research article compressive sensing approach in...

Research ArticleCompressive Sensing Approach in the HermiteTransform Domain

Srdjan StankoviT Ljubiša StankoviT and Irena OroviT

University of Montenegro Faculty of Electrical Engineering Dzordza Vasingtona bb 81000 Podgorica Montenegro

Correspondence should be addressed to Irena Orovic irenaoacme

Received 20 August 2015 Accepted 16 November 2015

Academic Editor Yuqiang Wu

Copyright copy 2015 Srdjan Stankovic et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

Compressive sensing has attracted significant interest of researchers providing an alternative way to sample and reconstruct thesignals This approach allows us to recover the entire signal from just a small set of random samples whenever the signal is sparsein certain transform domainTherefore exploring the possibilities of using different transform basis is an important task needed toextend the field of compressive sensing applications In this paper a compressive sensing approach based on the Hermite transformis proposed The Hermite transform by itself provides compressed signal representation based on a smaller number of Hermitecoefficients compared to the signal length Here it is shown that for a wide class of signals characterized by sparsity in the Hermitedomain accurate signal reconstruction can be achieved even if incomplete set ofmeasurements is used Advantages of the proposedmethod are demonstrated on numerical examples The presented concept is generalized for the short-time Hermite transform andcombined transform

1 Introduction

The Hermite polynomials and Hermite functions haveattracted the attention of researchers in various fields ofengineering and signal processing [1ndash7] such as in quantummechanics (harmonic oscillators) ultra-high band telecom-munication channels and ECG data compression using Her-mite functions representation of the QRS complexes A set ofHermite functions forming an orthonormal basis is suitablefor approximation classification and data compression tasks[3] Since the Hermite functions are eigenfunctions of theFourier transform time and frequency spectra are simulta-neously approximated Here we are especially interested ina class of signals that are sparse in the Hermite transformdomain Note that generally such signals are not sparse inthe Fourier transform domain In the light of compressivesensing (CS) theory [8ndash12] we propose the method for effi-cient reconstruction of signals from its incomplete set of sam-ples using the Hermite transform The number of Hermitefunctions used in this approach is much smaller comparedto the original length of the signal The proposed approach isuseful in the applications where the significant informationis missing and the total signal reconstruction is required

Note that the large amount of missing signal samples mayoccur as a consequence of the compressed sampling strategybut also as a consequence of discarding damaged signalparts [13ndash15] The theory is illustrated through examplesshowing that the Hermite transform based CS for certaintypes of signals can outperform the Fourier transform relatedreconstructions Furthermore in analogy with the time-frequency analysis based on the Fourier transform the short-time Hermite transform is defined as a linear representationthat reveals the local behavior of windowed signal parts If thesignal components are of short duration then the short-timeHermite transform is more suitable for compressive sensingthan the standard Hermite transform Finally the possibilityof combining different transforms depending on the signalcharacteristics is explored

The paper is organized as follows The theory behindthe Hermite transform and the fast method for Hermitecoefficients calculation is given in Section 2The formulationof compressive sensing approach in the Hermite transformdomain is given in Section 3The possibilities to exploit othersparsity domains based on the short-time Hermite transformand combined transform are presented in Section 4 Theexperimental evaluation of the proposed approach is given

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015 Article ID 286590 9 pageshttpdxdoiorg1011552015286590

2 Mathematical Problems in Engineering

in Section 4 while the concluding remarks are given inSection 5

2 Hermite Transform

The Hermite functions provide good localization and thecompact support in both time and frequency domain [4]The119894th order Hermite function is defined as follows

120595119894 (119905) =

(minus1)11989411989011990522

radic2119894119894radic120587

sdot

119889119894(119890minus1199052

)

119889119905119894 (1)

The Hermite functions provide an orthonormal basis set foran optimal representation of different signals using the fewestnumber of basis functions Signal expansion into Hermitefunctions known as theHermite transform has been used forboth 1D and 2D signals in various applications The Hermiteexpansion for a signal 119891(119905) can be defined as follows

119891 (119905) =

119873minus1

sum

119894=0

119862119894120595119894 (119905) (2)

where 120595119894(119905) are the Hermite functions and 119873 is the number

of functions used for the approximation The number ofHermite functions 119873 could be usually much smaller thanthe number of signal samples 119872 (119873 le 119872) The Hermitecoefficients can be defined by using the Hermite polynomialsas follows

119862119894=

1

radic2119894119894radic120587

int

infin

minusinfin

119890minus1199052

(119891 (119905) 1198901199052

)119867119894 (119905) 119889119905 (3)

where

119867119894 (119905) = (minus1)

1198941198901199052119889119894(119890minus1199052

)

119889119905119894(4)

represents the Hermite polynomial An efficient procedurefor calculation of Hermite coefficients can be done by apply-ing the Gauss-Hermite quadrature [5]

119862119894

=1

radic2119894119894radic120587

119872

sum

119898=1

2119872minus1

119872radic120587

11987221198672

119872minus1(119905119898)(119891 (119905119898) 1198901199052

1198982)119867119894(119905119898)

(5)

where 119905119898

are zeros of Hermite polynomials By usingthe Hermite functions instead of polynomials a simplifiedexpression is obtained

119862119894asymp1

119872

119872

sum

119898=1

120583119894

119872minus1(119905119898) 119891 (119905119898) (6)

The constants 120583119894119872minus1

(119905119898) are calculated as follows

120583119894

119872minus1(119905119898) =

120595119894(119905119898)

(120595119872minus1

(119905119898))2 (7)

3 Compressive Sensing Formulation inthe Hermite Transform Domain

Generally the compressive sensing scenarios are focused onthe new sampling strategy which results in a large numberof randomly missing samples compared to the standard sam-pling methods [8] Hence based on a small set of acquiredmeasurements the entire signal needs to be reconstructedThemissing samples in compressive sensing generally cannotbe recovered using standard interpolation methods due tothe complexity of nonstationary signals in real applicationsNamely the interpolation methods such as polynomial fitcubic spline interpolation or similar usually assume certainmodel function which is mostly inappropriate for timedomain signal modeling Therefore the compressive sensingreconstruction is formulated in the literature as an optimiza-tion problem (rather than interpolation) which reconstructsthe signal by finding the sparsest transform domain solutioncorresponding to the available small set of samples

In the CS context we are dealing with a small set ofrandomly chosen samples of 119891(119905) Let us assume that we haveonly119872

119860out of119872 available samples (119872 is the total number

of signal samples and 119872 ≫ 119872119860) The vector of available

measurements is denoted as y Now we may write

y = Φf (8)

where f is original full signal (of length 119872) written in thevector form while Φ (119872

119860times 119872) is the measurement matrix

The original signal f can be expressed using the Hermitetransform as follows

f = ΨC (9)

whereC is the vector of Hermite transform coefficients whileΨ is the inverseHermite transformmatrix of size119872times119873 (119873 le

119872)

Ψ =

[[[[[[[[

[

1205950 (0) 120595

1 (0) sdot sdot sdot 120595119873minus1 (0)

1205950 (1) 120595


sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot

1205950 (119872 minus 1) 1205951 (119872 minus 1) sdot sdot sdot 120595119873minus1 (119872 minus 1)

]]]]]]]]

]

H =1

119872

sdot

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

[

1205950 (0)

(120595119873minus1 (0))

2

1205950 (1)

(120595119873minus1 (1))

2sdot sdot sdot

1205950 (119872 minus 1)

(120595119873minus1 (119872 minus 1))

2

1205951 (0)

(120595119873minus1 (0))

2

1205951 (1)

(120595119873minus1 (1))

2sdot sdot sdot

1205951 (119872 minus 1)

(120595119873minus1 (119872 minus 1))

2


120595119873minus1 (0)

(120595119873minus1 (0))

2

120595119873minus1 (1)

(120595119873minus1 (1))

2sdot sdot sdot

120595119873minus1 (119872 minus 1)

(120595119873minus1 (119872 minus 1))

2

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

]

(10)

Mathematical Problems in Engineering 3

The direct Hermite transform matrix is given by H In theextended form (9) can be written as follows

[[[[[

[

119891 (0)

119891 (1)

sdot sdot sdot

119891 (119872 minus 1)

]]]]]

]⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

f

=

[[[[[

[

1205950 (0) 120595


1205950 (1) 120595




]]]]]

]⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

Ψ

[[[[[

[

119862 (0)

119862 (1)

sdot sdot sdot

119862 (119873 minus 1)

]]]]]

]⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

C

(11)

The Hermite basis functions are calculated using the fastrecursive realization defined as follows

