# Steady-State Performance Analysis of Tracking Filter Using LFM Waveforms and Range-Rate Measurement

Authors: Kenshi Saho
Journal: Mathematical Problems in Engineering
 (2018)

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## Abstract

The steady-state performance of a moving-object tracking filter is theoretically analyzed, assuming the simultaneous measurement of the range and range-rate (RRM system), and the use of linear frequency modulated (LFM) waveforms (RRM-LFM filter). An efficient analytical steady-state performance index, called an RMS index, is derived for the RRM-LFM filter to clarify the steady-state range prediction errors, theoretically. Using the derived RMS index, the optimal performance of the RRM-LFM filter is analyzed. The performance variation due to the use of LFM waveforms is clarified for the RRM tracking system. The theoretical performance analysis verifies that the measured range-rate significantly improves the tracking accuracy, compared to the conventional range-only measuring LFM tracking filter. Furthermore, the quantitative relationships among the measurement accuracy, degree of target maneuvering, and steady-state range prediction errors are clarified to validate the effectiveness of the RRM-LFM filter.


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## Body

## 1. Introduction

Tracking filters for robots and aircrafts predict the trajectory and velocity of a maneuvering target based on the measurements by remote sensors, such as radars, lidars, and/or sonars. For certain applications in such fields, linear frequency-modulated (LFM) waveforms are used for improving the resolution of the range measurements and imaging [1–4]. In measurements using LFM waveforms, a bias error occurs due to the Doppler effect, known as range-Doppler coupling [4, 5], which must be compensated. To reduce such bias errors, Kalman [6, 7] and α-β filters [8–10] modified for LFM waveforms are used. The analysis of the α-β filter for LFM waveforms (LFM α-β filter) is important because it theoretically clarifies the mathematical formulation of the steady-state performances of tracking filters [5, 9, 10]. Based on the theoretical performance analysis, an optimal tracking filter that minimizes the steady-state range prediction errors for LFM waveforms is derived and validated [8].Most conventional studies on steady-state LFM tracking filters have considered the measurement of the range (distance) alone [5, 8–10], neglecting the measurement of the range-rate. However, owing to technical advancements, recent LFM radar systems can generally measure the range and range-rate with sufficient accuracy [1–4, 11]. In [4], a measurement model incorporating compensation for range-Doppler coupling using the measured range-rate was presented and applied to radar ship tracking. In addition, the effectiveness of this tracking technique was validated in various real environments [12, 13]. Therefore, the performance analysis of tracking filters for range-rate-measured (RRM) systems is important for their appropriate design [14–17]. Although Bar-Shalom [14] presented numerical examples of an LFM tracking filter for RRM measurement, a stringent theoretical analysis was not provided. Recently, some studies have clarified the performance of LFM radar tracking by numerical simulations [15–17]; however, their practicalities were confirmed only in limited situations. Hence, these studies have empirically designed the tracking-filter parameters, and the performance was not sufficiently investigated. In view of the above, a theoretical performance analysis of tracking filters for LFM waveforms, assuming an RRM system, is required for their appropriate design.For the steady-state performance analysis of RRM systems, without using LFM waveforms, our previous work had proposed an efficient analytical performance index that expresses the steady-state error in range prediction, called the root-mean-square index (RMS index) [8, 18]. We had presented an RMS index-based optimal design strategy for a steady-state second-order tracking filter using an RRM system, called the α-β-η-θ filter [18, 19]. Moreover, the previous research [8] had verified that the tracking accuracy of an LFM α-β filter designed using the RMS index is better than that designed using a representative Kalman-filter equation-based approach [9, 10]. Based on these studies, we believe that an RMS index-based analysis can reveal the theoretical performance and enable the optimal design of a steady-state tracking filter using an RRM system with LFM waveforms.In this paper, the theoretical analysis of a tracking filter, using LFM waveforms and an RRM system (RRM-LFM filter), in steady-state is presented. Theα-β-η-θ filter design [18] is extended to the use of LFM waveforms through the derivation of the RMS index. The derived RMS index-based analysis clarifies the relationship between the parameters of the LFM waveforms and the tracking accuracy of the RRM-LFM filter. The theoretical steady-state performance of the RRM-LFM filter is compared to that of the conventional LFM α-β and α-β-η-θ filters to reveal the effects of the use of LFM waveforms in an RRM system and demonstrate the effectiveness of range-rate measurement for tracking with LFM waveforms.


