065

70

75

80

85

90

95

100


Clas

sifica

tion

rate

() o

f SRC



50 100 150 200 250 300 350 400 45070

75

80

85

90

95

100


Clas

sifica

tion

rate

() o

f CRC







5 Conclusion












Acknowledgments


References
















RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation



















Acknowledgments


References
















RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









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