Review Article Compressive Sensing in Signal Processing...

Review ArticleCompressive Sensing in Signal Processing Algorithms andTransform Domain Formulations

Irena OroviT1 Vladan PapiT2 Cornel Ioana3 Xiumei Li4 and Srdjan StankoviT1

1Faculty of Electrical Engineering University of Montenegro Dzordza Vasingtona bb 81000 Podgorica Montenegro2Faculty of Electrical Engineering Mechanical Engineering amp Naval Architecture University of Split Split Croatia3Grenoble Institute of Technology GIPSA-Lab Saint-Martin-drsquoHeres France4School of Information Science and Engineering Hangzhou Normal University Zhejiang China

Correspondence should be addressed to Irena Orovic irenaoacme

Received 26 March 2016 Revised 23 July 2016 Accepted 2 August 2016

Academic Editor Francesco Franco

Copyright copy 2016 Irena Orovic et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Compressive sensing has emerged as an area that opens new perspectives in signal acquisition and processing It appears asan alternative to the traditional sampling theory endeavoring to reduce the required number of samples for successful signalreconstruction In practice compressive sensing aims to provide saving in sensing resources transmission and storage capacitiesand to facilitate signal processing in the circumstances when certain data are unavailable To that end compressive sensing relieson the mathematical algorithms solving the problem of data reconstruction from a greatly reduced number of measurements byexploring the properties of sparsity and incoherenceTherefore this concept includes the optimization procedures aiming to providethe sparsest solution in a suitable representation domain This work therefore offers a survey of the compressive sensing idea andprerequisites together with the commonly used reconstruction methods Moreover the compressive sensing problem formulationis considered in signal processing applications assuming some of the commonly used transformation domains namely the Fouriertransform domain the polynomial Fourier transform domain Hermite transform domain and combined time-frequency domain

1 Introduction

The fundamental approach for signal reconstruction from itsmeasurements is defined by the Shannon-Nyquist samplingtheorem stating that the sampling rate needs to be at leasttwice the maximal signal frequency In the discrete case thenumber of measurements should be at least equal to thesignal length in order to be exactly reconstructed Howeverthis approach may require large storage space significantsensing time heavy power consumption and large number ofsensors Compressive sensing (CS) is a novel theory that goesbeyond the traditional approach [1ndash4] It shows that a sparsesignal can be reconstructed from much fewer incoherentmeasurements The basic assumption in CS approach isthat most of the signals in real applications have a conciserepresentation in a certain transform domain where onlyfew of them are significant while the rest are zero ornegligible [5ndash7]This requirement is defined as signal sparsity

Another important requirement is the incoherent natureof measurements (observations) in the signal acquisitiondomain Therefore the main objective of CS is to provide anestimate of the original signal from a small number of linearincoherent measurements by exploiting the sparsity property[3 4]

The CS theory covers not only the signal acquisitionstrategy but also the signal reconstruction possibilities anddifferent algorithms [8ndash17] Several approaches for CS signalreconstruction have been developed andmost of thembelongto one of three main approaches convex optimizations [8ndash11] such as basis pursuit Dantzig selector and gradient-basedalgorithms greedy algorithms like matching pursuit [14] andorthogonal matching pursuit [15] and hybrid methods suchas compressive sampling matching pursuit [16] and stage-wise OMP [17] When comparing these algorithms convexprogramming provides the best reconstruction accuracy butat the cost of high computational complexity The greedy

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2016 Article ID 7616393 16 pageshttpdxdoiorg10115520167616393

2 Mathematical Problems in Engineering

algorithms bring about low computation complexity whilethe hybrid methods try to provide a compromise betweenthese two requirements [18]

The proposed work provides a survey of the generalcompressive sensing concept supplemented with the severalexisting approaches and methods for signal reconstructionwhich are briefly explained and summarized in the formof algorithms with the aim of providing the readers withan easier and practical insight into the state of the art inthis field Apart from the standard CS algorithms a fewrecent solutions have been included as well Furthermorethe paper provides an overview of different sparsity domainsand the possibilities of employing them in the CS problemformulation Additional contribution is provided through theexamples showing the efficiency of the presented methods inpractical applications

The paper is organized as follows In Section 2 a briefreview of the general compressive sensing idea is providedtogether with the conditions for successful signal recon-struction from reduced set of measurements and the signalrecovery formulations using minimization approaches InSection 3 the commonly used CS algorithms are reviewedThe commonly used domains for CS strategy implementationare given in Section 4 while some of the examples inreal applications are provided in Section 5 The concludingremarks are given in Section 6

2 Compressive Sensing A General Overview

21 Sparsity and Compressibility Reducing the sampling rateusing CS is possible for the case of sparse signals that can berepresented by a small number of significant coefficients inan appropriate transform basis A signal having K nonzerocoefficients is called 119870-sparse Assume that signal 119904 exhibitssparsity in certain orthonormal basis Ψ defined by the basisvectors 1205951 1205952 120595119873The signal 119904 can be represented usingits sparse transform domain vector x as follows

119904 (119899) = 119873sum119894=1

119909119894120595119894 (119899) (1)

In matrix notation the previous relation can be written as

s = Ψx (2)

Commonly the sparsity is measured using the ℓ0-normwhich represents the cardinality of the support of x

x0 = card supp (x) = 119870 (3)

In real applications the signals are usually not strictly sparsebut only approximately sparse Therefore instead of beingsparse these signals are often called compressible meaningthat the amplitudes of coefficients decrease rapidly whenarranged in descending order For instance if we considercoefficients |1199091| ge |1199092| ge sdot sdot sdot ge |119909119899| then the magnitudedecays with a power law if there exist constants 1198621 and 119902 gt 0satisfying [19]

10038161003816100381610038161199091198941003816100381610038161003816 le 1198621119894minus119902 (4)

where larger 119902 means faster decay and consequently morecompressible signal The signal compressibility can be quan-tified using the minimal error between the original and spar-sified signal (obtained by keeping only119870 largest coefficients)

120590119870 (x)119901 = min 1003817100381710038171003817x minus x1198701003817100381710038171003817119901 (5)

22 Conditions on the CS Matrix Null Space PropertyRestricted Isometry Property and Incoherence Instead ofacquiring a full set of signal samples of length 119873 in CSscenario we deal with a quite reduced set of measurementsy of length 119872 where 119872 lt 119873 The measurement procedurecan be modeled by projections of the signal s onto vectors1206011 1206012 120601119872 constituting the measurement matrixΦ

y = Φs (6)

Using the sparse transform domain representation of vectors given by (2) we have

y = ΦΨx = Ax (7)

where A will be referred to as CS matrixIn order to define some requirements for the CS matrix

A which are important for successful signal reconstructionlet us introduce the null space of matrixThe null space of CSmatrix A contains all vectors x that are mapped to 0

N (A) = x Ax = 0 (8)

In order to provide a unique solution it is necessary toprovide the notion that two K-sparse vectors x and x1015840 do notresult in the same measurement vector In other words theirdifference should not be part of the null space of CS matrixA

A (x minus x1015840) = 0 (9)

Since the difference between two K-sparse vectors is at most2K-sparse then aK-sparse vector x is uniquely defined if nullspace of A contains no 2K-sparse vectors This correspondsto the condition that any 2K columns of A are linearlyindependent that is

spark (A) gt 2119870 (10)

and since spark(A) isin [2119872+ 1] we obtain a lower bound onthe number of measurements

119872 gt 2119870 (11)

In the case of strictly sparse signals the spark can providereliable information about the exact reconstruction possi-bility However in the case of approximately sparse signalsthis condition is not sufficient and does not guarantee stablerecovery Hence there is another property called null spaceproperty that measures the concentration of the null spaceof matrix A The null space property is satisfied if there is aconstant 120574 isin (0 1) such that [19]

1003817100381710038171003817a11987810038171003817100381710038171 = 120574 1003817100381710038171003817aSc10038171003817100381710038171 foralla isin N (A) (12)

Mathematical Problems in Engineering 3

for all sets 119878 sub 1 119873 with cardinality K and theircomplements Sc = 1 119873 119878 If the null space propertyis satisfied then a strictly 119870-sparse signal can be perfectlyreconstructed by using ℓ1-minimization For approximately119870-sparse signals an upper bound of the ℓ1-minimizationerror can be defined as follows [20]

x minus x1 le 21 + 1205741 minus 120574120590119870 (x)1 (13)

where 120590119870(x)1 is defined in (5) for 119901 = 1 as the minimal errorinduced by the best 119870-sparse approximation

The null space property is necessary and sufficient forestablishing guarantees for recovery A stronger condition isrequired in the presence of noise (and approximately sparsesignals) Hence in [8] the restricted isometry property (RIP)of CSmatrixA has been introducedTheCSmatrixA satisfiesthe RIP property with constant 120575119870 isin (0 1) if

(1 minus 120575119870) x22 le Ax22 le (1 + 120575119870) x22 (14)for every119870-sparse vector xThis property shows howwell thedistances are preserved by a certain linear transformationWemight now say that if the RIP is satisfied for 2K with 1205752119870 lt 1then there are no two K-sparse vectors x that can correspondto the same measurement vector y

Finally the incoherence condition mentioned beforewhich is also related to the RIP of matrix A refers to theincoherence of the projection basis Φ and the sparsifyingbasisΨThemutual coherence can be simply defined by usingthe combined CS matrix A as follows [21]

120583 (A) = max119894 =1198951le119894119895le119873

100381610038161003816100381610038161003816100381610038161003816100381610038161003816⟨119860 119894 119860119895⟩1003817100381710038171003817119860 11989410038171003817100381710038172 10038171003817100381710038171003817119860119895100381710038171003817100381710038172

100381610038161003816100381610038161003816100381610038161003816100381610038161003816 (15)

The mutual coherence is related to the restricted isometryconstant using the following bound [22]

120575119870 = (119870 minus 1) 120583 (16)

23 Signal Recovery Using Minimization Approach The sig-nal recovery problem is defined as the reconstruction ofvector x from the measurements y = Ax This problem canbe generally seen as a problemof solving an underdeterminedset of linear equations However in the circumstances whenx is sparse the problem can be reduced to the followingminimization

min x0st y = Ax (17)

The ℓ0-minimization requires an exhaustive search over all(119873119870 ) possible sparse combinations which is computationallyintractable Hence the ℓ0-minimization is replaced by convexℓ1-minimization which will provide the sparse result withhigh probability if the measurement matrix satisfies theprevious conditionsThe ℓ1-minimization problem is definedas follows


and it has been known as the basis pursuit

In the situation when the measurements are corrupted bythe noise of level e y = ΦΨx + e = Ax + e and e2 le 120576 thereconstruction problem can be defined in a form

min x1st 1003817100381710038171003817y minus Ax10038171003817100381710038172 le 120576 (19)

called basis pursuit denoising The error bound for thesolution of (19) where A satisfies the RIP of order 2119870 with1205752119870 lt radic2 minus 1 and y = Ax + e is given by

x minus x2 le 1198620120590119870 (x)1 + 11986212120576radic1 + 21205752119870 (20)

where the constants 1198620 and 1198621 are defined as [19]

1198620 = 21 minus (1 minus radic2) 12057521198701 minus (1 + radic2) 1205752119870

1198621 = 2 11 minus (1 + radic2) 1205752119870

(21)

For a particular regularization parameter 120582 gt 0 theminimization problem (19) can be defined using the uncon-strained version as follows

min119909

12 1003817100381710038171003817y minus Ax100381710038171003817100381722 + 120582 x1 (22)

which is known as the Lagrangian form of the basis pursuitdenoising These algorithms are commonly solved usingprimal-dual interior-point methods [22]

Another form of basis pursuit denoising is solved usingthe least absolute shrinkage and selection operator (LASSO)and it is defined as follows

min119909

12 1003817100381710038171003817y minus Ax100381710038171003817100381722

st x1 lt 120591(23)

where 120591 is a nonnegative real parameter The convex opti-mization methods usually require high computational com-plexity and high numerical precision

When the noise is unbounded one may apply the convexprogram based on Dantzig selector (it is assumed that thenoise variance is 1205902 per measurement ie the total varianceis1198721205902)

min x1st 1003817100381710038171003817Alowast (y minus Ax)1003817100381710038171003817infin le radic2 log119873120590 (24)

which (for enough measurements) reconstructs a signal withthe error bound

x minus x2 le 120590radic2 log (119873)119870 (25)

The norm ℓinfin is infinite norm called also supreme (maxi-mum) norm


Input998835Measurement matrix A (of size119872times119873)998835Measurement vector y (of length119872)Output998835 A signal estimate x (of length N)(1) = 0 1199030 larr y 119894 = 0(2) while halting criterion false do(3) 119894 = 119894 + 1(4) 120596119894 larr argmax119895=1119873|⟨r119894minus1 119860119895⟩| (max correlation column)(5) x119894 larr ⟨r119894minus1 120596119894⟩ (new signal estimate)(6) r119894 larr r119894minus1 minus 120596119894x119894 (update residual)(7) end while(8) return x larr x119894

Algorithm 1 Matching pursuit

Besides the ℓ1-norm minimization there exist someapproaches using the ℓ119901-normminimization with 0 lt 119901 lt 1


or using ℓ2-normminimization in which case the solution isnot rigorously sparse enough [23]


