j new approaches for chirplet approximation - math.cmu.edu · nov. 2002, vol. 1, pp. 42–46, 2002....

734 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 55, NO. 2, FEBRUARY 2007

The power of J1q + J2q can be obtained through direct calculations.

E jJ1qj2

= Es;h;�1

N2

N�1

l =0

N�1

k =0

N�1

n =0

sn ej

� hl �k �l e�j

�

N�1

l =0

N�1

k =0

N�1

n =0

s�

n e�j

hl �k �l ej

=�2�

N

N�1

k=0

min(k+V�1;N�1)

l=k

�2h;l�k: (28)

Similarly

E jJ2qj2 =

�2�

N

N�1

k=0

k�N+V�1

l=0

�2h;l+N�k: (29)

Due to the independence assumption on different paths, it is straight-forward to check also that

E J1qJ�

2q = E J�

1qJ2q = 0: (30)

REFERENCES

[1] “First report and order, revision of Part 15 of the commission’s rulesregarding ultra-wideband transmission systems,” FCC, ET Docket98–153, Feb. 14, 2002.

[2] ——, “Design of a multiband OFDM system for realistic UWBchannel environments,” IEEE Trans. Microw. Theory Tech., vol. 52,pp. 2123–2138, Sep. 2004.

[3] A. Batra, Multi-band OFDM physical layer proposal for IEEE 802.15Task Group 3a IEEE P802.15-04/0493r1, Sep. 2004.

[4] A. Bahai, B. Saltzberg, and M. Ergen, Multi-Carrier Digital Commu-nications: Theory and Applications of OFDM. New York: Kluwer,2004.

[5] B. Muquet, “Cyclic prefixing or zero padding for wireless multicarriertransmissions?,” IEEE Trans. Commun., vol. 50, pp. 2136–2148, Dec.2002.

[6] C. R. Athaudage, “BER sensitivity of OFDM systems to time synchro-nization error,” in Proc. 8th Int. Conf. Communication Systems (ICCS),Nov. 2002, vol. 1, pp. 42–46, 2002.

[7] Y. Mostofi and D. C. Cox, “Analysis of the effect of timing synchro-nization errors on pilot-aided OFDM systems,” in Proc. 37th AsilomarConf. Signals, Systems, Computers, Nov. 2003, vol. 1, pp. 638–642.

[8] Y. Zhao and S. G. Haggman, “Sensitivity to Doppler shift and carrierfrequency errors in OFDM systems-the consequences and solutions,”in Proc. IEEE 46th Vehicular Technology Conf. (VTC), Apr. 1996, vol.3, pp. 1564–1568.

[9] X. Wang, T. T. Tjhung, Y. Wu, and B. Caron, “SER performance eval-uation and optimization of OFDM system with residual frequency andtiming offsets from imperfect synchronization,” IEEE Trans. Broad-cast., vol. 49, no. 2, pp. 170–177, Jun. 2003.

[10] A. Saleh and R. Valenzuela, “A statistical model for indoor multipathpropagation,” IEEE J. Sel. Areas Commun., vol. SAC-5, pp. 128–137,Feb. 1987.

[11] J. Foerster, “Channel Modeling Sub-Committee Report Final,” IEEEP802.15-02/368r5-SG3a, Dec. 2002.

New Approaches for Chirplet Approximation

J. M. Greenberg, Zhisong Wang, and Jian Li

Abstract—This correspondence proposes efficient algorithms for ap-proximating complex-valued and real-valued signals as a weighted sumof multiple chirplet atoms, which are characterized by four parameters,namely the scale, time, frequency, and chirp rate. Direct sequential esti-mation of the parameters of multiple chirplets causes error propagations,i.e., the estimated parameters of the initial chirplets significantly affectthe parameter estimation of the subsequent chirplets and may result inlarge chirplet approximation errors. To deal with this problem, we furtherexploit a relaxation (RELAX) method for recursive chirplet parameterestimation. RELAX can be used in conjunction with time-domain or fre-quency-domain algorithms to improve the parameter estimation accuracyfor multiple chirplets. Unlike previous methods, our chirplet approx-imations require neither any a priori complete dictionary of chirpletsnor complicated multidimensional searches to obtain suitable choicesof chirplet parameters. The effectiveness of the proposed approaches isdemonstrated via a number of simulated examples.

