Abstract
ABSTRACT In radar signal processing, the detection and parameter estimation of high-speed maneuvering targets, which often utilize a coherent pulse train signal with linear frequency modulation, have been receiving increasing attention. Fundamentally and significantly, the Cramer-Rao lower bound (CRLB), as a cornerstone for evaluating the estimation performance of high-order kinematic parameters, has been derived and investigated here. In this paper, a 2-D echo signal model expressed in the fast-frequency and slow-time domains is adopted. The scaled orthogonal Legendre polynomials are deliberately introduced to solve the inverse problem of the Fisher information matrix, and then, a linear mapping relationship between different polynomial parameters can be used to obtain the analytical CRLB expressions. The main contributions included are: 1) the CRLBs, which are exact and closed form, have been extended to arbitrary motion model orders and reference time instants; 2) the influences of the motion model order, the reference time instant, as well as the radar parameters on the CRLBs are exploited comprehensively; and 3) some specific cases, including four low-order motion models and two preferred reference times, are also presented to better demonstrate the CRLB performance relationships. It highlights the fact that the reference time instant corresponding to the middle of the pulse train is a reasonable and compromised choice for parameter estimation, although it is not necessarily optimal for the kinematic parameters of all models and orders. The above research results are illustrated with numerical simulations and further verified using the maximum likelihood estimation method combined with Monte Carlo experiments.
Abstract
INDEX TERMS Cramer-Rao lower bound, kinematic parameter estimation, motion model order, reference time instant.
I. INTRODUCTION
The polynomial phase signal (PPS) parameter estimation problem appears in various engineering applications such as biomedicine, speech, communication, and radar signal processing and has attracted substantial research interest in the past few decades [1] . Based on the fundamental signal model with constant amplitude and polynomial phase of arbitrary degree, numerous effective estimation algorithms have been proposed and analyzed in the literature [2] - [9] . One of the most popular PPS estimators, the polynomial-phase transform (PPT) [2] , also called the high-order ambiguity function (HAF) [3] , accomplishes phase order decrementing via phase differentiation (PD) techniques. Subsequently, the integrated generalized ambiguity function (IGAF) [4] and the product high-order ambiguity function (PHAF) [5] can be used to suppress the cross-terms of multicomponent PPSs.
I. INTRODUCTION
In addition, time-frequency analysis [6] and time-frequency rate analysis [7] for nonstationary signals offer alternative solutions such as the polynomial Wigner-Ville distribution (PWVD) [6] , [8] , the high-order Wigner distribution (HO-WD) [6] , [9] , and the high-order cubic phase function (HO-CPF) [7] , [9] . Along with the complications presented by practical application scenarios, the signal model has even been extended to the time-varying-amplitude case. It is often treated as multiplicative noise, and some cyclostationary approaches have been employed [10] , [11] .
I. INTRODUCTION
To reasonably evaluate these algorithms' performance, the theoretical lower bound for unbiased estimators, well known as the Cramer-Rao Lower Bound (CRLB), provides an excellent benchmark for performance comparisons and therefore is of great importance. Peleg and Porat [12] first presented approximate but general CRLB expressions for the constant-amplitude PPS embedded in additive white Gaussian noise. Ristic and Boashash [13] then noted in their comment that the CRLB results also depend on the observation interval over which the signal is defined. Rytel-Andrianik [14] expanded on this research and investigated the critical influence of changes in the measurement interval center on the CRLBs of PPS. Legg and Gray [15] further evaluated the bounds for polynomial phase parameter estimation with nonuniform and random sampling schemes. In addition, performance analyses of deterministic signals under both additive and multiplicative noise can be found in [16] - [18] .