3 Review of Some SignalReconstruction Algorithms

The ℓ1-minimization problems in CS signal reconstructionare usually solved using the convex optimization methodsIn addition there exist greedy methods for sparse signalrecovery which allow faster computation compared to ℓ1-minimization Greedy algorithms can be divided into twomajor groups greedy pursuit methods and thresholding-based methods In practical applications the widely usedones are the orthogonal matching pursuit (OMP) and com-pressive sampling matching pursuit (CoSaMP) from thegroup of greedy pursuit methods while from the threshold-ing group the iterative hard thresholding (IHT) is commonlyused due to its simplicity although it may not be alwaysefficient in providing an exact solution Some of thesealgorithms are discussed in detail in this section

31 Matching Pursuit The matching pursuit algorithm hasbeen known for its simplicity and was first introduced in [14]This is the first algorithm from the class of iterative greedymethods that decomposes a signal into a linear set of basisfunctionsThrough the iterations this algorithm chooses in agreedy manner the basis functions that best match the signalAlso in each iteration the algorithm removes the signalcomponent having the form of the selected basis functionand obtains the residual This procedure is repeated until the

norm of the residual becomes lower than a certain predefinedthreshold value (halting criterion) (Algorithm 1)

The matching pursuit algorithm however has a slow con-vergence property and generally does not provide efficientlysparse results

32 Orthogonal Matching Pursuit The orthogonal matchingpursuit (OMP) has been introduced [15] as an improvedversion to overcome the limitations of the matching pursuitalgorithm OMP is based on principle of orthogonalizationIt computes the inner product of the residue and the mea-surement matrix and then selects the index of the maximumcorrelation column and extracts this column (in each itera-tion) The extracted columns are included into the selectedset of atomsThen the OMP performs orthogonal projectionover the subspace of previously selected atoms providing anew approximation vector used to update the residual Herethe residual is always orthogonal to the columns of the CSmatrix so there will be no columns selected twice and theset of selected columns is increased through the iterationsOMP provides better asymptotic convergence compared tothe previous matching pursuit version (Algorithm 2)

33 CoSaMP Compressive sampling matching pursuit(CoSaMP) is an extended version of the OMP algorithm[16]

In each iteration the residual vector r is correlated withthe columns of the CS matrix A forming a ldquoproxy signalrdquo zThen the algorithm selects 2119870 columns of A correspondingto the 2119870 largest absolute values of z where 119870 defines thesignal sparsity (number of nonzero components) Namely allbut the largest 2119870 elements of z are set to zero and the 2119870support Ω is obtained by finding the positions of nonzeroelements The indices of the selected columns (2119870 in total)are then added to the current estimate of the support ofthe unknown vector After solving the least squares a 3119870-sparse estimate of the unknown vector is obtained Then the3119870-sparse vector is pruned to obtain the 119870-sparse estimateof the unknown vector (by setting all but the 119870 largestelements to zero) Thus the estimate of the unknown vectorremains 119870-sparse and all columns that do not correspond


Input998835 Transform matrixΨ Measurement matrixΦ998835 CS matrix A A = ΦΨ998835Measurement vector yOutput998835 A signal estimate x larr x119870998835 An approximation to the measurements y by a119870998835 A residual r119870 = y minus a119870998835 A setΩ119870 with positions of non-zero elements of x(1) r0 larr yΩ0 larr Θ0 larr [](2) for 119894 = 1 119870(3) 120596119894 larr argmax119895=1119873|⟨r119894minus1 119860119895⟩| (max correlation column)(4)Ω119894 larr Ω119894minus1 cup 120596119894 (Update set of indices)(5)Θ119894 larr [Θ119894minus1 119860120596119894

] (Update set of atoms)(6) x119894 = argminxr119894minus1 minusΘ119894x22(7) a119894 larr Θ119894x119894 (new approximation)(8) r119894 larr y minus a119894 (update residual)(9) end for(10) return x119870 a119870 r119870Ω119870

Algorithm 2 Orthogonal matching pursuit

to the true signal components are removed which is animprovement over the OMP Namely if OMP selects in someiteration a column that does not correspond to the true signalcomponent the index of this column will remain in the finalsignal support and cannot be removed (Algorithm 3)

The CoSaMP can be observed through five crucial steps

(i) Identification (Line 5) It finds the largest 2s compo-nents of the signal proxy

(ii) Support Merge (Line 6) It merges the support of thesignal proxy with the support of the solution from theprevious iteration

(iii) Estimation (Line 7) It estimates a solution via leastsquares where the solution lies within a support T

(iv) Pruning (Line 8) It takes the solution estimate andcompresses it to the required support

(v) Sample Update (Line 9) It updates the ldquosamplerdquomeaning the residual as the part of signal that has notbeen approximated

34 Iterative Hard Thresholding Algorithm Another groupof algorithms for signal reconstruction from a small set ofmeasurements is based on iterative thresholding Generallythese algorithms are composed of two main steps The firstone is the optimization of the least squares term whichis done by solving the optimization problem without ℓ1-minimizationThe other one is the decreasing of the ℓ1-normwhich is done by applying the thresholding operator to themagnitude of entries in x In each iteration the sparse vectorx119894 is estimated by the previous version x119894minus1 using negativegradient of the objective function defined as

119869 (x) = 12 1003817100381710038171003817y minus Ax100381710038171003817100381722 (28)

while the negative gradient is then

minusnabla119869 (x) = A119879 (y minus Ax) (29)

Generally the obtained estimate x119894 is sparse and we needto set all but the K largest components to zero using thethresholding operator Here we distinguish two types ofthresholding Hard thresholding [24 25] sets all but the Klargest magnitude values to zero where the thresholdingoperator can be written as

119867119870 (x) = x (119896) for |x (119896)| gt 1205850 otherwise (30)

where 120585 is the K largest component of x The algorithm issummarized in Algorithm 4

The stopping criterion for IHT can be a fixed number ofiterations or the algorithm terminates when the sparse vectordoes not change much between consecutive iterations

The soft-thresholding function can be defined as

119878120591 (x (119896)) =

x (119896) minus 120591 x gt 1205910 |x (119896)| lt 120591x (119896) + 120591 x lt minus120591

(31)

and it is applied to each element of x

35 Iterative Soft Thresholding for Solving LASSO Let usconsider the minimization of the function

119869 (x) = 1003817100381710038171003817y minus Ax10038171003817100381710038172 + 120582 x1 (32)

as given by the LASSOminimization problem (22)One of thealgorithms that has been used for solving (32) is the iterated


Input998835 Transform matrixΨ Measurement matrixΦ998835 CS matrix A A = ΦΨ998835Measurement vector y998835 K being the signal sparsity998835Halting criterionOutput998835 119870-sparse approximation x of the target signal(1) x0 larr 0 r larr y 119894 larr 0(2) while halting condition false do(3) 119894 larr 119894 + 1(4) z larr Alowastr (signal proxy)(5)Ω larr supp(z2119870) (support of best 2K-sparse approximation)(6) 119879 larr Ω cup supp(x119894minus1) (merge supports)(7) x = argminxsupp(x)=119879Ax minus y22 (solve least squares)(8) x119894 larr x119870 (Prune best 119870-sparse approximation)(9) r larr y minus Ax119894 (update current sample)(10) end while(11) x larr x119894(12) return x

Algorithm 3 CoSaMP

Input998835 Transform matrixΨ Measurement matrixΦ998835 CS matrix A A = ΦΨ998835Measurement vector y998835 K being the signal sparsityOutput998835 119870-sparse approximation x(1) x0 larr 0(2) while stopping criterion do(3) 119894 larr 119894 + 1(4) x119894 larr 119867119870(x119894minus1 + A119879(y minus Ax119894minus1))(5) end while(6) x larr x119894(7) return x

Algorithm 4 Iterative hard thresholding

soft-thresholding algorithm (ISTA) also called the thresh-olded Landweber algorithm In order to provide an iterativeprocedure for solving the considered minimization problemwe use the minimization-maximization [26] approach tominimize 119869(x) Hence we should create a function 119866119896(x)that is equal to 119869(x) at x119896 otherwise it upper-bounds 119869(x)Minimizing a majorizer 119866119896(x) is easier and avoids solving asystem of equations Hence [27]

119866119896 (x) = 119869 (x) + nonnegative function of x119866119896 (x) = 119869 (x) + (x minus x119896)119879 (120572I minus A119879A) (x minus x119896)

= 1003817100381710038171003817y minus Ax10038171003817100381710038172 + 120582 x1+ (x minus x119896)119879 (120572I minus A119879A) (x minus x119896)

(33)

and120572 is such that the added function is nonnegativemeaningthat 120572 must be equal to or greater than the maximumeigenvalue of A119879A

120572 gt max eig A119879A (34)

In order to minimize 119866119896(x) the gradient of 119866119896(x) should bezero

120597119866119896 (x)120597x119879 = 0 997904rArrminus 2A119879y + 2A119879Ax + 120582 sign (x)

+ 2 (120572I minus A119879A) (x minus x119896) = 0(35)

The solution is

x + 1205822120572 sign (x) = 1

120572Α119879 (y minus Ax119896) + x119896 (36)

The solution of the linear equation

119886 + 1205822120572 sign (119886) = 119887 (37)

is obtained using the soft-thresholding approach

119886 = 119878120582 (119887) =

119887 minus 120582 119887 gt 1205820 |119887| lt 120582119887 + 120582 119887 lt minus120582

(38)

Hence we may write the soft-thresholding-based solution of(36) as follows

x = 1198781205822120572 ( 1120572Α119879 (y minus Ax119896) + x119896) (39)


In terms of iteration 119896 + 1 the previous relation can berewritten as

x119896+1 = 1198781205822120572 ( 1120572Α119879 (y minus Ax119896) + x119896) (40)

36 AutomatedThreshold Based Iterative Solution This algo-rithm (see [12 13]) starts from the assumption that the miss-ing samples modeled by zero values in the signal domainproduce noise in the transform domain Namely the zerovalues at the positions of missing samples can be observed asa result of adding noise to these samples where the noise atthe unavailable positions has values of original signal sampleswith opposite sign For instance let us observe the signals thatare sparse in the Fourier transform domain The variance ofthe mentioned noise when observed in the discrete Fouriertransform (DFT) domain can be calculated as

1205902119872119878 larr997888 119872119873 minus119872119873 minus 1

119872sum119894=1

1003816100381610038161003816y (119894)10038161003816100381610038162119872 (41)

whereM is the number of available andN is the total numberof samples and y is a measurement vector Consequentlyusing the variance of noise it is possible to define a thresholdto separate signal components from the nonsignal compo-nents in the DFT domain

For a desired probability 119875(119879) the threshold is derived as119879 = radicminus1205902119872119878 log (1 minus 119875 (119879)1119873) (42)

The automated threshold based algorithm in each iterationdetects a certain number of DFT components above thethreshold A set of positions k119894 corresponding to the detectedcomponents is selected and the contribution of these compo-nents is removed from the signalrsquos DFT This will reveal theremaining components that are below the noise level Furtherit is necessary to update the noise variance and thresholdvalue for the new iteration Since the algorithmdetects a set ofcomponents in each iteration it usually needs just a couple ofiterations to recover the entire signal In the case that all signalcomponents are above the noise level in DFT the componentdetection and reconstruction are achieved in single iteration

37 Adaptive Gradient-Based Algorithm This algorithmbelongs to the group of convexminimization approaches [11]Unlike other convex minimization approaches this methodstarts from some initial values of unavailable samples (initialstate) which are changed through the iterations in a way toconstantly improve the concentration in sparsity domain Ingeneral it does not require the signal to be strictly sparsein certain transform domain which is an advantage overother methods Particularly themissing samples in the signaldomain can be considered as zero values In each iterationthe missing samples are changed for +Δ and for minusΔ Thenfor both changes the concentration is measured as ℓ1-normof the transform domain vectors x+ (for +Δ change) and xminus(for minusΔ change) while the gradient is determined using theirdifference (lines 8 9 and 10) Finally the gradient is used toupdate the values of missing samples Here it is important to

note that each sample is observed separately in this algorithmand one iteration is finished when all samples are processed

When the algorithm reaches the vicinity of the sparsitymeasureminimum the gradient changes direction for almost180 degrees (line 16) meaning that the step Δ needs to bedecreased (line 17)

The precision of the result in this iterative algorithm isestimated based on the change of the result in the last iteration(lines 18 and 19)

4 Exploring Different Transform Domains forCS Signal Reconstruction

41 CS in the Standard Fourier Transform Domain As thesimplest case of CS scenario we will assume a multicompo-nent signal that consists of K sinusoids that are sparse in theDFT domain

119904 (119899) = 119870sum119894=1

119860 1198941198901198952120587119896119894119899119873 (43)

where the sparsity level is 119870 ≪ 119873 where 119873 is total signallength while119860 119894 and 119896119894 denote amplitudes and frequencies ofsignal components respectively Since s is sparse in the DFTdomain then we can write

s = ΨF = Fminus1F (44)

where F is a vector of DFT coefficients where at most Kcoefficients are nonzero while Fminus1 is the inverse Fouriertransform matrix of size119873 times119873

If s is a signal in compressive sensing application thenonly the random measurements y sub s are available and theseare defined by the set of M positions 1198991 1198992 119899119872 Thereforethe measurement process can be modeled by matrixΦ

y = ΦFminus1F = AF (45)

