Radar Imaging of Non-Uniformly Rotating Targets via a Novel Approach for Multi-Component AM-FM Signal Parameter Estimation

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1 Sensors 5, 5, ; doi:.339/s53695 Article OPEN ACCESS sensors ISSN Radar Iaging of Non-Uniforly Rotating Targets via a Novel Approach for Multi-Coponent AM-FM Signal Paraeter Estiation Yong Wang Research Institute of Electronic Engineering Technology, Harbin Institute of Technology, Harbin 5, China; wangyong6@hit.edu.cn; Tel.: Acadeic Editor: Assefa M. Melesse Received: 3 January 5 / Accepted: 7 March 5 / Published: 3 March 5 Abstract: A novel radar iaging approach for non-uniforly rotating targets is proposed in this study. It is assued that the aneuverability of the non-cooperative target is severe, and the received signal in a range cell can be odeled as ulti-coponent aplitude-odulated and frequency-odulated (AM-FM) signals after otion copensation. Then, the odified version of Chirplet decoposition (MCD) based on the integrated high order abiguity function (IHAF) is presented for the paraeter estiation of AM-FM signals, and the corresponding high quality instantaneous ISAR iages can be obtained fro the estiated paraeters. Copared with the MCD algorith based on the generalized cubic phase function (GCPF) in the authors previous paper, the novel algorith presented in this paper is ore accurate and efficient, and the results with siulated and real data deonstrate the superiority of the proposed ethod. Keywords: radar iaging; odified version of Chirplet decoposition; IHAF. Introduction Radar iaging of non-uniforly rotating targets has developed for about two decades [ 6]. It is assued that the target is engaged in coplex aneuvers, and the classical Range-Doppler (RD) algorith is ineffective to generate a well-focused radar iage because of the tie varying character for the Doppler frequency of each scatterer contribution [7]. Then, the Range-Instantaneous-Doppler (RID) technique was proposed to deal with this proble. For the RID algorith, the aziuth focusing is ipleented by the tie frequency analysis for the non-stationary signal in a certain range cell. The

2 Sensors 5, perforance of non-stationary signal has been studied for a long tie, and any useful results have been obtained [8,9]. One ind of algoriths for tie frequency analysis is based on the high resolution tie frequency distribution (TFD) with reduced cross-ters, such as the soothed-pseudo-wigner-ville distribution (SPWVD) [], or the high order TFD [ 3], but these algoriths still suffer fro the tradeoff between the tie frequency resolution and cross-ters. The other ind of algoriths for tie frequency analysis is based on paraetric techniques. These algoriths odel the received signal in a range cell as ulti-coponent aplitude-odulated and frequency-odulated (AM-FM) signals after otion copensation. By estiating the paraeters of each coponent, the focused radar iages can be obtained with the RID technique. Considering the coplications of AM-FM signal paraeter estiation, soe approxiated signal odels have been investigated recently. In [3,4], the received signal in a range cell is odeled as a ulti-coponent linear frequency odulated (LFM) signal with constant aplitude, and soe efficient algoriths have been proposed, but these algoriths are only valid in the situation where the target s aneuverability is not too severe. For targets with significantly coplex otion, high order phase ters will exist in the aziuth echoes. Then, the received signal in a range cell can be odeled as a ulti-coponent cubic phase signal. Soe efficient algoriths for the paraeters estiation of cubic phase signal are proposed in [5 8], and the radar iage quality can be iproved copared with the LFM signal odel. For the AM-FM signal odel, an efficient algorith for the paraeters estiation of it is the Chirplet decoposition. By decoposing the AM-FM signal into paraetric, redundant well localized coponents, its energy curve in the tie-frequency plane can be approxiated as the cobination of a set of beelines. Soe Chirplet decoposition algoriths are proposed in [9 ], and ost of the have been used in the field of radar iaging successfully. The Chirplet ato has the for of LFM signal odel with Gaussian envelop, and it is inappropriate to characterize the coplicated tie varying perforance for the instantaneous frequency. Then, the odified version of Chirplet ato and polynoial Chirplet transfor are proposed in [3 6], where the Chirplet ato is extended to the for of polynoial phase signal. But the corresponding signal decoposition algoriths for the are coplicated and thus suffer fro the coputational load. In this paper, the odified version of Chirplet ato is used to characterize the AM-FM signal, and a novel signal decoposition algorith based on integrated high order abiguity function (IHAF) is proposed. This algorith requires only one-diensional (D) axiizations to estiate the third order coefficient for the odified version of Chirplet ato, and the other paraeters can be obtained by the Dechirp technique and Fourier transfor. The novel algorith is used in radar iaging of aneuvering target, and the high quality instantaneous radar iages can be obtained consequently. This paper is organized as follows: in Section, the ulti-coponent AM-FM signal odel for the aziuth received signal is established. In Section 3, the principle and ipleentation of odified version of Chirplet decoposition based on IHAF are proposed. The corresponding radar iaging algorith for the non-uniforly rotating target is presented in Section 4. Section 5 is the radar iaging results for siulated and real data. Section 6 is the conclusion for the paper.