## 2. LFM Tracking Filter with Range-Rate Measurement

### 2.1. Problem Definition

This paper considers a steady-state second-order tracking filter using LFM waveforms, assuming the simultaneous measurement of the range and range-rate (RRM system). Figure1 outlines a system for measurement using LFM waveforms and the tracking filter assumed in this study. First, a measurement model for the RRM system is described to define the tracking filter. This study assumes the measurement model presented in [4]. Moving-object tracking along the range direction using LFM waveforms is assumed, and the target state includes the range,r, and range-rate, v. At each time step k, the target range, rok, and the range-rate, vok, measured using LFM waveforms are input to the tracking filter. It is traditionally known that the range measured using the LFM waveforms includes not only the random errors due to sensor noise but also bias error due to the Doppler shift, which is modeled as [5] (1)r′ok=rtk+∆tvtk+wrk,where rtk and vtk are the true range and range-rate at timekT,T is the sampling interval, wrk is the white Gaussian noise in terms of the range with a standard deviation, σr, and ∆t is the coefficient of the range-Doppler coupling expressed as [4, 5, 10](2)∆t=τf0Bw,where τ, f0, and Bw are the pulse length, carrier frequency, and bandwidth of the LFM waveform, respectively. ∆t > 0 corresponds to an up-chirp waveform, while ∆t < 0 corresponds to a down-chirp one [5]. For the RRM system, the range-Doppler coupling, ∆tvtk, in the range measurement of (1) is compensated using the measured range-rate, vok, at each time step [4]. This compensation process is expressed as [4] (3)rok=r′ok-∆tvo=rtk+∆tvtk+wrk-∆tvok.We assume that the measured range-rate also contained the white Gaussian noise as [4](4)vok=vtk+wvk,where wvk is the white Gaussian noise in terms of the range-rate with a standard deviation, σv. Substituting (4) into (3), we have(5)rok=rtk+wrk-∆twvk,where wrk and wvk are independent. The measurement models of the RRM system for defining the filtering process are (4) and (5). As indicated in (5), although the bias error due to the Doppler shift is removed by the range-rate correction in (3), rok is affected by random errors in vok and the coefficient of the range-Doppler coupling. Thus, the reduction of these errors is required in the tracking filtering process.Figure 1
Outline of the assumed measurement/tracking system.From the measured parameters,rok and vok, the tracking filter calculates the predicted range, rpk, and the range-rate, vpk, as the outputs, as indicated in Figure 1. The tracking filter invokes an iterative prediction and smoothing process. The second-order tracking filter assumes a constant range-rate motion model for the prediction process, which is expressed as follows [8–10]:(6)rpk+1=rsk+Tvsk,(7)vpk+1=vsk,where rpk is the predicted target range, vpk is the predicted target range-rate, and rsk and vsk are the smoothed range and range-rate, respectively, obtained in the smoothing process. The smoothing process of the RRM system is given by [18](8)rsk=rpk+αrok-rpk+Tηvok-vpk,(9)vsk=vpk+βTrok-rpk+θvok-vpk,where α, β, η, and θ are the filter gains.The aim of this paper is to clarify the steady-state performance of the tracking filter, expressed by (6)–(9), whose measurement models are (4) and (5). We refer to this tracking filter as the RRM-LFM filter. The assumptions made are as follows:(i)
The measurement parametersσr, σv, and ∆t are known and constant.(ii)
The sampling interval,T, is known and constant.(iii)
As a steady-state is assumed, the filter gainsα, β, η, and θ are fixed.(iv)
Filter gains that minimize the RMS index [8, 18] are set. The RMS index is a performance index of the tracking filter that expresses the mean of the steady-state errors in the predicted range.In the rest of this paper, the RMS index of the RRM-LFM filter is derived. Then, the optimal gain is calculated using the derived RMS index. Finally, to clarify the effectiveness of the RRM-LFM filter, its optimal performance is compared with those of the conventional range-only measured LFM tracking filter (LFMα-β filter) [8–10] and the range and range-rate measured filter without considering LFM waveforms (α-β-η-θ filter) [18].


### 2.2. Relationship with the LFMα-β Filter

The LFMα-β filter is a steady-state tracking filter for LFM waveforms, assuming a range only measurement [8–10]. The difference between the LFM α-β and RRM-LFM filters is that the measurement range-rate, vo, is not obtained in the former. Thus, differing from (3), the range-Doppler coupling is not compensated in the measurement process. Instead, this is compensated in the smoothing process of the tracking filter. The smoothing process in the LFM α-β filter is given by [8–10](10)rsk=rpk+αrok-∆tvpk-rpk,(11)vsk=vpk+βTrok-∆tvpk-rpk.The prediction process is the same as (6) and (7). As indicated in (10) and (11), the bias error due to range-Doppler coupling in ro is compensated using the predicted range-rate, vp. The optimization of the LFM α-β filter is presented in [10]. The comparison of the RRM-LFM and LFM α-β filters reveals the effectiveness of range-rate measurements in LFM radars.


### 2.3. Relationship with theα-β-η-θ Filter

Theα-β-η-θ filter is a steady-state tracking filter, which considers the range and range-rate (position and velocity) measurements [18, 19]. The filtering process is the same as that of the RRM-LFM filter, i.e., (6)–(9). Although its performance analysis and optimization are presented in [18], the use of LFM waveforms is not considered. The conventional performance analysis of the α-β-η-θ filter assumes that there is no error covariance between the range and range-rate. The variances and covariance of the measurement noise assumed in [18] are as follows:(12)Erokconv-rtk2=σr2,Evokconv-vtk2=σv2,Erokconv-rtkvokconv-vtk=0,where E indicates the mean with respect tok and roconv and voconv are the measurement range and range-rate, respectively, assumed in conventional studies. The performance analysis and design of the optimal filter gains of the α-β-η-θ filter assume (12). However, for the RRM-LFM filter, the error covariances of ro and vo apparently exist, as indicated in (5). Thus, the analysis of the filter expressed by (6)–(9), considering the error covariance matrix of the measurements expressed by (4) and (5), is required for determining its performance. The comparison of the RRM-LFM and conventional α-β-η-θ filters reveals the effect of range-Doppler coupling on the RRM system.


