Novel atomic decomposition algorithm for parameter estimation of multiple superimposed Gaussian...

www.ietdl.org

Published in IET Radar, Sonar and NavigationReceived on 27th July 2010Revised on 24th March 2011doi: 10.1049/iet-rsn.2011.0012

ISSN 1751-8784

Novel atomic decomposition algorithm forparameter estimation of multiple superimposedGaussian chirpletsX. Wang J. Liu H. Meng Y. LiuElectronic Engineering Department, Tsinghua University, Beijing, People’s Republic of ChinaE-mail: [email protected]

Abstract: Atomic decomposition (AD) is an adaptive approximation technique that provides a sparse, flexible and physicallymeaningful representation of signals. Gaussian chirplet is suitable to represent radar signals because linear-frequencymodulation is very common for radar signals and chirplet exhibits good time-frequency concentration. Thus, the applicationof AD, for complex radar emitter detection and estimation, has been extensively investigated. To overcome the resolutionproblem and the ‘overfitting’ phenomenon caused when previous AD algorithms are used to deal with multiple chirplets thatare partially superimposed in the time domain, a novel algorithm using the subspace orthogonal matching pursuit technique ispresented in this study. This algorithm can also obtain higher parameter estimation accuracy, compared with other commonlyemployed AD algorithms, especially at low signal-to-noise ratio. Moreover, the parameter estimates can be further refined byan iterative alternating projection algorithm, which makes the estimation accuracy much closer to the Cramer–Rao bound.The simulation results illustrate the advantages of the two proposed methods.

1 Introduction

Joint time-frequency analysis of signals has been extensivelyinvestigated in the past several decades [1, 2]. However,commonly used time-frequency tools such as short-timeFourier transform, Gabor transform or wavelet transform,lack the flexibility to represent signals compactly with verydifferent time-frequency structures. Precise and compactinterpretation of complex signals requires decompositioninto parametric, redundant and well-localised components inthe time-frequency plane. These localised functions arecalled time-frequency atoms. Atomic decomposition (AD)[3, 4], also known as the adaptive Gabor representation[5–8], is a class of adaptive approximation techniques thatprovide a sparse, flexible and physically meaningfulrepresentation of signals with time-frequency atoms.Gaussian chirplet is suitable to represent radar signalsbecause linear frequency modulation is very common inradar signals and chirplet exhibits good time-frequencyconcentration. Such decomposition can give importantinformation about the inner structure of a signal, and theapplication of AD to signal detection, estimation andclassification has been extensively investigated in pastseveral decades [9–14].

Most AD methods are based on the matching pursuit (MP)technique [3, 4, 8–10, 15], which is used to extract atoms thatare most correlated to the chirplet signal one by one.Unfortunately, MP demonstrates poor resolution whenmultiple chirplets are close in the time domain [11, 16, 17].In this case, the extracted atoms do not correspond to the

854

& The Institution of Engineering and Technology 2011

real structure of the chirplet components and therepresentation loses its intended physical meaning. Thisproblem is inherent because of the fact that MP is a‘greedy’ algorithm in the sense that the atom extracted ateach step is the one that absorbs the most remaining energyin the signal. This results in an algorithm that sacrificeslocal fit for global fit. To overcome this limitation, the highresolution pursuit (HRP) algorithm has been proposed [16]which maximises a similar measure that is different frominner-product module in MP. HRP is able to resolvechirplets that are closely spaced in time. However, it isintended for real signals rather than complex ones and thecalculation cost of HRP is much greater than that for MP.Therefore another approach [11, 17] has been proposed thatconsists of a detector for the greedy atom after an atomicextraction by MP. The foundation of the greediness detector(GD) is similar to the HRP similarity measure. GD hasbeen proven to own higher resolution and lesscomputational load than HRP [11, 17]. However, GD isonly applicable at high ENR (signal energy to noise powerratio), and the parameter estimation performance decreasesrapidly as noise increases. This restricts its application formultiple signal detection and estimation in strong noisebackground.

