Hindawi Publishing CorporationEURASIP Journal on Advances in Signal ProcessingVolume 2008, Article ID 456598, 8 pagesdoi:10.1155/2008/456598

Research ArticleModified Adaptive Chirplet Decomposition withApplication in ISAR Imaging of Maneuvering Targets

YongWang and Yi-Cheng Jiang

Research Institute of Electronic Engineering Technology, Harbin Institute of Technology, Harbin 150001, China

Correspondence should be addressed to Yong Wang, wangyong6012@hit.edu.cn

Received 21 February 2008; Revised 21 July 2008; Accepted 20 October 2008

Recommended by Sharon Gannot

The modified adaptive Chirplet decomposition (MACD) is presented in this paper. It is based on the extension of the traditionalChirplet atoms to the form of the quadratic frequency-modulated signals. The accuracy of the signal decomposition is improvedcompared with the traditional Chirplet decomposition. A practical algorithm for the parameters estimation of monocomponentMACD atom and the decomposition method based on the CLEAN technique to the multicomponent signals are proposed. Finally,this algorithm is used in the ISAR imaging of maneuvering targets, and the quality of ISAR images improves greatly. The results ofreal data and simulated data demonstrate the validity of the method proposed.

Copyright © 2008 Y. Wang and Y.-C. Jiang. This is an open access article distributed under the Creative Commons AttributionLicense, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properlycited.

1. INTRODUCTION

In the inverse synthetic aperture radar (ISAR) imaging ofmaneuvering targets, the received signals can be consideredas multicomponent polynomial phase signals after motioncompensation. The parameters of each component shouldbe estimated accurately in order to obtain the high-qualityrange-instantaneous-Doppler images of the target [1]. Forthe parameters estimation of multicomponent signals, manymethods have been proposed in the past decades, such as thejoint time frequency algorithm [2], the CLEAN technique bydechirping, and the RELAX method [3, 4]. But, all of thesemethods are appropriate to deal with the linear frequency-modulated signals, which can be accepted when the maneu-verability of the target is not strong. In practice, the instancewhen the target has a high-maneuvering movement oftenoccurs. If we use the traditional methods to get the ISARimages, there will be many false scatters, and the qualityof the images will be influenced. A signal decompositionmethod-adaptive chirplet decomposition is proposed in[5], this algorithm decomposes a multicomponent signalsinto the combination of a series of chirplet atoms. Theparameters of the chirplet atoms are estimated adaptivelyby the principle of matching pursuit according to thecharacteristics of signals. This method improves the accuracyof the parameters estimation greatly, but the chirplet atoms

still have the form of linear frequency-modulated signalsand cannot characterize the complicated polynomial phasesignals accurately.

In this paper, a new algorithm of signal decomposition-modified adaptive chirplet decomposition (MACD) is pro-posed. The form of the chirplet atoms is extended tothe quadratic frequency-modulated signals in order tocharacterize the original signals more accurately. First, thedefinition of MACD is given, and the parameters estimationalgorithm for monocomponent signal is studied, Then, theCLEAN technique is used to estimate the parameters ofmulticomponent signals. Finally, this method is used in theISAR imaging of maneuvering targets, and the quality ofISAR images is improved greatly.

2. THE BRIEF INTRODUCTIONOF ADAPTIVECHIRPLET DECOMPOSITION

The adaptive chirplet decomposition is an approximationtechnique, providing representation of the signals, for a givensignal s(t), it can be represented by a sum of rectangle chirpletatoms gn(t) as follows:

s(t) =∞∑

n=0Cngn(t), (1)

2 EURASIP Journal on Advances in Signal Processing

gn(t) =(πσ2n

)−0.25exp

{−(t − tn

)2

2σ2n+ jωn

(t − tn

)

+ jβn(t − tn

)2}

,

(2)

where σn, tn, ωn, and βn indicate time extent, time center,initial frequency, and chirp rate, respectively.

The chirplet atoms can be estimated based on thefollowing principle:

∣∣Cn∣∣2 = max

gn

∣∣〈sn(t), gn(t)〉∣∣2, (3)

where

sn(t) = sn−1(t)− Cn−1gn−1(t), s0(t) = s(t), (4)

where sn(t) is the residual of orthogonal projection of sn−1(t)on gn−1(t), which is

sn(t) = s(t)−n−1∑

i=1, i /=nCigi(t). (5)

The decomposition of signals can be realized by comput-ing (3)–(5).