## 2.1. Problem Definition

This paper considers a steady-state second-order tracking filter using LFM waveforms, assuming the simultaneous measurement of the range and range-rate (RRM system). Figure1 outlines a system for measurement using LFM waveforms and the tracking filter assumed in this study. First, a measurement model for the RRM system is described to define the tracking filter. This study assumes the measurement model presented in [4]. Moving-object tracking along the range direction using LFM waveforms is assumed, and the target state includes the range,r, and range-rate, v. At each time step k, the target range, rok, and the range-rate, vok, measured using LFM waveforms are input to the tracking filter. It is traditionally known that the range measured using the LFM waveforms includes not only the random errors due to sensor noise but also bias error due to the Doppler shift, which is modeled as [5] (1)r′ok=rtk+∆tvtk+wrk,where rtk and vtk are the true range and range-rate at timekT,T is the sampling interval, wrk is the white Gaussian noise in terms of the range with a standard deviation, σr, and ∆t is the coefficient of the range-Doppler coupling expressed as [4, 5, 10](2)∆t=τf0Bw,where τ, f0, and Bw are the pulse length, carrier frequency, and bandwidth of the LFM waveform, respectively. ∆t > 0 corresponds to an up-chirp waveform, while ∆t < 0 corresponds to a down-chirp one [5]. For the RRM system, the range-Doppler coupling, ∆tvtk, in the range measurement of (1) is compensated using the measured range-rate, vok, at each time step [4]. This compensation process is expressed as [4] (3)rok=r′ok-∆tvo=rtk+∆tvtk+wrk-∆tvok.We assume that the measured range-rate also contained the white Gaussian noise as [4](4)vok=vtk+wvk,where wvk is the white Gaussian noise in terms of the range-rate with a standard deviation, σv. Substituting (4) into (3), we have(5)rok=rtk+wrk-∆twvk,where wrk and wvk are independent. The measurement models of the RRM system for defining the filtering process are (4) and (5). As indicated in (5), although the bias error due to the Doppler shift is removed by the range-rate correction in (3), rok is affected by random errors in vok and the coefficient of the range-Doppler coupling. Thus, the reduction of these errors is required in the tracking filtering process.Figure 1
Outline of the assumed measurement/tracking system.From the measured parameters,rok and vok, the tracking filter calculates the predicted range, rpk, and the range-rate, vpk, as the outputs, as indicated in Figure 1. The tracking filter invokes an iterative prediction and smoothing process. The second-order tracking filter assumes a constant range-rate motion model for the prediction process, which is expressed as follows [8–10]:(6)rpk+1=rsk+Tvsk,(7)vpk+1=vsk,where rpk is the predicted target range, vpk is the predicted target range-rate, and rsk and vsk are the smoothed range and range-rate, respectively, obtained in the smoothing process. The smoothing process of the RRM system is given by [18](8)rsk=rpk+αrok-rpk+Tηvok-vpk,(9)vsk=vpk+βTrok-rpk+θvok-vpk,where α, β, η, and θ are the filter gains.The aim of this paper is to clarify the steady-state performance of the tracking filter, expressed by (6)–(9), whose measurement models are (4) and (5). We refer to this tracking filter as the RRM-LFM filter. The assumptions made are as follows:(i)
The measurement parametersσr, σv, and ∆t are known and constant.(ii)
The sampling interval,T, is known and constant.(iii)
As a steady-state is assumed, the filter gainsα, β, η, and θ are fixed.(iv)
Filter gains that minimize the RMS index [8, 18] are set. The RMS index is a performance index of the tracking filter that expresses the mean of the steady-state errors in the predicted range.In the rest of this paper, the RMS index of the RRM-LFM filter is derived. Then, the optimal gain is calculated using the derived RMS index. Finally, to clarify the effectiveness of the RRM-LFM filter, its optimal performance is compared with those of the conventional range-only measured LFM tracking filter (LFMα-β filter) [8–10] and the range and range-rate measured filter without considering LFM waveforms (α-β-η-θ filter) [18].


## 2.2. Relationship with the LFMα-β Filter

The LFMα-β filter is a steady-state tracking filter for LFM waveforms, assuming a range only measurement [8–10]. The difference between the LFM α-β and RRM-LFM filters is that the measurement range-rate, vo, is not obtained in the former. Thus, differing from (3), the range-Doppler coupling is not compensated in the measurement process. Instead, this is compensated in the smoothing process of the tracking filter. The smoothing process in the LFM α-β filter is given by [8–10](10)rsk=rpk+αrok-∆tvpk-rpk,(11)vsk=vpk+βTrok-∆tvpk-rpk.The prediction process is the same as (6) and (7). As indicated in (10) and (11), the bias error due to range-Doppler coupling in ro is compensated using the predicted range-rate, vp. The optimization of the LFM α-β filter is presented in [10]. The comparison of the RRM-LFM and LFM α-β filters reveals the effectiveness of range-rate measurements in LFM radars.