Thus, a novel method, referred to as the subspaceorthogonal matching pursuit (SOMP) algorithm, ispresented in this paper, the core idea of which has beenbriefly introduced in the previous work of the authors [18].Moreover, a subspace iterative alternating projection (SIAP)algorithm is further proposed as an improvement of SOMP

IET Radar Sonar Navig., 2011, Vol. 5, Iss. 8, pp. 854–861doi: 10.1049/iet-rsn.2011.0012

www.ietdl.org

to refine the chirplet parameter estimates. Simulation resultshave shown that, compared with GD, the SOMP algorithmcan obtain better resolution and parameter estimationaccuracy at moderate ENRs in the close signal situation.The SIAP algorithm can be utilised to achieve highparameter estimation accuracy close to the Cramer–Raobound. In addition, the convergence speed of the proposedalgorithms is acceptable, making them quite suitable formultiple close signal detection and estimation in strongnoise background environments.

This paper is organised as follows. The basis model of ADand the MP algorithm are briefly introduced in Section 2. Theproposed SOMP algorithm is described in Section 3, and theSIAP algorithm is given in Section 4. Section 5 gives ananalysis of the performance of both algorithms withsimulation results. Finally, the conclusions are given inSection 6.

2 AD and MP

Let D ¼ (hg)g[G be a highly redundant dictionary of unit-energy atoms and f be the signal under analysis. Hence,AD can represent the signal by the expansion of atoms as

f ≃∑

n

bnhgn, n = 1, 2, . . . (1)

where the coefficient bn is usually a complex number and hgnthe atoms representing the time-frequency characteristics ofthe signal. Suppose Rnf is the nth order signal residual andR0f ¼ f. The MP algorithm is carried so that at eachiteration it extracts an atom gn that best matches the signalresidue Rn21f

gn = arg maxg

|kRn−1f , hgl| (2)

The coefficient bn equals the inner product between the signalresidue and the extracted atom

bn = kRn−1f , hgnl (3)

Then, the residue Rnf is sub-decomposed into

Rnf = Rn−1f − bnhgn(4)

Finally, the signal under analysis can be approximated as

f ≃∑

n

kRn−1f , hgnlhgn

, n = 1, 2, . . . (5)

As in [8], the highly redundant dictionary is composed ofchirplets (i.e. chirped Gabor functions), as linear frequencymodulation is very common for radar signals and chirpletsexhibit good time-frequency concentration [5, 6, 8]. Thus,in this paper, we all use Gaussian chirplets to representsignals. The basic chirplet function can be parameterisedthrough the following four-component vector

g = [a, b, T , f ]T [ {G = R+ × R3} (6)

in which a is inversely related to the atom duration d in thetime domain, d =

��2/a

√, T, f and b are the chirplet’s mean

time, mean frequency and chirp rate of the chirplet,


respectively, and every chirplet is defined as

hg(n) = a

p

( )1/4e−(a/2)(n−T)2

ej[2pf (n−T)+pb(n−T)2] (7)

The atom parameters can be discretised, as in [3], to form theovercomplete atom dictionary. Adaptive spectrogram (AS) [4,6] is usually used to represent the AD expansion in the time-frequency domain, defined as

AS(t, f ) =∑

n

|bn|2WVDhgn(t, f ) (8)

where WVDhgn(t, f ) is the Wigner–Ville distribution (WVD)

of every extracted atom. Thus, AS is a cross-term freedistribution that yields a better representation of signalcomponents in the time-frequency domain than the WVDof the original signal.

Suppose the original signal is the same as that used in [11,17], consisting of two chirplets of unit energy, with the samefrequency and scale, no chirp rate and being close in time as

f =∑2

q=1bqhgq

+ n (9)

Coefficient bq is usually a complex number and n is the zero-mean complex white Gaussian noise of power s2. The signallength is N ¼ 1024. The adaptive spectrogram and real part ofthe original signal free of noise are shown in Fig. 1. Thechirplet parameters are given in Table 1.

The AS and real part of the first extracted atom by MP arealso shown in Fig. 2, with estimated parameters listed inTable 2. As shown, when multiple chirplets partiallyoverlap in the time domain, MP always extracts a large-scale (i.e. long duration) atom first. This atom absorbs mostof the signal energy, and it does not correspond to any ofthe original components. The residue introduced by the firstextracted atom should be compensated by the second

Fig. 1 Adaptive spectrogram and real part of original signal(noise free)

Table 1 Parameters of the original signals

Atom1 [b1, a1, b1, T1, f1] Atom2 [b2, a2,b2, T2, f2]

[1, 1024, 0, 400, 0.125] [1, 1024, 0, 720, 0.125]

855


www.ietdl.org

atoms, and thus neither of them corresponds to the realchirplet component. The large-scale atom is always definedas a greedy atom [11, 17], and this phenomenon is called‘overfitting’.