I. INTRODUCTION
In the radar field, the PPS is common and usually due to a target's complex radial motion patterns. For example, when we transmit a simple sinusoidal signal, a constantvelocity motion will result in a single-tone echo signal, a constant-acceleration motion leads to a linear frequency modulation (LFM) or chirp signal, and a constant-jerk motion corresponds to a quadratic frequency modulation (QFM) signal. More generally, any continuous radial motion function on a closed time interval can be uniformly approximated by polynomials. With increasing coherent processing interval (CPI), more and more high-order motion models for maneuvering targets have been established and adopted; the relevant estimation problems of the high-order kinematic parameters are thus receiving increasing attention [19] , [20] . The LFM coherent pulse train is an important waveform and is widely used in practical radar scenarios such as the long-time integration for weak target detection [21] - [23] and radar imaging for target identification [24] - [26] . For high-speed maneuvering targets, there exist envelope migration effects from pulse to pulse. Then, unlike the pure PPS case, the targets' motion information is simultaneously coupled into the envelope and phase modulations of the two-dimensional echo signal. In addition, the instantaneous kinematic parameters may be time-varying during the CPI, and it is necessary to specify a reference time instant within the pulse train for the estimation process. Correspondingly, the possible influences of designated reference times on the estimation performance are interesting and can be helpful for the envelope alignment and phase focusing operations, etc.
I. INTRODUCTION
The maximum likelihood estimation (MLE) is one of the effective methods for the joint parameter estimation of LFM pulse train signals, and many fast implementations have been developed for low-order motion models [27] - [30] . Herein, we are concerned with the theoretical performance analysis and not a particular estimation algorithm. Abatzoglou and Gheen [27] already presented the CRLBs of range, radial velocity and acceleration using an LFM pulse train. Xu et al. [19] obtained the approximate CRLBs of higher order motion parameters based on the results in [12] , with the reference time instant fixed at the starting pulse. Deng et al. [28] derived the CRLBs of some low-order motion parameters under a more accurate high-speed echo model. Pan et al. [31] further considered the estimation performance of a two-order motion model using a frequency modulation coded LFM pulse train. However, the last two studies did not obtain intuitive CRLB expressions. As we can see, exact and closed-form CRLBs for high-order motion models remain absent in the literature, and the influence of the reference time instant within the pulse train is rarely considered. Motivated by these facts, we will derive the analytical CRLB expressions for the high-order kinematic parameters estimation using the LFM coherent pulse train signal. Compared with current results, the performance bounds will be extended to arbitrary motion model orders and reference time instants, so they are more general and extensive.
I. INTRODUCTION
The remainder of this paper is organized as follows. In Section 2, the echo model of an LFM coherent pulse train signal is established. The two-dimensional signal is then expressed in the fast-frequency and slow-time domains, with which the concerned estimation problem is stated. In Section 3, the scaled orthogonal Legendre polynomials are introduced to derive the Fisher information matrix. Since this matrix is close to diagonal, the exact CRLBs can be easily obtained by matrix inversion and linear mapping. In Section 4, the CRLB performance relationships are further analyzed and compared through some specific cases with different motion model orders and reference time instants. In Section 5, some numerical simulations are performed. The factor influences on the CRLBs of the kinematic parameters are illustrated, and the theoretical values are compared with the MLE results via Monte Carlo experiments. Finally, we conclude this paper and provide some future work in Section 6.
II. SIGNAL MODELING AND PROBLEM STATEMENT A. ECHO SIGNAL MODEL
The coherent pulse train signal transmitted by a radar can be expressed as
II. SIGNAL MODELING AND PROBLEM STATEMENT A. ECHO SIGNAL MODEL
where t is the full time, t m = mT r represents the slow-time domain,t = t − t m indicates the fast-time domain, T r is the pulse repetition period, m is the pulse index, with a total pulse number of (2M + 1), and the dwell time is T D = (2M + 1) T r . Moreover, f c is the signal carrier frequency, and p t (t) is the LFM waveform, with a form of