## 2.3. Relationship with theα-β-η-θ Filter

Theα-β-η-θ filter is a steady-state tracking filter, which considers the range and range-rate (position and velocity) measurements [18, 19]. The filtering process is the same as that of the RRM-LFM filter, i.e., (6)–(9). Although its performance analysis and optimization are presented in [18], the use of LFM waveforms is not considered. The conventional performance analysis of the α-β-η-θ filter assumes that there is no error covariance between the range and range-rate. The variances and covariance of the measurement noise assumed in [18] are as follows:(12)Erokconv-rtk2=σr2,Evokconv-vtk2=σv2,Erokconv-rtkvokconv-vtk=0,where E indicates the mean with respect tok and roconv and voconv are the measurement range and range-rate, respectively, assumed in conventional studies. The performance analysis and design of the optimal filter gains of the α-β-η-θ filter assume (12). However, for the RRM-LFM filter, the error covariances of ro and vo apparently exist, as indicated in (5). Thus, the analysis of the filter expressed by (6)–(9), considering the error covariance matrix of the measurements expressed by (4) and (5), is required for determining its performance. The comparison of the RRM-LFM and conventional α-β-η-θ filters reveals the effect of range-Doppler coupling on the RRM system.


## 3. Performance Index and Optimal Gain of the RRM-LFM Filter

This section derives the RMS index of the RRM-LFM filter in a closed form and presents the optimal calculation procedure for the filter gains. The definition of the RMS-index is given by [8](13)εp≡limk→∞⁡Ertkca-rpk2=σp2+LrpAmaxT22,where(14)rtkca=AmaxkT22is the true range of a target moving at constant acceleration, Amax; σp2 is the steady-state error variance in the range prediction assuming only sensor noise; and LrpAmaxT2 is the steady-state bias error in the range prediction [8, 10]. σp2 expresses the degree of random error due to sensor noise, and LrpAmaxT2 is the bias error caused by the difference between the assumed motion model (constant-range-rate model or constant-velocity model) and the target motion (accelerating motion). When we assume that the maximum acceleration of the target is Amax, the RMS index expresses the maximum RMS error of the range prediction in steady-state, for the second-order tracking filter (a detailed discussion is available in [18]).The optimal filter gains are calculated by the minimization ofεp. Thus, we need σp2 and Lrp to derive εp of the RRM-LFM filter. Moreover, to derive σp2, the variances and covariance of the measurement errors in the RRM system are required. Thus, in the following subsections, the variances and covariance of the measurement errors are derived. Using these, σp2 is derived. Further, Lrp and εp are determined, and the optimal gain calculation procedure is finally presented.


### 3.1. Derivation of the Measurement Error Variance/Covariance

In this section, the covariance matrix of the RRM-LFM filter is derived for the derivation of the RMS index. With (4) and (5), the measurement errors at time,kT, are(15)rok-rtk=wrk-∆twvk,(16)vok-vtk=wvk.Using (15) and (16), the variances and covariance of these errors are calculated as follows: (17)σro2≡Erok-rtk2=∆t2Ewvk2+Ewrk2+2∆tEwvkwrk=∆t2σv2+σr2,(18)σvo2≡Evok-vtk2=Ewvk2=σv2,(19)σxvo2≡Erok-rtkvok-vtk=-∆tEwvk2+Ewvkwrk=-∆tσv2.


