A. Scaglione, S. Barbarossa

Transcription

1 280 ESTIMATING MOTION PARAMETERS USING PARAMETRIC MODELING BASED ON TIME-FREQUENCY REPRESENTATIONS A. Scaglione, S. Barbarossa Univ. of Rome La Sapienza (ITALY) 1 ABSTRACT In this work we propose a method for estimating the instantaneous phase shift induced by the relative radartarget motion based on two main steps: 1) time-frequency representation of the received echoes; 2) estimation based on parametric modeling. The estimates can then be used to initialize a compensation procedure for obtaining the ISAR image or to extract features to be used for target classification directly from the received signal, without necessarily passing through the target image. In the paper we show some application examples of the proposed procedure in the presence of noise. 1. INTRODUCTION ISAR imaging requires some kind of motion compensation to form the synthetic aperture necessary to get the desired cross-range resolution [2],[7],[ll],[12]. Being the motion a-priori unknown, it must be estimated from the data. The information on the relative motion is contained on the instantaneous phase shift induced by the motion on the radar echo. Therefore, the basic step in the estimation of the relative motion passes through the estimation of the instantaneous phase of the received signals. This requires the presence of prominent scatterers on the target, whose radar echoes are sufficiently stronger than the background to allow a reliable estimate. The situation is complicated, however, when dominant scatterers occupy the same range cell, situation which is likely to occur in many practical circumstances. In fact, if no a-priori knowledge of the relative motion is available, at the beginning of the estimation procedure it is safer do not use a high range resolution to avoid the range migration problem. On the other hand, the worse is the range resolution This work was supported by Alenia-Elsag Sistemi Navali. the higher is the probability to observe more than one dominant scatterer on the same range cell. As proposed in [3], an iterative procedure can be followed in these cases: 1) the range resolution is initially kept low, e.g. without exploiting all the bandwidth, to avoid the range resolution problem; 2) the instantaneous phase of the echo from a dominant scatterer is estimated; 3) the change of distance is recovered from the estimate of the instantaneous phase and used for compensating the range migration, 4) the data are analyzed using the full range resolution capabilities. The estimation of the instantaneous phase is rather complicated when more dominant scatterers occupy the same range cell because, in such a case, we should separate the components corresponding to each echo before estimating their instantaneous phases. The separation is simple if the components have a linear phase, in which case an FFT-based approach may be sufficient. However, especially when long observation intervals are used to obtain a high cross-range resolution, the phase cannot be assumed to be simply a linear function of time. Time-frequency representations of the signal can provide an important analysis tool in such cases. In this paper we propose a method for separating the signal components and estimating their instantaneous phases using the so called Reassigned Smoothed Pseudo Wigner- Ville Distribution (RSPWVD) followed by a parametric estimation method. The paper is organized as follows: Section 2 describes a proper echo model for the backscattering from prominent points belonging to a rigid body; Section 3 proposes the use of the RSPWVD for the analysis of the radar echoes and Section 4 provides results concerning the application of parametric techniques. Radar 97, October 1997, Publication No. 449 OlEE 1997

2 ECHO FROM A ROTATING RIGID BODY We assume that the target is a rigid body lying in the far field of the radar antenna and is characterized by a certain number of dominant scatterers. Transmitting at a frequency fo = c/x, the echo from the k-th scatterer is: A k e jz?rfo(t-21r,-r;(t)l/c), k=o,..., K-1 (1) where r, is the vector indicating the radar position, ri(t) indicates the k-th scatterer and K is the number of scatterers. Under the far field hypothesis, the echo can be approximated as: A ke j2nfote-j4ar,/xej4ne?r~(t)/x) (2) where R, = ITo[. The motion of a rigid body can always be expressed in terms of translation of one point and rotation of the body around that point. Every imaging or classification procedure must apply some kind of motion compensation and, in general, the translational motion is compensated first. This operation is performed by multiplying the radar echo by a reference signal matched to the echo from one dominant scatterer, and resampling the data in range to remove any range migration of the scatterer assumed as a reference. Taking as a reference the echo from the 0-th scatterer in (2) (setting k = 0 and A0 = l), the signals after compensation assume the following form: Akej4nl'Tqk(t)/X, k=1,..., K-1, (3) where q k(t) := ri(t) - rb(t). Imposing the rigid body constraint, the vectors qk(t) can only rotate and the rotation matrix has to be the same for all the points belonging to the target. The differential equation characterizing the rotation of the generic vector qk (t) is: (4) where o(t) is the vector containing the instantaneous pitch, roll and yaw pulsations (wp(t),w,(t), wy(t)) and x denotes vectorial product. Making the assumption of a constant pulsation, i.e. w(t) = o, and indicating d - by R its modulus, i.e. R = w: + w," +U;, given an initial position qk(0) = qk of the vector at time to = 0, the solution of equation (4) is: ck. qk(t) = ak -k bkcos(rt) + -SZn(flt), R (5) where ak, bk, ck are the following vectors: Hence the echo from the generic k-th scatterer is: A j4te?qk(t)/x = A jl?r(mkcos(rt+~k)+cyk)/x, ke ke where: It is important to notice that the instantaneous phases of each echo contain a constant term plus sinusoidal contributions having the same frequencies for all the scatterers but different amplitudes and initial phases. This is a consequence of the rigid body constraint. 3. SIGNAL ANALYSIS BASED ON TIME-FREQUENCY REPRESENTATIONS The general model for the frequency modulation induced by the relative radar-target motion can be always decomposed in the sum of a slow component, which can be well approximated by a low order polynomial plus a possible fast component having a sinusoidal behavior. The echo can then be modeled as: where w(t) is additive noise. The slow component is mainly due to the translation whereas the fast component depends on the rotation. If the period T = 27~ f R of the sinusoidal component is much larger than the duration To of the observation interval, the sinusoidal component can also be approximated as a low order polynomial and the overall signal can then be modeled as a polynomial-phase signal (PPS). However, when T is much less than To, the sinusoidal component has to be estimated separately. On the other hand the velocity of variation is not known a-priori. Therefore it has to be estimated USing, at least initially, a non parametric approach. For this reason, we first analyze the received signal using time-frequency distributions (TFD) as a way to obtain preliminary information about the signal modulation law. In particular, we use the Reassigned Smoothed Pseudo Wigner-Ville Distribution (RSPWVD), introduced by Auger and Flandrin [l] for its good localization properties. The information extracted using the RSPWVD will then be used to properly initialize parametric methods aimed at improving the estimation accuracy. The smoothed pseudo WVD (SPWVD) of a (9)