3 Sensors 5, Signal Model A plane target with coplex otion is used here as an exaple to define the radar iaging geoetry, as shown in Figure. It is assued that the otion copensation has been ipleented, and the target can be considered as a turntable target with rotating center O in the ( x, y, z) Cartesian coordinate. It is assued that the unit vector of the radar line of sight (RLOS) is r, and the z -axis is deterined by it. Ω is the synthetic vector for the angular velocity of the rotating target, and the x -axis is deterined by the z -axis and Ω. That is to say, the synthetic vector Ω is located in the x z plane. Then, the y -axis is deterined by the z -axis and x -axis as y = z x, where denotes outer product. A R x, y, z fro the rando scatterer P on the target is selected, and it is represented by the vector ( ) rotating center O to the position of point P. The purpose of radar iaging is to obtain a focused two-diensional radar iage on the iage projection plane. The high range resolution is deterined by the large bandwidth for the transitted signal, and the high cross-range resolution is obtained by the relative rotation between the radar and target. Then, the otion copensation should be ipleented before the radar iaging procedure. The otion copensation includes the range alignent and phase adjustent. p p p Figure. Radar iaging geoetry of target with non-unifor rotation. The purpose of range alignent is to copensate the translational coponent of each scatterer after range copressing, and the purpose of phase adjustent is to reove the Doppler phase caused by the translation. After otion copensation, the iaging target can be considered as a turn table target rotating around a reference point. In this case, the received signal in a range cell can be odeled as ulti-coponent aplitude-odulated and frequency-odulated (AM-FM) signals, which can be illustrated as follows: Step : The Doppler frequency for scatterer P can be expressed as: [ Ω ( R r) ] where denotes the inner product, λ represents the wavelength. f d = () λ

4 Sensors 5, Step : For the tie varying character of angular velocity of the rotating target, the synthetic vector Ω can be expressed as: Ω = K α t () = where α =,,, K are coefficients of the first, second and high order ters of Ω. t is the aziuth tie, and K is the phase order. Step 3: Substitute Equation () into Equation (), the Doppler frequency can be further written as: α is the constant ter, ( ) f d K = α t λ = ( R r) Step 4: The distance fro scatterer P to radar can be coputed as: Dis λ t t () t = f dt = α ( R r) t d t K = t dt = R K + = α ( R r) t where R denotes the initial distance fro radar to the target center at the initial tie t. Step 5: Assue that there are Q scatterers in a range cell, and the received signal can be odeled as ulti-coponent AM-FM signals as follows: s () t = A () t exp j Dis () t = Q i= A Q i= i () t i 4π exp j λ 4π λ i ( R r) K + α i R + = where A i () t, i =,,, Q is the tie varying aplitude of the i th coponent. Fro the steps above, we can see that for the aneuvering target, the received signal in a range cell can be odeled as ulti-coponent AM-FM signals after otion copensation. In order to obtain the focused two-diensional radar iages in this case, the paraeters for the ulti-coponent AM-FM signals should be estiated with high precision. Hence, the odified version of Chirplet decoposition based on integrated high order abiguity function (IHAF) is presented for the paraeters estiation of AM-FM signals in this paper, and the corresponding radar iaging schee is presented siultaneously. Then, the radar iage quality can be iproved cobined with the RID technique. 3. Modified Version of Chirplet Decoposition Based on IHAF 3.. Principle of Modified Version of Chirplet Decoposition The odified version of Chirplet ato is defined in [3] as follows: g () t = 4 exp πσ ( t t ) σ + jω t ( t t ) + jβ ( t t ) + jγ ( t t ) where the paraeter ( t, ω ) R deterines the tie and frequency center, ( β, γ ) R denotes the + chirp rate and curvature, and the variance σ R controls the width for the odified version of 3 (3) (4) (5) (6)