The CS matrix A represents a partial random inverse Fouriertransform matrix obtained by omitting rows from Fminus1 thatcorresponds to unavailable samples positions

A =[[[[[[[

1205950 (1198991) 1205951 (1198991) sdot sdot sdot 120595119873minus1 (1198991)1205950 (1198992) 1205951 (1198992) sdot sdot sdot 120595119873minus1 (1198992)

d

1205950 (119899119872) 1205951 (119899119872) sdot sdot sdot 120595119873minus1 (119899119872)

]]]]]]] (46)

with the Fourier basis functions

120595119896 (119899) = 1198901198952120587119896119899119873 (47)

The CS problem formulation is now given as

min F1st y = AF (48)


42 CS in the Polynomial Fourier Transform Domain In thissection we consider the possibility of CS reconstruction ofpolynomial phase signals [28] Observe the multicomponentpolynomial phase signal vector s with elements s(n)

119904 (119899) = 119870sum119894=1

119904119894 (119899) =119870sum119894=1

119903119894119890119895(2120587119873)(1198991198861119894+119899211988621198942+sdotsdotsdot+119899119901119886119901119894119901) (49)

where the polynomial coefficients are assumed to be boundedintegers The assumption is that the signal can be consideredas sparse in the polynomial Fourier transform (PFT) domainHence the discrete PFT form is given by

119883(1198961119894 119896119901119894)= 119873minus1sum119899=0

119870sum119894=1

119903119894119890119895(2120587119873)(1198991198861119894+119899211988621198942+sdotsdotsdot+119899119901119886119901119894119901)

times 119890minus119895(2120587119873)(119899211989621198942+sdotsdotsdot+119899119901119896119901119894119901)119890minus119895(2120587119873)1198991198961 (50)

If we choose a set of parameters (1198962119894 1198963119894 119896119901119894) that matchthe polynomial phase coefficients (1198862119894 1198863119894 119886119901119894)

(1198962119894 1198963119894 119896119901119894) = (1198862119894 1198863119894 119886119901119894) (51)

then the 119894th signal component is demodulated and becomesa sinusoid 119903119894119890119895(2120587119873)(1198991198861119894) which is dominant in the PFT Inother words the spectrum is highly concentrated at 119896 = 1198861119894Therefore we might say that if (51) is satisfied then the PFTis compressible with the dominant 119894th component Note thatthe sparsity (compressibility) in the PFT domain is observedwith respect to the single demodulated component

In order to define the CS problem in the PFT domaininstead of the signal s itself we will consider amodified signalform obtained after the multiplication with the exponentialterm

120585 (119899) = 119890minus119895(2120587119873)(119899211989621198942+sdotsdotsdot+119899119901119896119901119894119901) (52)

given in the PFT definition (50) The new signal form isobtained as

119911 (119899) = 119904 (119899) 120585 (119899) = 119870sum119894=1

119903119894sdot 119890119895(2120587119873)(1198991198861119894+119899211988621198942+sdotsdotsdot+119899119901119886119901119894119901)119890minus119895(2120587119873)(119899211989621198942+sdotsdotsdot+119899119901119896119901119894119901)

(53)

or in the vector form

z = s120585 (54)

Then the PFT definition given by (50) can be rewritten inthe vector form as follows

X = Fz (55)

whereF is the discrete Fourier transform matrix (119873 times119873)For a chosen set of parameters (1198962119894 1198963119894 119896119901119894) in 120585 that is

equal to the set (1198862119894 1198863119894 119886119899119894) in s X = X119894 is characterizedby one dominant sinusoidal component at the frequency 1198861119894

421 CS Scenario Now assume that z is incomplete in thecompressive sensing sense and instead of z we are dealingwith119872 available measurements defined by vector y119911 and

y119911 = A119911X (56)

where A119911 is the partial random Fourier matrix obtained byomitting rows from F that correspond to the unavailablesamples When (1198962119894 1198963119894 119896119901119894) = (1198862119894 1198863119894 119886119901119894) then Xcan be observed as a demodulated version of the 119894th signalcomponent X119894 having the dominant 119894th component in thespectrum with the support 1198961119894 The rest of the componentsin spectrum are much lower than X119894 and could be observedas noise Hence we may write the minimization problem inthe form

min 1003817100381710038171003817X11989410038171003817100381710038171st 1003817100381710038171003817y119911 minus A119911X119894

10038171003817100381710038172 lt 119879 (57)

where T is certain threshold The dominant componentscan be detected using the threshold based algorithm (Algo-rithm 5) described in the previous section Using an iter-ative procedure one may change the values of parameters1198962119894 119896119901119894 between 119896min and 119896max forall119894 The algorithm willdetect the support 1198961119894 when the set (1198962119894 1198963119894 119896119901119894) matchesthe set (1198862119894 1198863119894 119886119901119894) otherwise no component support isdetected Hence as a result of this phase we have identifiedthe sets of signal phase parameters k119894 = (1198961119894 1198962119894 119896119899119894) =(1198861119894 1198862119894 119886119899119894)422 Reconstruction of Exact Components Amplitudes Thenext phase is reconstruction of components amplitudesDenote the set of measurements positions by (1198991 1198992 119899119872)such that [y(1) y(2) y(119872)] = [s(1198991) s(1198992) s(119899119872)]while the detected sets of signal phase parameters are k119894 =(1198961119894 1198962119894 119896119899119894) = (1198861119894 1198862119894 119886119899119894)

In order to calculate the exact amplitudes 1199031 1199032 119903119870 ofsignal components we observe the set of equations in theform

[[[[[

119904 (1198991)

119904 (119899119872)]]]]]

=[[[[[[[[[

119890119895(2120587119873)(119899111989611+sdotsdotsdot+11989911990111198961199011) sdot sdot sdot 119890119895(2120587119873)(11989911198961119870+sdotsdotsdot+1198991199011 119896119901119870)119890119895(2120587119873)(119899211989611+sdotsdotsdot+11989911990121198961199011) sdot sdot sdot 119890119895(2120587119873)(11989921198961119870+sdotsdotsdot+1198991199012 119896119901119870)

119890119895(2120587119873)(11989911987311989611+sdotsdotsdot+1198991199011198731198961199011) sdot sdot sdot 119890119895(2120587119873)(1198991198721198961119870+sdotsdotsdot+119899119901119873119896119901119870)

]]]]]]]]]

[[[[[

1199031119903119870

]]]]]

(58)

or in other words we have another system of equations givenby

y = AR (59)


Input998835 Transform matrixΨ Measurement matrixΦ998835 N119872 = 1198991 1198992 119899119872119872 ndashnumber of available samples119873-original signal length998835Matrix A A = ΦΨ (A is obtained as a partial random Fourier matrix A =ΦΨ =Ψ120599120599 = 1205951198991 1205951198992 120595119899119872998835Measurement vector y = ΦfOutput(1) 119894 = 1 N119872 = 1198991 1198992 119899119872(2) p = (3) 119875 larr 099 (Set desired probability 119875)(4) 1205902119872119878 larr 119872((119873 minus119872)(119873 minus 1))sum119872119894=1(|y(119894)|2119872) (Calculate variance)(5) 119879119894 larr radicminus1205902119872119878 log(1 minus 119875(119879)1119873) (Calculate threshold)(6) X119894 larr Aminus1y (Calculate initial DFT of y)(7) k119894 larr arg|X119894| gt 119879119894119873 (Find comp in X119894 above the 119879119894)(8) Update p larr p cup k119894(9) A119888119904119894 larr A(k119894) (CS matrix)(10) X119894 larr (A119888119904119894119879A119888119904119894)minus1A119888119904119894119879y(11) for forall119901 isin p(12) y larr y minus (119872119873)X119894(119901) exp(1198952120587119901N119872119873) (Update y)(13)Update 1205902119872119878larr (119872(119873 minus119872)(119873 minus 1))sum |y|2119872(14) If sum |y|2119872 lt 120575 break Else(15) Set 119894 = 119894 + 1 and go to (5)(16) return X119894 k

Algorithm 5 Automated threshold based iterative algorithm

where R = [1199031 119903119870]119879 contains the desired 119870 signalamplitudes The matrix A of size119873 times 119870 is based on the PFT

A

=[[[[[[[[[

119890119895(2120587119873)(119899111989611+sdotsdotsdot+1198991199011 1198961199011) sdot sdot sdot 119890119895(2120587119873)(11989911198961119870+sdotsdotsdot+1198991199011 119896119901119870)119890119895(2120587119873)(119899211989611+sdotsdotsdot+1198991199012 1198961199011) sdot sdot sdot 119890119895(2120587119873)(11989921198961119870+sdotsdotsdot+1198991199012 119896119901119870)


]]]]]]]]]

(60)

The rows of A correspond to positions of measurements(1198991 1198992 119899119872) and columns correspond to phase parame-ters k119894 = (1198961119894 1198962119894 119896119899119894) = (1198861119894 1198862119894 119886119899119894) for 119894 = 1 119870The solution of the observed problem can be obtained in theleast squares sense as follows

X = (AlowastA)minus1 Alowasty (61)

The resulting reconstructed signal is obtained as

119904 (119899) = 1199031119890minus119895(2120587119873)(11989911988611+sdotsdotsdot+1198991199011198861199011119901) + sdot sdot sdot+ 119903119870119890minus119895(2120587119873)(1198991198861119870+sdotsdotsdot+119899119901119886119901119870119901)

(62)

43 CS in the Hermite Transform Domain The Hermiteexpansion using119873Hermite functions can be written in terms

of the Hermite transform matrix First let us define theHermite transform matrixH (of size119873 times119873) [29 30]

H = 1119873

sdot

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

1205950 (1)(120595119873minus1 (1))2

1205950 (2)(120595119873minus1 (2))2 sdot sdot sdot 1205950 (119873)

(120595119873minus1 (119873))21205951 (1)(120595119873minus1 (1))2


(120595119873minus1 (119873))2 d

120595119873minus1 (1)(120595119873minus1 (1))2

120595119873minus1 (2)(120595119873minus1 (2))2 sdot sdot sdot 120595119873minus1 (119873)

(120595119873minus1 (119873))2

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

(63)

Here the 119901th order Hermite basis function is defined usingthe 119901th order Hermite polynomial as follows

120595119901 (119899) = (1205902119901119901radic120587)minus12 119890minus11989922119867119901 (119899120590) (64)

where 120590 is the scaling factor used to ldquostretchrdquo or ldquocompressrdquoHermite functions in order to match the signal

The Hermite functions are usually calculated using thefollowing fast recursive realization

1205950 (119899) = 14radic120587119890minus119899221205902

1205951 (119899) = radic21198994radic120587 119890minus119899221205902

120595119894 (119899) = 119899radic2119894 Ψ119894minus1 (119899) minus radic 119894 minus 1

119894 Ψ119894minus2 (119899) forall119894 ge 2

(65)


The Hermite transform coefficients can be calculated in thematrix form as follows

c = Hs (66)

where the vector of Hermite transform coefficients is c =[1198880 1198881 119888119873minus1]119879 while the vector of signal samples is s =[119904(1) 119904(2) 119904(119873)]119879Following the Gauss-Hermite approximation the inverse

matrixΨ = Hminus1 contains119873Hermite functions and it is givenby

Ψ =[[[[[[[

1205951 (1) 1205952 (1) sdot sdot sdot 120595119873 (1)1205951 (2) 1205952 (2) sdot sdot sdot 120595119873 (2)

d

1205951 (119873) 1205952 (119873) sdot sdot sdot 120595119873 (119873)

]]]]]]] (67)

Now in the context of CS let us assume that a signal s is K-sparse in the Hermite transform domain meaning that it canbe represented by a small set of nonzero Hermite coefficientsc

119904 (119899) = 119896119870sum119901=1198961

119888119901120595119901 (119899) (68)

where 119888119901 is the 119901th element from the vector c of Hermiteexpansion coefficients In the matrix form we may write

s = Ψc (69)

such thatmost of the coefficients in c are zero values Further-more assume that only the compressive sensing is done usingrandom selection of M signal values [1198991 1198992 119899119872] Thenthe CS matrix A can be defined as random partial inverseHermite transform matrix

A =[[[[[[[


d


]]]]]]] (70)

For a vector of available measurements vectory = (s(1198991) s(1198992) s(119899119872)) the linear system of equations(undetermined system of M linear equations and 119873unknowns) can be written as

y = Ac (71)

Since c has only 119870 nonzero components the reconstructionproblem can be defined as

min c1st y = Ac (72)

If we identify the signal support in the Hermite transformdomain by the set of indices [1198961 1198962 119896119870] then the problemcan be solved in the least squares sense as

c = (Θlowast119888119904Θ119888119904)minus1Θlowast119888119904y (73)

where (lowast) denotes the conjugate transpose operation while

Θ119888119904 =[[[[[[[[

1205951198961 (1198991) 1205951198961 (1198992) 1205951198961 (119899119872)1205951198962 (1198991) 1205951198962 (1198992) sdot sdot sdot 1205951198962 (119899119872)

d

120595119896119870 (1198991) 120595119896119870 (1198992) sdot sdot sdot 120595119896119870 (119899119872)

]]]]]]]] (74)

and c = [1198881198961 1198881198962 119888119896119870]119879 The estimated signal s can be nowobtained according to (68)