1205950 (119905) =

1

4radic120587

119890minus11990522

1205951 (119905) =

radic2119905

4radic120587

119890minus11990522

120595119894 (119905) = 119905

radic2

119894Ψ119894minus1 (119905) minus

radic119894 minus 1

119894Ψ119894minus2 (119905) forall119894 ge 2

(12)

According to (8) and (9) we have

y = ΦΨC = ΘC (13)

The reconstructed signal f can be obtained as a solutionof 119872119860linear equations with 119872 unknowns The system is

undetermined and can have infinitely many solutions Nowwe assume that the signal is sparse in the Hermite transformdomain It means that the observed signal can be efficientlyrepresented by a very small number119870 of Hermite expansioncoefficients such that 119870 lt 119872

119860 Therefore the optimization

basedmathematical algorithms should be employed to searchfor the sparsest solution A near optimal solution is achievedby using the 119897

1norm based minimization as follows

min 10038171003817100381710038171003817C100381710038171003817100381710038171198971

subject to y = ΘC(14)

where C is the Hermite transform vector of signal f In order to solve the previousminimization problem first

we need to calculate the initial Hermite transform using theavailable set of 119872

119860samples with the time support Ω y =

Φf = f(Ω) Therefore we observe the signal in the followingform

1198910 (119899) =

119910 (119899) for 119899 isin Ω

0 elsewhere(15)

The initial vector of Hermite transform coefficients can bethen calculated using 119873 Hermite functions (119873 le 119872) asfollows

C0= HΩy (16)

whereHΩcontains only the columns ofH that correspond to

instants 119899 isin Ω Alternatively we can write

1198620 (119894) =

1

119872

119872minus1

sum

119899=0

120595119894 (119899)

(120595119872minus1 (119899))

21198910 (119899) 119894 = 1 119873 (17)

In order to determine the signal support in the Hermitetransformdomain the initial vector of theHermite transformcoefficients C

0is compared by the threshold 119879

k = arg C0gt 119879 (18)

The exact values of Hermite coefficients at positions selectedin vector k are obtained as a solution of the CS problem

ΘCSC0 = y (19)

The CS matrix ΘCS is obtained from the inverse Hermitetransform matrix ΘCS = Ψ(Ω k) using columns thatcorrespond to frequencies k and rows corresponding tomeasurements with support Ω The system is solved in theleast square sense as follows

C = (ΘlowastCSΘCS)minus1Θlowast

CSy (20)

where (lowast) denotes the conjugate transpose operation

Analysis of Components Reconstruction Let us consider theisometry property of Hermite transform matrixΨ

ΨC22=10038171003817100381710038171198911003817100381710038171003817

2

2=1003816100381610038161003816119891 (0)

1003816100381610038161003816

2+ sdot sdot sdot +

1003816100381610038161003816119891 (119873 minus 1)1003816100381610038161003816

2

=

1003816100381610038161003816100381610038161003816100381610038161003816

119873minus1

sum

119896=0

119862 (119896) 120595119896 (0)

1003816100381610038161003816100381610038161003816100381610038161003816

2

+ sdot sdot sdot +

1003816100381610038161003816100381610038161003816100381610038161003816

119873minus1

sum

119896=0

119862 (119896) 120595119896 (119873 minus 1)

1003816100381610038161003816100381610038161003816100381610038161003816

2

=

119873minus1

sum

119896=0

119862 (119896)2

119873minus1

sum

119899=0

120595119896 (119899)2

+

119873minus2

sum

1198961=0

119873minus1

sum

1198962=1198961+1

2119862 (1198961) 119862 (119896

2)

119873minus1

sum

119899=0

1205951198961(119899) 120595119896

2(119899)

(21)

The previous equation can be written as follows

ΨC22= ((119862 (0))

2

119873minus1

sum

119899=0

(1205950 (119899))2+ sdot sdot sdot

+ (119862 (119873 minus 1))2

119873minus1

sum

119899=0

(120595119873minus1 (119899))

2)

+

119873minus2

sum

1198961=0

119873minus1

sum

1198962=1198961+1

2119862 (1198961) 119862 (119896

2)

119873minus1

sum

119899=0

1205951198961(119899) 120595119896

2(119899)

(22)

Now observe the first term on the right side given by119875 = 119862(0)

2sum119873minus1

119899=0(1205950(119899))2+ sdot sdot sdot + 119862(119873 minus 1)

2sum119873minus1

119899=0(120595119873minus1

(119899))2

particularly the sum of squared values of different Hermitefunctions Figure 1 illustrates the sum of squared values ofdifferentHermite functions calculated in the zeros ofHermite


0 20 40 60 80 100 1202

25

3

35

4

45

5

55

Hermite function order

Sum

of s

quar

ed v

alue

s

128 functions100 functions


Figure 1 Sum of squared values of Hermite functions for different119873

polynomials for119873 = (128 100 70 and 50) Unlike in the caseof the Fourier transform basis where the sum of absolutevalues of complex exponential basis would be constant forany frequency 119896 from Figure 1 we can note an approximatelow-pass characteristic of the curves meaning that the lower-order coefficients are favored compared to the higher ordercoefficients Consequently when applying the threshold forcomponents detection it would be easier to detect a set oflow-order coefficients than the high-order ones

4 Introducing the Short-TimeHermite Transform and Short-TimeCombined Transform

41 Short-Time Hermite Transform Let us assume that 119873 =

119872 in (10) and define an119872times119872Hermite matrix

H119872=1

119872

sdot

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

[

1205950 (0)

(120595119872minus1 (0))

2

1205950 (1)

(120595119872minus1 (1))

2sdot sdot sdot

1205950 (119872 minus 1)

(120595119872minus1 (119872 minus 1))

2

1205951 (0)

(120595119872minus1 (0))

2

1205951 (1)

(120595119872minus1 (1))

2sdot sdot sdot

1205951 (119872 minus 1)

(120595119872minus1 (119872 minus 1))

2


120595119872minus1 (0)

(120595119872minus1 (0))

2

120595119872minus1 (1)

(120595119872minus1 (1))

2sdot sdot sdot

120595119872minus1 (119872 minus 1)

(120595119872minus1 (119872 minus 1))

2

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

]

(23)

The short-time Hermite transform (STHT) can be defined asa composition of Hermite transform matrices whose size isdefined by the window width Without loss of generality wemay assume the nonoverlapping unit rectangular windows

(with the width of 119872 samples) The total transform matrixfor the STHT can be created as follows

W = I119871119872

otimesH119872=

[[[[[

[

H119872

0 sdot sdot sdot 00 H

119872sdot sdot sdot 0

sdot sdot sdot sdot sdot sdot sdot sdot sdot 00 0 sdot sdot sdot H

119872

]]]]]

]

(24)

where I is identity matrix of size (119871119872 times 119871119872) 0 is119872 times119872

zero matrix and 119871 is the total length of the signal while otimesdenotes the Kronecker product The STHT can be definedusing transform matrixW as follows

STHT =Wf =WΨ119871C (25)

where f is a time domain signal vector of size 119871 times 1 and Cis a Hermite transform vector of size 119871 times 1 while STHT isa column vector containing all STHT vectors STHT

119872119894

(119898119894)

119894 = 0 1 119875 (119875 is the number of windows) Furthermorewe may write

STHT = AC (26)

where A = WΨ119871is matrix of size 119871 times 119871 that maps the global

transform domain information in C into local informationin STHT In the case of windows with variable length overtime (time-varying windows) the smaller matrices withinWwill be of different sizes Particularly for a set of119875 rectangularwindows of sizes119872

11198722 119872

119875instead ofH

119872we will have

H1198721

H1198722

H119872119875

In the context of compressive sensing the STHT allows us

to define two types of CS problem A common CS problem isdefined by assuming that the missing samples appear in thetime domain while the sparsity is exploited in the transformdomainHowever themissing samplesmay also appear in theSTHT domain after applying certain filter forms such as the119871-estimate filtering in the presence of strong noisy pulsesThetwo CS optimization problems are discussed in the sequel

(1) Assume that CS procedure is done in the time domainsuch that the measurements are taken from each windowedsignal part y

119894= f119872119894

(Ω) Based on the definitions given inSection 3 the minimization problem is given by

min 1003817100381710038171003817C11989410038171003817100381710038171198971

subject to y119894= ΨΩ

119872C119894

(27)

for 119894 = 1 2 119875 where 119875 is the total number of windowswhile Ψ

119872is the inverse Hermite transform of size 119872 times 119872

where only the rows corresponding to the set of positions Ωare kept

(2) Let us observe the missing values in the STHTdomain where the measurements vector is denoted bySTHTCS The CS problem formulation is defined using thelinear relationship (26) The CS matrix denoted by ACSis formed by omitting the rows from A corresponding tothe positions of missing values in STHTCS Then a simplelinear relationship can be established between the reduced