5 Sensors 5, Chirplet ato. Figure shows the coparison between the traditional Chirplet ato and odified version of Chirplet ato. Figure a is the tie series for the Chirplet ato; Figure b is the Wigner-Ville distribution (WVD) for the Chirplet ato; Figure c is the tie series for the odified version of Chirplet ato; Figure d is the WVD for the odified version of Chirplet ato. We can see fro Figure that the curvature for the odified version of Chirplet ato has a bending effect on the traditional Chirplet ato, and thus it is suitable for the analysis of signals with strongly nonlinear instantaneous frequencies. Then, for an arbitrary analytic signal s () t, it can be expressed by the su of g () t as follows: where () t C g () t C is the weighted coefficient to be estiated. = s (7) = (c) (d) Figure. Coparison between Chirplet ato and odified version of Chirplet ato. Tie series for the Chirplet ato; WVD for the Chirplet ato; (c) Tie series for the odified version of Chirplet ato; (d) WVD for the odified version of Chirplet ato. Then, the basic principle of odified version of Chirplet decoposition can be illustrated as follows: Step : Design a odified version of Chirplet ato g () t with the condition that the distance between s () t and its orthogonal projection on g () t is iniu, which is equivalent to: C () t, g () t s () t = s() t, g = ax s (8)

6 Sensors 5, 5 69 Step : Copute the reainder signal () t s after g () t is obtained, just as follows: () t s () t C g () t s = (9) Step 3: Repeat the steps above until the residual energy satisfies a given threshold, we obtain: s () t s () t C g () t = () It is obvious fro Equation (8) that the estiation of C requires ulti-diensional axiizations, which suffers fro high coputational load. In this paper, a novel odified version of Chirplet decoposition algorith is proposed, which is based on the integrated high order abiguity function (IHAF) proposed in [7]. The novel odified version of Chirplet decoposition algorith requires only one-diensional axiizations with high precision, and it will be illustrated in the next section. 3.. Modified Version of Chirplet Decoposition Based on IHAF The IHAF algorith is based on the high order abiguity function (HAF) proposed in [7]. The HAF can estiate the paraeters of a cubic phase signal. But for ulti-coponent signals, the cross-ters will appear and the auto-ters can not be detected correctly. Hence, the IHAF is proposed to reduce the cross-ters between different coponents. In this section, the curvature γ for the odified version of Chirplet ato is estiated by the IHAF algorith, and then the other paraeters are estiated by the Dechirp technique and Fourier transfor. For a weighted odified version of Chirplet ato with the discrete for: s () t ( t t ) = D 4 exp + jω γ πσ σ ( t t ) + jβ ( t t ) ( ) } + j t t 3 The high order abiguity function (HAF) is defined in [7] as follows: ( Θ, ) = s( t + + ) s( t ) s ( t + ) s ( t + ) ( jθ t) S + exp, dt () () where and are lags different fro zero, () denotes the conjugate. Substituting Equation () into Equation (), we obtain: Let S 4 ( ) ( ) ( + ) Θ,, = D πσ exp + exp ( t t ) σ exp t t = t, we can rewrite Equation (3) as follows: + 8 j β [ 4 jγ ( t t )] exp[ j Θt] σ dt (3)