Index Terms—Atomic decomposition, chirplets, matching pursuit, relax-ation method, time-frequency analysis, Wigner–Ville distribution.

I. INTRODUCTION

In the last decade, a lot has been written about how to approximatea given complex-valued signal

P (t) = a(t)exp(�j�(t)); a(t) � 0 (1)

by K chirplets; that is, how to find parameters Ck 2 C , �k > 0, andreal-valued numbers tk , !k , and ck so that

a(t)exp(�j�(t))'

K�1

k=0

Ck

(��k)1=4

�exp �(t� tk)

2

2�k� j !k(t� tk) +

ck(t� tk)2

2: (2)

The parameters need to be chosen so that the two functions appearingin (2) are close in some prescribed sense. It should be noted that thefunctions

g(t� tk;�k; !k; ck)def=

1

(��k)1=4

�exp �(t� tk)

2

2�k� j !k(t� tk) +

ck(t� tk)2

2(3)

Manuscript received December 9, 2004; revised April 26, 2006. The asso-ciate editor coordinating the review of this manuscript and approving it for pub-lication was Dr. Hongya Ge. This work was supported in part by the NationalScience Foundation Grants CCR-0104887 and ECS-0097636.

J. M. Greenberg is with Carnegie Mellon University, Pittsburgh, PA 15213USA (e-mail: [email protected]).

Z. Wang was with the Department of Electrical and Computer Engineering,University of Florida, Gainesville, FL 32611 USA. He is now with the Schoolof Health Information Sciences, University of Texas Health Science Center atHouston, Houston, TX 77030 USA (e-mail: [email protected]).

J. Li is with the Department of Electrical and Computer Engineering, Univer-sity of Florida, Gainesville, FL 32611 USA (e-mail: [email protected]).

Color versions of Figs. 1–5 are available online at http://ieeexplore.ieee.org.Digital Object Identifier 10.1109/TSP.2006.885725

1053-587X/$25.00 © 2006 IEEE

IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 55, NO. 2, FEBRUARY 2007 735

are typically referred to as chirplet atoms that are characterized by thescale

p�k , time tk , frequency !k and chirp rate ck , and satisfy

kgk2 = hg; gi=

1

�1

g(t� tk; �k; !k; ck)g(t� tk;�k; !k; ck)dt = 1: (4)

Here, (�) denotes the complex conjugate.Such four-parameter chirplet approximation offers more efficient

representations of the radar, seismic, and biological signals than therepresentations obtained using short time Fourier transform, Gabortransform, wavelet transform, and wavelet packets. A sampling of theliterature on this subject may be found in [1]–[6] and the referencescontained therein. Applications of chirplet approximation includetime-frequency analysis, feature extraction, data compression, anddenoising. The central problem in establishing a formula like (2) isthe development of an efficient method for obtaining the parameters(Ck; �k; tk; !k; ck). The “matching-pursuit” and “ridge-pursuit”algorithms have been used by a number of authors (for details, see[2], [4], and [5]). They may be slow since they require complicatedmultidimensional searches to obtain chirplet parameter estimates.They also demand an a priori complete dictionary of chirplets. Inthis correspondence, we present new methods for obtaining chirpletapproximations for both complex-valued signals and real-valuedsignals. For a complex-valued signal a(t)exp(�j�(t)), we proposetwo time-domain algorithms to deal with functions of the formak(t)exp(�j�k(t)) by using local information about ak(t) and �k(t)in a neighborhood of a point tk , which is the location of a global max-imum of ak(t). Similarly, we can apply a frequency-domain algorithmto deal with the Fourier transform of the complex-valued signal byusing local information in the frequency domain. To save space, thisfrequency-domain algorithm is not presented in the correspondence.For a real-valued time-domain signal, we can decompose it intochirplets based on its analytic signal using a time-domain algorithm orbased on its Fourier transform using a frequency-domain algorithm.Unlike previously proposed methods, our chirplet approximations donot require any a priori complete dictionary of chirplets, and we donot have to apply complicated multidimensional searches to obtainsuitable choices of chirplet parameters.