II. SIGNAL MODELING AND PROBLEM STATEMENT A. ECHO SIGNAL MODEL
where rect (u) = 1, |u| ≤ 1/2 0, |u| > 1/2 , T p expresses the pulse width, γ is the signal chirp rate, and the signal bandwidth B is equal to γ T p . Without loss of generality, the observed moving target is treated as a point target, and its echo delay is denoted as τ . During the dwell time, τ is time varying with respect to the target's radial kinematic parameters. Consider that T p is rather small relative to T r for a typical pulse-Doppler radar, and the effect of inter-pulse modulation is more pronounced than that of intra-pulse modulation; thus, only the inter-pulse motion is considered here. Moreover, the Doppler shift and pulse spread effects caused by a high-speed target are also ignored [28] . Then, the echo delay τ from pulse m can be further simplified as
II. SIGNAL MODELING AND PROBLEM STATEMENT A. ECHO SIGNAL MODEL
where r (t m ) is the target's radial motion trajectory over a finite measurement window and c is the speed of light. Although the measurement window can be set arbitrarily like [14] , for convenience, we assume that it is symmetric
II. SIGNAL MODELING AND PROBLEM STATEMENT A. ECHO SIGNAL MODEL
to indicate the reference time instant within the pulse train for the high-order kinematic parameters estimation, and m 0 is used to mark the corresponding pulse index (not necessarily an integer). a i is the relevant kinematic parameter, and the maximum motion model order is I . According to (1)∼(3), the m-th baseband signal after pulse compression is modeled as follows
II. SIGNAL MODELING AND PROBLEM STATEMENT A. ECHO SIGNAL MODEL
where sinc (u) =
sin(πu)
πu and A r and ϕ represent the target's scattering amplitude and initial phase, respectively. We further assume that both A r and ϕ are constant during the dwell time. To facilitate the following derivations, the signal is expressed in the fast-frequency and slow-time domains [27] , which gives
sin(πu)
In particular, if the radar bandwidth is relatively narrow and the moving target's range migration during the dwell time is not so obvious, the envelope can not provide effective constraint information, and then the above echo signal model will degenerate into a pure PPS case. Equation (5) becomes
sin(πu)
As we know, the slow-time domain is naturally discretized with an interval of T r , yet the fast-frequency domain is discretized asf = k f , where f is the frequency interval and the frequency index k ∈ {−K , · · · , 0, · · · , K }. Obviously, B ≈ (2K + 1) f . Then, the discretization signal with noise yields
sin(πu)
where
sin(πu)
is the new amplitude defined in the frequency domain. n corresponds to the unique (k, m) pair, where
sin(πu)
The equivalent vector form of (6) is Z = S + W , where S is the above-mentioned deterministic signal, and it can be treated as a variant of the constant-amplitude PPS. W is the additive complex Gaussian noise with zero mean and known covariance. In this situation, the total number of sampled data points is N = (2M + 1) (2K + 1), and the parameters to be estimated are θ = {A, ϕ, a 0 , a 1 , · · · , a I }.
B. MAXIMUM LIKELIHOOD ESTIMATION
Based on the above echo signal model, the multidimensional joint probability density function (PDF) of Z is written as
B. MAXIMUM LIKELIHOOD ESTIMATION
where C w is the covariance matrix of W , ''det'' is the matrix determinant, and the superscript ''H'' means conjugate transpose. Equation (7) can also be considered as a likelihood function of θ , and hence, the corresponding log-likelihood function is
B. MAXIMUM LIKELIHOOD ESTIMATION
The classical MLE method determines the unknown parameters by maximizing the function in (8) , which meanŝ
B. MAXIMUM LIKELIHOOD ESTIMATION
It is well known that the MLE method is asymptotically effective, and evidently, its estimation performance can achieve the ideal CRLBs when the signal-to-noise ratio (SNR) is high enough. However, this method suffers from a serious computational problem because it often requires a multidimensional search over an unknown parameter space; the time consumption becomes unacceptable when the search dimension is large. Therefore, continued efforts are being made to improve the algorithm efficiency such as with the generalized Radon-Fourier transform (GRFT) [19] and the fast two-step search techniques combining the Hough transform (HT) and Newton's method [27] , [28] . Moreover, researchers are also seeking suboptimal but more efficient algorithms [20] , [23] , [26] . Next, based on the echo signal model Z expressed in the fast-frequency and slow-time domains, as well as the given log-likelihood function (θ ; Z), we continue to derive the exact and closed-form CRLB expressions.