### 3.2. Derivation of the Range Prediction Steady-State Error Variance

σ p 2 is defined as [8, 18](20)σp2≡limk→∞⁡Ertkcv-rpk2,where rtkcv is the true range of a target moving at a constant range-rate. Because the RRM-LFM filter assumes a constant range-rate model, random errors due to sensor noise alone are included in the predicted range of the target moving at a constant range-rate. Thus, the true target motion for the derivation of σp2 is the constant range-rate motion.σ p 2 of the RRM-LFM filter is derived as follows. rtkcv can be expressed as(21)rtkcv=rtk-1cv+Tvtcv.With (6) and (21), the mean squared predicted error is calculated as(22)Ertkcv-rpk2=Ertk-1cv-rsk-12+2TErtk-1cv-rsk-1vtcv-vsk-1+Evtcv-vsk-12.Because we assume a steady-state (k→∞), we can define following smoothing error variance/covariance: (23)σrs2≡Ertk-1cv-rsk-12=Ertkcv-rsk2,(24)σrvs2≡Ertk-1cv-rsk-1vtcv-vsk-1=Ertkcv-rskvtcv-vsk,(25)σvs2≡Evtcv-vsk-12=Evtcv-vsk2.With (20) and (22)–(25), (26)σp2=σxs2+2Tσxvs2+T2σvs2.Therefore, σxs2, σxvs2, and σvs2 are then derived to calculate (26). The true range and range-rate can be expressed similar to (8) and (9) as(27)rtkcv=rtkcv+αrtkcv-rtkcv+Tηvtcv-vtcv,(28)vtcv=vtcv+βTrtkcv-rtkcv+θvtcv-vtcv.The smoothing errors are calculated using (8), (9), (27), and (28) as(29)rtkcv-rsk=1-αrtkcv-rpk+αrtkcv-rok+Tηvtcv-vok-vtcv-vpk,(30)vtcv-vsk=1-θvtcv-vpk+θvtcv-vok+βTrtkcv-rok-rtkcv-rpk.Substituting (6), (7), and (21) into (29) and (30), (31)rtkcv-rsk=1-αrtk-1cv-rsk-1+Tvtcv-vsk-1+αrtkcv-rok+Tηvtcv-vok-vtcv-vsk-1,(32)vtcv-vsk=1-θvtcv-vsk-1+θvtcv-vok+βTrtkcv-rok-rtk-1cv-rsk-1-Tvtcv-vsk-1.Using (31), the error variance of the smoothed range is calculated as(33)Ertkcv-rsk2=1-α2Ertk-1cv-rsk-12+T2Evtcv-vsk-12+2TErtk-1cv-rsk-1vtcv-vsk-1+α2Ertkcv-rok2+T2η2Evtcv-vok2-Evtcv-vsk-12-2TEvtcv-vokvtcv-vsk-1+2α1-αErtk-1cv-rsk-1rtkcv-rok+TEvtcv-vsk-1rtkcv-rok+2TαηErtkcv-rokvtcv-vok-Ertkcv-rokvtcv-vsk-1+2Tη1-αErtk-1cv-rsk-1vtcv-vok-Ertkcv-rsk-1vtcv-vsk-1+TEvtcv-vsk-1vtcv-vok-TEvtcv-vsk-12.The following relations are satisfied because the smoothed parameters are a linear combination of the measured parameters: (34)Ertk-1cv-rsk-1rtkcv-rok=0,Evtcv-vokvtcv-vsk-1=0,Evtcv-vsk-1rtkcv-rok=0,Ertk-1cv-rsk-1vtcv-vok=0.Substituting (17)–(19), (23)–(25), and (34) into (33), and simplifying, we obtain a linear equation with respect to σxs2, σxvs2, and σvs2 as(35)α2-ασxs2+21-αα+η-1Tσxvs2-α+η-12T2σvs2=α2σr2+ηT-α∆t2σv2.Similar to the above calculation of Ertkcv-rsk2 using (32), we obtain another linear equation from the calculation of Evtcv-vsk2 as(36)-β2σxs2+2β1-β-θTσxvs2+2-β-θβ+θT2σvs2=β2σr2+θT-β∆t2σv2.With the calculation of Ertkcv-rskvtcv-vsk using (29) and (30), we also have(37)β1-ασxs2+α+2β+θ-αθ-βη-2αβTσxvs2+α+η-11-β-θT2σvs2=αβσr2+ηT-α∆tθT-β∆tσv2.Substituting the solution (σxs2, σxvs2, σvs2) of the linear system composed of (35)–(37) into (26), we arrive at σp2 as(38)σp2=g2α,β,η,θσr2+T2g3α,β,η,θ+2T∆tg4α,β,η,θ+∆t2g2α,β,η,θσv2g1α,β,η,θ,where(39)g1α,β,η,θ=α+θ+βη-αθ4-2α-β-2θ+αθ-βη,(40)g2α,β,η,θ=α2θ2-αθ2-2αβηθ+βηθ-αβθ-βθ-3α2θ+2αθ+β2η2+β2η+3αβη-2βη+αβ+2β+2α2,(41)g3α,β,η,θ=α2θ3-θ3-βηθ2+2αηθ2-αθ2+2θ2-2βη2θ+2αη2θ+2βηθ-2βη3+2βη2αθ-βη+β,(42)g4α,β,η,θ=θ2+αηθ-2θ-βη2-βη-2αη.Note that, when ∆t=0, σp2 of (38) is equal to the steady-state error variance of the predicted position in the conventional α-β-η-θ filter presented in [18], which does not consider range-Doppler coupling.


### 3.3. Derivation of the RMS Index

The coefficientLrp is also required for deriving the RMS index. However, this is equivalent to that of the conventional α-β-η-θ filter. This is because it is assumed that there are no random errors in the derivation of Lrp [18], and the bias error due to ∆t is completely compensated by vo in this situation, as indicated in (1). Thus, Lrp is (see Eq. (26) of [18])(43)Lrp=2-2η-θ2αθ-βη+β.The RMS index of the RRM-LFM filter is derived by substituting (38) and (43) into (13) as(44)εp=g2α,β,η,θσr2+T2g3α,β,η,θ+2T∆tg4α,β,η,θ+∆t2g2α,β,η,θσv2g1α,β,η,θ+2-2η-θ2αθ-βη+β2Amax2T41/2.Table 1 summarizes the features and RMS indices of the tracking filters, whose steady-state performances are analyzed in Section 4.Table 1
Summary of the tracking filters considered in this paper.
Filter Measurement parameter LFM waveform RMS index LFMα-β Range Yes See Eq.(17) of [8] Conventionalα-β-η-θ Range and range rate No Eq. (44), when ∆t=0 Proposed RRM-LFM Yes Eq. (44)


### 3.4. Optimal Gain Calculation

The optimal gains are calculated by minimizing the RMS index. Based on (44), an evaluating function is defined as(45)μα,β,η,θ,Rrv,cRD,ΓD≡εp2T2σv2=g2α,β,η,θRrv+g3α,β,η,θ+2cRDg4α,β,η,θ+cRD2g2α,β,η,θg1α,β,η,θ+2-2η-θ2αθ-βη+β2RrvΓD2,where(46)Rrv≡σr2T2σv2indicates the ratio of the measurement accuracies in the range and range-rate [18]; (47)cRD≡∆tTis the normalized coefficient of the range-Doppler coupling [5]; and(48)ΓD≡AmaxT2σris the deterministic tracking index defined in [10]. ΓD indicates the degree of maneuvering of the target. The optimal gains of the RRM-LFM filter are obtained by solving the following problem:(49)argminα,β,η,θμα,β,η,θ,Rrv,cRD,ΓDsub.  to1-ηβ+αθ>0and4-2α-β-2θ+αθ-βη>0andαθ-βη-α-θ+1<1,where the constraints are the stability conditions of the α-β-η-θ filter [18].