II. CHIRPLET APPROXIMATION FOR COMPLEX-VALUED SIGNALS

For k = 0, we let the zeroth-order residual R0(t) be the originalsignal; i.e.,

R0(t) = a(t)exp(�j�(t)); a(t) � 0 (5)

and for any integer k � 0 we assume that the kth residual has the form

Rk(t) = ak(t)exp(�j�k(t)); (6)

where ak(t) � 0 is the amplitude of the residual and �k(t) is the real-valued phase. We let tk be the location of a global maximum of ak(t)(what we have in mind is that ak(t) has a finite number of local maximaand that the global maximum may be found by a simple search) and we

assume that this maximum is nondegenerate; i.e., that a(2)k (tk) < 0.Here, (�)(m) denotes the mth-order derivative. In this situation, we let

�k=ak(tk)

ja(2)k (tk)j; �k=�k(tk); !k=�

(1)k (tk); and ck=�

(2)k (tk) (7)

and define the (k + 1)th residual by

Rk+1(t)

=Rk(t)� (��k)1=4ak(tk)exp(�j�k)g(t� tk;�k; !k; ck): (8)

Taylor’s Theorem implies that for t near to tk , Rk+1(t) =

O (t� tk)3 and thus if we write Rk+1(t) as

Rk+1(t) = ak+1(t)exp(�j�k+1(t)) (9)

the global maximum of ak+1(t) is unlikely to occur near tk but if itdoes, we are guaranteed that the (k+1)th residual is uniformly small.One could easily envision using a greedy version of the above algorithmwhere, instead of using only the global maximum at the kth stage, onealso uses some or all of the local maxima of ak(t), namely, the numberstk;1 < tk;2 < � � � < tk;N(k).

An alternative algorithm is to choose the parameters (tk; !k; ck) aswe did in (7) but to instead write the (k + 1)th residual as

Rk+1(t) = Rk(t)�1

�1

Rk(s)g (s� tk;�k; !k; ck)ds

�g (t� tk;�k; !k; ck) (10)

where �k > 0 is chosen to maximize

�k(�k) =1

�1

Rk(s)g (s� tk; �k; !k; ck)ds : (11)

We note that if the original signalF (t) = R0(t) is inL1\L2, whereLpdenotes a space composed of the measureable functions f(t) for which1

�1

jf(t)jpdt1=p

< 1, then the same will be true of each of the

residuals Rk(t). This guarantees that the numbers �k(�k) defined in(11) satisfy two properties. First, when �k goes towards 0+, �k(�k) isasymptotic to 21=2(��k)1=4ak(tk). Second, when �k goes towards1,�k(�k) is less than or equal to 1

�1

ak(s)ds=(��k)1=4. This follows

from (after substituting (3) and (6) into (11))

�k(�k) �1

�1

ak(s)1

(��k)1=4exp � (s� tk)

2

2�kds

� 21=2(��k)1=4

�1

�1

ak(s)1

(2��k)1=2exp � (s� tk)

2

2�kds (12)

and

1

�1

1

(2��k)1=2exp � (s� tk)

2

2�kds = 1: (13)


Thus, there is a point ��k < 1 such that �k(�k) attains its globalmaximum. With this alternative algorithm, we choose �k = ��k . We nolonger have the pointwise estimate Rk+1(t) = O (t� tk)

3 but wedo have

kRk+1k22 = kRkk

22 � �

2k(�k): (14)

Thus, our alternative algorithm is norm reducing in L2 with �k(�k) aslarge as possible.