III. DERIVATION OF THE ANALYTICAL CRLBS A. FISHER INFORMATION MATRIX AND ITS INVERSION
In statistical signal processing, the Fisher information provides a way of measuring the amount of information that an observable random variable carries about an unknown VOLUME 6, 2018 parameter, and the Fisher information matrix (FIM) is generally used to calculate the covariance matrices associated with MLE. By definition, the FIM can be written as
III. DERIVATION OF THE ANALYTICAL CRLBS A. FISHER INFORMATION MATRIX AND ITS INVERSION
where ''E{}'' represents the mathematical expectation and the superscript ''T'' denotes the matrix transpose operation. In this paper, the CRLB results under the white Gaussian noise (WGN) condition are mainly considered, and hence, C w = σ 2 I, where σ 2 is the known noise variance and I is an identity matrix. Then, the FIM elements in (10) can be simplified as
III. DERIVATION OF THE ANALYTICAL CRLBS A. FISHER INFORMATION MATRIX AND ITS INVERSION
where ''Re{}'' is to obtain the real part of a complex value, and θ i , θ j ∈ θ . Moreover, the information matrix is symmetrical, and
III. DERIVATION OF THE ANALYTICAL CRLBS A. FISHER INFORMATION MATRIX AND ITS INVERSION
For writing simplicity, we preliminarily define
and a new matrix spanned by T n,i with respect to different n, which is
where ''diag{}'' expresses a vector diagonalization operation. More intuitively, when it is specific to each estimated parameter of θ in (11), we have
and a new matrix spanned by T n,i with respect to different n, which is
where the operation '' n '' means a two-dimensional summa-
and a new matrix spanned by T n,i with respect to different n, which is
Thereafter, the general FIM can be summarized as (19) , as shown at the top of the next page.
and a new matrix spanned by T n,i with respect to different n, which is
By inverting the information matrix in (19) , the covariance matrix of the estimated parametersθ can be deduced. Then, the CRLB of the i-th parameter is the i-th element on the diagonal of that covariance matrix. We further obtain
and a new matrix spanned by T n,i with respect to different n, which is
It is not difficult to find that the estimation performance with respect to the signal amplitude is independent of that with respect to the phase parameters. As a result, the CRLB of the parameterÂ can simply be written as var(Â) ≥ σ 2 2N , and this parameter is no longer discussed here. However, for those polynomial phase parameters, the direct inversion of their partial FIM seems to be mathematically intractable. In this situation, although accurate CRLB results can still be obtained by numerical evaluation, the parameter dependency relationships are not so intuitive. Considering that the partial FIM is extremely ill conditioned, some authors [12] , [19] have resorted to approximations. However, the derived CRLB results are tightly based on the inverse form of the Hilbert matrix, and not surprisingly, they are valid only for the starting position of the sampled signal sequence. In this research, the arbitrariness of the reference time t m 0 makes the matrix inversion problem more complex, and thus, we need to seek new solutions.
B. ORTHOGONAL LEGENDRE POLYNOMIALS AND THE CRLBS
To derive the exact and closed-form CRLBs, we introduce orthogonal polynomials, such as the Legendre polynomials [14] , [32] or the Chebyschev polynomials [33] , [34] , to represent the PPS of large order. According to the properties of these polynomials, the information matrix can be reduced to an approximately diagonal form and it is much easier to find the matrix inversion. Furthermore, using the linear mapping relationship between different polynomial parameters, the required CRLBs for the high-order kinematic parameters can be obtained.
B. ORTHOGONAL LEGENDRE POLYNOMIALS AND THE CRLBS
The standard, i-th Legendre polynomials P i (x) are orthogonal on x ∈ [−1, 1]. The polynomials satisfy
B. ORTHOGONAL LEGENDRE POLYNOMIALS AND THE CRLBS
and 1 −1 P i (x)dx can also be calculated through (21) when we assume P j (x) = P 0 (x) = 1. On this basis, we define the scaled Legendre polynomials over the interval −
B. ORTHOGONAL LEGENDRE POLYNOMIALS AND THE CRLBS
The slow-time trajectory r(t m ) can be represented simultaneously with two types of polynomial forms as
B. ORTHOGONAL LEGENDRE POLYNOMIALS AND THE CRLBS
where α i,0≤i≤I are the relevant coefficients of the new scaled Legendre polynomials. Note that the discrete Legendre polynomials in (23) can maintain their orthogonality with respect to t m if the number of sampling points in the slow-time domain is sufficiently large. Then, we can repeat the previous calculation steps, and a reconstructed information matrix J for the phase parameters θ = {ϕ, α 0 , · · · , α I } is written as (24) , as shown at the top of this page, where the auxiliary factors β 1 and β 2 are defined as follows:
B. ORTHOGONAL LEGENDRE POLYNOMIALS AND THE CRLBS
and
B. ORTHOGONAL LEGENDRE POLYNOMIALS AND THE CRLBS
Inverting the near-diagonal information matrix in (24), we can subsequently obtain
B. ORTHOGONAL LEGENDRE POLYNOMIALS AND THE CRLBS
and
B. ORTHOGONAL LEGENDRE POLYNOMIALS AND THE CRLBS
Notice that the estimation performance of the initial phasê ϕ is coupled with that of the range-related parameterα 0 . Despite having the same order 0, their identifiability is based VOLUME 6, 2018 on the fact thatφ is determined only by the phase information, butα 0 is constrained by both the envelope information and the phase information. For 1 ≤ i ≤ I , the estimation performance of the parameterα i is independent of each other, and its CRLB is a linear function of the order i.