0 20 40 60 80 100 120

Original sparse signal

Time (samples)

minus2

minus1

0

1

2

3A

mpl

itude

(a)

0 10 20 30 40 50 60

Hermite coefficients of a sparse signal

Hermite functioncoefficient order

0

02

04

06

08

1

Am

plitu

de

(b)

Figure 2 The original signal (a) time domain representation (b) Hermite transform of signal calculated using119873 = 60Hermite functions

observations in STHTCS and the sparse Hermite transformvector C (corresponding to the entire signal) Hence the CSproblem is given in the following form

min C1subject to STHTCS = ACSC

(28)

It means that the missing samples in the STHT can bereconstructed such as to provide the best concentratedCTheabove CS optimization problem can be solved using variousexisting CS reconstruction algorithms

42 Short-Time Combined Transform On the basis of theSTHT we can also define the short-time combined transformas follows

X = Zf (29)

where the transform matrix Z is made as a combination ofHermite transform matrices and other transforms

Z =[[[[[

[

Z1198721

0 sdot sdot sdot 00 Z

1198722

sdot sdot sdot 0sdot sdot sdot sdot sdot sdot sdot sdot sdot 00 0 sdot sdot sdot Z

119872119875

]]]]]

]

(30)

where Z119872119894

can be either the119872119894times 119872119894Hermite transform or

some other suitable transform matrix of the same size Forinstance we may observe one example of the combination ofHermite transform and Fourier transform as a special case

Z =[[[[[

[

H119872

0 sdot sdot sdot 00 F119872

sdot sdot sdot 0sdot sdot sdot sdot sdot sdot F

1198720

0 0 sdot sdot sdot H119872

]]]]]

]

(31)

The combined matrix Z is thus composed of Hermite trans-form matrix H

119872of size 119872 times 119872 two sequential Fourier

transformmatrices F119872of size119872times119872 and again oneHermite

transformmatrix of the same size In the case ofwindowswith

variable length over time the short-time combined transformcan be written as

[[[[[

[

X1198721

X1198722

sdot sdot sdot

X119872119875

]]]]]

]

=

[[[[[

[

Z1198721


1198722


119872119875

]]]]]

]

[[[[[

[

f1198721

f1198722

sdot sdot sdot

f119872119875

]]]]]

]

(32)

where f1198721

represents the vector containing first1198721samples

in f and vector f1198722

contains the next1198722samples from f while

f119872119875

contains the last119872119875samples from f such that119872

1+1198722+

sdot sdot sdot + 119872119875= 119871

We can again observe the set of measurements corre-sponding to different signal parts y

119894= f119872119894

(Ω) where Θ119872119894

=

inv(Z119872119894

)Therefore the CSminimization problem is given by

min 10038171003817100381710038171003817X119872119894

100381710038171003817100381710038171

subject to y119894= ΘΩ

119872119894

X119872119894

(33)

for 119894 = 1 2 119875 where Ω in superscript denotes thatwe use only the rows corresponding to available samplesdefined by the set Ω The inverse transform denoted by ΘΩ

119872119894

can be changed in each window depending on the signalcharacteristics

5 Experimental Evaluation

Example 1 In order to illustrate efficiency of the proposedmethod let us observe a signal that is sparse in the Hermitetransform domain The time domain signal is shown inFigure 2(a) The Hermite transform of the observed signalconsists of ten components with unit amplitudes as shownin Figure 2(b) The available number of samples is 60 ofthe total signal length while the maximal order of Hermitefunctions used in the expansion is limited to119873 = 60

After calculating the initial Hermite transform usingavailable signal samples the threshold is applied to selectsignal support in the Hermite domain The threshold is setempirically to119879 = 120572maxC120572 = 04 after performing a largenumber of experiments Moreover the results are not verysensitive on threshold settings and even lower thresholds(eg 120572 = 03 120572 = 02) can be used because all additional


minus04

minus02

0

02

04

06

08

1

Am

plitu

de

10 20 30 40 50 600Hermite functioncoefficient order

Figure 3 Hermite coefficients of signal with missing samples -Iand the reconstructed Hermite coefficients -times

(false) components selected by such a threshold would obtaintheir true zero values afterwards The suitable threshold canbe even theoretically derived (under certain assumptions)Namely the effects caused bymissing samples in time domaincan be modeled by the noise in the Hermite transformdomain The noise can be described using two randomvariables the first one corresponds to the noise appearing atnonsignal components (noise alone) while the second onecorresponds to the noise appearing at the signal componentspositions The two random variables can be described bythe corresponding mean values and variances that could befurther employed to define the threshold level Namely withcertain probability we may define the threshold value thatis just above the level of noise components Consequentlysuch a threshold would select only the signal componentsdetermining the signal support used in the reconstructionprocedure Since the threshold derivation would requiresignificant space and analyses it could be a topic of somefurther work

The selected components in the Hermite transformdomain are shown in Figure 3 Using the detected positionsof components the exact values of Hermite coefficients arecalculated according to (20) (Figure 3)The reconstructed sig-nal and the absolute error between original and reconstructedsignal are shown in Figures 4(a) and 4(b) respectively Forthe comparison the reconstruction based on the Fouriertransform is consideredHowever it is shown that the Fouriertransform based reconstruction cannot be used in this casebecause it produces a significant error (Figure 4(c) 10 largestFourier transform components are used for reconstruction)The reason is in the fact that the signals that are sparsein Hermite transform domain do not exhibit sparsity inthe Fourier transform domain (or some other transformdomains) This means that the exact results in this casecan be obtained only using the Hermite transform providingthe sparse signal representation Similar conclusion holdsfor other transform domains where the signal is not sparse(Fractional Fourier Wavelet etc)

Example 2 In this example we have observed different sparsesignals (having 10 components) in the Hermite domain

through the 1000 repetitions of the previously described pro-cedure The aim is to test how the accuracy of the proposedmethod changes for different number of missing samplesNamely the number of missing samples was increased insteps of 10 starting from zero up to 80 of missingsamples For each instance on the 119910-axes we performed 1000iterations with different 10-component signals and the meansquare errors between original and reconstructed signals arecalculated and shown in Figure 5(a) We can observe that upto 70 of missing samples the MSE is still low but it startsto increase significantly afterwards (for values higher than70) This means that the reliable reconstruction cannot beguaranteed for such a large number of missing samples (suchas 80 of missing samples)

Additionally the proposed approach is tested in thepresence of external additive Gaussian noise Namely theMSE is calculated for different values of SNR (for each SNRwe assume 1000 realizations of random noise)The results arepresented in Figure 5(b) showing that the proposed approachcan be efficiently used for SNR gt 8 dB For SNR lt 8 dB theMSE starts to increase rapidly

Example 3 This example illustrates the concept of combinedHermite-Fourier transform and CS reconstruction Observethe signal with the total length of119871 = 32 samples composed oftwo parts first part exhibits sparsity in theHermite transformdomain with 119870 = 2 components (HT components) atpositions 7 and 12 with amplitudes 08 and 065 secondpart exhibits sparsity in the Fourier transform domain with119870 = 2 components (FT components) at positions minus3 and4 with amplitudes 1 and 08 respectively Now consider thecompressive sensing scenario with only 50 of samples avail-able (original full length signal and available measurementsare shown in Figure 6) The original short-time combinedtransform obtained using windows width equal to 16 samplesis shown in Figure 7(a) where the white fields represent 0value Thus the signal represented by a full set of samples issparse meaning that it is fully concentrated on a few nonzerocomponents In the case when we deal with 50 of missingsamples the corresponding short-time combined transformis shown in Figure 7(b) Note that in this case the sparsity isdisturbed due to the missing samples Instead of zero valuesat nonsignal position the noise appears as a consequence ofmissing samples

The threshold based components selection and signalreconstruction (in analogy to (18) and (20)) is applied to bothparts of the signal FT components andHTcomponents (Fig-ures 8(a) and 8(b) resp) Based on the identified positions ofcomponents the exact signal reconstruction is done using theprocedure presented in Section 3 The reconstructed signalcomponents are shown in Figure 9

6 Conclusion

Thepossibility of using theHermite transform in compressivesensing applications was explored in this work The com-pressive sensing setup and signal reconstruction approach inthe Hermite transform domain were defined A simple and


Reconstructed signal

20 40 60 80 1000 120Time (samples)

minus2

minus1

0

1

2

3

Am

plitu

de

(a)

Error between original and reconstructed signal

minus4

minus2

0

2

4

Am

plitu

de

20 40 60 80 100 1200Time (samples)

times10minus15

(b)

Reconstructed signal and error between original and reconstructed signal

minus3

minus2

minus1

0

1

2

Am

plitu

de

20 40 60 80 100 1200Time (samples)