7 Sensors 5, 5 69 S 4 ( ) ( ) ( ) Θ,, = D πσ exp + 8 j β exp[ j Θt ] + = D + t exp σ 4 [ 4 jγ t ] exp[ j Θ t ] ( ) ( ) + πσ exp + 8 j β exp[ j Θt ] t exp σ exp exp σ σ + [ j t( Θ 4γ )] dt dt (4) By using the following forula: where We obtain: + A. A and Re( ) S exp ( ) ± + = B π At Bt C dt exp + C A A + π 4 (6) σ 8 4 ( ) ( ) ( ) Θ, = D σ exp exp σ ( Θ γ ) Thus, the (, ), S Θ, yields a pea at Chirplet ato can be readily obtained as: (5) Θ = 4γ, and the curvature γ for the odified version of ( Θ,, ) 4 γ = arg ax S (7) Ω Fro the definition of HAF in Equation (), we can see that the HAF has the nonlinearity character. Hence, the cross-ters will appear for ulti-coponent odified version of Chirplet atos. Then, we can use the IHAF algorith to reduce the cross-ters, and the auto-ters can be aplified by the integration operation siultaneously. The IHAF is defined as follows: + + ( ) = S ( Θ,, ) d d G Θ (8) So, for ulti-coponent odified version of Chirplet atos, the curvature γ for the signal coponent with axiu energy should be estiated as follows: ( ) 4 γ = arg ax G Θ (9) Ω After the curvature γ is estiated, the other paraeters can be estiated by the Dechirp technique and Fourier transfor. The paraeters for other odified version of Chirplet coponents can be estiated cobined with the CLEAN technique [8] Nuerical Exaple A two coponents odified version of Chirplet atos is considered in this section to deonstrate the perforance of HAF and IHAF algoriths for the signal decoposition procedure. The paraeters for

8 Sensors 5, 5 69 the two coponents odified version of Chirplet atos with discrete for are shown in Table, where it is assued that the sapling rate is unity, and t [ 7,7]. Table. Paraeters of the siulated signal. Coponents ( ) D σ t ω β γ The siulated signal in tie doain is shown in Figure 3a. Figure 3b is the HAF for the signal, and the odified version of Chirplet atos can not be detected for the nonlinearity character of HAF. Figure 3c is the IHAF for the signal, where the lags and are selected as [: ] and [: ]. We can see that there exist two peas in the IHAF for the signal, and the odified version of Chirplet atos can be detected by the pea positions. Figure 3d is the IHAF for the signal with lags [: ] and [: 4]. It is obvious that with the increase of lag nubers in IHAF, the resolution of curvature estiation is iprove siultaneously. 5 5 (c) (d) Figure 3. Results of the nuerical exaple. Siulated signal; HAF for the signal; (c) IHAF with lags [: ] and [: ] ; (d) IHAF with lags [: ] and [: 4]. 4. Radar Iaging Based on Modified Version of Chirplet Decoposition For radar iaging of non-uniforly rotating target, the received signal in a range cell can be odeled as ulti-coponent AM-FM signals after otion copensation, which has been illustrated in Section.

9 Sensors 5, Then, the odified version of Chirplet decoposition based on IHAF algorith is proposed in Section 3 to analyze the ulti-coponent AM-FM signals. In this section, the corresponding novel radar iaging algorith is presented as follows: Step : Assue that the received signal in a range cell is Q coponents AM-FM signals, as shown in Equation (5). Step : Approxiate the ulti-coponent AM-FM signals in a range cell as a weighted su of N coponents odified version of Chirplet atos of the for: N () t C g () t = s () = where g () t has the for of Equation (6). Step 3: Initialize =, and s () t = s() t. Step 4: Estiate γ for s () t by finding the pea of G ( Θ) as Equation (9). Step 5: Dechirp the th odified version of Chirplet coponent s () t to the for of traditional 3 Chirplet ato by constructing the reference signal s () t = exp( jγ t ). We obtain: s dechirp exp.5 ( t t ) () t = s () t s () t = C ( πσ ) exp ref 3 [ j( ω β t + 3γ t ) t] exp[ j( ω t + β t γ t )] ref σ exp [ j( β 3γ t ) t ] Step 6: For the traditional Chirplet ato s dechirp () t, the chirp rate ˆ β = β 3γ t can be estiated by the existing ethods, such as the integrated cubic phase function (ICPF) algorith proposed in [9]. Then, the Chirplet ato can be further dechirped to a sinusoidal signal, and the tie center t can be estiated by the Wigner-Ville distribution (WVD) for the sinusoidal signal with the pea position. The detailed ipleentation of it can be found in [3]. Step 7: The chirp rate for the first odified version of Chirplet ato can be estiated as: () β = ˆ β + 3γ t () Step 8: The other paraeters, including ω, σ and C can be estiated by the Fourier transfor and the siilar algorith as in [3]; Step 9: Subtract the estiated th coponent fro s () t based on CLEAN technique; Step : Set = +, and repeat the above steps until = N or the residual signal energy is less than a threshold. Then, the instantaneous radar iages can be obtained by the procedure above cobined with the RID technique. Figure 4 is the flowchart for the radar iaging algorith proposed in this paper, where M is the nuber of range bins.