## 3.1. Derivation of the Measurement Error Variance/Covariance

In this section, the covariance matrix of the RRM-LFM filter is derived for the derivation of the RMS index. With (4) and (5), the measurement errors at time,kT, are(15)rok-rtk=wrk-∆twvk,(16)vok-vtk=wvk.Using (15) and (16), the variances and covariance of these errors are calculated as follows: (17)σro2≡Erok-rtk2=∆t2Ewvk2+Ewrk2+2∆tEwvkwrk=∆t2σv2+σr2,(18)σvo2≡Evok-vtk2=Ewvk2=σv2,(19)σxvo2≡Erok-rtkvok-vtk=-∆tEwvk2+Ewvkwrk=-∆tσv2.


## 3.2. Derivation of the Range Prediction Steady-State Error Variance

σ p 2 is defined as [8, 18](20)σp2≡limk→∞⁡Ertkcv-rpk2,where rtkcv is the true range of a target moving at a constant range-rate. Because the RRM-LFM filter assumes a constant range-rate model, random errors due to sensor noise alone are included in the predicted range of the target moving at a constant range-rate. Thus, the true target motion for the derivation of σp2 is the constant range-rate motion.σ p 2 of the RRM-LFM filter is derived as follows. rtkcv can be expressed as(21)rtkcv=rtk-1cv+Tvtcv.With (6) and (21), the mean squared predicted error is calculated as(22)Ertkcv-rpk2=Ertk-1cv-rsk-12+2TErtk-1cv-rsk-1vtcv-vsk-1+Evtcv-vsk-12.Because we assume a steady-state (k→∞), we can define following smoothing error variance/covariance: (23)σrs2≡Ertk-1cv-rsk-12=Ertkcv-rsk2,(24)σrvs2≡Ertk-1cv-rsk-1vtcv-vsk-1=Ertkcv-rskvtcv-vsk,(25)σvs2≡Evtcv-vsk-12=Evtcv-vsk2.With (20) and (22)–(25), (26)σp2=σxs2+2Tσxvs2+T2σvs2.Therefore, σxs2, σxvs2, and σvs2 are then derived to calculate (26). The true range and range-rate can be expressed similar to (8) and (9) as(27)rtkcv=rtkcv+αrtkcv-rtkcv+Tηvtcv-vtcv,(28)vtcv=vtcv+βTrtkcv-rtkcv+θvtcv-vtcv.The smoothing errors are calculated using (8), (9), (27), and (28) as(29)rtkcv-rsk=1-αrtkcv-rpk+αrtkcv-rok+Tηvtcv-vok-vtcv-vpk,(30)vtcv-vsk=1-θvtcv-vpk+θvtcv-vok+βTrtkcv-rok-rtkcv-rpk.Substituting (6), (7), and (21) into (29) and (30), (31)rtkcv-rsk=1-αrtk-1cv-rsk-1+Tvtcv-vsk-1+αrtkcv-rok+Tηvtcv-vok-vtcv-vsk-1,(32)vtcv-vsk=1-θvtcv-vsk-1+θvtcv-vok+βTrtkcv-rok-rtk-1cv-rsk-1-Tvtcv-vsk-1.Using (31), the error variance of the smoothed range is calculated as(33)Ertkcv-rsk2=1-α2Ertk-1cv-rsk-12+T2Evtcv-vsk-12+2TErtk-1cv-rsk-1vtcv-vsk-1+α2Ertkcv-rok2+T2η2Evtcv-vok2-Evtcv-vsk-12-2TEvtcv-vokvtcv-vsk-1+2α1-αErtk-1cv-rsk-1rtkcv-rok+TEvtcv-vsk-1rtkcv-rok+2TαηErtkcv-rokvtcv-vok-Ertkcv-rokvtcv-vsk-1+2Tη1-αErtk-1cv-rsk-1vtcv-vok-Ertkcv-rsk-1vtcv-vsk-1+TEvtcv-vsk-1vtcv-vok-TEvtcv-vsk-12.The following relations are satisfied because the smoothed parameters are a linear combination of the measured parameters: (34)Ertk-1cv-rsk-1rtkcv-rok=0,Evtcv-vokvtcv-vsk-1=0,Evtcv-vsk-1rtkcv-rok=0,Ertk-1cv-rsk-1vtcv-vok=0.Substituting (17)–(19), (23)–(25), and (34) into (33), and simplifying, we obtain a linear equation with respect to σxs2, σxvs2, and σvs2 as(35)α2-ασxs2+21-αα+η-1Tσxvs2-α+η-12T2σvs2=α2σr2+ηT-α∆t2σv2.Similar to the above calculation of Ertkcv-rsk2 using (32), we obtain another linear equation from the calculation of Evtcv-vsk2 as(36)-β2σxs2+2β1-β-θTσxvs2+2-β-θβ+θT2σvs2=β2σr2+θT-β∆t2σv2.With the calculation of Ertkcv-rskvtcv-vsk using (29) and (30), we also have(37)β1-ασxs2+α+2β+θ-αθ-βη-2αβTσxvs2+α+η-11-β-θT2σvs2=αβσr2+ηT-α∆tθT-β∆tσv2.Substituting the solution (σxs2, σxvs2, σvs2) of the linear system composed of (35)–(37) into (26), we arrive at σp2 as(38)σp2=g2α,β,η,θσr2+T2g3α,β,η,θ+2T∆tg4α,β,η,θ+∆t2g2α,β,η,θσv2g1α,β,η,θ,where(39)g1α,β,η,θ=α+θ+βη-αθ4-2α-β-2θ+αθ-βη,(40)g2α,β,η,θ=α2θ2-αθ2-2αβηθ+βηθ-αβθ-βθ-3α2θ+2αθ+β2η2+β2η+3αβη-2βη+αβ+2β+2α2,(41)g3α,β,η,θ=α2θ3-θ3-βηθ2+2αηθ2-αθ2+2θ2-2βη2θ+2αη2θ+2βηθ-2βη3+2βη2αθ-βη+β,(42)g4α,β,η,θ=θ2+αηθ-2θ-βη2-βη-2αη.Note that, when ∆t=0, σp2 of (38) is equal to the steady-state error variance of the predicted position in the conventional α-β-η-θ filter presented in [18], which does not consider range-Doppler coupling.