(c)

Figure 4 The reconstructed signal (a) time domain representation (b) the absolute error between original and reconstructed signal (c) forcomparison reconstructed signal based on the Fourier transform reconstruction approach (-∘) and the corresponding error (solid line)

0 05 1 15 2MSE

0

10

20

30

40

50

60

70

80

Miss

ing

sam

ples

()

(a)

6

10

15

20

25

SNR

(dB)

1 2 3 4 50MSE (in 1000 realizations)

(b)

Figure 5 (a) MSE calculated for 1000 realizations and for different number of missing samples (b) MSE calculated for different values ofSNR in 1000 realizations of external noise

0 5 10 15 20 25 30

05

1

15

Missing samples -o blue available samples -o red

Figure 6 Missing samples and available measurements in time domain


0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30

HT16 (24 7)

HT16 (24 15)

HT16 (24 14)

HT16 (24 13)

HT16 (24 12)

HT16 (24 11)

HT16 (24 10)

HT16 (24 9)

HT16 (24 6)

HT16 (24 5)

HT16 (24 4)

HT16 (24 3)

HT16 (24 2)

HT16 (24 1)

HT16 (24 0)

HT16 (24 8)

FT16 (8 7)

FT16 (8 6)

FT16 (8 5)FT16 (8 4)

FT16 (8 3)

FT16 (8 2)

FT16 (8 1)

FT16 (8 minus1)

FT16 (8 minus2)

FT16 (8 minus3)

FT16 (8 minus4)

FT16 (8 minus5)

FT16 (8 minus6)

FT16 (8 minus7)

FT16 (8 minus8)

FT16 (8 0)

minus16minus14minus12minus10


02468

101214

(a)0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30

HT16 (24 7)

HT16 (24 15)

HT16 (24 14)

HT16 (24 13)

HT16 (24 12)

HT16 (24 11)

HT16 (24 10)

HT16 (24 9)

HT16 (24 6)

HT16 (24 5)

HT16 (24 4)

HT16 (24 3)

HT16 (24 2)

HT16 (24 1)

HT16 (24 0)

HT16 (24 8)

FT16 (8 7)

FT16 (8 6)

FT16 (8 5)FT16 (8 4)

FT16 (8 3)

FT16 (8 2)

FT16 (8 1)

FT16 (8 minus1)

FT16 (8 minus2)

FT16 (8 minus3)

FT16 (8 minus4)

FT16 (8 minus5)

FT16 (8 minus6)

FT16 (8 minus7)

FT16 (8 minus8)

FT16 (8 0)



02468

101214

(b)

Figure 7 (a) Original short-time combined transform of full length signal (FT Fourier transform HT Hermite transform) (b) short-timecombined transform of available measurements

minus8 minus6 minus4 minus2 0 2 4 6

02

04

06

08

1Fourier coefficients with missing samples -o red original -o blue

(a)2 4 6 8 10 12 14 16

02

04

06

08Hermitian coefficients with missing samples -o red original -o blue

(b)

Figure 8 Selecting the components of interest by applying threshold to the following (a) FT components and (b) HT components

2 4 6 8 10 12 14 16

Reconstructed Hermite transform components

minus8 minus6 minus4 minus2 0 2 4 6 80

02

04

06

08

0

02

04

06

081Reconstructed Fourier transform components

Figure 9 Reconstructed signal components

fast procedure for the total reconstruction of signals usingselected Hermite transform coefficients provides the resultsvery close to the original signal even when we deal withsignificant missing information It is important to emphasizethat the proposed concept in the Hermite transform domaincan be also combined with other known compressive sensingsolvers The entire concept is generalized and extended bydefining the short-time Hermite transform as well as theshort-time combined transform These two transforms openmore possibilities to apply the compressive sensing approachin different scenarios

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This work is supported by the Montenegrin Ministry ofScience Project Grant ldquoNew ICTCompressive Sensing BasedTrends Applied to Multimedia Biomedicine and Communi-cationsrdquo (ACRONYM CS-ICT)


References

[1] A Sandryhaila S Saba M Puschel and J Kovacevic ldquoEfficientcompression of QRS complexes using hermite expansionrdquo IEEETransactions on Signal Processing vol 60 no 2 pp 947ndash9552012

[2] M Lagerholm G Peterson G Braccini L Edenbrandt andL L Sornmo ldquoClustering ECG complexes using hermitefunctions and self-organizing mapsrdquo IEEE Transactions onBiomedical Engineering vol 47 no 7 pp 838ndash848 2000

[3] B Beliczynski ldquoApproximation of functions by multivariablehermite basis a hybrid methodrdquo in Adaptive and NaturalComputing Algorithms vol 6593 of Lecture Notes in ComputerScience pp 130ndash139 Springer 2011

[4] Y K Alp and O Arıkan ldquoTime-frequency analysis of signalsusing support adaptive Hermite-Gaussian expansionsrdquo DigitalSignal Processing vol 22 no 6 pp 1010ndash1023 2012

[5] A Krylov and D Kortchagine ldquoFast Hermite projectionmethodrdquo inProceedings of the International Conference on ImageAnalysis and Recognition (ICIAR rsquo06) pp 329ndash338 Povoa deVarzim Portugal September 2006

[6] D Kortchagine and A Krylov ldquoProjection filtering in imageprocessingrdquo in Proceedings of the International Conferenceon Computer Graphics and Applications pp 42ndash45 MoscowRussia August-September 2000

[7] I Orovi S Stankovi T Chau CM Steele and E Sejdic ldquoTime-frequency analysis and hermite projection method appliedto swallowing accelerometry signalsrdquo EURASIP Journal onAdvances in Signal Processing vol 2010 Article ID 323125 2010

[8] R G Baraniuk ldquoCompressive sensingrdquo IEEE Signal ProcessingMagazine vol 24 no 4 pp 118ndash124 2007

[9] D L Donoho ldquoCompressed sensingrdquo IEEE Transactions onInformation Theory vol 52 no 4 pp 1289ndash1306 2006

[10] E J Candes J Romberg and T Tao ldquoRobust uncertaintyprinciples exact signal reconstruction from highly incompletefrequency informationrdquo IEEE Transactions on InformationThe-ory vol 52 no 2 pp 489ndash509 2006

[11] D Vukobratovic and A Pizurica ldquoCompressed sensing usingsparse adaptive measurementsrdquo in Proceedings of the Sympo-sium on Information Theory in the Benelux (SITB rsquo14) Eind-hoven The Netherlands May 2014

[12] D Angelosante G B Giannakis and E Grossi ldquoCompressedsensing of time-varying signalsrdquo in Proceedings of the 16thInternational Conference on Digital Signal Processing (DSP rsquo09)pp 1ndash8 Santorini-Hellas Greece July 2009

[13] L Stankovic S Stankovic andMAmin ldquoMissing samples anal-ysis in signals for applications to L-estimation and compressivesensingrdquo Signal Processing vol 94 no 1 pp 401ndash408 2014

[14] S Stankovic I Orovic and L Stankovic ldquoAn automated signalreconstructionmethod based on analysis of compressive sensedsignals in noisy environmentrdquo Signal Processing vol 104 pp 43ndash50 2014

[15] S Stankovic L Stankovic and I Orovic ldquoRelationship betweenthe robust statistics theory and sparse compressive sensedsignals reconstructionrdquo IET Signal Processing vol 8 no 3 pp223ndash229 2014

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


in Section 4 while the concluding remarks are given inSection 5

2 Hermite Transform

The Hermite functions provide good localization and thecompact support in both time and frequency domain [4]The119894th order Hermite function is defined as follows

120595119894 (119905) =

(minus1)11989411989011990522

radic2119894119894radic120587

sdot

119889119894(119890minus1199052

)

119889119905119894 (1)

The Hermite functions provide an orthonormal basis set foran optimal representation of different signals using the fewestnumber of basis functions Signal expansion into Hermitefunctions known as theHermite transform has been used forboth 1D and 2D signals in various applications The Hermiteexpansion for a signal 119891(119905) can be defined as follows

119891 (119905) =

119873minus1

sum

119894=0

119862119894120595119894 (119905) (2)

where 120595119894(119905) are the Hermite functions and 119873 is the number

of functions used for the approximation The number ofHermite functions 119873 could be usually much smaller thanthe number of signal samples 119872 (119873 le 119872) The Hermitecoefficients can be defined by using the Hermite polynomialsas follows

119862119894=

1

radic2119894119894radic120587

int

infin

minusinfin

119890minus1199052

(119891 (119905) 1198901199052

)119867119894 (119905) 119889119905 (3)

where

119867119894 (119905) = (minus1)