10 Sensors 5, Radar Iaging Results Figure 4. Flowchart of radar iaging algorith in this paper. In this section, the radar iaging results for siulated and real data are provided to deonstrate the effectiveness of the IHAF algorith for odified version of Chirplet decoposition for radar iaging of aneuvering target. 5.. Siulated Data The paraeters for the siulated data are shown as follows: the carrier frequency for the transitted LFM signal is f = 5.5 GHz, the bandwidth is B = 4 Hz, the pulse width is 5.6 µs. After otion copensation, it is assued that the target is rotating with equal changing acceleration, and the rotating paraeters are as follows: the initial velocity is. rad/s, the acceleration is.5 rad/s, and the acceleration rate is.6 rad/s 3. Figure 5 shows the siulated target odel, and it consists of 93 scatterers. Figure 5. Siulated target odel.

11 Sensors 5, Figure 6 is the radar iage based on the RD algorith. It is obvious that the iage has been blurred severely for the high aneuverability of the target. Figure 6. Radar iage based on the RD algorith. The WVD for the received signal in the 5th range bin is shown in Figure 7a, and the nonlinear character for the Doppler frequency is obvious. Figure 7b is the WVD for the two LFM signal coponents estiated fro the original signal, Figure 7c is the WVD for the two Chirplet coponents estiated fro the original signals, and Figure 7d is the WVD for the two odified version of Chirplet coponents estiated fro the original signals. We can see that the odified version of Chirplet decoposition algorith has better perforance in the presentation for the original signal. (c) (d) Figure 7. Tie frequency representations for the received signal in a range bin. WVD for the original signal; WVD for two LFM signal coponents; (c) WVD for two Chirplet coponents; (d) WVD for two odified version of Chirplet coponents.

12 Sensors 5, The instantaneous radar iages at different tie positions based on the LFM signal odel are shown in Figure 8. It can be seen that the iages quality has been iproved greatly copared with Figure 6. Then, the instantaneous radar iages at the sae tie positions as Figure 8 by the traditional Chirplet decoposition algorith are shown in Figure 9. Figure 8. Radar iages based on LFM signal odel. Radar iage at tie t =.7 s; Radar iage at tie t =. s. Figure 9. Radar iage based on Chirplet decoposition algorith. Radar iage at tie t =.7 s; Radar iage at tie t =. s. We can see that that iages quality has been further iproved. Figure are the instantaneous radar iages at the sae tie positions as Figures 8 and 9 based on the odified version of Chirplet decoposition algorith proposed in [3], and the instantaneous radar iages based on the novel odified version of Chirplet decoposition algorith proposed in this paper are shown in Figure.

13 Sensors 5, Figure. Radar iage based on odified version of Chirplet decoposition algorith proposed in [3]. Radar iage at tie t =.7 s; Radar iage at tie t =. s. Figure. Radar iage based on odified version of Chirplet decoposition algorith proposed in this paper. Radar iage at tie t =.7 s; Radar iage at tie t =. s. Coparing Figures 8, it is obvious that the iage quality in Figure is better, especially in the part of the wings of the target. This deonstrates that the iages quality for the odified version of Chirplet decoposition algorith proposed in this paper is better than the traditional radar iaging algoriths. Here, we give a quantitative coparison of the radar iages in Figures 8 by the entropy criterion with the conclusion that better focused iage has saller entropy [3]. The entropy is coputed by the definition in [3] (where the entropy is calculated without the noralization procedure for the iage, this is equivalent to the original definition and thus it is a negative nuber), and the corresponding results are shown in Table as follows: Table. Entropies of radar iages in Figures 8. Figures Figure Figure Figure Figure