Our algorithms are different from the “matching-pursuit” type algo-rithms [2], [4]. There, one attempts to pick parameters tk , �k , !k , andck so that

F (tk; �k; !k; ck)def= jhRk(t); g(t� tk; �k; !k; ck)ij

�1

�1

Rk(t)g(t� tk; �k; !k; ck)dt

(15)

is a maximum. Typically, this involves a complicated four-dimensionalsearch procedure. Once the parameters (tk; �k; !k; ck) have been de-termined, the (k + 1)th residual is defined by

Rk+1(t)=Rk(t)�hRk(t);g(t� tk; �k; !k; ck)ig(t�tk; �k; !k; ck)

(16)

and the residual satisfies

kRk+1k22 = kRkk

22 � jhRk(t); g(t� tk; �k; !k; ck)ij

2: (17)

III. CHIRPLET APPROXIMATION FOR REAL-VALUED SIGNALS

In Section II, we dealt with chirplet approximations for a complex-valued signal, while in this section, we focus on chirplet approximationfor a real-valued signal f(t), whose Fourier transform

f(!)def=

1

�1

exp (�j!t)f(t)dt (18)

satisfies

f(!) = f(�!): (19)

The identity in (19) guarantees that if we write f(!) as

f(!) = A0(!)exp (�j 0(!)) (20)

where A0(!) � 0 is the amplitude of the transform and 0(!) is itsreal-valued phase, then

0 � A0(!) = A0(�!) and 0(�!) = � 0(!): (21)

The symmetry ofA0(!) guarantees that if!0 > 0 is a global maximumof A0(!), then so is �!0. We will use both of these points next. Wedefine the first residual R1(!) by

R1(!) = f(!)�D0h0(!)

1

�1h0(!)h0(!)d!

1=2(22)

with

D0 =

1

�1A0(!)exp(�j 0(!))h0(!)d!

1

�1h0(!)h0(!)d!

1=2(23)

and

h0(!) =1

(��0)1=4exp

�(! � !0)2

2�0

�j 0+t0(!� !0)+�0

2(!� !0)

2

+ exp�(! + !0)

2

2�0+ j 0 � t0(! + !0)

+�0

2(! + !0)

2

(24)

where!0 > 0 is the location of the positive global maximum ofA0(!),the parameters ( 0; t0; �0) are given by

0 = 0(!0); t0 = (1)0 (!0); and �0 =

(2)0 (!0) (25)

and �0 is chosen to maximize jD0j.The fact that A0(!) and 0(!) satisfy (21) along with the identity

that h0(!) = h0(�!) guarantees that D0 is real, that R1(!) satisfiesR1(!) = R1(�!), and that R1(!) may be written as

R1(!) = A1(!)exp(�j 1(!)) (26)

whereA1(!) and 1(!) satisfy (21). The residualR1(!) also satisfies

1

�1

A21(!)d! =

1

�1

A20(!)d!�D

20: (27)

One now proceeds inductively in going from the kth residual

to the (k + 1)th. The key fact is that the kth residual Rk(!) =

Ak(!)exp (�j k(!)) satisfies Rk(!) = Rk(�!). This guarantees

that

0 � Ak(!) = Ak(�!) and k(�!) = � k(!) (28)


Fig. 1. Root mean-square errors (RMSEs) of t , ! , c , and � as functions of the noise standard deviation. The signal consists of one chirplet and additivecomplex Gaussian noise. (a) RMSE of t . (b) RMSE of ! . (c) RMSE of c . (d) RMSE of � .

and thus implies that if !k > 0 is the location of a global maximum

of Ak(!) then so is �!k . One then repeats (22)–(23) with the index

0 replaced by k and arrives at the formula for the (k + 1)th residual.