44 CS in the Time-Frequency Domain Starting from thedefinition of the short-time Fourier transform

STFT (119899 119896) = 119872minus1sum119898=0

119909 (119899 + 119898) 119890minus1198952120587119898119896119872 (75)

we can define the following STFT vector and signal vector

STFT119872 (119899) = [STFT (119899 0) STFT (119899119872 minus 1)]119879 x (119899) = [119909 (119899) 119909 (119899 + 1) 119909 (119899 +119872 minus 1)]119879 (76)

respectively The STFT at an instant n can be now rewrittenin the matrix form

STFT119872 (119899) = F119872x (119899) (77)

where F119872 is the standard Fourier transform matrix of size119872times119872 with elements

F119872 (119898 119896) = 119890minus1198952120587119896119898119872 (78)

If we consider the nonoverlapping signal segments for theSTFT calculation then 119899 takes the values from the set0119872 2119872 119873 minus119872 where119872 is the window length while119873 is the total signal length The STFT vector for 119899 isin0119872 2119872 119873 minus119872 is given by

STFT =[[[[[[[

STFT119872 (0)STFT119872 (119872)

STFT119872 (119873 minus119872)

]]]]]]] (79)

Consequently (77) can be extended to the entire STFT for allconsidered values of 119899 isin 0119872 2119872 119873 minus 119872 as follows[31 32]

[[[[[[[

STFT119872 (0)STFT119872 (119872)

STFT119872 (119873 minus119872)

]]]]]]]

=[[[[[[

F119872 0119872 sdot sdot sdot 01198720119872 F119872 sdot sdot sdot 0119872sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot0119872 0119872 sdot sdot sdot F119872

]]]]]]

[[[[[[[

x (0)x (119872)

x (119873 minus119872)

]]]]]]]

(80)


0 10 20 30 40 50minus300

minus200

minus100

0

100Hermite transform

(a)0 10 20 30 40 50

Discrete Fourier transform

0

500

1000

1500

(b)

Figure 1 (a) Hermite transform coefficients of the selected QRS complex (120590 = 68038) and (b) Fourier transform coefficients (absolutevalues)

or in the compact form

STFT = Fx (81)

In the case of time-varying windows with lengths 11987211198722 119872119876 the previous system of equations becomes

[[[[[[[[

STFT1198720 (0)STFT1198721 (1198720)

STFT119872119876 (119873 minus119872119876)

]]]]]]]]

=[[[[[[

F11987200 sdot sdot sdot 0

0119872 F1198721sdot sdot sdot 0

sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot0 0 sdot sdot sdot F119872119876

]]]]]]

[[[[[[[

x (0)x (119872)

x (119873 minus119872)

]]]]]]]

(82)

Note that the signal vector is x = [x(0)119879 x(119872)119879 x(119873 minus119872)119879]119879 = [119909(0) 119909(1) 119909(119873 minus 1)]119879 and can be expressedusing the119873-point inverse Fourier transform as follows

x = Fminus1119873X (83)

where X is a sparse vector of DFT coefficients By combining(81) and (83) we obtain [31]

STFT = FFminus1119873X (84)

Furthermore if we assume that STFT vector is not completelyavailable but has only a small percent of available samplesthen we are dealing with the CS problem in the form

ySTFT = AX (85)

where ySTFT is a vector of available measurements fromSTFT whileA is the CS matrix obtained as partial combined

transform matrix Ψ = FFminus1119873 with rows corresponding to

the available samples ySTFT In order to reconstruct the entireSTFT we can define the minimization problem in the form

min X1st ySTFT = AX (86)

which can be solved using any of the algorithms provided inthe previous section

5 Experimental Evaluation

Example 1 Let us observe the QRS complex extracted fromthe real ECG signal It has been known that these types ofsignals are sparse in the Hermite transform (HT) domain[33 34] As an illustration we present the Hermite transformand discrete Fourier transform (DFT) in Figure 1 fromwhichit can be observed that HT provides sparse representationwhile DFT is dense The analyzed signal is obtained fromthe MIT-ECG Compression Test Database [35] originallysampled uniformly with 119879 = 1250 [s] being the samplingperiod with amplitude gain 1400 Extracted QRS is of length119873 = 51 centered at the 119877 peak Note that in order toprovide the sparsest HT representation the scaling factor 120590should be set to the optimal value [33 34] which is 68038 inthis example that is 120590Δ119905 = 0027 in seconds proportionalto the scaling factor 0017 presented in [34] for signal with27 samples when it is scaled with the signal lengths ratioAlso the QRS signal is resampled at the zeros of Hermitepolynomial using sinc interpolation functions as described in[33]

The QRS signal is subject to compressed sensingapproach meaning that it is represented by 50 of availablemeasurements The available measurements and the corre-sponding HT (which is not sparse in the CS conditions) areshown in Figure 2

In order to reconstruct the entire QRS signal fromavailable measurements two of the presented algorithms


0

minus02

0

02

200100minus100minus200

(a)0 10 20 30 40 50

minus200

minus150

minus100

minus50

0

50

100

(b)

Figure 2 (a) Measurements of the selected QRS complex (50 of the total samples number) and (b) Hermite transform of measurementsvector

minus04

minus02

0

02

minus100 0 100 200minus200

Time (ms)

(a)

20 30 40 5010Order of Hermite coefficients

minus1

minus08

minus06

minus04

minus02

0

02

(b)

minus04

minus02

0

02

minus100 0 100 200minus200

Time (ms)

(c)

minus1

minus08

minus06

minus04

minus02

0

02


(d)

Figure 3 Results of signal reconstruction using gradient-based algorithms (a) original signal (solid line) and reconstructed signal (dashline) and (b) Hermite coefficients of the original signal (solid line) and the reconstructed signal (squares) Reconstruction results using OMPalgorithm (c) original signal (solid) and reconstructed signal (dash) in time domain and (d) Hermite coefficients of original signal (solidline) and reconstructed signal (squares)

are employed namely the OMP and gradient-based recon-struction algorithm

The reconstruction results are shown in Figure 3 inboth time and transform domain It can be observed thatboth algorithms provide quite successful reconstruction

performance which is certainly better in the case of gradient-based approach In order to measure and compare thereconstruction performance of these two algorithms themean square error (MSE) is calculated showing that thegradient-based algorithm in the considered case provides


Input998835 Available samples position N119872 = 1198991 1198992 119899119872998835Missing samples position N N119872 where N = 1 2 119873998835 Available samples f(119899) 119899 isin N119872998835Measurement vector y(119899) = f(119899) for 119899 isin N119872 0 for 119899 isin N N119872Output(1) Set 119894 = 0 y(0)(119899) larr y(119899) Δ larr max |y(0)(119899)|(2) repeat(3) Set y119901(119899) = y(119894)(119899)(4) repeat(5) 119894 larr 119894 + 1(6) for 119899119904 larr 0 to 119873 minus 1 do(7) if 119899119904 isin N N119872(8) x+(119896) larr Iy(119894)(119899) + Δ120575(119899 minus 119899119904)(9) xminus(119896) larr Iy(119894)(119899) minus Δ120575(119899 minus 119899119904)(10) G(119894)(119899119904) larr (1119873)(x+1 minus xminus1)(11) else G(119894)(119899119904) larr 0(12) end if(13) y(119894+1)(119899119904) larr y(119894)(119899119904) minus G(119894)(119899119904)(14) end for(15) 120573119894 = arcos(⟨G119894minus1G119894⟩G119894minus122G11989422)(16) until 120573119894 lt 170∘(17) Δ larr Δradic10(18) Ξ = 10 log10(sum119899 isinNN119872 |y119901(119899) minus y(119894)(119899)|2sum119899 isinNN119872 |y(119894)(119899)|2)(19) until Ξ lt Ξmax(20) return y(i)(119899)Reconstructed signal x119877(119899) = y(119894)(119899)

Algorithm 6 Adaptive gradient-based algorithm

MSE = 1385 much lower than MSE = 3164 for the OMPalgorithm

Example 2 In this example the reconstruction of imagesusing adaptive gradient-based algorithm (Algorithm 6) isconsidered The images are generally not strictly sparse inany domain and as such could be quite demanding forreconstruction For that reason as stated in Section 37the adaptive gradient-based algorithm is employed since itdoes not require the strict sparsity condition As a suitabletransform domain representation for natural image the two-dimensional DCT (I rarr 2DDCT) is used We might saythat the image can be observed as approximately sparse in the2DDCT domain

The image with missing pixels is divided into 16 times 16blocks where 40 of the pixels are missing in each block(missing pixels are denoted in white in Figures 4(a) 4(c)and 4(e)) Hence the algorithm is applied on a block-by-block basis where the measurement vector y is created foreach observed image block using the available block pixelsThe value of gradient step Δ = 128 is used The resultingreconstructed images are shown in Figures 4(b) 4(d) and4(f)

The quality of the reconstructed image is measured usingthe structural similarity index (SSIM) [36] between theoriginal image (with full set of pixels) and reconstructedimageThe SSIM ranges between 0 and 1 1 being identical and

0 being not similar It is designed to improve the traditionalmetrics such as peak signal-to-noise ratio (PSNR) in order tobe more consistent with the perception of the human visualsystem The achieved values of SSIM for the test images inFigure 4 are SSIM = 09 SSIM = 094 and SSIM = 095

Example 3 Let us assume that a radar signal consists ofstationary (sinusoidal) components belonging to the rigidbody and nonstationary components belonging to the distur-bances caused by rotating target parts and noise The Fouriertransform and the STFT of the considered radar signal typeare given in Figure 5

The total signal length is 4096 samples The STFT iscalculated for the window width 119908 = 64 samples (STFT is ofsize 64times64)The aim of this experiment is to extract the rigidbody components which are presented as a set of stationarysinusoids

The STFT values of the original full signal along columnsand then the 75 of the largest values (possibly belongingto rotating components and noise) are set to zero or inother words discarded from the STFT plane Also a smallpercentage of the smallest values along the sorted columnsare also set to zero with the aim of discarding small valuesbelonging to noise The remaining STFT values are unsortedandplaced to their original positions in the STFTplaneThesevalues represent a small amount of availablemeasurements inthe STFT domain and we are faced with the CS problem In


(a) (b)

(c) (d)

(e) (f)

Figure 4 ((a) (c) and (e)) Available measurementspixels (missing values are denoted by white pixels) ((b) (d) and (f)) Reconstructedimages


20 30 40 50 6010Frequency

60

50

40

30

20

10

Tim

e

(a)

0

1000

2000

3000

4000

5000

6000

7000

8000

Am

plitu

des

500 1000 1500 2000 2500 3000 3500 40000Frequency

(b)

Figure 5 (a) STFT of the observed signal (b) DFT of the observed signal

20 30 40 50 6010Frequency

60

50

40

30

20

10

Tim

e

(a)

Am

plitu

des

0

1000

2000

3000

4000

5000

6000

7000

8000

1000 2000 3000 40000Frequency

(b)


Figure 6 we show the available measurements from the STFTafter discarding all unwanted components

In order to reconstruct the stationary rigid body com-ponents from the available measurements the CS matrix isdefined as partial combined transformmatrix with rows cor-responding to the available samples in the STFT (accordingto the procedure described in Section 44) The STFT of thereconstructed stationary components is given in Figure 6(a)while the corresponding DFT is given in Figure 6(b)

6 Conclusion

The paper reviews the fundamental concepts of compressivesensing theory comprising the main requirements con-ditions and common optimization problem formulationsSeveral algorithms for signal reconstruction from incompletemeasurements are summarized allowing an insight into thediversity of approaches the related complexity assumptionsand efficiency As a special contribution the paper presentssome interesting compressive sensing formulations for dif-ferent types of signals and sparsity domains such as the

Fourier transform domain polynomial Fourier transformdomain Hermite transform and time-frequency domainParticularly this part aims to reveal the possibilities andperspectives of using compressive sensing in different signalprocessing scenarios

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

This work is supported by project CS-ICT ldquoNew ICTCompressive Sensing Based Trends Applied to MultimediaBiomedicine and Communicationsrdquo with a grant from theMinistry of Science (World Bank loan) bilateral project(Montenegro-Croatia) ldquoCompressive Sensing and Superreso-lution in Surveillance Systems Based on Optical Sensors andUAVsrdquo bilateral project (Montenegro-China) ldquoCompressiveSensing and Time-Frequency Analysis with Applicationsrdquo


and the National Natural Science Foundation of China(61571174)

References

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

[2] E J Candes and M B Wakin ldquoAn introduction to compressivesamplingrdquo IEEE Signal Processing Magazine vol 25 no 2 pp21ndash30 2008

[3] Y Eldar and G Kutyniok Compressive Sensing Theory andApplications Cambridge University Press 2012

[4] S Foucart and H Rauhut A Mathematical Introduction toCompressive Sensing Springer New York NY USA 2013

[5] E Candes and J Romberg ldquoSparsity and incoherence incompressive samplingrdquo Inverse Problems vol 23 no 3 pp 969ndash985 2007

[6] L Stankovic M Dakovic S Stankovic and I Orovic ldquoSparsesignal processingrdquo inDigital Signal Processing L Stankovic EdCreateSpace 2015

[7] S Stankovic I Orovic and E Sejdic Multimedia Signals andSystems Basic and Advanced Algorithms for Signal ProcessingSpringer New York NY USA 2016