3 SOMP algorithm

To avoid the ‘overfitting’ phenomenon and extract the trueatoms corresponding to the chirplet components, it isnatural to consider that if we can construct a subspacespanned by both the greedy atom (large-scale) and the trueatoms (small-scale) corresponding to the chirpletcomponents, the projection of signals on the subspaceobtained by the least squares (LS) method can tend towardsthe true atoms, but not concentrate on the greedy one.Based on this consideration, a subspace extraction methodaimed at overfitting is first proposed so that the selectedatoms to span a subspace at each iteration should satisfy thefollowing conditions. First, the subspace should containdifferent scale atoms with relatively large inner products.Besides, to guarantee the projection performance of the LS,once an atom exists in a subspace, other atoms with similarscales and strong correlation to it should be excluded fromthe subspace. Balancing these two requests, the subspaceshould be designed so that the atoms at similar scalesshould be weakly correlated, whereas the correlationbetween atoms at different scales has no limitation. Thesubspace extraction method can be summarised as follows:

1. Discretise the atom dictionary to obtain K atoms. N is thesignal length. Set q ¼ 1 and the correlation threshold Th,scale similarity r and subspace atom number m. Calculatethe modules of the inner product between the current signal

Fig. 2 Adaptive spectrogram and real part of the first extractedatom by MP (noise free)

Table 2 Parameters of the first two extracted atoms by the MP

algorithm

Atom 1 [b1, a1, b1,T1, f1]

[1.396, 1.9953 × 1025, 0, 560, 0.125]

Atom 2 [b2, a2, b2, T2, f2]

[20.272, 5.012 × 1024, 0, 560, 0.125]

856


residue Rn21f and every atom as

rk = |kRn−1f , hgk=(ak ,bk ,Tk ,fk )l| k = 1, 2, . . . , K (10)

Inner product means kx(n), y(n)l ¼ Snx(n)y∗(n).2. Choose the atom that is best correlated with the signalresidue

kq = arg maxk

{rk , k = 1, 2, . . . , K} (11)

3. Calculate the modules of the inner product between thebest correlated atom and atoms at a similar scale

gk =|khgkq

, hgkl|, if ak , r ·akq

0, else

{k = 1, 2, . . . , K (12)

4. Eliminate the atoms strongly correlated to hgkq

{rk = 0, if gk ≥ Th}, k = 1, 2, . . . , K (13)

5. q ¼ q + 1 and repeat steps (2)–(5) to determine k1, k2, . . . ,km and construct the subspace as S = [hgk1

hgk2· · · hgkm

].

From the above steps, we know how to design a subspacespanned by m atoms for the current signal residue.Subsequently, the SOMP algorithm for close chirpletrepresentation can be elaborated as follows.

1. The signal under analysis is f. Set n ¼ 1, R0f ¼ f, andsupport set V0 ¼ f and parameter 1 to end the iteration.2. Choose m atoms for signal residue Rn21f to constructsubspace Sn = [hn

gk1hngk2

· · · hngkm

] by the subspace extractionmethod stated above.3. Calculate the orthogonal projection of Rn21f on Sn

(cn1, cn

2, . . . , cnm) = arg min

c(q)||Rn−1f −

∑m

q=1c(q)hn

gkq||22

(14)This can be solved by the LS method as

(cn1, cn

2, . . . , cnm)T = (SH

n Sn)−1SHn Rn−1f (15)

4. Choose the atom hggnwith the maximal projection

coefficient module as

gn = arg maxk

{|cnk |, k = 1, . . . , m} (16)

and merge it into the support set Vn = [Vn−1, hggn].

5. Calculate the signal residue Rnf by projecting signal fonto the orthogonal complement of Vn as

Rnf = (I − PVn)f = f −Vn(VH

n Vn)−1VHn f (17)

where PVn= Vn(VH

n Vn)−1VHn is the projection matrix.

6. Determine the terminal condition. If ||Rnf ||22 ≤ 1||f ||22, thealgorithm ends and the approximation of signal is

f ≃ Vn[VHn Vn]−1

VHn f (18)

with the extracted atoms {hgg1, hgg2

, . . . , hggn}. Otherwise, set


www.ietdl.org

n ¼ n + 1 and repeat steps 2–6 until the terminal condition isachieved.

Up to now, the SOMP algorithm has been detailed. For theexample in Section 2, the AS and the real part of the first twoextracted atoms by the SOMP algorithm are shown in Fig. 3,with parameters listed in Table 3. For the SOMP, theparameters are defined as m ¼ 3, r ¼ 5, Th ¼ 0.2 and1 ¼ 0.05.