One also has (27) with the index 0 replaced by k and 1 by k + 1. The

identity that (k + 1)th residual in the time domain is given by

Rk+1(t)def=

1

2�

1

�1

exp (j!t)Rk+1(!)d!

= f(t)�1

2�

k

p=0

Dp

1

�1

exp (j!t)hp(!)d!

1

�1

hp(s)hp(s)ds1=2

(29)

and the identity (30), shown at the bottom of the next page, yields the

chirplet expansion for the real-valued signal f(t).In addition to using the above frequency-domain algorithm for

chirplet approximation, we can employ the time-domain algorithm

in Section II, provided that the real-valued signal is properly pre-processed. Specifically, in the preprocessing stage, we compute thecomplex-valued analytic signal of the real-valued signal based onthe Hilbert transform. We then apply the time-domain algorithm anddecompose the analytic signal into several chirplets. The sum of thesechirplets and their complex conjugates constitute the approximationof the original real-valued signal.

IV. RECURSIVE CHIRPLET PARAMETER ESTIMATION

We can implement the algorithms in the previous sections forchirplet approximation in a direct sequential way. After the parametersof the first chirplet are estimated, the first chirplet is reconstructed andsubtracted out from the original signal. Based on the remaining signal,the parameters of the second chirplet are estimated. Then, the secondchirplet is reconstructed and subtracted out from the remaining signal.This process goes on until all the parameters of the existing chirpletshave been estimated.


The problem with direct sequential estimation is that the estimationerrors of the initial chirplets may significantly affect the estimationaccuracy of the subsequent chirplets. To reduce such an error propa-gation effect, we consider recursive chirplet parameter estimation byexploiting a relaxation (RELAX)-based algorithm, which was origi-nally proposed for estimating sinusoidal parameters in the presence ofnoise [7] and later was used for angle and waveform estimation [8]. TheRELAX algorithm can be used in conjunction with either the time-do-main algorithm or the frequency-domain algorithm to refine the chirpletparameter estimates.

We use the time-domain algorithm as an example to illustrate thesteps of RELAX. Let K denote the number of chirplets we wish toestimate in the signal.

Define

Yk(t) = R0(t)�

K�1

i=0;i6=k

Cig(t� ti; �i; !i; ci) (31)

where fti; !i; ci; �i; CigK�1i=0 are the estimated chirplet parameters.

The RELAX algorithm consists of the following steps.

Step 1: Assume K = 1. Obtain t0, !0, c0, �0 and C0 fromR0(t).

Step 2: Assume K = 2. Compute Y1(t) with (31) by using t0,!0, c0, �0, and C0 obtained in Step 1. Obtain t1, !1,c1, �1, and C1 from Y1(t). Next, compute Y0(t) with(31) by using t1, !1, c1, �1, and C1 and redetermine t0,!0, c0, �0, and C0 from Y0(t). Iterate the previous twosubsteps until “practical convergence” is achieved (to bediscussed later on).

Step 3: Assume K = 3. Compute Y2(t) with (31) by usingfti; !i; ci; �i; Cig

1i=0 obtained in Step 2. Obtain t2, !2,

c2, �2 and C2 from Y2(t). Next, compute Y0(t) with(31) by using fti; !i; ci; �i; Cig

2i=1 and redetermine t0,

!0, c0, �0, and C0 from Y0(t). Then, compute Y1(t)with (31) by using fti; !i; ci; �i; Cigi=0;2 and redeter-mine t1, !1, c1, �1 and C1 from Y1(t). Iterate the pre-vious three substeps until “practical convergence.”

RemainingSteps:

Continue similarly until K is equal to the desired orestimated number of chirplets.

The “practical convergence” in the iterations of the RELAX algo-rithm may be determined by checking if the relative changes of the pa-rameter estimates are sufficiently small or if a preset maximum numberof iterations is reached.