1198941198901199052119889119894(119890minus1199052

)

119889119905119894(4)

represents the Hermite polynomial An efficient procedurefor calculation of Hermite coefficients can be done by apply-ing the Gauss-Hermite quadrature [5]

119862119894

=1

radic2119894119894radic120587

119872

sum

119898=1

2119872minus1

119872radic120587

11987221198672

119872minus1(119905119898)(119891 (119905119898) 1198901199052

1198982)119867119894(119905119898)

(5)

where 119905119898

are zeros of Hermite polynomials By usingthe Hermite functions instead of polynomials a simplifiedexpression is obtained

119862119894asymp1

119872

119872

sum

119898=1

120583119894

119872minus1(119905119898) 119891 (119905119898) (6)

The constants 120583119894119872minus1

(119905119898) are calculated as follows

120583119894

119872minus1(119905119898) =

120595119894(119905119898)

(120595119872minus1

(119905119898))2 (7)

3 Compressive Sensing Formulation inthe Hermite Transform Domain

Generally the compressive sensing scenarios are focused onthe new sampling strategy which results in a large numberof randomly missing samples compared to the standard sam-pling methods [8] Hence based on a small set of acquiredmeasurements the entire signal needs to be reconstructedThemissing samples in compressive sensing generally cannotbe recovered using standard interpolation methods due tothe complexity of nonstationary signals in real applicationsNamely the interpolation methods such as polynomial fitcubic spline interpolation or similar usually assume certainmodel function which is mostly inappropriate for timedomain signal modeling Therefore the compressive sensingreconstruction is formulated in the literature as an optimiza-tion problem (rather than interpolation) which reconstructsthe signal by finding the sparsest transform domain solutioncorresponding to the available small set of samples

In the CS context we are dealing with a small set ofrandomly chosen samples of 119891(119905) Let us assume that we haveonly119872

119860out of119872 available samples (119872 is the total number

of signal samples and 119872 ≫ 119872119860) The vector of available

measurements is denoted as y Now we may write

y = Φf (8)

where f is original full signal (of length 119872) written in thevector form while Φ (119872

119860times 119872) is the measurement matrix

The original signal f can be expressed using the Hermitetransform as follows

f = ΨC (9)

whereC is the vector of Hermite transform coefficients whileΨ is the inverseHermite transformmatrix of size119872times119873 (119873 le

119872)

Ψ =

[[[[[[[[

[

1205950 (0) 120595


1205950 (1) 120595




]]]]]]]]

]

H =1

119872

sdot

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

[

1205950 (0)

(120595119873minus1 (0))

2

1205950 (1)

(120595119873minus1 (1))

2sdot sdot sdot

1205950 (119872 minus 1)

(120595119873minus1 (119872 minus 1))

2

1205951 (0)

(120595119873minus1 (0))

2

1205951 (1)

(120595119873minus1 (1))

2sdot sdot sdot

1205951 (119872 minus 1)

(120595119873minus1 (119872 minus 1))

2


120595119873minus1 (0)

(120595119873minus1 (0))

2

120595119873minus1 (1)

(120595119873minus1 (1))

2sdot sdot sdot

120595119873minus1 (119872 minus 1)

(120595119873minus1 (119872 minus 1))

2

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

]

(10)



[[[[[

[

119891 (0)

119891 (1)

sdot sdot sdot

119891 (119872 minus 1)

]]]]]

]⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

f

=

[[[[[

[

1205950 (0) 120595


1205950 (1) 120595




]]]]]

]⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

Ψ

[[[[[

[

119862 (0)

119862 (1)

sdot sdot sdot

119862 (119873 minus 1)

]]]]]

]⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

C

(11)


1205950 (119905) =

1

4radic120587

119890minus11990522

1205951 (119905) =

radic2119905

4radic120587

119890minus11990522

120595119894 (119905) = 119905

radic2

119894Ψ119894minus1 (119905) minus

radic119894 minus 1

119894Ψ119894minus2 (119905) forall119894 ge 2

(12)


y = ΦΨC = ΘC (13)






min 10038171003817100381710038171003817C100381710038171003817100381710038171198971






1198910 (119899) =

119910 (119899) for 119899 isin Ω

0 elsewhere(15)


C0= HΩy (16)



1198620 (119894) =

1

119872

119872minus1

sum

119899=0

120595119894 (119899)

(120595119872minus1 (119899))

21198910 (119899) 119894 = 1 119873 (17)



k = arg C0gt 119879 (18)


ΘCSC0 = y (19)



CSy (20)



ΨC22=10038171003817100381710038171198911003817100381710038171003817

2

2=1003816100381610038161003816119891 (0)

1003816100381610038161003816

2+ sdot sdot sdot +

1003816100381610038161003816119891 (119873 minus 1)1003816100381610038161003816

2

=

1003816100381610038161003816100381610038161003816100381610038161003816

119873minus1

sum

119896=0

119862 (119896) 120595119896 (0)

1003816100381610038161003816100381610038161003816100381610038161003816

2

+ sdot sdot sdot +

1003816100381610038161003816100381610038161003816100381610038161003816

119873minus1

sum

119896=0

119862 (119896) 120595119896 (119873 minus 1)

1003816100381610038161003816100381610038161003816100381610038161003816

2

=

119873minus1

sum

119896=0

119862 (119896)2

119873minus1

sum

119899=0

120595119896 (119899)2

+

119873minus2

sum

1198961=0

119873minus1

sum

1198962=1198961+1

2119862 (1198961) 119862 (119896

2)

119873minus1

sum

119899=0

1205951198961(119899) 120595119896

2(119899)

(21)


ΨC22= ((119862 (0))

2

119873minus1

sum

119899=0

(1205950 (119899))2+ sdot sdot sdot

+ (119862 (119873 minus 1))2

119873minus1

sum

119899=0

(120595119873minus1 (119899))

2)

+

119873minus2

sum

1198961=0

119873minus1

sum

1198962=1198961+1

2119862 (1198961) 119862 (119896

2)

119873minus1

sum

119899=0

1205951198961(119899) 120595119896

2(119899)

(22)


2sum119873minus1


2sum119873minus1

119899=0(120595119873minus1

(119899))2



0 20 40 60 80 100 1202

25

3

35

4

45

5

55


Sum

of s

quar

ed v

alue

s








H119872=1

119872

sdot

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

[

1205950 (0)

(120595119872minus1 (0))

2

1205950 (1)

(120595119872minus1 (1))

2sdot sdot sdot

1205950 (119872 minus 1)

(120595119872minus1 (119872 minus 1))

2

1205951 (0)

(120595119872minus1 (0))

2

1205951 (1)

(120595119872minus1 (1))

2sdot sdot sdot

1205951 (119872 minus 1)

(120595119872minus1 (119872 minus 1))

2


120595119872minus1 (0)

(120595119872minus1 (0))

2

120595119872minus1 (1)

(120595119872minus1 (1))

2sdot sdot sdot

120595119872minus1 (119872 minus 1)

(120595119872minus1 (119872 minus 1))

2

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

]

(23)



W = I119871119872

otimesH119872=

[[[[[

[

H119872




119872

]]]]]

]

(24)



STHT =Wf =WΨ119871C (25)


119872119894

(119898119894)


STHT = AC (26)



11198722 119872

119875instead ofH

119872we will have

H1198721

H1198722

H119872119875




119894= f119872119894


min 1003817100381710038171003817C11989410038171003817100381710038171198971


119872C119894

(27)






0 20 40 60 80 100 120


Time (samples)

minus2

minus1

0

1

2

3A

mpl

itude

(a)

0 10 20 30 40 50 60



0

02

04

06

08

1

Am

plitu

de

(b)




(28)



X = Zf (29)


Z =[[[[[

[

Z1198721


1198722


119872119875

]]]]]

]

(30)

where Z119872119894



Z =[[[[[

[

H119872



1198720


]]]]]

]

(31)






[[[[[

[

X1198721

X1198722

sdot sdot sdot

X119872119875

]]]]]

]

=

[[[[[

[

Z1198721


1198722


119872119875

]]]]]

]

[[[[[

[

f1198721

f1198722

sdot sdot sdot

f119872119875

]]]]]

]

(32)

where f1198721




f119872119875


1+1198722+

sdot sdot sdot + 119872119875= 119871


119894= f119872119894

(Ω) where Θ119872119894

=

inv(Z119872119894


min 10038171003817100381710038171003817X119872119894

100381710038171003817100381710038171


119872119894

X119872119894

(33)


119872119894






minus04

minus02

0

02

04

06

08

1

Am

plitu

de










6 Conclusion




20 40 60 80 1000 120Time (samples)

minus2

minus1

0

1

2

3

Am

plitu

de

(a)


minus4

minus2

0

2

4

Am

plitu

de

20 40 60 80 100 1200Time (samples)

times10minus15

(b)


minus3

minus2

minus1

0

1

2

Am

plitu

de

20 40 60 80 100 1200Time (samples)