B. ORTHOGONAL LEGENDRE POLYNOMIALS AND THE CRLBS
According to (23) , there exists a linear mapping relationship between the two types of polynomial coefficients:
B. ORTHOGONAL LEGENDRE POLYNOMIALS AND THE CRLBS
where a = {a 0 , a 1 , · · · , a I } T and α = {α 0 , α 1 , · · · , α I } T . P is an upper triangular matrix with the element p ij (j ≥ i) in row i and column j. The standard Legendre polynomials can also be expanded
B. ORTHOGONAL LEGENDRE POLYNOMIALS AND THE CRLBS
, where '' '' denotes the integer floor function. Substituting it into the above formulas, we can obtain the specific expression of p ij . It shows that
B. ORTHOGONAL LEGENDRE POLYNOMIALS AND THE CRLBS
Consequently, the estimation covariance ofâ can be transformed from that ofα with the above affine matrix P, which means
B. ORTHOGONAL LEGENDRE POLYNOMIALS AND THE CRLBS
Finally, with (28)∼(32), the exact and closed-form CRLBs for estimating the high-order kinematic parameters (1 ≤ i ≤ I ) are expressed as
B. ORTHOGONAL LEGENDRE POLYNOMIALS AND THE CRLBS
where SNR N = . The above CRLBs are related to both the signal frequency f c and the signal bandwidth B. Similarly, the CRLBs for the pure PPS case can also be derived, and the multiplicative factor λ a in (33) will be replaced by λ c which is only connected with the signal frequency f c .
B. ORTHOGONAL LEGENDRE POLYNOMIALS AND THE CRLBS
For the special case of the range parameterâ 0 , because the CRLB ofα 0 in (28) does not conform to the general form ofα i in (29), we need to make some minor corrections in (33) . Specifically,
B. ORTHOGONAL LEGENDRE POLYNOMIALS AND THE CRLBS
where the additional quantity is denoted â 0 here. Since â 0 > 0, the actual estimation performance ofâ 0 degrades due to the negative influence of the unknown initial phase ϕ. For a typical narrowband radar, B f c , λ b λ a , and thus, â 0 plays a major role in (34) . Moreover, â 0 also decreases with increasing SNR N . While for the pure PPS, the range parameter a 0 can not be distinguished from the initial phase parameter ϕ by the phase information.
B. ORTHOGONAL LEGENDRE POLYNOMIALS AND THE CRLBS
In general, (33) and (34) provide more extensive performance bounds for the high-order kinematic parameters estimation. All the relevant factors have clear physical meanings, and their relationships with the CRLBs can be studied analytically and quantitatively. Preliminarily, the derived analytical expressions reveal some useful properties of the CRLBs:
B. ORTHOGONAL LEGENDRE POLYNOMIALS AND THE CRLBS
• The CRLBs of high-order kinematic parameters are independent of the parameter values themselves.
B. ORTHOGONAL LEGENDRE POLYNOMIALS AND THE CRLBS
• The large model order I normally increases the CRLBs of the kinematic parameters, which implies that the estimation performance is degraded. This is easy to understand because the number of unknown parameters to be estimated increases.