Taking the Wigner-Ville distribution of the chirpletatom, (2) yields

WVDgn(t,ω) = 2 exp{−(t − tn

)2

σ2n

− 4σ2n[(

ω− ωn)

2− βn

(t − tn

)]2}.

(6)

By calculating the partial derivative with respect to ωand by equaling it to zero, we can see that the Wigner-Villedistribution of the chirplet atom gn(t) is concentrated on thecurve

ω = 2βnt +(ωn − 2βntn

) = 2βnt + �n, (7)

where �n = ωn − 2βntn.

3. MODIFIED ADAPTIVE CHIRPLETDECOMPOSITION (MACD)

The modified adaptive chirplet decomposition (MACD) isproposed based on the adaptive chirplet decomposition, andthe atoms has the form of

hn(t) =(πσ2n

)−0.25exp

{−(t − tn)2

2σ2n+ jωn

(t − tn

)

+ jβn(t − tn

)2+ jγn

(t − tn

)3}

,

(8)

where σn, tn, ωn, and βn indicate time extent, time center,initial frequency, and chirp rate, respectively. γn is defined asthe curvature of the atom, which makes the time frequency

Freq

uen

cy

Time

(a) Traditional chirplet atom

Freq

uen

cy

Time

(b) MACD atom

Figure 1: Time frequency distributions of the atoms.

distribution of the atom become a curve, as shown inFigure 1.

The MACD is an approximation technique providingrepresentation of the signals. For a given signal s(t), it canbe represented by a sum of new atom hn(t), that is,

s(t) =∞∑

n=0Dnhn(t). (9)

The MACD algorithm finds the atom hn(t) most similar withsn(t):

∣∣Dn∣∣2 = max

hn

∣∣〈sn(t),hn(t)〉∣∣2, (10)

where

sn(t) = sn−1(t)−Dn−1hn−1(t), (11)

where sn(t) is the residual of orthogonal projection of sn−1(t)on hn−1(t), s0(t) = s(t).

As we can see from (10) that the computation of thecoefficient Dn is a multidimensional nonlinear optimizationproblem, and we cannot get the analytic results. If we esti-mate the parameters by searching throughout the parametersspace, the computation burden will be very heavy. Somealgorithms for the parameters estimation are proposed forthe linear frequency-modulated chirplet atoms [6, 7], butstill need pursuit in a certain range or adaptive recursivecomputation. In this paper, a new practical parametersestimation algorithm of MACD is proposed.

3.1. The parameters estimation ofmonocomponent signal

For a monocomponent signal, we can use the parametersinformation and the characteristics of the modified chirpletatoms to get the estimation of the parameters by somemature methods in the signal processing. The steps are givenas follows.

Step 1. Suppose the original signal s(t) = D0h0(t), constructthe reference signal sref(t) = exp(− jγt3 − jβt2), where γ and

Y. Wang and Y.-C. Jiang 3

β are the variables to be determined. Multiply the referencesignal with s(t), we have

s(t)× sref(t) = D0(πσ20

)−0.25 × exp{−(t − t0)2

2σ20

}

× exp { j(− ω0t0 + β0t20 − γ0t30)}

× exp { j(ω0 − 2β0t0 + 3γ0t20)t}

× exp { j(β0 − β − 3γ0t0)t2}

× exp { j(γ0 − γ)t3}.

(12)

We can see from (12) that when γ = γ0 and β = β0−3γ0t0,the s(t) × sref(t) will be a sinusoidal signal of variable t. Ifwe compute the Fourier transform of it, there will be a peakin the γ − β plane, and the other undetermined parameters(such as ω0, t0, and σ0) will not influence it. Hence, we canget the estimated values γ̂0 and β̂ by searching the peak of(12).