14 Sensors 5, It is obvious fro Table that, the entropies for the radar iages in Figure are saller than those in Figures 8. This also deonstrates the superiority of the odified version of Chirplet decoposition algorith in this paper. 5.. Real Data For the real data, the radar paraeters are the sae with the siulated data. The raw data is collected by the radar receiver for the Ya-4 plane with the length of 36.38, the width of and the height of An optical picture of the target is shown in Figure. Figure. Optical picture of the plane. The blurred radar iage based on RD algorith is shown in Figure 3. Figure 3. Radar iage based on the RD algorith. The WVD for the received signal in the 35th range bin is shown in Figure 4a. Figure 4b is the WVD for the LFM signal odel estiated fro the original signal, Figure 4c is the WVD for the Chirplet coponent estiated fro the original signals, and Figure 4d is the WVD for the odified version of Chirplet coponent estiated fro the original signals. We can see that the odified version of Chirplet decoposition algorith has better perforance in the presentation for the original signal. Figure 4. Cont.

15 Sensors 5, (c) (d) Figure 4. Tie frequency representations for the received signal in a range bin. WVD for the original signal; WVD for the LFM signal coponent; (c) WVD for the Chirplet coponent; (d) WVD for the odified version of Chirplet coponent. Figure 5. Radar iages based on LFM signal odel. Radar iage at tie t =. s; Radar iage at tie t =.3 s. Figure 5a,b are the instantaneous radar iages at different tie positions based on the LFM signal odel. Figure 6a,b are the radar iages at the sae tie positions as Figure 5 based on the traditional Chirplet decoposition algorith. It is obvious that the iages quality has been iproved copared with Figure 3. Figure 6. Radar iages based on Chirplet decoposition algorith. Radar iage at tie t =. s; Radar iage at tie t =.3 s. Figure 7a,b are the instantaneous radar iages at the sae tie positions as Figures 5 and 6 based on the odified version of Chirplet decoposition algorith proposed in [3], and the instantaneous

16 Sensors 5, 5 69 radar iages based on the novel odified version of Chirplet decoposition algorith are shown in Figure 8a,b, respectively. Figure 7. Radar iage based on odified version of Chirplet decoposition algorith proposed in [3]. Radar iage at tie t =. s; Radar iage at tie t =.3 s. Figure 8. Radar iages based on odified version of Chirplet decoposition algorith proposed in this paper. Radar iage at tie t =. s; Radar iage at tie t =.3 s. It can be seen fro Figures 5 8 that the focus perforance of the odified version of Chirplet decoposition algorith in this paper is better than other algoriths, especially in the elliptical parts of the ISAR iages. Table 3. Entropies of radar iages in Figures 5 8. Figures Figure Figure Figure Figure The entropies for the radar iages in Figures 5 8 are shown in Table 3. This deonstrates the superiority of the novel algorith in this paper. 6. Conclusions For radar iaging of aneuvering targets, the received signal in a range cell can be odeled as ulti-coponent AM-FM signals. In this paper, the odified version of Chirplet decoposition based on the IHAF algorith is proposed to analyze the AM-FM signals. This algorith decoposes the AM-FM signals into the cobination of a series of odified version of Chirplet atos, and when

17 Sensors 5, 5 69 cobined with the RID technique, high quality instantaneous radar iages can be obtained. Results of siulated and real data deonstrate the validity of the novel algorith proposed in this paper. Acnowledgeents This wor was supported in part by the National Natural Science Foundation of China under grant 64749, the Progra for New Century Excellent Talents in University under grant NCET--49, the National Science Foundation for Post-doctoral Scientists of China under grant 3M549, the postdoctoral science-research developental foundation of Heilongjiang province under grant LBH-Q9 and the Heilongjiang Postdoctoral Specialized Research Fund. Noenclature O Rotating center of the target r Unit vector of the RLOS Outer product Inner product P Rando scatterer on the target λ Wavelength Ω Synthetic vector for the angular velocity of the rotating target K Polynoial phase order α Constant ter R Initial distance fro radar to the target center t Initial tie Q Nuber of scatterers in a range cell t Tie center of the odified version of Chirplet ato ω Frequency center of the odified version of Chirplet ato β Chirp rate of the odified version of Chirplet ato γ Curvature of the odified version of Chirplet ato σ Width for the odified version of Chirplet ato C Weighted coefficient () Conjugate Tie lags Tie lags Conflicts of Interest The author declares no conflict of interest.

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