[8] E J Candes and T Tao ldquoDecoding by linear programmingrdquoIEEE Transactions on Information Theory vol 51 no 12 pp4203ndash4215 2005

[9] 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

[10] G Pope Compressive Sensing a Summary of ReconstructionAlgorithms Eidgenossische Technische Hochschule ZurichSwitzerland 2008

[11] L StankovicMDakovic and SVujovic ldquoAdaptive variable stepalgorithm for missing samples recovery in sparse signalsrdquo IETSignal Processing vol 8 no 3 pp 246ndash256 2014

[12] 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

[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 G Mallat and Z Zhang ldquoMatching pursuits with time-frequency dictionariesrdquo IEEE Transactions on Signal Processingvol 41 no 12 pp 3397ndash3415 1993

[15] J A Tropp and A C Gilbert ldquoSignal recovery from randommeasurements via orthogonal matching pursuitrdquo IEEE Trans-actions on Information Theory vol 53 no 12 pp 4655ndash46662007

[16] D Needell and J A Tropp ldquoCoSaMP iterative signal recoveryfrom incomplete and inaccurate samplesrdquo Applied and Compu-tational Harmonic Analysis vol 26 no 3 pp 301ndash321 2009

[17] D L Donoho Y Tsaig I Drori and J-L Starck ldquoSparsesolution of underdetermined systems of linear equations bystagewise orthogonal matching pursuitrdquo IEEE Transactions onInformation Theory vol 58 no 2 pp 1094ndash1121 2012

[18] Y Liu I Gligorijevic V Matic M De Vos and S Van HuffelldquoMulti-sparse signal recovery for compressive sensingrdquo inProceedings of the IEEE International Conference on Engineering

in Medicine and Biology Society (EMBC rsquo12) pp 1053ndash1056 SanDiego Calif USA August-September 2012

[19] Y C Eldar Sampling Theory Beyond Bandlimited SystemsCambridge University Press New York NY USA 2015

[20] P Machler VLSI Architectures for Compressive Sensing andSparse Signal Recovery Hartung Gorre Konstanz Germany2013

[21] M F Duarte and Y C Eldar ldquoStructured compressed sensingfrom theory to applicationsrdquo IEEE Transactions on SignalProcessing vol 59 no 9 pp 4053ndash4085 2011

[22] S J Wright Primal-Dual Interior-Point Methods SIAMPhiladelphia Pa USA 1997

[23] Z Zhang Y Xu J Yang X Li andD Zhang ldquoA survey of sparserepresentation algorithms and applicationsrdquo IEEEAccess vol 3pp 490ndash530 2015

[24] T Blumensath and M E Davies ldquoIterative thresholding forsparse approximationsrdquo The Journal of Fourier Analysis andApplications vol 14 no 5-6 pp 629ndash654 2008

[25] T Blumensath and M E Davies ldquoIterative hard thresholdingfor compressed sensingrdquo Applied and Computational HarmonicAnalysis vol 27 no 3 pp 265ndash274 2009

[26] M A Figueiredo J M Bioucas-Dias and R D NowakldquoMajorization-minimization algorithms for wavelet-basedimage restorationrdquo IEEE Transactions on Image Processing vol16 no 12 pp 2980ndash2991 2007

[27] I Selesnick ldquoSparse signal restorationrdquo Connexions 2009httpcnxorgcontentm32168

[28] S Stankovic I Orovic and L Stankovic ldquoPolynomial Fourierdomain as a domain of signal sparsityrdquo Signal Processing vol130 pp 243ndash253 2017

[29] S Stankovic L Stankovic and I Orovic ldquoCompressive sensingapproach in the Hermite transform domainrdquo MathematicalProblems in Engineering vol 2015 Article ID 286590 9 pages2015

[30] M Brajovic I OrovicM Dakovic and S Stankovic ldquoGradient-based signal reconstruction algorithm in Hermite transformdomainrdquo Electronics Letters vol 52 no 1 pp 41ndash43 2016

[31] L Stankovı S Stankovı TThayaparanMDakovı and I OrovıldquoSeparation and reconstruction of the rigid body and micro-Doppler signal in ISAR part II-statistical analysisrdquo IET RadarSonar and Navigation vol 9 no 9 pp 1155ndash1161 2015

[32] L Stankovic S Stankovic T Thayaparan M Dakovic and IOrovic ldquoSeparation and reconstruction of the rigid body andmicro-Doppler signal in ISARpart Imdashtheoryrdquo IETRadar Sonaramp Navigation vol 9 no 9 pp 1147ndash1154 2015

[33] A Sandryhaila S Saba M Puschel and J Kovacevic ldquoEfficientcompression of QRS complexes using Hermite expansionrdquoIEEE Transactions on Signal Processing vol 60 no 2 pp 947ndash955 2012

[34] A Sandryhaila J Kovacevic and M Puschel ldquoCompression ofQRS complexes using Hermite expansionrdquo in Proceedings of the36th IEEE International Conference on Acoustics Speech andSignal Processing (ICASSP rsquo11) pp 581ndash584 IEEE Prague CzechRepublic May 2011

[35] MIT-BIH ECG Compression Test Database httpwwwphysi-onetorgphysiobankdatabasecdb

[36] ZWang A C Bovik H R Sheikh and E P Simoncelli ldquoImagequality assessment from error visibility to structural similarityrdquoIEEE Transactions on Image Processing vol 13 no 4 pp 600ndash612 2004

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


algorithms bring about low computation complexity whilethe hybrid methods try to provide a compromise betweenthese two requirements [18]

The proposed work provides a survey of the generalcompressive sensing concept supplemented with the severalexisting approaches and methods for signal reconstructionwhich are briefly explained and summarized in the formof algorithms with the aim of providing the readers withan easier and practical insight into the state of the art inthis field Apart from the standard CS algorithms a fewrecent solutions have been included as well Furthermorethe paper provides an overview of different sparsity domainsand the possibilities of employing them in the CS problemformulation Additional contribution is provided through theexamples showing the efficiency of the presented methods inpractical applications

The paper is organized as follows In Section 2 a briefreview of the general compressive sensing idea is providedtogether with the conditions for successful signal recon-struction from reduced set of measurements and the signalrecovery formulations using minimization approaches InSection 3 the commonly used CS algorithms are reviewedThe commonly used domains for CS strategy implementationare given in Section 4 while some of the examples inreal applications are provided in Section 5 The concludingremarks are given in Section 6

2 Compressive Sensing A General Overview

21 Sparsity and Compressibility Reducing the sampling rateusing CS is possible for the case of sparse signals that can berepresented by a small number of significant coefficients inan appropriate transform basis A signal having K nonzerocoefficients is called 119870-sparse Assume that signal 119904 exhibitssparsity in certain orthonormal basis Ψ defined by the basisvectors 1205951 1205952 120595119873The signal 119904 can be represented usingits sparse transform domain vector x as follows

119904 (119899) = 119873sum119894=1

119909119894120595119894 (119899) (1)

In matrix notation the previous relation can be written as

s = Ψx (2)

Commonly the sparsity is measured using the ℓ0-normwhich represents the cardinality of the support of x

x0 = card supp (x) = 119870 (3)

In real applications the signals are usually not strictly sparsebut only approximately sparse Therefore instead of beingsparse these signals are often called compressible meaningthat the amplitudes of coefficients decrease rapidly whenarranged in descending order For instance if we considercoefficients |1199091| ge |1199092| ge sdot sdot sdot ge |119909119899| then the magnitudedecays with a power law if there exist constants 1198621 and 119902 gt 0satisfying [19]

10038161003816100381610038161199091198941003816100381610038161003816 le 1198621119894minus119902 (4)

where larger 119902 means faster decay and consequently morecompressible signal The signal compressibility can be quan-tified using the minimal error between the original and spar-sified signal (obtained by keeping only119870 largest coefficients)

120590119870 (x)119901 = min 1003817100381710038171003817x minus x1198701003817100381710038171003817119901 (5)

22 Conditions on the CS Matrix Null Space PropertyRestricted Isometry Property and Incoherence Instead ofacquiring a full set of signal samples of length 119873 in CSscenario we deal with a quite reduced set of measurementsy of length 119872 where 119872 lt 119873 The measurement procedurecan be modeled by projections of the signal s onto vectors1206011 1206012 120601119872 constituting the measurement matrixΦ

y = Φs (6)

Using the sparse transform domain representation of vectors given by (2) we have

y = ΦΨx = Ax (7)

where A will be referred to as CS matrixIn order to define some requirements for the CS matrix

A which are important for successful signal reconstructionlet us introduce the null space of matrixThe null space of CSmatrix A contains all vectors x that are mapped to 0

N (A) = x Ax = 0 (8)

In order to provide a unique solution it is necessary toprovide the notion that two K-sparse vectors x and x1015840 do notresult in the same measurement vector In other words theirdifference should not be part of the null space of CS matrixA

A (x minus x1015840) = 0 (9)

Since the difference between two K-sparse vectors is at most2K-sparse then aK-sparse vector x is uniquely defined if nullspace of A contains no 2K-sparse vectors This correspondsto the condition that any 2K columns of A are linearlyindependent that is

spark (A) gt 2119870 (10)

and since spark(A) isin [2119872+ 1] we obtain a lower bound onthe number of measurements

119872 gt 2119870 (11)

In the case of strictly sparse signals the spark can providereliable information about the exact reconstruction possi-bility However in the case of approximately sparse signalsthis condition is not sufficient and does not guarantee stablerecovery Hence there is another property called null spaceproperty that measures the concentration of the null spaceof matrix A The null space property is satisfied if there is aconstant 120574 isin (0 1) such that [19]

1003817100381710038171003817a11987810038171003817100381710038171 = 120574 1003817100381710038171003817aSc10038171003817100381710038171 foralla isin N (A) (12)








120583 (A) = max119894 =1198951le119894119895le119873

100381610038161003816100381610038161003816100381610038161003816100381610038161003816⟨119860 119894 119860119895⟩1003817100381710038171003817119860 11989410038171003817100381710038172 10038171003817100381710038171003817119860119895100381710038171003817100381710038172

100381610038161003816100381610038161003816100381610038161003816100381610038161003816 (15)


120575119870 = (119870 minus 1) 120583 (16)












1198621 = 2 11 minus (1 + radic2) 1205752119870

(21)


min119909

12 1003817100381710038171003817y minus Ax100381710038171003817100381722 + 120582 x1 (22)



min119909

12 1003817100381710038171003817y minus Ax100381710038171003817100381722

st x1 lt 120591(23)


































119869 (x) = 12 1003817100381710038171003817y minus Ax100381710038171003817100381722 (28)








119878120591 (x (119896)) =


(31)



119869 (x) = 1003817100381710038171003817y minus Ax10038171003817100381710038172 + 120582 x1 (32)




Algorithm 3 CoSaMP






(33)


120572 gt max eig A119879A (34)




The solution is

x + 1205822120572 sign (x) = 1

120572Α119879 (y minus Ax119896) + x119896 (36)


119886 + 1205822120572 sign (119886) = 119887 (37)


119886 = 119878120582 (119887) =


(38)


x = 1198781205822120572 ( 1120572Α119879 (y minus Ax119896) + x119896) (39)



x119896+1 = 1198781205822120572 ( 1120572Α119879 (y minus Ax119896) + x119896) (40)


1205902119872119878 larr997888 119872119873 minus119872119873 minus 1

119872sum119894=1

1003816100381610038161003816y (119894)10038161003816100381610038162119872 (41)










119904 (119899) = 119870sum119894=1

119860 1198941198901198952120587119896119894119899119873 (43)







A =[[[[[[[


d


]]]]]]] (46)


120595119896 (119899) = 1198901198952120587119896119899119873 (47)





119904 (119899) = 119870sum119894=1

119904119894 (119899) =119870sum119894=1

119903119894119890119895(2120587119873)(1198991198861119894+119899211988621198942+sdotsdotsdot+119899119901119886119901119894119901) (49)


119883(1198961119894 119896119901119894)= 119873minus1sum119899=0

119870sum119894=1

119903119894119890119895(2120587119873)(1198991198861119894+119899211988621198942+sdotsdotsdot+119899119901119886119901119894119901)



(1198962119894 1198963119894 119896119901119894) = (1198862119894 1198863119894 119886119901119894) (51)





119911 (119899) = 119904 (119899) 120585 (119899) = 119870sum119894=1


(53)


z = s120585 (54)


X = Fz (55)




y119911 = A119911X (56)


min 1003817100381710038171003817X11989410038171003817100381710038171st 1003817100381710038171003817y119911 minus A119911X119894

10038171003817100381710038172 lt 119879 (57)



[[[[[

119904 (1198991)

119904 (119899119872)]]]]]

=[[[[[[[[[



]]]]]]]]]

[[[[[

1199031119903119870

]]]]]

(58)


y = AR (59)





A

=[[[[[[[[[



]]]]]]]]]

(60)





(62)



H = 1119873

sdot

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

1205950 (1)(120595119873minus1 (1))2


(120595119873minus1 (119873))21205951 (1)(120595119873minus1 (1))2


(120595119873minus1 (119873))2 d

120595119873minus1 (1)(120595119873minus1 (1))2


(120595119873minus1 (119873))2

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

(63)





1205950 (119899) = 14radic120587119890minus119899221205902



119894 Ψ119894minus2 (119899) forall119894 ge 2

(65)



c = Hs (66)