Step 2. Construct a new reference signal based on γ̂0 and β̂:

ŝref(t) = exp{− jγ̂0t3 − jβ̂t2

}. (13)

Step 3. Compute the following equation:

�̂ = max�

{FFT

[s(t)× ŝref(t)

]}, (14)

where FFT[·] denotes the Fourier transform.By designing a filter with narrow bandwidth and filtering

out the narrow spectrum nearby �, we can get an approx-imated sinusoid u(t) by computing the inverse Fouriertransform of the spectrum, that is,

u(t) = IFFT{window(�)× FFT[s(t)× ŝref(t)]}

, (15)

where

window(�) ={

1, �L < � < �R,

0, else,(16)

is the filter, �L and �R are determined by the width ofthe spectrum. Hence, IFFT[·] denotes the inverse Fouriertransform. The width of the window will affect the result ofsignal decomposition, if the width is large, other componentwill be filtered out combined with the narrow spectrum andwe cannot get an approximated sinusoid u(t) by computingthe inverse Fourier transform of it. Hence, the width of thewindow must be small. The precise implementation of thewindow width may be computationally expensive, but inthe condition of ISAR imaging, the accuracy of the windowwidth is not the main factor and it can be easily realized bythe observation of the width of the narrow spectrum nearby�. Compared with the Gaussian filter presented in [6], thismethod is more suitable to the ISAR imaging.

Step 4. Compute the Wigner-Ville distribution (WVD) ofu(t), we can get t̂0 of s(t) as

t̂0 = arg maxt

[WVDu(t,ω)

]. (17)

We have to illustrate that the estimation of t̂0 based on(17) will produce error with low signal to-noise-ratio (SNR),and the derivation of (17) is based on the high SNR forthe signals, which is appropriate for the ISAR imaging ofmaneuvering targets in most cases. Furthermore, the valueof t̂0 will not influence the quality of the ISAR image greatly.

Then, we can get the estimated value of β0 as

β̂0 = β̂ + 3γ̂0 t̂0. (18)Step 5. Construct the reference signal as

s′ref(t) = exp{− jβ̂0

(t − t̂0

)2 − jγ̂0(t − t̂0

)3}, (19)

and we can get the value ω̂0 by

ω̂0 = maxω

{FFT

[s(t)× s′ref(t)

]}. (20)

Step 6. Compute the value σ̂0 according to

σ̂0 = arg maxσ

∣∣〈s(t),h(t)〉∣∣2, (21)

where

h(t) = (πσ2)−0.25 exp{−(t − t̂0

)2

2σ2+ jω̂0

(t − t̂0

)

+ jβ̂0(t − t̂0

)2+ jγ̂0

(t − t̂0

)3}.

(22)

Step 7. Get the value D̂0 by (10).It is obvious that the estimation of D̂0 will suffer from the

error propagation problem. But, in this paper, the parametersestimation of MACD atoms can be treated as a choice of theinitial values, which is accurate enough in the ISAR imaging.The method to improve the accuracy of the parametersestimation can be found in [8] as an example.

We estimate the parameters of monocomponent signalby the steps described above, since the process of compu-tation requires only simple pursuit, Fourier transform, andmultiplication, the computation efficiency is high, and thealgorithm can be realized easily. Next, we will study theparameters estimation algorithm for the multicomponentsignal.

3.2. Parameters estimation ofthemulticomponent signal

We use the CLEAN technique [9–11] to estimate theparameters of the multicomponent signal. The steps aregiven as follows.

Step 1. Estimate the curve gene γ̂1, chirp rate β̂1, time centert̂1, initial frequency ω̂1, chirp width σ̂1, and the coefficient D̂1of the first signal s1(t) by (12), (17), (18), (20), (21), and (10).

Step 2. Subtract the estimated signal from the original signalaccording to

s2(t) = IFFT{

[1−window(ω)]× FFT[s(t)× e− jφ̂1]}e jφ̂1 ,(23)

4 EURASIP Journal on Advances in Signal Processing

Am

plit

ude

−0.3

−0.2

−0.1

0

0.1

0.2

Time

0 100 200 300 400

(a)

Freq

uen

cy

400

300

200

100

Time

100 200 300 400

(b)

Freq

uen

cy

400

300

200

100

Time

100 200 300 400

(c)

Freq

uen

cy

400

300

200

100

Time

100 200 300 400

(d)

Figure 2: The bat signal and its time frequency distributions: (a) the bat signal, (b) Wigner-Ville distribution, (c) traditional chirpletdecomposition, and (d) MACD algorithm.