Fig. 2. Residue mean as a function of the noise standard deviation. The signalconsists of one chirplet and additive complex Gaussian noise.

V. NUMERICAL EXAMPLES

We provide several simulated examples to demonstrate the effec-tiveness of the chirplet approximation algorithms. In all of the exam-ples considered below, we assume that the signal is a discrete time se-quence of N = 128 samples composed of several chirplets and ad-ditive zero-mean circularly symmetric complex Gaussian noise. Ac-cordingly, we use difference and summation operations instead of dif-ferentiation and integration operations, respectively, in the algorithms.For complex-valued signals, we have proposed two time-domain algo-rithms. The first one is faster than the second one since it does not re-quire a one-dimensional search for �k . However, numerical examplesshow that in the presence of noise or multiple chirplets, the estimatesof f�kg obtained via the first algorithm are not as accurate as those ob-tained via the second. In the examples below, whenever we deal withchirplet approximation for complex-valued signals in the time domain,the second algorithm is used. We define the residue at the kth stage asthe Euclidean norm of the remaining signal after k chirplets are recon-structed and subtracted out from the original signal.

A. Example 1

In the first example, we consider a signal consisting of one chirpletand additive complex Gaussian noise. The chirplet has the followingparameters:C1 exp(�j�1) = 1, t1 = 54,!1 = 40�=N , c1 = 2�=N ,

1

�1

exp(j!t)hp(!)d!

=23=2(��p)1=4Re

exp�� (t�t )

2(1+� � )+ j ( p � !ptp)� !p(t� tp)�

� � (t�t )

2(1+� � )

(1� j�p�p)1=2(30)


Fig. 3. Comparison of our method and O’Neill and Flandrin’s method in termsof (a) residue and (b) computational time. The signal consists of one chirpletand additive complex Gaussian noise.

and �1 = 4. We examine the effect of the noise standard deviationon the chirplet parameter estimates obtained from 100 Monte Carlosimulations. In Fig. 1(a)–(d), the root mean-square errors (RMSEs) ofthe chirplet parameters t1, !1, c1, and �1 are shown. The logarithmscale is not used in the y axis for Fig. 1(a) and (d) due to the fact thatzero-valued RMSEs occur when the noise standard deviation is verysmall. As expected, the larger the noise standard deviation, the worsethe parameter estimates. Fig. 2 shows the residue mean E [Rk(t)] asa function of the noise standard deviation after the estimated chirplettime sequence is subtracted out from the signal, where E[�] denotes theexpectation operator. Comparing Figs. 1 and 2, we see that the residuemean changes with respect to the noise standard deviation in a similarway as the RMSEs of the chirplet parameters. Hence, it can be usedas a performance measure of the chirplet approximation. In Fig. 3, wecompare the residue and computational time obtained via our methodand O’Neill and Flandrin’s method [3] for each of the above 100 MonteCarlo simulations when the noise standard deviation is equal to 10�3.

Fig. 4. Comparison of the residue means as functions of the second chirplet pa-rameter t . The signal consists of two chirplets and additive complex Gaussiannoise.

TABLE IPARAMETERS OF THE THREE CHIRPLETS IN THE THIRD EXAMPLE

Fig. 5. Comparison of the residues as functions of the number of the estimatedchirplets. The signal consists of three chirplets and additive complex Gaussiannoise.

Neither method requires a priori complete dictionary of chirplets. Thecomputational time measured in Matlab can indicate roughly the com-putational complexity of the chirplet decomposition. We see that ourmethod not only requires less time than O’Neill and Flandrin’s methodbut also has smaller residue than O’Neill and Flandrin’s method.


TABLE IIPARAMETER ESTIMATION ERRORS OF THE THREE CHIRPLETS IN THE

THIRD EXAMPLE

Fig. 6. (a) Wigner–Ville distribution of the real-valued signal, and (b) the sumof the Wigner–Ville distributions of the estimated pair of complex conjugatechirplets. Chirplet approximation is done in the frequency domain.