(c)


0 05 1 15 2MSE

0

10

20

30

40

50

60

70

80

Miss

ing

sam

ples

()

(a)

6

10

15

20

25

SNR

(dB)


(b)


0 5 10 15 20 25 30

05

1

15




0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30

HT16 (24 7)

HT16 (24 15)

HT16 (24 14)

HT16 (24 13)

HT16 (24 12)

HT16 (24 11)

HT16 (24 10)

HT16 (24 9)

HT16 (24 6)

HT16 (24 5)

HT16 (24 4)

HT16 (24 3)

HT16 (24 2)

HT16 (24 1)

HT16 (24 0)

HT16 (24 8)

FT16 (8 7)

FT16 (8 6)

FT16 (8 5)FT16 (8 4)

FT16 (8 3)

FT16 (8 2)

FT16 (8 1)

FT16 (8 minus1)

FT16 (8 minus2)

FT16 (8 minus3)

FT16 (8 minus4)

FT16 (8 minus5)

FT16 (8 minus6)

FT16 (8 minus7)

FT16 (8 minus8)

FT16 (8 0)



02468

101214

(a)0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30

HT16 (24 7)

HT16 (24 15)

HT16 (24 14)

HT16 (24 13)

HT16 (24 12)

HT16 (24 11)

HT16 (24 10)

HT16 (24 9)

HT16 (24 6)

HT16 (24 5)

HT16 (24 4)

HT16 (24 3)

HT16 (24 2)

HT16 (24 1)

HT16 (24 0)

HT16 (24 8)

FT16 (8 7)

FT16 (8 6)

FT16 (8 5)FT16 (8 4)

FT16 (8 3)

FT16 (8 2)

FT16 (8 1)

FT16 (8 minus1)

FT16 (8 minus2)

FT16 (8 minus3)

FT16 (8 minus4)

FT16 (8 minus5)

FT16 (8 minus6)

FT16 (8 minus7)

FT16 (8 minus8)

FT16 (8 0)



02468

101214

(b)



02

04

06

08


(a)2 4 6 8 10 12 14 16

02

04

06


(b)


2 4 6 8 10 12 14 16



02

04

06

08

0

02

04

06






Acknowledgment



References























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











[[[[[

[

119891 (0)

119891 (1)

sdot sdot sdot

119891 (119872 minus 1)

]]]]]

]⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

f

=

[[[[[

[

1205950 (0) 120595


1205950 (1) 120595




]]]]]

]⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

Ψ

[[[[[

[

119862 (0)

119862 (1)

sdot sdot sdot

119862 (119873 minus 1)

]]]]]

]⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

C

(11)


1205950 (119905) =

1

4radic120587

119890minus11990522

1205951 (119905) =

radic2119905

4radic120587

119890minus11990522

120595119894 (119905) = 119905

radic2

119894Ψ119894minus1 (119905) minus

radic119894 minus 1

119894Ψ119894minus2 (119905) forall119894 ge 2

(12)


y = ΦΨC = ΘC (13)






min 10038171003817100381710038171003817C100381710038171003817100381710038171198971






1198910 (119899) =

119910 (119899) for 119899 isin Ω

0 elsewhere(15)


C0= HΩy (16)



1198620 (119894) =

1

119872

119872minus1

sum

119899=0

120595119894 (119899)

(120595119872minus1 (119899))

21198910 (119899) 119894 = 1 119873 (17)



k = arg C0gt 119879 (18)


ΘCSC0 = y (19)



CSy (20)



ΨC22=10038171003817100381710038171198911003817100381710038171003817

2

2=1003816100381610038161003816119891 (0)

1003816100381610038161003816

2+ sdot sdot sdot +

1003816100381610038161003816119891 (119873 minus 1)1003816100381610038161003816

2

=

1003816100381610038161003816100381610038161003816100381610038161003816

119873minus1

sum

119896=0

119862 (119896) 120595119896 (0)

1003816100381610038161003816100381610038161003816100381610038161003816

2

+ sdot sdot sdot +

1003816100381610038161003816100381610038161003816100381610038161003816

119873minus1

sum

119896=0

119862 (119896) 120595119896 (119873 minus 1)

1003816100381610038161003816100381610038161003816100381610038161003816

2

=

119873minus1

sum

119896=0

119862 (119896)2

119873minus1

sum

119899=0

120595119896 (119899)2

+

119873minus2

sum

1198961=0

119873minus1

sum

1198962=1198961+1

2119862 (1198961) 119862 (119896

2)

119873minus1

sum

119899=0

1205951198961(119899) 120595119896

2(119899)

(21)


ΨC22= ((119862 (0))

2

119873minus1

sum

119899=0

(1205950 (119899))2+ sdot sdot sdot

+ (119862 (119873 minus 1))2

119873minus1

sum

119899=0

(120595119873minus1 (119899))

2)

+

119873minus2

sum

1198961=0

119873minus1

sum

1198962=1198961+1

2119862 (1198961) 119862 (119896

2)

119873minus1

sum

119899=0

1205951198961(119899) 120595119896

2(119899)

(22)


2sum119873minus1


2sum119873minus1

119899=0(120595119873minus1

(119899))2



0 20 40 60 80 100 1202

25

3

35

4

45

5

55


Sum

of s

quar

ed v

alue

s








H119872=1

119872

sdot

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

[

1205950 (0)

(120595119872minus1 (0))

2

1205950 (1)

(120595119872minus1 (1))

2sdot sdot sdot

1205950 (119872 minus 1)

(120595119872minus1 (119872 minus 1))

2

1205951 (0)

(120595119872minus1 (0))

2

1205951 (1)

(120595119872minus1 (1))

2sdot sdot sdot

1205951 (119872 minus 1)

(120595119872minus1 (119872 minus 1))

2


120595119872minus1 (0)

(120595119872minus1 (0))

2

120595119872minus1 (1)

(120595119872minus1 (1))

2sdot sdot sdot

120595119872minus1 (119872 minus 1)

(120595119872minus1 (119872 minus 1))

2

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

]

(23)



W = I119871119872

otimesH119872=

[[[[[

[

H119872




119872

]]]]]

]

(24)



STHT =Wf =WΨ119871C (25)


119872119894

(119898119894)


STHT = AC (26)



11198722 119872

119875instead ofH

119872we will have

H1198721

H1198722

H119872119875




119894= f119872119894


min 1003817100381710038171003817C11989410038171003817100381710038171198971


119872C119894

(27)






0 20 40 60 80 100 120


Time (samples)

minus2

minus1

0

1

2

3A

mpl

itude

(a)

0 10 20 30 40 50 60



0

02

04

06

08

1

Am

plitu

de

(b)




(28)



X = Zf (29)


Z =[[[[[

[

Z1198721


1198722


119872119875

]]]]]

]

(30)

where Z119872119894



Z =[[[[[

[

H119872



1198720


]]]]]

]

(31)






[[[[[

[

X1198721

X1198722

sdot sdot sdot

X119872119875

]]]]]

]

=

[[[[[

[

Z1198721


1198722


119872119875

]]]]]

]

[[[[[

[

f1198721

f1198722

sdot sdot sdot

f119872119875

]]]]]

]

(32)

where f1198721




f119872119875


1+1198722+

sdot sdot sdot + 119872119875= 119871


119894= f119872119894

(Ω) where Θ119872119894

=

inv(Z119872119894


min 10038171003817100381710038171003817X119872119894

100381710038171003817100381710038171


119872119894

X119872119894

(33)


119872119894






minus04

minus02

0

02

04

06

08

1

Am

plitu

de










6 Conclusion




20 40 60 80 1000 120Time (samples)

minus2

minus1

0

1

2

3

Am

plitu

de

(a)


minus4

minus2

0

2

4

Am

plitu

de

20 40 60 80 100 1200Time (samples)

times10minus15

(b)


minus3

minus2

minus1

0

1

2

Am

plitu

de

20 40 60 80 100 1200Time (samples)

(c)


0 05 1 15 2MSE

0

10

20

30

40

50

60

70

80

Miss

ing

sam

ples

()

(a)

6

10

15

20

25

SNR

(dB)


(b)


0 5 10 15 20 25 30

05

1

15




0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30

HT16 (24 7)

HT16 (24 15)

HT16 (24 14)

HT16 (24 13)

HT16 (24 12)

HT16 (24 11)

HT16 (24 10)

HT16 (24 9)

HT16 (24 6)

HT16 (24 5)

HT16 (24 4)

HT16 (24 3)