Actually, it is important for the subspace extraction methodto choose the parameters. All the parameters in the SOMPalgorithm cannot be determined in theory, and theseparameters are chosen jointly to ensure that the effectiveatoms can be extracted in the subspace. r is related to thescale similarity between the greedy atom and the trueatoms. If r is set too large, the true atom will be regardedas the scale similar atom with the greedy atom andeliminated because of the high correlation. Usually, to beon the safe side, r is usually chosen to be a little small tomake sure that the effective atoms will not be eliminatedduring iterations. m is related to the number of chirpletcomponents. Owing to the lack of prior knowledge, m isusually chosen to be large enough to make sure that theeffective atoms can be extracted in the subspace. Th isdecided by the correlation between the true atoms andgreedy atom.

It can be seen that the SOMP algorithm can overcomethe ‘overfitting’ phenomenon and distinguish two chirpletcomponents successfully. The greedy atom has the largerinner product module with the original signal comparedwith any of the true atoms. However, when it coexists inthe subspace with the true atoms, the projection coefficientsby LS can reflect the real structure of the chirpletcomponents. In other words, LS can give rise to a subspaceprojection that minimises the distance to the signal, as in(16). However, the inner product only reflects thecorrelation between the signal and atoms. Experimental

Fig. 3 Adaptive spectrogram and real part of the first twoextracted atoms by SOMP (noise free)

Table 3 Parameters of the first two extracted atoms by SOMP

Atom 1 [b1, a1, b1, T1, f1]

[1.02, 9.534 × 1025, 0, 696, 0.125]

Atom 2 [b2, a2, b2, T2, f2]

[0.96, 1.258 × 1024, 0, 392, 0.125]


results show that the parameters b and f have exactestimates, but the estimates of a and T have relatively largedeviations. This is because the overlapped chirpletcomponents can cause the first extracted atom to absorb asmall part of the energy of the second chirplet component,causing it to become longer than it really is (i.e. a becomessmaller). Accordingly, the second extracted atom becomesshorter and has a smaller projection coefficient than thefirst one. Thus, in the next section, another algorithmis introduced to improve the parameter-estimationperformance of the SOMP algorithm.

4 SIAP algorithm

The SOMP algorithm described in the previous section can beimplemented to distinguish multiple superimposed chirplets.However, because of two main reasons, it is impossible toachieve parameter-estimation accuracy high enough. First,the overlapped chirplet components mean that any extractedatom can hardly correspond to only one chirpletcomponent. Second, the SOMP algorithm is implementedin a direct sequence manner, meaning that the atoms areextracted one by one. Once an atom is extracted, it willnever be changed. This will result in that an estimationerror of the initial atoms may significantly affect thesubsequent atoms. Li [19] has proposed a RELAX-basedrecursive algorithm to reduce the error propagation by usingcyclic chirplet approximation iterations. Based on the ideaof iteration, an SIAP algorithm is proposed in this sectionto refine the parameter estimates of the close chirpletcomponents by the SOMP algorithm.

Suppose the extracted atoms are {hgk, k = 1, 2, . . . , n}

and that hgiis the last extracted atom. The following

S\i = [hg1· · · hgi−1

hgi+1· · · hgn

] (19)

is defined as a subspace spanned by the given (n 2 1) atoms,except for hgi

. Thus, the optimal approximation of signal fby subspace S\i is well known to be

f = S\i(SH\i S\i)

−1Sf (20)

PS\i= S\i(S

H\i S\i)

−1SH\i is defined as the projection matrix, and

P⊥S\i

= I − PS\iis the orthogonal projection matrix. The

residue can be calculated as

R\if = P⊥S\i

f = (I − PS\i)f = f − S\i[S

H\i S\i]

−1SH\i f (21)

Hence, the next atom hgito be extracted should be the one

corresponding to the maximal projection coefficient modulein the subspace constructed for signal residue R\if by thesubspace extraction algorithm. In theory, the next extractedatom hgi

should be the same as hgi. Otherwise, the extracted

atoms should be revised to represent the chirpletcomponents more adequately. Based on this consideration,an SIAP algorithm is proposed as follows to refine theparameter estimate.