## 3.3. Derivation of the RMS Index

The coefficientLrp is also required for deriving the RMS index. However, this is equivalent to that of the conventional α-β-η-θ filter. This is because it is assumed that there are no random errors in the derivation of Lrp [18], and the bias error due to ∆t is completely compensated by vo in this situation, as indicated in (1). Thus, Lrp is (see Eq. (26) of [18])(43)Lrp=2-2η-θ2αθ-βη+β.The RMS index of the RRM-LFM filter is derived by substituting (38) and (43) into (13) as(44)εp=g2α,β,η,θσr2+T2g3α,β,η,θ+2T∆tg4α,β,η,θ+∆t2g2α,β,η,θσv2g1α,β,η,θ+2-2η-θ2αθ-βη+β2Amax2T41/2.Table 1 summarizes the features and RMS indices of the tracking filters, whose steady-state performances are analyzed in Section 4.Table 1
Summary of the tracking filters considered in this paper.
Filter Measurement parameter LFM waveform RMS index LFMα-β Range Yes See Eq.(17) of [8] Conventionalα-β-η-θ Range and range rate No Eq. (44), when ∆t=0 Proposed RRM-LFM Yes Eq. (44)


## 3.4. Optimal Gain Calculation

The optimal gains are calculated by minimizing the RMS index. Based on (44), an evaluating function is defined as(45)μα,β,η,θ,Rrv,cRD,ΓD≡εp2T2σv2=g2α,β,η,θRrv+g3α,β,η,θ+2cRDg4α,β,η,θ+cRD2g2α,β,η,θg1α,β,η,θ+2-2η-θ2αθ-βη+β2RrvΓD2,where(46)Rrv≡σr2T2σv2indicates the ratio of the measurement accuracies in the range and range-rate [18]; (47)cRD≡∆tTis the normalized coefficient of the range-Doppler coupling [5]; and(48)ΓD≡AmaxT2σris the deterministic tracking index defined in [10]. ΓD indicates the degree of maneuvering of the target. The optimal gains of the RRM-LFM filter are obtained by solving the following problem:(49)argminα,β,η,θμα,β,η,θ,Rrv,cRD,ΓDsub.  to1-ηβ+αθ>0and4-2α-β-2θ+αθ-βη>0andαθ-βη-α-θ+1<1,where the constraints are the stability conditions of the α-β-η-θ filter [18].


## 4. Performance Analysis

### 4.1. Analysis Method

The minimum RMS index of the RRM-LFM, conventionalα-β-η-θ, and LFM α-β filters are evaluated and compared by theoretical analyses. The minimization in (49) is solved by a simple gradient descent method, and the minimum RMS index, εp,min, is obtained by substituting the solutions of (49) into (44). εp,min of the other conventional filters are similarly calculated, using their RMS indices (see Table 1). We assume that σr and T are normalized to unity and Amax is known. Varying Rrv, cRD, and ΓD, the performance is analyzed to quantitatively investigate the effect of range-Doppler coupling in the RRM system and the performance improvement by the application of the measured range-rate in LFM radar tracking.