HT16 (24 2)

HT16 (24 1)

HT16 (24 0)

HT16 (24 8)

FT16 (8 7)

FT16 (8 6)

FT16 (8 5)FT16 (8 4)

FT16 (8 3)

FT16 (8 2)

FT16 (8 1)

FT16 (8 minus1)

FT16 (8 minus2)

FT16 (8 minus3)

FT16 (8 minus4)

FT16 (8 minus5)

FT16 (8 minus6)

FT16 (8 minus7)

FT16 (8 minus8)

FT16 (8 0)



02468

101214

(a)0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30

HT16 (24 7)

HT16 (24 15)

HT16 (24 14)

HT16 (24 13)

HT16 (24 12)

HT16 (24 11)

HT16 (24 10)

HT16 (24 9)

HT16 (24 6)

HT16 (24 5)

HT16 (24 4)

HT16 (24 3)

HT16 (24 2)

HT16 (24 1)

HT16 (24 0)

HT16 (24 8)

FT16 (8 7)

FT16 (8 6)

FT16 (8 5)FT16 (8 4)

FT16 (8 3)

FT16 (8 2)

FT16 (8 1)

FT16 (8 minus1)

FT16 (8 minus2)

FT16 (8 minus3)

FT16 (8 minus4)

FT16 (8 minus5)

FT16 (8 minus6)

FT16 (8 minus7)

FT16 (8 minus8)

FT16 (8 0)



02468

101214

(b)



02

04

06

08


(a)2 4 6 8 10 12 14 16

02

04

06


(b)


2 4 6 8 10 12 14 16



02

04

06

08

0

02

04

06






Acknowledgment



References























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 20 40 60 80 100 1202

25

3

35

4

45

5

55


Sum

of s

quar

ed v

alue

s








H119872=1

119872

sdot

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

[

1205950 (0)

(120595119872minus1 (0))

2

1205950 (1)

(120595119872minus1 (1))

2sdot sdot sdot

1205950 (119872 minus 1)

(120595119872minus1 (119872 minus 1))

2

1205951 (0)

(120595119872minus1 (0))

2

1205951 (1)

(120595119872minus1 (1))

2sdot sdot sdot

1205951 (119872 minus 1)

(120595119872minus1 (119872 minus 1))

2


120595119872minus1 (0)

(120595119872minus1 (0))

2

120595119872minus1 (1)

(120595119872minus1 (1))

2sdot sdot sdot

120595119872minus1 (119872 minus 1)

(120595119872minus1 (119872 minus 1))

2

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

]

(23)



W = I119871119872

otimesH119872=

[[[[[

[

H119872




119872

]]]]]

]

(24)



STHT =Wf =WΨ119871C (25)


119872119894

(119898119894)


STHT = AC (26)



11198722 119872

119875instead ofH

119872we will have

H1198721

H1198722

H119872119875




119894= f119872119894


min 1003817100381710038171003817C11989410038171003817100381710038171198971


119872C119894

(27)






0 20 40 60 80 100 120


Time (samples)

minus2

minus1

0

1

2

3A

mpl

itude

(a)

0 10 20 30 40 50 60



0

02

04

06

08

1

Am

plitu

de

(b)




(28)



X = Zf (29)


Z =[[[[[

[

Z1198721


1198722


119872119875

]]]]]

]

(30)

where Z119872119894



Z =[[[[[

[

H119872



1198720


]]]]]

]

(31)






[[[[[

[

X1198721

X1198722

sdot sdot sdot

X119872119875

]]]]]

]

=

[[[[[

[

Z1198721


1198722


119872119875

]]]]]

]

[[[[[

[

f1198721

f1198722

sdot sdot sdot

f119872119875

]]]]]

]

(32)

where f1198721




f119872119875


1+1198722+

sdot sdot sdot + 119872119875= 119871


119894= f119872119894

(Ω) where Θ119872119894

=

inv(Z119872119894


min 10038171003817100381710038171003817X119872119894

100381710038171003817100381710038171


119872119894

X119872119894

(33)


119872119894






minus04

minus02

0

02

04

06

08

1

Am

plitu

de










6 Conclusion




20 40 60 80 1000 120Time (samples)

minus2

minus1

0

1

2

3

Am

plitu

de

(a)


minus4

minus2

0

2

4

Am

plitu

de

20 40 60 80 100 1200Time (samples)

times10minus15

(b)


minus3

minus2

minus1

0

1

2

Am

plitu

de

20 40 60 80 100 1200Time (samples)

(c)


0 05 1 15 2MSE

0

10

20

30

40

50

60

70

80

Miss

ing

sam

ples

()

(a)

6

10

15

20

25

SNR

(dB)


(b)


0 5 10 15 20 25 30

05

1

15




0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30

HT16 (24 7)

HT16 (24 15)

HT16 (24 14)

HT16 (24 13)

HT16 (24 12)

HT16 (24 11)

HT16 (24 10)

HT16 (24 9)

HT16 (24 6)

HT16 (24 5)

HT16 (24 4)

HT16 (24 3)

HT16 (24 2)

HT16 (24 1)

HT16 (24 0)

HT16 (24 8)

FT16 (8 7)

FT16 (8 6)

FT16 (8 5)FT16 (8 4)

FT16 (8 3)

FT16 (8 2)

FT16 (8 1)

FT16 (8 minus1)

FT16 (8 minus2)

FT16 (8 minus3)

FT16 (8 minus4)

FT16 (8 minus5)

FT16 (8 minus6)

FT16 (8 minus7)

FT16 (8 minus8)

FT16 (8 0)



02468

101214

(a)0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30

HT16 (24 7)

HT16 (24 15)

HT16 (24 14)

HT16 (24 13)

HT16 (24 12)

HT16 (24 11)

HT16 (24 10)

HT16 (24 9)

HT16 (24 6)

HT16 (24 5)

HT16 (24 4)

HT16 (24 3)

HT16 (24 2)

HT16 (24 1)

HT16 (24 0)

HT16 (24 8)

FT16 (8 7)

FT16 (8 6)

FT16 (8 5)FT16 (8 4)

FT16 (8 3)

FT16 (8 2)

FT16 (8 1)

FT16 (8 minus1)

FT16 (8 minus2)

FT16 (8 minus3)

FT16 (8 minus4)

FT16 (8 minus5)

FT16 (8 minus6)

FT16 (8 minus7)

FT16 (8 minus8)

FT16 (8 0)



02468

101214

(b)



02

04

06

08


(a)2 4 6 8 10 12 14 16

02

04

06


(b)


2 4 6 8 10 12 14 16



02

04

06

08

0

02

04

06






Acknowledgment



References























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 20 40 60 80 100 120


Time (samples)

minus2

minus1

0

1

2

3A

mpl

itude

(a)

0 10 20 30 40 50 60



0

02

04

06

08

1

Am

plitu

de

(b)




(28)



X = Zf (29)


Z =[[[[[

[

Z1198721


1198722


119872119875

]]]]]

]

(30)

where Z119872119894



Z =[[[[[

[

H119872



1198720


]]]]]

]

(31)






[[[[[

[

X1198721

X1198722

sdot sdot sdot

X119872119875

]]]]]

]

=

[[[[[

[

Z1198721


1198722


119872119875

]]]]]

]

[[[[[

[

f1198721

f1198722

sdot sdot sdot

f119872119875

]]]]]

]

(32)

where f1198721




f119872119875


1+1198722+

sdot sdot sdot + 119872119875= 119871


119894= f119872119894

(Ω) where Θ119872119894

=

inv(Z119872119894


min 10038171003817100381710038171003817X119872119894

100381710038171003817100381710038171


119872119894

X119872119894

(33)


119872119894






minus04

minus02

0

02

04

06

08

1

Am

plitu

de










6 Conclusion




20 40 60 80 1000 120Time (samples)

minus2

minus1

0

1

2

3

Am

plitu

de

(a)


minus4

minus2

0

2

4

Am

plitu

de

20 40 60 80 100 1200Time (samples)

times10minus15

(b)


minus3

minus2

minus1

0

1

2

Am

plitu

de

20 40 60 80 100 1200Time (samples)

(c)


0 05 1 15 2MSE

0

10

20

30

40

50

60

70

80

Miss

ing

sam

ples

()

(a)

6

10

15

20

25

SNR

(dB)


(b)


0 5 10 15 20 25 30

05

1

15




0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30

HT16 (24 7)

HT16 (24 15)

HT16 (24 14)

HT16 (24 13)

HT16 (24 12)

HT16 (24 11)

HT16 (24 10)

HT16 (24 9)

HT16 (24 6)

HT16 (24 5)

HT16 (24 4)

HT16 (24 3)

HT16 (24 2)