1. Suppose the number of atoms is n ¼ 1. Construct subspaceS1 = [hgk1

hgk2· · · hgkm

] for signal residue R0f by the subspaceextraction algorithm. Choose the atom g1 = (a1, b1, T1, f1)that corresponds to the maximal projection coefficientmodule, as in (16).

857


www.ietdl.org

2. Obtain the support set Vn = [hg1, . . . , hgn

] and calculatethe signal residue Rnf, as in (17).3. Let n ¼ n + 1. Choose the next atom gn = (an, bn, Tn, fn)for signal residue Rnf, as (16).4. For i ¼ 1, . . . , n, construct a subspace with the renewedatoms, except hgi

, as S\i = [hg1· · · hgi−1

hgi+1· · · hgn

] andcalculate the signal residue Rn

\if as

Rn\if = f − S\i[S

H\i S\i]

−1SH\i f (22)

Redetermine gi = (ai, bi, Ti, fi) for the signal residue Rn\if via

(16).5. Iterate step 4 until ‘practical convergence’ is achieved.‘Practical convergence’ means that the relative change of{gi = (ai, bi, Ti, fi) i = 1, . . . , n} between two consecutiveiterations is less than a predefined threshold or that a presetmaximum number of iterations is reached.6. Repeat steps 2–5 until ||Rnf ||22 ≤ 1||f ||22.

The SIAP algorithm is also used for the example in Section2. The AS and the real part of the first two extracted atoms bySIAP are shown in Fig. 4, with parameters listed in Table 4.

It can be seen that the SIAP algorithm can extract twochirplets that exactly correspond to the chirplet componentswith no noise. This is because the cyclic approximationiterations can reduce the estimate error gradually until thealgorithm converges. In other words, the SIAP algorithmcan improve the precision of the parameter estimatessignificantly. As the noise increases, the ability of the twoalgorithms in resolving multiple close chirplets andestimating chirplet parameters still needs to be verified, asdiscussed in the next section.

5 Numerical examples

In this section, numerical examples are provided todemonstrate the performance of the SOMP and SIAP

Fig. 4 Adaptive spectrogram and real part of the first twoextracted atoms by SIAP (noise free)

Table 4 Parameters of the first two extracted atoms by SIAP

Atom 1 [b1, a1, b1, T1, f1] Atom 2 [b2, a2, b2, T2, f2]

[1, 1024, 0, 400, 0.125] [1, 1024, 0, 720, 0.125]

858


algorithms. Some traditional algorithms, like MP and GD,are used for comparison. The signal under consideration isthe same as that used above, but with noise. Noise energy isvalued by ENR, where ENR ¼ |b|2/s2 and signal to noiseratio (SNR) ¼ ENR/d. ENR is usually used [11, 17], as itdoes not depend on the signal parameters.

5.1 RMSEs of parameter estimates

First, the RMSEs of the estimation of the atom scale a1, a2

and atom time centre T1, T2 by different algorithms againstENR are illustrated, as shown in Figs. 5 and 6. Theparameters b and f can be estimated accurately and are notdiscussed here. The Cramer–Rao bound, calculated as in[20], is also depicted as the benchmark. The RMSEs wereobtained through 100 Monte-Carlo trials. As shown inFig. 5, the RMSE of a1 of the first extracted atom by MP isalways large, approaching the true value of a1 (a1 ¼ 1024).This means that the estimate of a1 is much smaller than thetrue value. In other words, because a1 is inversely related tothe scale or duration, the scale of the first extracted atom isalways much longer than it really is. Accordingly, the scaleof the second extracted atom by MP is much smaller andthe estimate of a2 is larger than the true value (a2 ¼ 1024).This ‘overfitting’ phenomenon is inherently because of thegreedy character of the MP algorithm. For the GDalgorithm, the greedy atom can be rightly distinguished andabandoned at high ENR (.25 dB). Subsequently, a newextraction by MP can be carried out in a reduced dictionary.However, the greedy atom will still be extracted at lowENR (,25 dB) and the RMSEs of a1 and a2 are as largeas MP. This is because the judgment of the atom’sgreediness by GD is only efficient at high ENR (to beanalysed in Section 5.2), whereas, for the SOMP algorithm,the estimation of a1 and a2 is quite robust to noise, and thegreedy atom will never be extracted at moderate ENR(.13 dB). By overcoming the overfitting problem causedmainly by the simple use of an inner product, the subspaceprojection by LS can reflect the real structure of the chirpletcomponents. Moreover, the projections of noise on differentatoms in the subspace are approximately equivalent. Thus,the noise will not destroy the relative size of the projectioncoefficients. The amplitude of the projection coefficients ofdifferent atoms will be analysed in Section 5.2. Finally, theresults of the SIAP algorithm have shown that theparameter estimation accuracy of a1 and a2 can tend closerto the corresponding Cramer–Rao bound, as the recursivealgorithm can refine the parameter estimate to some extent.From Fig. 6, we obtained a similar conclusion for theestimates of T1 and T2. The MP algorithm will alwaysextract the greedy atom, whose mean time is at the centre ofthe entire signal (T1 = 500 − 600). Thus, the residue willconcentrate on the middle part of the signal, and the timecentre of the second extracted atom will still remain at thecentral part (T2 = 500 − 600) to make up for the errorcaused by the first greedy atom. The estimation accuraciesof T1 and T2 by GD are as low as MP at low ENR (,25).However, at high ENR the estimation performance of GD issimilar to the SOMP algorithm with small discrepanciesfrom the true value. The discrepancies are caused by theoverlapped chirplet components, which can be significantlyovercome using the SIAP algorithm because the iterationprocess revises the estimates continuously untilconvergence. In brief, both SOMP and SIAP are morerobust to noise than MP and GD, and they are applicablefor close signal resolution and parameter estimation.