Ψ =[[[[[[[


d

1205951 (119873) 1205952 (119873) sdot sdot sdot 120595119873 (119873)

]]]]]]] (67)


119904 (119899) = 119896119870sum119901=1198961

119888119901120595119901 (119899) (68)


s = Ψc (69)


A =[[[[[[[


d


]]]]]]] (70)


y = Ac (71)






Θ119888119904 =[[[[[[[[

1205951198961 (1198991) 1205951198961 (1198992) 1205951198961 (119899119872)1205951198962 (1198991) 1205951198962 (1198992) sdot sdot sdot 1205951198962 (119899119872)

d

120595119896119870 (1198991) 120595119896119870 (1198992) sdot sdot sdot 120595119896119870 (119899119872)

]]]]]]]] (74)



STFT (119899 119896) = 119872minus1sum119898=0

119909 (119899 + 119898) 119890minus1198952120587119898119896119872 (75)




STFT119872 (119899) = F119872x (119899) (77)


F119872 (119898 119896) = 119890minus1198952120587119896119898119872 (78)


STFT =[[[[[[[

STFT119872 (0)STFT119872 (119872)

STFT119872 (119873 minus119872)

]]]]]]] (79)


[[[[[[[

STFT119872 (0)STFT119872 (119872)

STFT119872 (119873 minus119872)

]]]]]]]

=[[[[[[


]]]]]]

[[[[[[[

x (0)x (119872)

x (119873 minus119872)

]]]]]]]

(80)


0 10 20 30 40 50minus300

minus200

minus100

0


(a)0 10 20 30 40 50


0

500

1000

1500

(b)



STFT = Fx (81)


[[[[[[[[

STFT1198720 (0)STFT1198721 (1198720)

STFT119872119876 (119873 minus119872119876)

]]]]]]]]

=[[[[[[




]]]]]]

[[[[[[[

x (0)x (119872)

x (119873 minus119872)

]]]]]]]

(82)


x = Fminus1119873X (83)




ySTFT = AX (85)











0

minus02

0

02


(a)0 10 20 30 40 50

minus200

minus150

minus100

minus50

0

50

100

(b)


minus04

minus02

0

02

minus100 0 100 200minus200

Time (ms)

(a)


minus1

minus08

minus06

minus04

minus02

0

02

(b)

minus04

minus02

0

02

minus100 0 100 200minus200

Time (ms)

(c)

minus1

minus08

minus06

minus04

minus02

0

02


(d)

















(a) (b)

(c) (d)

(e) (f)



20 30 40 50 6010Frequency

60

50

40

30

20

10

Tim

e

(a)

0

1000

2000

3000

4000

5000

6000

7000

8000

Am

plitu

des

500 1000 1500 2000 2500 3000 3500 40000Frequency

(b)


20 30 40 50 6010Frequency

60

50

40

30

20

10

Tim

e

(a)

Am

plitu

des

0

1000

2000

3000

4000

5000

6000

7000

8000

1000 2000 3000 40000Frequency

(b)




6 Conclusion



Competing Interests


Acknowledgments




References













































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















120583 (A) = max119894 =1198951le119894119895le119873

100381610038161003816100381610038161003816100381610038161003816100381610038161003816⟨119860 119894 119860119895⟩1003817100381710038171003817119860 11989410038171003817100381710038172 10038171003817100381710038171003817119860119895100381710038171003817100381710038172

100381610038161003816100381610038161003816100381610038161003816100381610038161003816 (15)


120575119870 = (119870 minus 1) 120583 (16)












1198621 = 2 11 minus (1 + radic2) 1205752119870

(21)


min119909

12 1003817100381710038171003817y minus Ax100381710038171003817100381722 + 120582 x1 (22)



min119909

12 1003817100381710038171003817y minus Ax100381710038171003817100381722

st x1 lt 120591(23)


































119869 (x) = 12 1003817100381710038171003817y minus Ax100381710038171003817100381722 (28)








119878120591 (x (119896)) =


(31)



119869 (x) = 1003817100381710038171003817y minus Ax10038171003817100381710038172 + 120582 x1 (32)




Algorithm 3 CoSaMP






(33)


120572 gt max eig A119879A (34)




The solution is

x + 1205822120572 sign (x) = 1

120572Α119879 (y minus Ax119896) + x119896 (36)


119886 + 1205822120572 sign (119886) = 119887 (37)


119886 = 119878120582 (119887) =


(38)


x = 1198781205822120572 ( 1120572Α119879 (y minus Ax119896) + x119896) (39)



x119896+1 = 1198781205822120572 ( 1120572Α119879 (y minus Ax119896) + x119896) (40)


1205902119872119878 larr997888 119872119873 minus119872119873 minus 1

119872sum119894=1

1003816100381610038161003816y (119894)10038161003816100381610038162119872 (41)










119904 (119899) = 119870sum119894=1

119860 1198941198901198952120587119896119894119899119873 (43)







A =[[[[[[[


d


]]]]]]] (46)


120595119896 (119899) = 1198901198952120587119896119899119873 (47)





119904 (119899) = 119870sum119894=1

119904119894 (119899) =119870sum119894=1

119903119894119890119895(2120587119873)(1198991198861119894+119899211988621198942+sdotsdotsdot+119899119901119886119901119894119901) (49)


119883(1198961119894 119896119901119894)= 119873minus1sum119899=0

119870sum119894=1

119903119894119890119895(2120587119873)(1198991198861119894+119899211988621198942+sdotsdotsdot+119899119901119886119901119894119901)



(1198962119894 1198963119894 119896119901119894) = (1198862119894 1198863119894 119886119901119894) (51)





119911 (119899) = 119904 (119899) 120585 (119899) = 119870sum119894=1


(53)


z = s120585 (54)


X = Fz (55)




y119911 = A119911X (56)


min 1003817100381710038171003817X11989410038171003817100381710038171st 1003817100381710038171003817y119911 minus A119911X119894

10038171003817100381710038172 lt 119879 (57)



[[[[[

119904 (1198991)

119904 (119899119872)]]]]]

=[[[[[[[[[



]]]]]]]]]

[[[[[

1199031119903119870

]]]]]

(58)


y = AR (59)





A

=[[[[[[[[[



]]]]]]]]]

(60)





(62)



H = 1119873

sdot

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

1205950 (1)(120595119873minus1 (1))2


(120595119873minus1 (119873))21205951 (1)(120595119873minus1 (1))2


(120595119873minus1 (119873))2 d

120595119873minus1 (1)(120595119873minus1 (1))2


(120595119873minus1 (119873))2

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

(63)





1205950 (119899) = 14radic120587119890minus119899221205902



119894 Ψ119894minus2 (119899) forall119894 ge 2

(65)



c = Hs (66)



Ψ =[[[[[[[


d

1205951 (119873) 1205952 (119873) sdot sdot sdot 120595119873 (119873)

]]]]]]] (67)


119904 (119899) = 119896119870sum119901=1198961

119888119901120595119901 (119899) (68)


s = Ψc (69)


A =[[[[[[[


d


]]]]]]] (70)


y = Ac (71)






Θ119888119904 =[[[[[[[[

1205951198961 (1198991) 1205951198961 (1198992) 1205951198961 (119899119872)1205951198962 (1198991) 1205951198962 (1198992) sdot sdot sdot 1205951198962 (119899119872)

d

120595119896119870 (1198991) 120595119896119870 (1198992) sdot sdot sdot 120595119896119870 (119899119872)

]]]]]]]] (74)



STFT (119899 119896) = 119872minus1sum119898=0

119909 (119899 + 119898) 119890minus1198952120587119898119896119872 (75)




STFT119872 (119899) = F119872x (119899) (77)


F119872 (119898 119896) = 119890minus1198952120587119896119898119872 (78)


STFT =[[[[[[[

STFT119872 (0)STFT119872 (119872)

STFT119872 (119873 minus119872)

]]]]]]] (79)


[[[[[[[

STFT119872 (0)STFT119872 (119872)

STFT119872 (119873 minus119872)

]]]]]]]

=[[[[[[


]]]]]]

[[[[[[[

x (0)x (119872)

x (119873 minus119872)

]]]]]]]

(80)


0 10 20 30 40 50minus300

minus200

minus100

0


(a)0 10 20 30 40 50


0

500

1000

1500

(b)



STFT = Fx (81)


[[[[[[[[

STFT1198720 (0)STFT1198721 (1198720)

STFT119872119876 (119873 minus119872119876)

]]]]]]]]

=[[[[[[




]]]]]]

[[[[[[[

x (0)x (119872)

x (119873 minus119872)

]]]]]]]

(82)


x = Fminus1119873X (83)




ySTFT = AX (85)











0

minus02

0

02


(a)0 10 20 30 40 50

minus200

minus150

minus100

minus50

0

50

100

(b)


minus04

minus02

0

02

minus100 0 100 200minus200

Time (ms)

(a)


minus1

minus08

minus06

minus04

minus02

0

02

(b)

minus04

minus02

0

02

minus100 0 100 200minus200

Time (ms)

(c)

minus1

minus08

minus06

minus04

minus02

0

02


(d)

















(a) (b)

(c) (d)

(e) (f)



20 30 40 50 6010Frequency

60

50

40

30

20

10

Tim

e

(a)

0

1000

2000

3000

4000

5000

6000

7000

8000

Am

plitu

des

500 1000 1500 2000 2500 3000 3500 40000Frequency

(b)


20 30 40 50 6010Frequency

60

50

40

30

20

10

Tim

e

(a)

Am

plitu

des

0

1000

2000

3000

4000

5000

6000

7000

8000

1000 2000 3000 40000Frequency

(b)




6 Conclusion



Competing Interests


Acknowledgments




References













































Volume 2014




Journal of











Journal of


Function Spaces






Algebra




































119869 (x) = 12 1003817100381710038171003817y minus Ax100381710038171003817100381722 (28)








119878120591 (x (119896)) =


(31)



119869 (x) = 1003817100381710038171003817y minus Ax10038171003817100381710038172 + 120582 x1 (32)




Algorithm 3 CoSaMP






(33)


120572 gt max eig A119879A (34)




The solution is

x + 1205822120572 sign (x) = 1

120572Α119879 (y minus Ax119896) + x119896 (36)


119886 + 1205822120572 sign (119886) = 119887 (37)


119886 = 119878120582 (119887) =


(38)


x = 1198781205822120572 ( 1120572Α119879 (y minus Ax119896) + x119896) (39)



x119896+1 = 1198781205822120572 ( 1120572Α119879 (y minus Ax119896) + x119896) (40)


1205902119872119878 larr997888 119872119873 minus119872119873 minus 1

119872sum119894=1

1003816100381610038161003816y (119894)10038161003816100381610038162119872 (41)










119904 (119899) = 119870sum119894=1

119860 1198941198901198952120587119896119894119899119873 (43)







A =[[[[[[[


d


]]]]]]] (46)


120595119896 (119899) = 1198901198952120587119896119899119873 (47)





119904 (119899) = 119870sum119894=1

119904119894 (119899) =119870sum119894=1

119903119894119890119895(2120587119873)(1198991198861119894+119899211988621198942+sdotsdotsdot+119899119901119886119901119894119901) (49)


119883(1198961119894 119896119901119894)= 119873minus1sum119899=0

119870sum119894=1

119903119894119890119895(2120587119873)(1198991198861119894+119899211988621198942+sdotsdotsdot+119899119901119886119901119894119901)



(1198962119894 1198963119894 119896119901119894) = (1198862119894 1198863119894 119886119901119894) (51)





119911 (119899) = 119904 (119899) 120585 (119899) = 119870sum119894=1


(53)


z = s120585 (54)


X = Fz (55)




y119911 = A119911X (56)


min 1003817100381710038171003817X11989410038171003817100381710038171st 1003817100381710038171003817y119911 minus A119911X119894

10038171003817100381710038172 lt 119879 (57)



[[[[[

119904 (1198991)

119904 (119899119872)]]]]]

=[[[[[[[[[



]]]]]]]]]

[[[[[

1199031119903119870

]]]]]

(58)


y = AR (59)





A

=[[[[[[[[[



]]]]]]]]]

(60)





(62)



H = 1119873

sdot

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

1205950 (1)(120595119873minus1 (1))2


(120595119873minus1 (119873))21205951 (1)(120595119873minus1 (1))2


(120595119873minus1 (119873))2 d

120595119873minus1 (1)(120595119873minus1 (1))2


(120595119873minus1 (119873))2

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

(63)





1205950 (119899) = 14radic120587119890minus119899221205902



119894 Ψ119894minus2 (119899) forall119894 ge 2

(65)



c = Hs (66)



Ψ =[[[[[[[


d

1205951 (119873) 1205952 (119873) sdot sdot sdot 120595119873 (119873)

]]]]]]] (67)


119904 (119899) = 119896119870sum119901=1198961

119888119901120595119901 (119899) (68)


s = Ψc (69)


A =[[[[[[[


d


]]]]]]] (70)


y = Ac (71)






Θ119888119904 =[[[[[[[[

1205951198961 (1198991) 1205951198961 (1198992) 1205951198961 (119899119872)1205951198962 (1198991) 1205951198962 (1198992) sdot sdot sdot 1205951198962 (119899119872)

d

120595119896119870 (1198991) 120595119896119870 (1198992) sdot sdot sdot 120595119896119870 (119899119872)

]]]]]]]] (74)