Ran

gebi

n

140

120

100

80

60

40

20

Doppler bin

20 40 60 80 100 120 140

Figure 3: ISAR images of real data based on the range-Doppleralgorithm.

where window(ω) is as (15), s(t) is the original signal, and

φ̂1 = ω̂1(t − t̂1

)+ β̂1

(t − t̂1

)2+ γ̂1

(t − t̂1

)3. (24)

Step 3. Estimate the curve gene γ̂2, chirp rate β̂2, time centert̂2, initial frequency ω̂2, chirp width σ̂2, and the coefficient D̂2of the second signal s2(t) by (12), (17), (18), (20), (21), and(10).

Step 4. Repeat the steps above until the energy of the residualsignal is less than a threshold ε:

∫ +∞

−∞

∣∣sres(t)∣∣2dt ≤ ε. (25)

In practice, such as the ISAR imaging of maneuveringtargets, the threshold ε can be selected as about 10% of theenergy for the original signal, which will be suitable to theISAR imaging, and the selection of ε is not the same indifferent circumstances.

We use the echolocation signals from brown bat toillustrate the effectiveness of the MACD algorithm, andthe results are shown in Figure 2. Figure 2(a) is the batsignal in time domain, Figure 2(b) is the Wigner-Villedistribution of the signal, and the influence of the cross-terms is obvious. Figure 2(c) is the result of the traditional

Y. Wang and Y.-C. Jiang 5

Ran

gebi

n

140

120

100

80

60

40

20

Doppler bin

20 40 60 80 100 120 140

(a)

Ran

gebi

n

140

120

100

80

60

40

20

Doppler bin

20 40 60 80 100 120 140

(b)

Ran

gebi

n

140

120

100

80

60

40

20

Doppler bin

20 40 60 80 100 120 140

(c)

Ran

gebi

n

140

120

100

80

60

40

20

Doppler bin

20 40 60 80 100 120 140

(d)

Figure 4: ISAR images of Yak-42: (a) chirplet decomposition, (b) MACD, (c) chirplet decomposition, and (d) MACD.

Ran

gebi

n

60

40

20

Doppler bin

50 100 150 200 250

Figure 5: ISAR images of simulated data based on the range-Doppler algorithm.

chirplet decomposition. The cross-terms have been reducedcompletely, but the nonlinear time varying property of thesignal has not been characterized. Figure 2(d) is obtained by

the MACD algorithm proposed in this paper, it is obviousthat the MACD depicts the nonlinear and linear componentsof the bat signal and can demonstrate the time varyingcharacter of the signal more accurately.

4. ISAR IMAGING OFMANEUVERING TARGETSBASED ONMACD

When the target has a maneuvering movement, the rota-tion axis and velocity both vary with time. Hence, thereceived signals are time-varying and the traditional range-Doppler algorithm will not be appropriate. Since the conceptof range-instantaneous-Doppler algorithm was proposedrecently, this algorithm is based on the assumption that thereceived signals of the scatters are multicomponent linearfrequency-modulated signals. But, in some circumstances,the maneuverability of the target is high, and the receivedsignals are high-order polynomial phase signals. In order toget the clear ISAR image in this case, a method based on theproduct high-order ambiguity function for multicomponentpolynomial phase signals is proposed in [12], but thisalgorithm suffers from the computational burden. Another

6 EURASIP Journal on Advances in Signal ProcessingR

ange

bin

60

40

20

Doppler bin

50 100 150 200 250

(a)R

ange

bin

60

40

20

Doppler bin

50 100 150 200 250

(b)

Ran

gebi

n

60

40

20

Doppler bin

50 100 150 200 250

(c)

Ran

gebi

n

60

40

20

Doppler bin

50 100 150 200 250

(d)

Ran

gebi

n

60

40

20

Doppler bin

50 100 150 200 250

(e)

Ran

gebi

n

60

40

20

Doppler bin

50 100 150 200 250

(f)

Figure 6: ISAR images of simulated data based on the chirplet decomposition.

algorithm based on the traditional chirplet decompositionis proposed in [13], this method approximates the receivedsignals as a series of linear frequency modulated signals. But,the accuracy is not high, and there are many false scattersin the image. So, a new ISAR imaging algorithm based onMACD is proposed in this paper, and the quality of ISARimages is improved greatly.

After the motion compensation of the received signals,the received signals in a range bin can be approximated asmulticomponent polynomial phase signals with changingamplitude, the model is as follows:

s(t) =K∑

k=1ak(t) exp

(jφk(t)

),

φk(t) = φ0,k + 2πM∑

m=1ck,mt

m,

(26)

where ak(t) is the amplitude of the kth signal, and ϕk(t) is thephase.