3 282 signal ~ (t) is defined as: SPWVg,h(Z; t, w) = (14) JJg(u)h(.r)z(t - U + $)x*(t - U - S)e-jw dudt, where g(u) and h(u) are two proper weighting functions with H(0) = 1 and G(0) = 1 (G(f) and H (f) denote the Fourier transforms of g(t) and h(t), respec- tively). The reassignment method consists in taking a local average of the WVD, to attenuate the undesired cross terms, but, instead of assigning the average to the center of the weighting window, e.g. to the generic point of coordinates t and w, the SPWVD is assigned to a point of coordinates related to the center of gravity of the distribution, that is to the point of coordinates PI: A possible way for extracting informations about the sinusoidal component is to estimate the coordinate w of the maxima of RSPWVD,,h(x;t,w) for each t. Therefore we define the following function: Umaz(t) = argmaz,{rspwvd,,h(z; t, U)}. (17) The sinusoidal component can then be extracted computing the FFT of Q(t). For example, the FFT of the function Q(t) relative to the RSPWVD of Fig.1 is shown in Fig.2 (the mean value has been subtracted before computing the FFT). We can see that, in spite 1200 where 7g(t) := tg(t) and Dh(t) := dh(t)/dt. The final distribution is then: RSPWVD,,h (2; t; U) = J- SPWVg,h(Z; t, U )@ - i(t,w))b(w - &(t,w)dt dw. As an example, Fig.1 shows the RSPWVD of a signal modeled as in (13), where the polynomial component is a second order PPS and the period of the sinusoidal component is one fourth of the duration interval. The 0.4 I f 0- n k Figure 1: RSPWVD of a signal having a phase modulation expressed as the sum of a polynomial plus a sinusoidal component. Signal-to-Noise Ratio (SNR) is 3 db and the number of samples is 128. We can see that, in spite of the low SNR, the sinusoidal component is clearly visible. time Figure 2: Spectral analysis of the lieu of the maxima ofthe RSPWVD. of the high noise, the peak due to the sinusoidal component is clearly visible. The low frequency spectral components are due to the superimposed polynomial contribution. Being the RSPWVD a nonlinear transformation, it suffers from the creation of cross terms when applied to multiple component signals. However, the smoothing implicit in the computation of the RSPWVD provides a consistent attenuation of the cross terms, at least as far as they are not too overlapped in the time-frequency plane. For example, in the case of two echoes having same amplitude and a sinusoidal FM, with the same period, the result is shown in Fig.3. The two components are clearly visible. Of course, if the frequency components are not well separated the discrimination capability decreases. We assume that the rotating object has two dominant scatterers, having the same backscattering coefficients and that the observation interval is equal to the equivalent rotation period, i.e. T = 27r /R. 4. PARAMETRIC ESTIMATION OF INSTANTANEOUS PHASES The echo modeling shown in Section 2 can be exploited to improve the performance of the estimation method, 5