STFT (119899 119896) = 119872minus1sum119898=0

119909 (119899 + 119898) 119890minus1198952120587119898119896119872 (75)




STFT119872 (119899) = F119872x (119899) (77)


F119872 (119898 119896) = 119890minus1198952120587119896119898119872 (78)


STFT =[[[[[[[

STFT119872 (0)STFT119872 (119872)

STFT119872 (119873 minus119872)

]]]]]]] (79)


[[[[[[[

STFT119872 (0)STFT119872 (119872)

STFT119872 (119873 minus119872)

]]]]]]]

=[[[[[[


]]]]]]

[[[[[[[

x (0)x (119872)

x (119873 minus119872)

]]]]]]]

(80)


0 10 20 30 40 50minus300

minus200

minus100

0


(a)0 10 20 30 40 50


0

500

1000

1500

(b)



STFT = Fx (81)


[[[[[[[[

STFT1198720 (0)STFT1198721 (1198720)

STFT119872119876 (119873 minus119872119876)

]]]]]]]]

=[[[[[[




]]]]]]

[[[[[[[

x (0)x (119872)

x (119873 minus119872)

]]]]]]]

(82)


x = Fminus1119873X (83)




ySTFT = AX (85)











0

minus02

0

02


(a)0 10 20 30 40 50

minus200

minus150

minus100

minus50

0

50

100

(b)


minus04

minus02

0

02

minus100 0 100 200minus200

Time (ms)

(a)


minus1

minus08

minus06

minus04

minus02

0

02

(b)

minus04

minus02

0

02

minus100 0 100 200minus200

Time (ms)

(c)

minus1

minus08

minus06

minus04

minus02

0

02


(d)

















(a) (b)

(c) (d)

(e) (f)



20 30 40 50 6010Frequency

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Tim

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(a)

0

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Am

plitu

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500 1000 1500 2000 2500 3000 3500 40000Frequency

(b)


20 30 40 50 6010Frequency

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6 Conclusion



Competing Interests


Acknowledgments




References













































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











Algorithm 3 CoSaMP






(33)


120572 gt max eig A119879A (34)




The solution is

x + 1205822120572 sign (x) = 1

120572Α119879 (y minus Ax119896) + x119896 (36)


119886 + 1205822120572 sign (119886) = 119887 (37)


119886 = 119878120582 (119887) =


(38)


x = 1198781205822120572 ( 1120572Α119879 (y minus Ax119896) + x119896) (39)



x119896+1 = 1198781205822120572 ( 1120572Α119879 (y minus Ax119896) + x119896) (40)


1205902119872119878 larr997888 119872119873 minus119872119873 minus 1

119872sum119894=1

1003816100381610038161003816y (119894)10038161003816100381610038162119872 (41)










119904 (119899) = 119870sum119894=1

119860 1198941198901198952120587119896119894119899119873 (43)







A =[[[[[[[


d


]]]]]]] (46)


120595119896 (119899) = 1198901198952120587119896119899119873 (47)





119904 (119899) = 119870sum119894=1

119904119894 (119899) =119870sum119894=1

119903119894119890119895(2120587119873)(1198991198861119894+119899211988621198942+sdotsdotsdot+119899119901119886119901119894119901) (49)


119883(1198961119894 119896119901119894)= 119873minus1sum119899=0

119870sum119894=1

119903119894119890119895(2120587119873)(1198991198861119894+119899211988621198942+sdotsdotsdot+119899119901119886119901119894119901)



(1198962119894 1198963119894 119896119901119894) = (1198862119894 1198863119894 119886119901119894) (51)





119911 (119899) = 119904 (119899) 120585 (119899) = 119870sum119894=1


(53)


z = s120585 (54)


X = Fz (55)




y119911 = A119911X (56)


min 1003817100381710038171003817X11989410038171003817100381710038171st 1003817100381710038171003817y119911 minus A119911X119894

10038171003817100381710038172 lt 119879 (57)



[[[[[

119904 (1198991)

119904 (119899119872)]]]]]

=[[[[[[[[[



]]]]]]]]]

[[[[[

1199031119903119870

]]]]]

(58)


y = AR (59)





A

=[[[[[[[[[



]]]]]]]]]

(60)





(62)



H = 1119873

sdot

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

1205950 (1)(120595119873minus1 (1))2


(120595119873minus1 (119873))21205951 (1)(120595119873minus1 (1))2


(120595119873minus1 (119873))2 d

120595119873minus1 (1)(120595119873minus1 (1))2


(120595119873minus1 (119873))2

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

(63)





1205950 (119899) = 14radic120587119890minus119899221205902



119894 Ψ119894minus2 (119899) forall119894 ge 2

(65)



c = Hs (66)



Ψ =[[[[[[[


d

1205951 (119873) 1205952 (119873) sdot sdot sdot 120595119873 (119873)

]]]]]]] (67)


119904 (119899) = 119896119870sum119901=1198961

119888119901120595119901 (119899) (68)


s = Ψc (69)


A =[[[[[[[


d


]]]]]]] (70)


y = Ac (71)






Θ119888119904 =[[[[[[[[

1205951198961 (1198991) 1205951198961 (1198992) 1205951198961 (119899119872)1205951198962 (1198991) 1205951198962 (1198992) sdot sdot sdot 1205951198962 (119899119872)

d

120595119896119870 (1198991) 120595119896119870 (1198992) sdot sdot sdot 120595119896119870 (119899119872)

]]]]]]]] (74)



STFT (119899 119896) = 119872minus1sum119898=0

119909 (119899 + 119898) 119890minus1198952120587119898119896119872 (75)




STFT119872 (119899) = F119872x (119899) (77)


F119872 (119898 119896) = 119890minus1198952120587119896119898119872 (78)


STFT =[[[[[[[

STFT119872 (0)STFT119872 (119872)

STFT119872 (119873 minus119872)

]]]]]]] (79)


[[[[[[[

STFT119872 (0)STFT119872 (119872)

STFT119872 (119873 minus119872)

]]]]]]]

=[[[[[[


]]]]]]

[[[[[[[

x (0)x (119872)

x (119873 minus119872)

]]]]]]]

(80)


0 10 20 30 40 50minus300

minus200

minus100

0


(a)0 10 20 30 40 50


0

500

1000

1500

(b)



STFT = Fx (81)


[[[[[[[[

STFT1198720 (0)STFT1198721 (1198720)

STFT119872119876 (119873 minus119872119876)

]]]]]]]]

=[[[[[[




]]]]]]

[[[[[[[

x (0)x (119872)

x (119873 minus119872)

]]]]]]]

(82)


x = Fminus1119873X (83)




ySTFT = AX (85)











0

minus02

0

02


(a)0 10 20 30 40 50

minus200

minus150

minus100

minus50

0

50

100

(b)


minus04

minus02

0

02

minus100 0 100 200minus200

Time (ms)

(a)


minus1

minus08

minus06

minus04

minus02

0

02

(b)

minus04

minus02

0

02

minus100 0 100 200minus200

Time (ms)

(c)

minus1

minus08

minus06

minus04

minus02

0

02


(d)

















(a) (b)

(c) (d)

(e) (f)



20 30 40 50 6010Frequency

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Tim

e

(a)

0

1000

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Am

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500 1000 1500 2000 2500 3000 3500 40000Frequency

(b)


20 30 40 50 6010Frequency

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(a)

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1000 2000 3000 40000Frequency

(b)




6 Conclusion



Competing Interests


Acknowledgments




References













































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











x119896+1 = 1198781205822120572 ( 1120572Α119879 (y minus Ax119896) + x119896) (40)


1205902119872119878 larr997888 119872119873 minus119872119873 minus 1

119872sum119894=1

1003816100381610038161003816y (119894)10038161003816100381610038162119872 (41)










119904 (119899) = 119870sum119894=1

119860 1198941198901198952120587119896119894119899119873 (43)







A =[[[[[[[


d


]]]]]]] (46)


120595119896 (119899) = 1198901198952120587119896119899119873 (47)





119904 (119899) = 119870sum119894=1

119904119894 (119899) =119870sum119894=1

119903119894119890119895(2120587119873)(1198991198861119894+119899211988621198942+sdotsdotsdot+119899119901119886119901119894119901) (49)


119883(1198961119894 119896119901119894)= 119873minus1sum119899=0

119870sum119894=1

119903119894119890119895(2120587119873)(1198991198861119894+119899211988621198942+sdotsdotsdot+119899119901119886119901119894119901)



(1198962119894 1198963119894 119896119901119894) = (1198862119894 1198863119894 119886119901119894) (51)





119911 (119899) = 119904 (119899) 120585 (119899) = 119870sum119894=1


(53)


z = s120585 (54)


X = Fz (55)




y119911 = A119911X (56)


min 1003817100381710038171003817X11989410038171003817100381710038171st 1003817100381710038171003817y119911 minus A119911X119894

10038171003817100381710038172 lt 119879 (57)



[[[[[

119904 (1198991)

119904 (119899119872)]]]]]

=[[[[[[[[[



]]]]]]]]]

[[[[[

1199031119903119870

]]]]]

(58)


y = AR (59)





A

=[[[[[[[[[



]]]]]]]]]

(60)





(62)



H = 1119873

sdot

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

1205950 (1)(120595119873minus1 (1))2


(120595119873minus1 (119873))21205951 (1)(120595119873minus1 (1))2


(120595119873minus1 (119873))2 d

120595119873minus1 (1)(120595119873minus1 (1))2


(120595119873minus1 (119873))2

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

(63)





1205950 (119899) = 14radic120587119890minus119899221205902



119894 Ψ119894minus2 (119899) forall119894 ge 2

(65)



c = Hs (66)



Ψ =[[[[[[[


d

1205951 (119873) 1205952 (119873) sdot sdot sdot 120595119873 (119873)

]]]]]]] (67)


119904 (119899) = 119896119870sum119901=1198961

119888119901120595119901 (119899) (68)


s = Ψc (69)


A =[[[[[[[


d


]]]]]]] (70)


y = Ac (71)






Θ119888119904 =[[[[[[[[

1205951198961 (1198991) 1205951198961 (1198992) 1205951198961 (119899119872)1205951198962 (1198991) 1205951198962 (1198992) sdot sdot sdot 1205951198962 (119899119872)

d

120595119896119870 (1198991) 120595119896119870 (1198992) sdot sdot sdot 120595119896119870 (119899119872)

]]]]]]]] (74)



STFT (119899 119896) = 119872minus1sum119898=0

119909 (119899 + 119898) 119890minus1198952120587119898119896119872 (75)




STFT119872 (119899) = F119872x (119899) (77)


F119872 (119898 119896) = 119890minus1198952120587119896119898119872 (78)


STFT =[[[[[[[

STFT119872 (0)STFT119872 (119872)

STFT119872 (119873 minus119872)

]]]]]]] (79)


[[[[[[[

STFT119872 (0)STFT119872 (119872)

STFT119872 (119873 minus119872)

]]]]]]]

=[[[[[[


]]]]]]

[[[[[[[

x (0)x (119872)

x (119873 minus119872)

]]]]]]]

(80)


0 10 20 30 40 50minus300

minus200

minus100

0


(a)0 10 20 30 40 50


0

500

1000

1500

(b)



STFT = Fx (81)


[[[[[[[[

STFT1198720 (0)STFT1198721 (1198720)

STFT119872119876 (119873 minus119872119876)

]]]]]]]]

=[[[[[[




]]]]]]

[[[[[[[

x (0)x (119872)

x (119873 minus119872)

]]]]]]]

(82)


x = Fminus1119873X (83)




ySTFT = AX (85)











0

minus02

0

02


(a)0 10 20 30 40 50

minus200

minus150

minus100

minus50

0

50

100

(b)


minus04

minus02

0

02

minus100 0 100 200minus200

Time (ms)

(a)


minus1

minus08

minus06

minus04

minus02

0

02

(b)

minus04

minus02

0

02

minus100 0 100 200minus200

Time (ms)

(c)

minus1

minus08

minus06

minus04

minus02

0

02


(d)

















(a) (b)

(c) (d)

(e) (f)



20 30 40 50 6010Frequency

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Tim

e

(a)

0

1000

2000

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6000

7000

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Am

plitu

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500 1000 1500 2000 2500 3000 3500 40000Frequency

(b)


20 30 40 50 6010Frequency

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(b)




6 Conclusion



Competing Interests


Acknowledgments




References













































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119904 (119899) = 119870sum119894=1

119904119894 (119899) =119870sum119894=1

119903119894119890119895(2120587119873)(1198991198861119894+119899211988621198942+sdotsdotsdot+119899119901119886119901119894119901) (49)


119883(1198961119894 119896119901119894)= 119873minus1sum119899=0

119870sum119894=1

119903119894119890119895(2120587119873)(1198991198861119894+119899211988621198942+sdotsdotsdot+119899119901119886119901119894119901)



(1198962119894 1198963119894 119896119901119894) = (1198862119894 1198863119894 119886119901119894) (51)





119911 (119899) = 119904 (119899) 120585 (119899) = 119870sum119894=1


(53)


z = s120585 (54)


X = Fz (55)




y119911 = A119911X (56)


min 1003817100381710038171003817X11989410038171003817100381710038171st 1003817100381710038171003817y119911 minus A119911X119894

10038171003817100381710038172 lt 119879 (57)



[[[[[

119904 (1198991)