www.ietdl.org

Fig. 6 RMSEs of T1 and T2 estimate obtained by different algorithms against ENR

Fig. 5 RMSEs of a1 and a2 estimate obtained by different algorithms against ENR

5.2 Key points of GD and SOMP

In this section, the reason for the effectiveness of the SOMPalgorithm at low ENR, compared with the GD algorithm, isanalysed. First, a brief introduction of the GD algorithm isgiven [8, 11]. GD is based on the similarity measure, whichis the same as the HRP algorithm [16]. For the pthextracted atom (hgp

) by MP, a set of subatoms ({hgpi}) are

defined with the same frequency law, finer scale and timesupport inside the time support of hgp

. GD is defined as

LGREED(hgp) = max

i{|Xpi|}

.Greedy atom

,Non-greedy atom

Th (23)

where Xpi is a normalised version of Qgpi

Xpi =Qgpi

− |kRp−1f , hgpl|

m+ sQgpi

(24)

The ratio Qgpiis

Qgpi=

|kRp−1f , hgpil|

|khgp, hgpi

l| ≃ |b + ri| (25)

where ri is the zero-mean complex Gaussian random noise,with the standard deviation calculated as

sQgpi= (

��||Rpf ||2/N

√)/(

��2

√× |khgp

, hgpil|). In theory, Qgpi

is a Rice random variable only if a high ENR is assumed.In this case, Xpi can be approximated as a real Gaussian


random variable with zero mean and unit variance [8], ifhgp

is the true atom. Thus, the variable maxi{|Xpi|} can beused for determination of the atom greediness. The variablemaxi{|Xpi|} for both the true atoms and the greedy atomagainst ENR is plotted in Fig. 7. The number of subatomsis 100.

As shown in Fig. 7, the true atoms approximately satisfythe distribution Xpi � N(0, 1), and the variable maxi{|Xpi|}is always below three or four under any ENR. Therefore thedetector is quite suitable to verify the true atoms. However,for the greedy atom, the variable maxi{|Xpi|} is amonotonically increasing function of the ENR and theassumption maxi{|Xpi|} ≥ Th is not true at low ENR. In

Fig. 7 maxi{|Xpi|} of greedy atom and true atoms along with ENR

859


www.ietdl.org

other words, the determination of atom greediness by Th isunreliable at low ENR, and it is a contradiction to set theTh. If Th is high, the greedy atom will first be extracted anddetermined as a non-greedy atom at low ENR to representthe signal. Otherwise, the true atom will also be determinedas greedy and abandoned. Th is usually set between twoand four to ensure that the true atoms be determinedcorrectly. This is why the algorithm always fails at ENRvalues smaller than 25 dB.

Next, for the SOMP algorithm, the extracted subspace at eachiteration should contain not only the atom with the maximuminner product module but also some other atoms withrelatively large inner products. The projection coefficients |cl

q|of the first extracted subspace spanned by three atoms areshown in Fig. 8.