B. ORTHOGONAL LEGENDRE POLYNOMIALS AND THE CRLBS
• Except for the highest order parameterâ I , the CRLBs of other order kinematic parametersâ i are always 2 (I − i)-order polynomial functions of the reference time factor η, and these functions are also even symmetrical.
B. ORTHOGONAL LEGENDRE POLYNOMIALS AND THE CRLBS
• The CRLBs are inversely proportional to the accumulated SNR, and the i-th parameter is inversely proportional to the 2i power of the accumulated time T D .
B. ORTHOGONAL LEGENDRE POLYNOMIALS AND THE CRLBS
• The CRLBs are also related to the signal frequency f c and its bandwidth B, and increasing them helps to achieve better estimation performance. In the following section, we will further analyze the CRLB performance relationships through some more specific cases. The CRLB results with respect to different motion model orders and reference time instants are demonstrated and compared.
IV. CRLB ANALYSIS AND COMPARISONS THROUGH SOME SPECIFIC CASES A. WITH DIFFERENT MOTION MODEL ORDERS
Considering the radar observation scenarios most frequently encountered in practical applications, we mainly present here the analytical CRLB expressions of different order kinematic parameters with I = 1, 2, 3, 4. I = 1 corresponds to a constant-velocity motion scenario. Then, the CRLBs of the radial rangeâ 1,0 and velocityâ 1,1 are simplified as
IV. CRLB ANALYSIS AND COMPARISONS THROUGH SOME SPECIFIC CASES A. WITH DIFFERENT MOTION MODEL ORDERS
Obviously, only the CRLB ofâ 1,0 is related to the reference time factor η, and it obtains the minimum CRLB value at η = 0. For convenience, the optimal reference time factor is denoted η opt here. I = 2 corresponds to a constant-acceleration motion scenario. The CRLBs of the radial rangeâ 2,0 , velocityâ 2,1 , and accelerationâ 2,2 are
IV. CRLB ANALYSIS AND COMPARISONS THROUGH SOME SPECIFIC CASES A. WITH DIFFERENT MOTION MODEL ORDERS
Now, the CRLBs ofâ 2,0 andâ 2,1 are related to η, and the optimal reference time factors for the radial range and velocity estimation are η opt = ± , 0. It is not difficult to find that the CRLBs derived in [27] are special cases of the above formulas when η = 0. I = 3 represents what has recently become a relatively popular constant-jerk motion model. The CRLBs of the radial rangeâ 3,0 , velocityâ 3,1 , accelerationâ 3,2 , and jerkâ 3,3 are written as
IV. CRLB ANALYSIS AND COMPARISONS THROUGH SOME SPECIFIC CASES A. WITH DIFFERENT MOTION MODEL ORDERS
Under this model, the optimal reference time factors for estimating the radial range, velocity, and acceleration become η opt = 0, ± 
IV. CRLB ANALYSIS AND COMPARISONS THROUGH SOME SPECIFIC CASES A. WITH DIFFERENT MOTION MODEL ORDERS
Similarly, we can also find that the CRLBs ofâ 4,0 ,â 4,1 , a 4,2 , andâ 4, 3 in (38) 
B. WITH DIFFERENT REFERENCE TIME INSTANTS
In practice, the starting and middle pulse positions are the two most preferred reference time instants to align the pulse train envelope and to estimate the relevant kinematic parameters. For the starting pulse case, we know that m 0 = −M , and the reference time factor η = − M 2M +1 . It approximates to − 1 2 if M 1; for the middle pulse case, we know that m 0 = 0, and the reference time factor η = 0. Then, with the results in [14] , the simplified CRLB expressions at η = − 1 2 and η = 0 can be easily written.
B. WITH DIFFERENT REFERENCE TIME INSTANTS
Substituting η = − 1 2 into (33), we can obtain
B. WITH DIFFERENT REFERENCE TIME INSTANTS
where the combination number n m = n! m!(n−m)! . On the one hand, (39) is quite similar in form to the frequently cited CRLB results in [12] , but the derived analytical expressions are exact, making it superior. On the other hand, as preliminarily discussed in [19] , it is noted that the performance differences between the LFM coherent pulse train signal and the pure PPS are mainly reflected in their multiplicative coefficients (related to λ 2 a and λ 2 c , respectively, here). It is VOLUME 6, 2018 known that λ 2 a = λ 2 c 1 −
B. WITH DIFFERENT REFERENCE TIME INSTANTS
, and thus, λ 2 a < λ 2 c . This means that the estimation performance based on the LFM coherent pulse train signal is better since it introduces extra envelope constraints into the phase information. Furthermore, the large signal bandwidth also facilities better estimation results. As expected, a larger bandwidth B implies a higher range resolution c 2B such that the target's migration effect is more obvious, and then, the envelope information provides more benefits to the parameter estimation. The above analysis can be easily generalized to other η as well.