B. Example 2

We consider in the second example a signal consisting of twochirplets and additive complex Gaussian noise. The first chirplet hasthe following parameters: C1 exp(�j�1) = exp(�j5�=N), t1 = 54,!1 = 0:2�, c1 = 3�=N , and �1 = 3. The second chirplet has thesame parameters as the first one except for t2. In Fig. 4, we comparethe residue means obtained via two methods as functions of t2 obtained

Fig. 7. (a) Wigner–Ville distribution of the analytic signal, and (b) the sumof the Wigner–Ville distributions of the estimated pair of complex conjugatechirplets. Chirplet approximation is done in the time domain.

via 100 Monte Carlo simulations with the noise standard deviationequal to 10�4. For the first method, we estimate the parameters ofone chirplet first, subtract it out from the signal, and then estimatethe parameters of the second chirplet. For the second method, weapply the RELAX based recursive algorithm by using cyclic chirpletapproximation iterations to refine the parameter estimates. Fig. 4demonstrates that RELAX can help reduce the residue significantlyand hence improve the accuracy of the parameter estimates.

C. Example 3

In the third example, we consider a signal consisting of threechirplets and additive complex Gaussian noise. The parameters of thethree chirplets are shown in Table I. The noise standard deviation isequal to 10�3 and we only use one realization of it. In Fig. 5, wecompare the residues of these two methods as functions of the numberof the estimated chirplets. The absolute values of the parameter


estimation errors of the two methods are listed in Table II. It is worthnoting that RELAX can contribute to smaller residues and moreaccurate parameter estimates.

D. Example 4

Finally, we consider in the fourth example a real-valued signal,which is obtained by taking the real part of one realization of thesignal in the first example with the noise standard deviation equal to10�4. Fig. 6(a) shows the Wigner–Ville distribution of the real-valued

signal. The artifacts near the zero frequency is due to the cross-terminterferences of the Wigner–Ville distribution. To determine theparameters of the real-valued signal, we can explore two options. Oneway is to first use discrete Fourier transform (DFT) to transform thesignal into the frequency domain and then use the frequency-domainchirplet approximation algorithm. Note that a pair of complex conju-gate chirplet should be used as the bases. An alternative way for thechirplet approximation of the real-valued signal is to first compute itscomplex-valued analytic signal based on the Hilbert transform. Then,we can apply the time-domain chirplet approximation algorithm tothe analytic signal to determine the corresponding chirplet. After thatits complex conjugate counterpart can be obtained easily. Fig. 6(b)shows the sum of the Wigner–Ville distributions of the reconstructedpair of chirplets obtained via the first method. Fig. 7(a) shows theWigner–Ville distribution of the analytic signal. Fig. 7(b) shows thesum of the Wigner–Ville distributions of the reconstructed pair ofchirplets obtained via the second method. Note from Figs. 7(b) and6(b) that the time-frequency distribution of the real-valued signalis readily observable and the artifacts near the zero frequency areeliminated. This demonstrates that both approaches can be used toestimate the chirplet parameters for a real-valued signal.

VI. CONCLUSION

We have presented several time- and frequency-domain algorithmsfor efficient chirplet approximation of complex-valued and real-valuedsignals. We have also shown how a recursive relaxation (RELAX)based procedure can be used in conjunction with either the time-domain algorithm or the frequency-domain algorithm to improve theparameter estimation accuracy for multiple chirplets. Unlike previousmethods, our chirplet approximations do not require any a prioricomplete dictionary of chirplets and complicated multidimensionalsearches to obtain suitable choices of chirplet parameters. Simu-lation results have demonstrated the effectiveness of the proposedapproaches.

REFERENCES

[1] S. Mann and S. Haykin, “The chirplet transform: Physical considera-tions,” IEEE Trans. Signal Process., vol. 43, no. 11, pp. 2745–2761,Nov. 1995.

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