It can be seen that the projection coefficient module ofdifferent atoms remains invariable at high ENRs. Neither ofthe projection coefficients for the small-scale atoms is equalto 1, as both have discrepancies in their time-centreestimates. The projection coefficient module of the greedyatom is always smaller than the small-scale atoms. Thisfurther proves the feasibility and effectiveness of the SOMPalgorithm. Moreover, the noise projections on differentatoms in subspace are approximately equivalent and havelittle effect on the relative size of the projection coefficientsmodule. In other words, the projection coefficients of noisysignals on valid atoms corresponding to the chirpletcomponents are always larger than that of the greedy atomat moderate ENR, until the noise is large enough to destroythe basic signal structure. When the ENR is smaller than8 dB, the noise will destroy the envelope shape of twochirplet components entirely and make the signal appear asa large envelope. In this case, the projection coefficient ofthe greedy atom is the largest, and the greedy atom will beextracted first to represent the signal.

From above, we know why the GD and SOMP algorithmsare effective for different ENR ranges. The recovered signalsat low ENRs using the first two extracted atoms by differentalgorithms are shown in Figs. 9b–e. The original signal isat an ENR of 20 dB (SNR ¼ 210.2 dB), as shown in Fig. 9a.

It is obvious that at an ENR value of 20 dB, the signal isentirely covered by noise and cannot be detected easily.Based on different AD methods, the signal can berepresented in different forms. The first atom extracted byMP and GD is the greedy atom. Thus, the second atom is

Fig. 8 Projection coefficients |cql | of the first subspace spanned by

three atoms

860


chosen to compensate the residue caused by the first atom.Yet, using the SOMP algorithm, the two atoms can becorrectly detected to represent the noisy signal, but withgreat discrepancies in the estimates of scale and mean time.The SIAP algorithm can successfully improve the parameterestimation performance and perfectly recover the originalsignal. This also demonstrates that a good AD method canbe regarded as a denoising technique and a featureextraction tool.

5.3 Resolution of the SOMP algorithm

It is important to discuss about the resolution of thealgorithms for parameter estimation of multiple chirpletcomponents. In fact, two chirplet components in thesimulation are entirely the same as in the time domain,except the different time centre. Thus, heavy overlap willlead to large distortion of both chirplets. The resolution ofour proposed algorithm could be obtained approximately asfollows. The interval of the time centre of two chirplets isnormalised by the signal length N ¼ 1024. Noise is free.The other parameters are the same as above. Then, theestimation error of parameter T can be shown as Fig. 10. Itcould be obviously seen that the estimation error is smallenough at the normalised interval larger than 0.35. But atsmall interval, the estimation error is large and not

Fig. 9 Real part of

a Original signal at ENR of 20 dBb Extracted atoms by MPc Extracted atoms by GDd Extracted atoms by SOMPe Extracted atoms by SIAP


www.ietdl.org

monotonous, because different intervals will lead to differentenvelope shapes.

5.4 Comparison of computational complexity

Finally, Table 5 shows the computational time for extractingthe first two atoms by different methods. All simulationswere run on a PC with an IntelwCoreTM2 Duo [email protected] GHz processor.

It can be seen that MP is rapid because it uses only thesimple MP and omits the influence of overlapped chirplets.Thus, the result is wrong. GD is a little faster than SOMP,since SOMP has to calculate inner-product many times. Butat low ENR, GD does not work. SIAP is obviously theslowest algorithm since the precise estimation of chirpletparameters is at the cost of many iterations. Therefore itcould be said that our algorithm needs an acceptableincrease of computational cost to achieve robust performance.

6 Conclusion

The MP algorithm for AD lacks resolution when the signalconsists of several components that are close in the timedomain. To solve this problem, a SOMP algorithm isproposed that replaces the extracted atom by a subspace ateach iteration. The subspace is constructed considering boththe LS coefficient matrix and the ‘overfitting’ phenomenon.The atom in the subspace corresponding to the maximumprojection efficient module is chosen iteratively to representthe signal. Next, a SIAP algorithm is proposed to improvethe parameter estimation performance of SOMP caused by

Table 5 Computational time for extracting the first two atoms

MP GD SOMP SIAP

0.136167s 0.403337s 0.962504 s 1.502937s

Fig. 10 RMSEs of T estimate obtained by SOMP againstnormalised interval


error propagation in a directly sequential way. Simulationresults prove that the close atoms comprising the signal canbe correctly detected and estimated by these algorithms, andthe parameter estimation performance is much better thanthe GD and MP algorithms, especially at low ENRs. Theconvergence speed of the algorithms is acceptable, and it isquite suitable for the detection and estimation of multiplesignals that are close in time.