B. WITH DIFFERENT REFERENCE TIME INSTANTS
Substituting η = 0 into (33), we also have
B. WITH DIFFERENT REFERENCE TIME INSTANTS
where s = I −i 2 . Unlike the former case, the CRLBs in (40) are associated with the motion model order I through an indirect relationship
B. WITH DIFFERENT REFERENCE TIME INSTANTS
2 . This indicates that the estimation performance of the same parameterâ i remains unchanged when the order (I − i) increases from even to odd. Moreover, with (39) and (40), it can be observed that the CRLB results obtained at η = 0 are relatively close to optimality, which are generally much smaller than those obtained at η = − is 16 times that with respect to η = 0.In contrast, the latter is only 1.75 times that with respect to η opt . More detailed CRLB comparisons with respect to different motion model orders and reference time factors are summarized in Tab. 1. Combined with the previous discussions, it is recommended to use the reference time instant corresponding to the middle of the pulse train (η = 0) for the parameter estimation, although it is not necessarily optimal for the kinematic parameters of all models and orders.
B. WITH DIFFERENT REFERENCE TIME INSTANTS
With regard to potential applications, the above CRLB results can provide a quantitative aid in determining the radar operating parameters, such as the signal frequency, the signal bandwidth and the accumulated time. As a performance benchmark, the analytical CRLBs can also be used to define the asymptotic relative efficiency (ARE) for evaluating the estimation performance of various algorithms [3] . And a smaller ARE denotes a better parameter estimation performance. At high SNR, if ARE is equal to 1, then the estimation performance achieves the CRLB, which means that the estimation method is asymptotically effective, just like the MLE method. In addition, the estimation performance of the highest order kinematic parameterâ I is constant and independent of the reference time factor η, and thus, its CRLB can be written as
B. WITH DIFFERENT REFERENCE TIME INSTANTS
In practical processing, the maximum order I may be unknown as well. Through the parameter estimation and comparisons with this theoretical result, a reasonable motion model order can be determined similar to [2] .
V. NUMERICAL SIMULATIONS
In this section, we further verify the above theoretical results with some numerical simulations. The influences of relevant factors on the CRLBs for estimating the high-order kinematic parameters are illustrated and discussed. Then, the derived CRLBs under four motion models are also confirmed by First, we will illustrate the main influences of the reference time factor η on the parameter estimation performance. Two specific situations are highlighted here: the same parameter estimation results under different order motion models and the estimation results of different parameters under the same motion model. In this simulation, we suppose that the signal carrier frequency f c = 1 GHz, its bandwidth B = 5 MHz, the accumulated time is 1.5 s, and the accumulated SNR is 15 dB. η ranges from − Fig. 1 (a) presents the CRLBs of the radial velocity (i = 1) versus η when the model order I is equal to 1, 2, 3, 4 and 5. Fig. 1 (b) shows the respective CRLBs of the radial range, velocity, acceleration, jerk and snap versus η when the model order I = 4. The simulation results clearly reveal the existing inconsistency of the optimal η for the parameter estimation with respect to different i and I . And the CRLBs at η = 0 are relatively close to those at η opt , usually no more than twice. Comprehensively, η = 0 is a more reasonable and suitable choice for the LFM pulse train signal processing.