HT16 (24 1)

HT16 (24 0)

HT16 (24 8)

FT16 (8 7)

FT16 (8 6)

FT16 (8 5)FT16 (8 4)

FT16 (8 3)

FT16 (8 2)

FT16 (8 1)

FT16 (8 minus1)

FT16 (8 minus2)

FT16 (8 minus3)

FT16 (8 minus4)

FT16 (8 minus5)

FT16 (8 minus6)

FT16 (8 minus7)

FT16 (8 minus8)

FT16 (8 0)



02468

101214

(a)0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30

HT16 (24 7)

HT16 (24 15)

HT16 (24 14)

HT16 (24 13)

HT16 (24 12)

HT16 (24 11)

HT16 (24 10)

HT16 (24 9)

HT16 (24 6)

HT16 (24 5)

HT16 (24 4)

HT16 (24 3)

HT16 (24 2)

HT16 (24 1)

HT16 (24 0)

HT16 (24 8)

FT16 (8 7)

FT16 (8 6)

FT16 (8 5)FT16 (8 4)

FT16 (8 3)

FT16 (8 2)

FT16 (8 1)

FT16 (8 minus1)

FT16 (8 minus2)

FT16 (8 minus3)

FT16 (8 minus4)

FT16 (8 minus5)

FT16 (8 minus6)

FT16 (8 minus7)

FT16 (8 minus8)

FT16 (8 0)



02468

101214

(b)



02

04

06

08


(a)2 4 6 8 10 12 14 16

02

04

06


(b)


2 4 6 8 10 12 14 16



02

04

06

08

0

02

04

06






Acknowledgment



References























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










minus04

minus02

0

02

04

06

08

1

Am

plitu

de










6 Conclusion




20 40 60 80 1000 120Time (samples)

minus2

minus1

0

1

2

3

Am

plitu

de

(a)


minus4

minus2

0

2

4

Am

plitu

de

20 40 60 80 100 1200Time (samples)

times10minus15

(b)


minus3

minus2

minus1

0

1

2

Am

plitu

de

20 40 60 80 100 1200Time (samples)

(c)


0 05 1 15 2MSE

0

10

20

30

40

50

60

70

80

Miss

ing

sam

ples

()

(a)

6

10

15

20

25

SNR

(dB)


(b)


0 5 10 15 20 25 30

05

1

15




0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30

HT16 (24 7)

HT16 (24 15)

HT16 (24 14)

HT16 (24 13)

HT16 (24 12)

HT16 (24 11)

HT16 (24 10)

HT16 (24 9)

HT16 (24 6)

HT16 (24 5)

HT16 (24 4)

HT16 (24 3)

HT16 (24 2)

HT16 (24 1)

HT16 (24 0)

HT16 (24 8)

FT16 (8 7)

FT16 (8 6)

FT16 (8 5)FT16 (8 4)

FT16 (8 3)

FT16 (8 2)

FT16 (8 1)

FT16 (8 minus1)

FT16 (8 minus2)

FT16 (8 minus3)

FT16 (8 minus4)

FT16 (8 minus5)

FT16 (8 minus6)

FT16 (8 minus7)

FT16 (8 minus8)

FT16 (8 0)



02468

101214

(a)0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30

HT16 (24 7)

HT16 (24 15)

HT16 (24 14)

HT16 (24 13)

HT16 (24 12)

HT16 (24 11)

HT16 (24 10)

HT16 (24 9)

HT16 (24 6)

HT16 (24 5)

HT16 (24 4)

HT16 (24 3)

HT16 (24 2)

HT16 (24 1)

HT16 (24 0)

HT16 (24 8)

FT16 (8 7)

FT16 (8 6)

FT16 (8 5)FT16 (8 4)

FT16 (8 3)

FT16 (8 2)

FT16 (8 1)

FT16 (8 minus1)

FT16 (8 minus2)

FT16 (8 minus3)

FT16 (8 minus4)

FT16 (8 minus5)

FT16 (8 minus6)

FT16 (8 minus7)

FT16 (8 minus8)

FT16 (8 0)



02468

101214

(b)



02

04

06

08


(a)2 4 6 8 10 12 14 16

02

04

06


(b)


2 4 6 8 10 12 14 16



02

04

06

08

0

02

04

06






Acknowledgment



References























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











20 40 60 80 1000 120Time (samples)

minus2

minus1

0

1

2

3

Am

plitu

de

(a)


minus4

minus2

0

2

4

Am

plitu

de

20 40 60 80 100 1200Time (samples)

times10minus15

(b)


minus3

minus2

minus1

0

1

2

Am

plitu

de

20 40 60 80 100 1200Time (samples)

(c)


0 05 1 15 2MSE

0

10

20

30

40

50

60

70

80

Miss

ing

sam

ples

()

(a)

6

10

15

20

25

SNR

(dB)


(b)


0 5 10 15 20 25 30

05

1

15




0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30

HT16 (24 7)

HT16 (24 15)

HT16 (24 14)

HT16 (24 13)

HT16 (24 12)

HT16 (24 11)

HT16 (24 10)

HT16 (24 9)

HT16 (24 6)

HT16 (24 5)

HT16 (24 4)

HT16 (24 3)

HT16 (24 2)

HT16 (24 1)

HT16 (24 0)

HT16 (24 8)

FT16 (8 7)

FT16 (8 6)

FT16 (8 5)FT16 (8 4)

FT16 (8 3)

FT16 (8 2)

FT16 (8 1)

FT16 (8 minus1)

FT16 (8 minus2)

FT16 (8 minus3)

FT16 (8 minus4)

FT16 (8 minus5)

FT16 (8 minus6)

FT16 (8 minus7)

FT16 (8 minus8)

FT16 (8 0)



02468

101214

(a)0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30

HT16 (24 7)

HT16 (24 15)

HT16 (24 14)

HT16 (24 13)

HT16 (24 12)

HT16 (24 11)

HT16 (24 10)

HT16 (24 9)

HT16 (24 6)

HT16 (24 5)

HT16 (24 4)

HT16 (24 3)

HT16 (24 2)

HT16 (24 1)

HT16 (24 0)

HT16 (24 8)

FT16 (8 7)

FT16 (8 6)

FT16 (8 5)FT16 (8 4)

FT16 (8 3)

FT16 (8 2)

FT16 (8 1)

FT16 (8 minus1)

FT16 (8 minus2)

FT16 (8 minus3)

FT16 (8 minus4)

FT16 (8 minus5)

FT16 (8 minus6)

FT16 (8 minus7)

FT16 (8 minus8)

FT16 (8 0)



02468

101214

(b)



02

04

06

08


(a)2 4 6 8 10 12 14 16

02

04

06


(b)


2 4 6 8 10 12 14 16



02

04

06

08

0

02

04

06






Acknowledgment



References























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30

HT16 (24 7)

HT16 (24 15)

HT16 (24 14)

HT16 (24 13)

HT16 (24 12)

HT16 (24 11)

HT16 (24 10)

HT16 (24 9)

HT16 (24 6)

HT16 (24 5)

HT16 (24 4)

HT16 (24 3)

HT16 (24 2)

HT16 (24 1)

HT16 (24 0)

HT16 (24 8)

FT16 (8 7)

FT16 (8 6)

FT16 (8 5)FT16 (8 4)

FT16 (8 3)

FT16 (8 2)

FT16 (8 1)

FT16 (8 minus1)

FT16 (8 minus2)

FT16 (8 minus3)

FT16 (8 minus4)

FT16 (8 minus5)

FT16 (8 minus6)

FT16 (8 minus7)

FT16 (8 minus8)

FT16 (8 0)



02468

101214

(a)0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30

HT16 (24 7)

HT16 (24 15)

HT16 (24 14)

HT16 (24 13)

HT16 (24 12)

HT16 (24 11)

HT16 (24 10)

HT16 (24 9)

HT16 (24 6)

HT16 (24 5)

HT16 (24 4)

HT16 (24 3)

HT16 (24 2)

HT16 (24 1)

HT16 (24 0)

HT16 (24 8)

FT16 (8 7)

FT16 (8 6)

FT16 (8 5)FT16 (8 4)

FT16 (8 3)

FT16 (8 2)

FT16 (8 1)

FT16 (8 minus1)

FT16 (8 minus2)

FT16 (8 minus3)

FT16 (8 minus4)

FT16 (8 minus5)

FT16 (8 minus6)

FT16 (8 minus7)

FT16 (8 minus8)

FT16 (8 0)



02468

101214

(b)



02

04

06

08


(a)2 4 6 8 10 12 14 16

02

04

06


(b)


2 4 6 8 10 12 14 16



02

04

06

08

0

02

04

06






Acknowledgment



References























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










References























Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









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