7 References

1 Frisch, M., Messer, H.: ‘The use of the wavelet transform in thedetection of an unknown transient signal’, IEEE Trans. Inf. Theory,1992, 38, (2), pp. 892–897

2 Friedlander, D., Porat, B.: ‘Detection of transient signals by Gaborrepresentation’, IEEE Trans. Acoust. Speech Signal Process., 1989,37, (2), pp. 169–180

3 Mallat, S.G., Zhang, Z.: ‘Matching pursuits with time-frequencydictionaries’, IEEE Trans. Signal Process., 1993, 41, (12),pp. 3397–3415

4 Mallat, S.G.: ‘A wavelet tour of signal processing’ (Academic Press,1998)

5 O’Neill, J.C., Flandrin, P.: ‘Chirp hunting’. IEEE Int. Symp. Time-Frequency and Time-Scale Analysis, 1998

6 Qian, S., Chen, D.: ‘Joint time-frequency analysis. Methods andapplications’ (Prentice Hall, 1996)

7 Yin, Q., Qian, S., Feng, A.: ‘A fast refinement for adaptive Gaussianchirplet decomposition’, IEEE Trans. Signal Process., 2002, 50, (6),pp. 1298–1306

8 Bultan, A.: ‘A four-parameter atomic decomposition of chirplets’, IEEETrans. Signal Process., 1999, 57, (3), pp. 731–745

9 Lopez-Risueno, G., Grajal, J.: ‘Unknown signal detection via atomicdecomposition’. Proc. 11th IEEE Workshop on Statistical SignalProcessing, SSP 2001, Singapore, August 2001, pp. 174–177

10 Lopez-Risueno, G., Grajal, J., Yeste-Ojeda, O.A.: ‘Signal detection andclassification using atomic decomposition’. Proc. European SignalProcessing Conf., EUPSICO 2002, Toulouse, France, 2002, vol. 1,pp. 129–132

11 Lopez-Risueno, G., Grajal, J., Yeste-Ojeda, O.A.: ‘Atomicdecomposition-based radar complex signal interception’. IEE Proc.Radar, Sonar and Navigation, 2003, vol. 150, pp. 323–331

12 Lopez-Risueno, G., Grajal, J.: ‘Multiple signal detection and estimationusing atomic decomposition and EM’, IEEE Trans. Aerosp. Electron.Syst., 2006, 42, pp. 84–102

13 Lopez-Risueno, G., Grajal, J., Sanz-Osorio, A.: ‘Digital channelizedreceiver based on time-frequency analysis for signal interception’,IEEE Trans. Aerosp. Electron. Syst., 2005, 41, pp. 879–898

14 Yeste-Ojeda, O.A., Grajal, J., Lopez-Risueno, G.: ‘Atomicdecomposition for radar applications’, IEEE Trans. Aerosp. Electron.Syst., 2008, 44, (1), pp. 187–200

15 Laura, R.N., David, L.: ‘Optimized orthogonal matching pursuitapproach’, IEEE Signal Process. Lett., 2002, 9, (4), pp. 137–140

16 Jaggi, S., Karl, W.C., Mallat, S.G., Willsky, A.S.: ‘High resolutionpursuit for parameter extraction’, Appl. Comput. Harmon. Anal., 1998,5, (4), pp. 428–449

17 Lopez-Risueno, G., Grajal, J., Yeste-Ojeda, O.A.: ‘Greediness detectionin atomic decomposition. Coordination of two dictionaries’. Proc. IEEEInt. Conf. Acoustic, Speech and Signal Processing, ICASSP 2002,Orlando, FL, USA, 2002, pp. 1441–1444

18 Liu, J.Y., Meng, Y.M., Wang, X.Q.: ‘Atomic decomposition-basedparameter estimation of multiple overlapped signals’, Electron. Lett.,2010, 46, (14), pp. 1024–1025

19 Greenberg, J.M., Wang, Z., Li, J.: ‘New approaches for Chirpletapproximation’, IEEE Trans. Signal Process., 2007, 55, (2),pp. 734–741

20 O’Neill, J.C., Flandrin, P.: ‘Sparse representations with chirplets viamaximum likelihood estimation’, IEEE Trans. Signal Process., 2006,42, pp. 84–102

861


Novel atomic decomposition algorithm for parameter estimation of multiple superimposed Gaussian...

Documents

Transcript of Novel atomic decomposition algorithm for parameter estimation of multiple superimposed Gaussian...