4 283 only a 2D function: t 120 \.os Figure 3: Reassigned Smoothed Pseudo Wigner Ville Distribution of the sum of two echoes from a rotating object. with respect to the straightforward use of the RSP- WVD. The RSPWVD is based on a smoothing of the TFD independent of the signal model. However, if the signals can be expressed in a parametric form, an improvement can be obtained by matching the smoothing to the class of signals under analysis. In principle, considering signals expressed as in (9), we could smooth the WVD integrating it over all possible sinusoids in the time-frequency plane, as a function of the sinusoidal parameters. Clearly this operation would be quite troublesome from the computational point of view. However the operation is almost equivalent to compute the square modulus of the scalar product between the received signal and the signal model (9), with Ak = 1. Hence the detection and estimation of FM signals satisfying (9) can be carried out by searching for the peaks of the following function: where N is the number of samples: If a peak exceeds a suitable threshold, a sinusoidal FM signal is detected and its parameters are estimated as the coordinates of the peak. Of course the straightforward application of (18) is also quite troublesome from the computational point of view. However, a great simplification comes from observing that all signal components have the same pulsation 52. Therefore, estimating 52 from the analysis of the RSPWVD as shown in the previous section, we can use that estimate in (18) and compute Figure 4: P(m, q5) of a three components sinusoidal FM signal. where 6 is the estimate obtained from the RSP- WVD. An estimation example is shown in Fig.4, relative to the case in which three sinusoidal FM signals occupy the same range cell. Fig.4 shows the function P(m,q5) given in (19). We can clearly observe the presence of three peaks. A further simplification of the proposed procedure consists in analyzing time intervals smaller than the rotation period. In these sub-intervals, the instantaneous phases can be approximated by polynomials (the first terms of their Taylor s series expansion). In this case we can use specific algorithms devised for the detection and parameter estimation of multicomponent polynomial phase signals embedded in noise, based on the so called high order ambiguity function [SI. Once the parameters of the polynomials have been estimated, we can exploit the relationship among homologous coefficients pertaining to different scatterers to improve the estimation accuracy. In fact we know that the frequency of the sinusoidal component has to be the same for all scatterers. More specifically, the basic steps are the following: we estimate the k-th order polynomial phase parameters Ui,k for each i-th component and then evaluate the sinusoidal parameters, defined in (9), as follows:

5 284 Trans. on Signal Processing, Vo1.43, No.5, May 1995, pp An application example of this method to the echoes from rotating objects is reported in Fig.5, where the instantaneous frequency is modeled as a third order polynomial. D.Ausherman, A.Kozma, J.L. Walker, H.M. Jones, E.C.Poggio, Developments in radar imaging, IEEE Trans. on Aerospace and Electronics Systems, Vol.AES-26, No.4, pp , S.Barbarossa, Detection and imaging of moving objects with SAR - Part I: Optimal detection and parameter estimation, IEE Proceedings, Part F, pp.79-87, Vo1.138, No.2, Febr. 92. S.Barbarossa, A.Scaglione, Target recognition using a set of scale, rotation and translation invariant features extracted directly from the radar echoes, Proc. of Radar 97, Edinburgh (UK), Oct SBarbarossa, A.Scaglione, Motion compensation techniques for target classification based on parametric modeling of the radar echoes, submitted to IEE Proc., I 5O time Figure 5: Estimation of the instantaneous phases using the high order ambiguity function; true instantaneous phase (solid line), estimations (dashed line). 5. CONCLUSION In this paper we have shown some advanced signal processing tools for tracking the instantaneous frequency induced on the radar echoes by the relative radar-target motion. The proposed approach is able to track arbitrary frequency modulation laws. The rigid body constraint has been explicitly taken into account. The proposed method is then particularly suitable for inverse SAR applications, as a motion compensation basic tool. In a companion paper [4] we use the algorithms proposed in this paper to extract features necessary for the classification of ship targets directly on the radar data, without the need of forming a synthetic aperture image. The details about the combination of the two methods are provided in the journal version [5]. 6. REFERENCES [l] F. Auger and P. Flandrin, Improving the Readability of Time-Frequency and Time-Scale Representations by the Reassignment Method, IEEE I S. Barbarossa, A. Porchia, and A. Scaglione, Multiplicative multilag higher-order ambiguity function Proc. of Int. Conf. on Acoustics, Speech, and Signal Processing, Atlanta, GA, vol. 5, pp ,May 7-11, W. G. Carrara, R. S. Goodman, R. M. Majewski, Spotlight Synthetic Aperture Radar, Artech House (Norwood, MA), October T.Crimmins, Private communications (1988). S. Werness, M. Stuff and J. Fienup, Twodimensional imaging of moving targets in SAR data, 24th Asilomar Conference on Circuit, Systems, and Computers, Monterey, CA, Nov [lo] M. Stuff, R. Sullivan et al, Automated two and three-dimensional fine resolution radar imaging of rigid targets with arbitrary unknown motions, SPIE International Symposium: Algorithms for SAR Imagery, Orlando, FLA. April [ll] N.J.Porter, R.J.Tough, K.D.Ward, Hybrid SAR/ISAR: The synthetic aperture imaging of rocking ship targets, Proc. of SAR 93, Gif sur Yvette (FR), April [12] K.D.Ward,, R.J.Tough, B. Haywood, Hybrid SAR/ISAR imaging of ships, Proc. of IEEE Int. Radar Conf., Washington, May 1990.