119904 (119899119872)]]]]]

=[[[[[[[[[



]]]]]]]]]

[[[[[

1199031119903119870

]]]]]

(58)


y = AR (59)





A

=[[[[[[[[[



]]]]]]]]]

(60)





(62)



H = 1119873

sdot

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

1205950 (1)(120595119873minus1 (1))2


(120595119873minus1 (119873))21205951 (1)(120595119873minus1 (1))2


(120595119873minus1 (119873))2 d

120595119873minus1 (1)(120595119873minus1 (1))2


(120595119873minus1 (119873))2

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

(63)





1205950 (119899) = 14radic120587119890minus119899221205902



119894 Ψ119894minus2 (119899) forall119894 ge 2

(65)



c = Hs (66)



Ψ =[[[[[[[


d

1205951 (119873) 1205952 (119873) sdot sdot sdot 120595119873 (119873)

]]]]]]] (67)


119904 (119899) = 119896119870sum119901=1198961

119888119901120595119901 (119899) (68)


s = Ψc (69)


A =[[[[[[[


d


]]]]]]] (70)


y = Ac (71)






Θ119888119904 =[[[[[[[[

1205951198961 (1198991) 1205951198961 (1198992) 1205951198961 (119899119872)1205951198962 (1198991) 1205951198962 (1198992) sdot sdot sdot 1205951198962 (119899119872)

d

120595119896119870 (1198991) 120595119896119870 (1198992) sdot sdot sdot 120595119896119870 (119899119872)

]]]]]]]] (74)



STFT (119899 119896) = 119872minus1sum119898=0

119909 (119899 + 119898) 119890minus1198952120587119898119896119872 (75)




STFT119872 (119899) = F119872x (119899) (77)


F119872 (119898 119896) = 119890minus1198952120587119896119898119872 (78)


STFT =[[[[[[[

STFT119872 (0)STFT119872 (119872)

STFT119872 (119873 minus119872)

]]]]]]] (79)


[[[[[[[

STFT119872 (0)STFT119872 (119872)

STFT119872 (119873 minus119872)

]]]]]]]

=[[[[[[


]]]]]]

[[[[[[[

x (0)x (119872)

x (119873 minus119872)

]]]]]]]

(80)


0 10 20 30 40 50minus300

minus200

minus100

0


(a)0 10 20 30 40 50


0

500

1000

1500

(b)



STFT = Fx (81)


[[[[[[[[

STFT1198720 (0)STFT1198721 (1198720)

STFT119872119876 (119873 minus119872119876)

]]]]]]]]

=[[[[[[




]]]]]]

[[[[[[[

x (0)x (119872)

x (119873 minus119872)

]]]]]]]

(82)


x = Fminus1119873X (83)




ySTFT = AX (85)











0

minus02

0

02


(a)0 10 20 30 40 50

minus200

minus150

minus100

minus50

0

50

100

(b)


minus04

minus02

0

02

minus100 0 100 200minus200

Time (ms)

(a)


minus1

minus08

minus06

minus04

minus02

0

02

(b)

minus04

minus02

0

02

minus100 0 100 200minus200

Time (ms)

(c)

minus1

minus08

minus06

minus04

minus02

0

02


(d)

















(a) (b)

(c) (d)

(e) (f)



20 30 40 50 6010Frequency

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1000 2000 3000 40000Frequency

(b)




6 Conclusion



Competing Interests


Acknowledgments




References













































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













A

=[[[[[[[[[



]]]]]]]]]

(60)





(62)



H = 1119873

sdot

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

1205950 (1)(120595119873minus1 (1))2


(120595119873minus1 (119873))21205951 (1)(120595119873minus1 (1))2


(120595119873minus1 (119873))2 d

120595119873minus1 (1)(120595119873minus1 (1))2


(120595119873minus1 (119873))2

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

(63)





1205950 (119899) = 14radic120587119890minus119899221205902



119894 Ψ119894minus2 (119899) forall119894 ge 2

(65)



c = Hs (66)



Ψ =[[[[[[[


d

1205951 (119873) 1205952 (119873) sdot sdot sdot 120595119873 (119873)

]]]]]]] (67)


119904 (119899) = 119896119870sum119901=1198961

119888119901120595119901 (119899) (68)


s = Ψc (69)


A =[[[[[[[


d


]]]]]]] (70)


y = Ac (71)






Θ119888119904 =[[[[[[[[

1205951198961 (1198991) 1205951198961 (1198992) 1205951198961 (119899119872)1205951198962 (1198991) 1205951198962 (1198992) sdot sdot sdot 1205951198962 (119899119872)

d

120595119896119870 (1198991) 120595119896119870 (1198992) sdot sdot sdot 120595119896119870 (119899119872)

]]]]]]]] (74)



STFT (119899 119896) = 119872minus1sum119898=0

119909 (119899 + 119898) 119890minus1198952120587119898119896119872 (75)




STFT119872 (119899) = F119872x (119899) (77)


F119872 (119898 119896) = 119890minus1198952120587119896119898119872 (78)


STFT =[[[[[[[

STFT119872 (0)STFT119872 (119872)

STFT119872 (119873 minus119872)

]]]]]]] (79)


[[[[[[[

STFT119872 (0)STFT119872 (119872)

STFT119872 (119873 minus119872)

]]]]]]]

=[[[[[[


]]]]]]

[[[[[[[

x (0)x (119872)

x (119873 minus119872)

]]]]]]]

(80)


0 10 20 30 40 50minus300

minus200

minus100

0


(a)0 10 20 30 40 50


0

500

1000

1500

(b)



STFT = Fx (81)


[[[[[[[[

STFT1198720 (0)STFT1198721 (1198720)

STFT119872119876 (119873 minus119872119876)

]]]]]]]]

=[[[[[[




]]]]]]

[[[[[[[

x (0)x (119872)

x (119873 minus119872)

]]]]]]]

(82)


x = Fminus1119873X (83)




ySTFT = AX (85)











0

minus02

0

02


(a)0 10 20 30 40 50

minus200

minus150

minus100

minus50

0

50

100

(b)


minus04

minus02

0

02

minus100 0 100 200minus200

Time (ms)

(a)


minus1

minus08

minus06

minus04

minus02

0

02

(b)

minus04

minus02

0

02

minus100 0 100 200minus200

Time (ms)

(c)

minus1

minus08

minus06

minus04

minus02

0

02


(d)

















(a) (b)

(c) (d)

(e) (f)



20 30 40 50 6010Frequency

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Tim

e

(a)

0

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Am

plitu

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500 1000 1500 2000 2500 3000 3500 40000Frequency

(b)


20 30 40 50 6010Frequency

60

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Tim

e

(a)

Am

plitu

des

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(b)




6 Conclusion



Competing Interests


Acknowledgments




References













































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











c = Hs (66)



Ψ =[[[[[[[


d

1205951 (119873) 1205952 (119873) sdot sdot sdot 120595119873 (119873)

]]]]]]] (67)


119904 (119899) = 119896119870sum119901=1198961

119888119901120595119901 (119899) (68)


s = Ψc (69)


A =[[[[[[[


d


]]]]]]] (70)


y = Ac (71)






Θ119888119904 =[[[[[[[[

1205951198961 (1198991) 1205951198961 (1198992) 1205951198961 (119899119872)1205951198962 (1198991) 1205951198962 (1198992) sdot sdot sdot 1205951198962 (119899119872)

d

120595119896119870 (1198991) 120595119896119870 (1198992) sdot sdot sdot 120595119896119870 (119899119872)

]]]]]]]] (74)



STFT (119899 119896) = 119872minus1sum119898=0

119909 (119899 + 119898) 119890minus1198952120587119898119896119872 (75)




STFT119872 (119899) = F119872x (119899) (77)


F119872 (119898 119896) = 119890minus1198952120587119896119898119872 (78)


STFT =[[[[[[[

STFT119872 (0)STFT119872 (119872)

STFT119872 (119873 minus119872)

]]]]]]] (79)


[[[[[[[

STFT119872 (0)STFT119872 (119872)

STFT119872 (119873 minus119872)

]]]]]]]

=[[[[[[


]]]]]]

[[[[[[[

x (0)x (119872)

x (119873 minus119872)

]]]]]]]

(80)


0 10 20 30 40 50minus300

minus200

minus100

0


(a)0 10 20 30 40 50


0

500

1000

1500

(b)



STFT = Fx (81)


[[[[[[[[

STFT1198720 (0)STFT1198721 (1198720)

STFT119872119876 (119873 minus119872119876)

]]]]]]]]

=[[[[[[




]]]]]]

[[[[[[[

x (0)x (119872)

x (119873 minus119872)

]]]]]]]

(82)


x = Fminus1119873X (83)




ySTFT = AX (85)











0

minus02

0

02


(a)0 10 20 30 40 50

minus200

minus150

minus100

minus50

0

50

100

(b)


minus04

minus02

0

02

minus100 0 100 200minus200

Time (ms)

(a)


minus1

minus08

minus06

minus04

minus02

0

02

(b)

minus04

minus02

0

02

minus100 0 100 200minus200

Time (ms)

(c)

minus1

minus08

minus06

minus04

minus02

0

02


(d)

















(a) (b)

(c) (d)

(e) (f)



20 30 40 50 6010Frequency

60

50

40

30

20

10

Tim

e

(a)

0

1000

2000

3000

4000

5000

6000

7000

8000

Am

plitu

des

500 1000 1500 2000 2500 3000 3500 40000Frequency

(b)


20 30 40 50 6010Frequency

60

50

40

30

20

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Tim

e

(a)

Am

plitu

des

0

1000

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7000

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1000 2000 3000 40000Frequency

(b)




6 Conclusion



Competing Interests


Acknowledgments




References













































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 10 20 30 40 50minus300

minus200

minus100

0


(a)0 10 20 30 40 50


0

500

1000

1500

(b)



STFT = Fx (81)


[[[[[[[[

STFT1198720 (0)STFT1198721 (1198720)

STFT119872119876 (119873 minus119872119876)

]]]]]]]]

=[[[[[[




]]]]]]

[[[[[[[

x (0)x (119872)

x (119873 minus119872)

]]]]]]]

(82)


x = Fminus1119873X (83)




ySTFT = AX (85)











0

minus02

0

02


(a)0 10 20 30 40 50

minus200

minus150

minus100

minus50

0

50

100

(b)


minus04

minus02

0

02

minus100 0 100 200minus200

Time (ms)

(a)


minus1

minus08

minus06

minus04

minus02

0

02

(b)

minus04

minus02

0

02

minus100 0 100 200minus200

Time (ms)

(c)

minus1

minus08

minus06

minus04

minus02

0

02


(d)

















(a) (b)

(c) (d)

(e) (f)



20 30 40 50 6010Frequency

60

50

40

30

20

10

Tim

e

(a)

0

1000

2000

3000

4000

5000

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Am

plitu

des

500 1000 1500 2000 2500 3000 3500 40000Frequency

(b)


20 30 40 50 6010Frequency

60

50

40

30

20

10

Tim

e

(a)

Am

plitu

des

0

1000

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3000

4000

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1000 2000 3000 40000Frequency

(b)




6 Conclusion



Competing Interests


Acknowledgments




References













































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0

minus02

0

02


(a)0 10 20 30 40 50

minus200

minus150

minus100

minus50

0

50

100

(b)


minus04

minus02

0

02

minus100 0 100 200minus200

Time (ms)

(a)


minus1

minus08

minus06

minus04

minus02

0

02

(b)

minus04

minus02

0

02

minus100 0 100 200minus200

Time (ms)

(c)

minus1

minus08

minus06

minus04

minus02

0

02


(d)

















(a) (b)

(c) (d)

(e) (f)



20 30 40 50 6010Frequency

60

50

40

30

20

10

Tim

e

(a)

0

1000

2000

3000

4000

5000

6000

7000

8000

Am

plitu

des

500 1000 1500 2000 2500 3000 3500 40000Frequency

(b)


20 30 40 50 6010Frequency

60

50

40

30

20

10

Tim

e

(a)

Am

plitu

des

0

1000

2000

3000

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1000 2000 3000 40000Frequency

(b)




6 Conclusion



Competing Interests


Acknowledgments




References













































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





















(a) (b)

(c) (d)

(e) (f)



20 30 40 50 6010Frequency

60

50

40

30

20

10

Tim

e

(a)

0

1000

2000

3000

4000

5000

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7000

8000

Am

plitu

des

500 1000 1500 2000 2500 3000 3500 40000Frequency

(b)


20 30 40 50 6010Frequency

60

50

40

30

20

10

Tim

e

(a)

Am

plitu

des

0

1000

2000

3000

4000

5000

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7000

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1000 2000 3000 40000Frequency

(b)




6 Conclusion



Competing Interests


Acknowledgments




References













































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










20 30 40 50 6010Frequency

60

50

40

30

20

10

Tim

e

(a)

0

1000

2000

3000

4000

5000

6000

7000

8000

Am

plitu

des

500 1000 1500 2000 2500 3000 3500 40000Frequency

(b)


20 30 40 50 6010Frequency

60

50

40

30

20

10

Tim

e

(a)

Am

plitu

des

0

1000

2000

3000

4000

5000

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7000

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1000 2000 3000 40000Frequency

(b)




6 Conclusion



Competing Interests


Acknowledgments




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









Review Article Compressive Sensing in Signal Processing...

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