V. NUMERICAL SIMULATIONS
Second, we also exploit the influences of the order i and I on the parameter estimation performance. Two representative reference time instants, the starting pulse case (η = − 1 2 ) and the middle pulse case (η = 0), are mainly considered and compared here. Similarly, the signal frequency is set to 1 GHz, the signal bandwidth is 5 MHz, the accumulated time is 1.5 s, and the accumulated SNR is 15 dB. The parameter order i varies from 0 to 4, and the model order I ranges from 1 to 10 with a step size of 1. Fig. 2 (a) and (b) present the CRLBs of the parameterâ i versus I at η = − 1 2 and η = 0, respectively. It can be observed that, for the same i, the CRLB values ofâ i generally increase with the motion order I . In particular, there is a value jump phenomenon (stage change) in the case of η = 0, which agrees with the results obtained from (38). In most situations, the estimation performance of the high-order kinematic parameters at η = 0 is better than that at η = − 1 2 , even over several orders of magnitude.
V. NUMERICAL SIMULATIONS
Third, we further investigate the comprehensive influences of the accumulated time T D and the signal bandwidth B on the parameter estimation performance. In this simulation, the signal frequency is also set to 1 GHz and the accumulated SNR is 15 dB. A fourth order motion model (I = 4) is considered and the optional bandwidth B = 5, 10, 50, 100 MHz. T D ranges from 0.1 s to 1.5 s with a step size of 0.2 s. The corresponding CRLB results versus T D and B are illustrated in Fig. 3 . It can be observed that the signal bandwidth B has a more significant impact on improving the range estimation performance. In contrast, the signal time length T D has a more significant impact on improving the estimation performance of other high-order kinematic parameters. As mentioned before, due to the interference of the initial phase ϕ, the estimation performance of the radial rangeâ 0 is very different from that of other order kinematic parameters. This is also why the curve change trend ofâ 0 is not as obvious in Fig. 1 and Fig. 2 , although it is a polynomial function of η.
V. NUMERICAL SIMULATIONS
Finally, the MLE method is used to estimate the target's motion parameters at different SNR levels. It is implemented with a ''coarse'' search followed by a ''fine'' search. We assume that the radar carrier frequency f c = 1 GHz, the transmitting LFM waveform has a bandwidth B = 5 MHz, the coherent accumulated time T D = 1 s, and the number of data points M = K = 16. Then, we have N = 1089. Four low-order motion models (I = 1, 2, 3, 4) are investigated separately. Under all the motion models, we choose a 0 = 10km, a 1 = 100m/s, a 2 = 20m/s 2 , a 3 = 15m/s 3 , and a 4 = 8m/s 4 as required. The signal amplitude and its initial phase A = 1 and ϕ = π 4 . The accumulated SNR varies from −5 dB to 30 dB with a step size of 3 dB. The number of Monte Carlo simulations for each experiment is 500. Then, the average mean square error (MSE) of the MLE is calculated and compared with the corresponding CRLB, as illustrated from Fig. 4 to Fig. 7 . It can be observed that the simulation results are quite consistent with the theoretical values for the kinematic parameters of all models and orders at high SNR. As the number of estimated parameters increases, a higher SNR threshold may be required.
VI. CONCLUSION
In this paper, Legendre polynomials and a linear mapping are effectively combined to derive the general CRLB expressions for jointly estimating the high-order kinematic parameters of maneuvering targets using an LFM coherent pulse train signal. And then, the CRLB performance relationships with relevant influencing factors, such as the motion model order, the reference time instant and the radar parameters, can be investigated analytically and quantitatively. The research shows that the derived CRLB results for the LFM pulse train signal differ from the pure PPS case in terms of the multiplicative factor and the initial phase. More importantly, it reveals the existing inconsistency of the optimal reference time instants for estimating the kinematic parameters of different models and orders. The reference time instant corresponding to the middle of the pulse train can be taken as a compromise choice, and in this situation, the CRLB of the same kinematic parameter experiences a stage change with the increasing of model order. As a result, the above exact and closed-form performance bounds can offer considerable potential in many applications.
VI. CONCLUSION
We will conduct our future research from the following perspectives: The time-varying-amplitude case needs to be investigated due to the actual echo fluctuation of the target between pulses. The possible influences of other sampling schemes, such as nonuniform and random sampling, on the above CRLB results deserve in-depth and careful analysis. A more accurate echo signal model may be required, especially considering the Doppler shift and pulse spread effects caused by the target's high-speed motion. Moreover, this research can be further extended to other signal waveforms and some new application scenarios. 

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