As mentioned previously, the MACD algorithm cancharacterize the multicomponent polynomial phase signalsmore accurately than the traditional chirplet decomposition.Now, we compare the ISAR imaging results of real data andsimulated data based on the two algorithms.

(1) Real data

We choose a segment of real data when the Yak-42 plane hasa high-maneuvering flight. The radar works in the C band,with a bandwidth of 400 MHz. Figure 3 is the ISAR imagebased on the range-Doppler algorithm; the image is blurredand cannot be recognized. Figures 4(a) and 4(c) are the ISARimages based on the traditional chirplet decomposition atdifferent time, and the quality is improved, the change ofthe target’s motion is also reflected. Figures 4(b) and 4(d)are the ISAR images based on the MACD algorithm at thesame time with Figures 4(a) and 4(c), we can see that thequality is improved furthermore, and the false scatters aredecreased. In both cases, we decompose the received signalsof each range bin into three chirplet atoms or MACD atoms,and the energy of the residual signals is less than 10% of theoriginal signals, that is, the energy threshold can be chosenas 10%.

(2) Simulated data

We choose the simulated data of B727 plane, which hasa slight maneuvering flight. The radar transmits steppedfrequency signals, the carrier frequency f0 = 9 GHz,bandwidth Bd = 150 MHz, the number of pulses is 256.

Y. Wang and Y.-C. Jiang 7R

ange

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Doppler bin

50 100 150 200 250

(a)R

ange

bin

60

40

20

Doppler bin

50 100 150 200 250

(b)

Ran

gebi

n

60

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Doppler bin

50 100 150 200 250

(c)

Ran

gebi

n

60

40

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Doppler bin

50 100 150 200 250

(d)

Ran

gebi

n

60

40

20

Doppler bin

50 100 150 200 250

(e)

Ran

gebi

n

60

40

20

Doppler bin

50 100 150 200 250

(f)

Figure 7: ISAR images of simulated data based on the MACD algorithm.

Figure 5 is the ISAR image based on the range-Doppleralgorithm, the image is blurred but the shape of the plane canbe recognized for the slight maneuverability. Figures 6(a)–6(f) are the ISAR images based on the traditional chirpletdecomposition at different time, and the quality is improvedgreatly. Figures 7(a)–7(f) are the ISAR images based on theMACD algorithm at the same time with Figures 6(a)–6(f).In both cases, we decompose the received signals of eachrange bin into two chirplet atoms or MACD atoms. Forthe slight maneuverability of the targets, the quality of theimages based on the MACD algorithm is analogous withor a bit better than the images based on the traditionalchirplet decomposition, which indicates the robustness ofthe proposed algorithm.

From the two examples above, we can see that theMACD algorithm can generate clear instantaneous ISARimages for common maneuvering targets. In the condition offast maneuvering movements, the long coherent processingintervals (CPIs) will be used to obtain finer resolutions,and the image plane is nonstationary. This causes severedefocusing which cannot be corrected by using higher-orderpolynomial phase models, and this is the main limitation forthe technique in this paper. An efficient way to solve thisproblem is the subaperture and super-resolution techniques.

5. CONCLUSION

A new algorithm of signal decomposition-modified adaptivechirplet decomposition (MACD) is proposed in this paper.The traditional chirplet decomposition cannot characterizethe original signal accurately, and this disadvantage isovercome by extending the chirplet atom to the form ofquadratic frequency-modulated signal. Then, the parametersestimation algorithm of monocomponent atom is given,and the CLEAN technique is used in order to estimate theparameters of multicomponent signals. Finally, we use thismethod in the ISAR imaging of maneuvering targets, and thequality of ISAR images improves greatly.

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1. INTRODUCTION2. THE BRIEF INTRODUCTION OF ADAPTIVE CHIRPLET DECOMPOSITION3. MODIFIED ADAPTIVE CHIRPLET DECOMPOSITION (MACD)3.1. The parameters estimation of monocomponent signal3.2. Parameters estimation of the multicomponent signal

4. ISAR IMAGING OF MANEUVERING TARGETS BASED ONMACD(1) Real data(2) Simulated data

5. CONCLUSIONREFERENCES