### 4.2. Results and Discussion

Figure2 shows the analysis results of the relationship between Rrv and εp,min for cRD = -0.5 (down-chirp), 0 (conventional α-β-η-θ filter without LFM waveforms), and 0.5 (up-chirp); ΓD = 0.1 and 1. As shown in these results, a larger Rrv improves the steady-state tracking accuracy. On comparing the results for cRD= 0.5 and 0, the improvement in the tracking accuracy by use of the up-chirp is confirmed. Although this improvement is known for range-only measured tracking systems [5], these results confirm a similar improvement in the RRM system, as well. As indicated in Figure 2(b), the performance improvement by using the up-chirp becomes considerable, when Rrv is small and ΓD = 1. This establishes that the use of the up-chirp achieves accurate tracking, even when the target maneuvering is relatively large and the measurement accuracy of the range-rate is worse. In contrast, the performance deterioration due to the use of the down-chirp is also confirmed. When Rrv becomes small, indicating that the accuracy of the range-rate measurement is worse, the performance deterioration for cRD = -0.5, compared to the results for cRD= 0, becomes considerable. Moreover, by comparing the results in Figures 2(a) and 2(b), the deterioration of the tracking accuracy for cRD = -0.5 is relatively large for ΓD=1. These results indicate that the down-chirp significantly deteriorates the tracking accuracy, when the range-rate measurement is unreliable, and the accelerating motion of the target is relatively large. In contrast, as indicated by the results for a relatively large Rrv, the tracking accuracy depending on cRD becomes small. This is because the compensation of the bias-error due to range-Doppler coupling is correctly performed in (3) by using a sufficiently accurate vo (sufficiently small σv).Figure 2
Relationship betweenRrv and εp,min for cRD = -0.5, 0, 0.5: (a) ΓD = 0.1. (b) ΓD = 1.
(a) (b)Figure3 shows the analysis results of the relationship between cRD and εp,min for the LFM α-β and RRM-LFM filters with Rrv = 0.1, 1, and 10 and ΓD = 0.1 and 1. The results for Rrv = 1 and 10 clearly indicate the enhancement of the steady-state tracking accuracy by using range-rate measurements. However, the results for Rrv = 0.1 at cRD<0 and ΓD = 0.1 demonstrate that the performance of the RRM-LFM filter is worse than that of the LFM α-β filter. This is because the compensation of the bias error in (3) is not accurate due to the poorer accuracy of the range-rate measurements. The LFM α-β filter compensates such errors using the estimated range-rate acquired in the filtering process, as indicated in (10) and (11). Moreover, errors due to the down-chirp also occur for cRD< 0, and these are also not sufficiently suppressed in the RRM system. Furthermore, when ΓD is small, the effects of a smaller Rrv and cRD for the minimization of the RMS index become considerable because the first term of (38) becomes relatively large. Thus, there are cases where the performance of the RRM-LFM filter is worse than that of the range-only measured LFM α-β filter. This is a notable result because accuracy deterioration due to range-rate measurements does not occur in the conventional α-β-η-θ filter as proved in [18].Figure 3
Relationship betweencRD and εp,min of the LFM α-β and RRM-LFM filters with Rrv = 0.1, 1, and 10: (a) ΓD = 0.1. (b) ΓD = 1.
(a) (b)Figure4 shows the analysis results of relationship between ΓD and εp,min for Rrv = 0.1, 1, and 10 and cRD = -0.5 and 0.5. Although the errors in the LFM α-β filter deteriorate the tracking accuracy with the increase in ΓD, those in the RRM-LFM filter converge because the RRM system effectively compensates the bias error using accurate range-rate information, when Rrv is relatively large. However, we can confirm that the tracking accuracy for Rrv = 0.1 is similar to that of the LFM α-β filter, when cRD = 0.5 because the range-rate measurement accuracy is insufficient for improving the tracking accuracy. On the other hand, the RRM-LFM filter with Rrv = 0.1 and cRD = -0.5 improves the accuracy for a relatively large ΓD because the measured range-rate (or velocity) information is effectively used to compensate the bias errors due to target acceleration. It is relatively difficult for a range-only measured system to compensate the errors due to acceleration because acceleration is the second-derivative of the range. In addition, the performance of the RRM-LFM filter is worse than that of the LFM α-β filter for relatively small ΓD, as discussed in the previous paragraph.Figure 4
Relationship betweenΓD and εp,min of the LFM α-β and RRM-LFM filters with Rrv = 0.1, 1, and 10: (a) cRD= 0.5. (b) cRD= -0.5.
(a) (b)The summary of the analyses results is as follows. All the analysis results prove that the steady-state performance improvement is significant on using an RRM system for tracking with LFM waveforms, when the measurement accuracy of the range-rate is sufficiently large. Even forRrv=1, when the reliabilities of the range and range-rate measurements are the same, the LFM tracking performance is improved. Furthermore, the effects of range-Doppler coupling and its compensation in the RRM system are quantitatively clarified. It is also revealed that there are cases where the performance of the RRM-LFM filter is worse than that of the conventional range-only measured LFM α-β filter, when down-chirp is used (cRD < 0), and Rrv and ΓD are both relatively small. However, for real applications of RRM-LFM radars, this might not be a serious problem due to the following reasons:(i)
For applications, assuming relatively small range measurements, whose order is smaller than 10 m, such as intelligent vehicles, moving robots, and indoor surveillance systems, the sampling interval,T, is less than order of 100 ms. Thus, for such applications, Rrv becomes large, as indicated in (46).(ii)
For applications, assuming relatively large range measurements, whose order is greater than 100 m, such as aircraft tracking, the assumed maximum target acceleration,Amax, and T are large. Thus, ΓD becomes large.Therefore, the representative applications of LFM radars do not consider a parameter setting, whereΓD and Rrv are both small.


## 4.1. Analysis Method

The minimum RMS index of the RRM-LFM, conventionalα-β-η-θ, and LFM α-β filters are evaluated and compared by theoretical analyses. The minimization in (49) is solved by a simple gradient descent method, and the minimum RMS index, εp,min, is obtained by substituting the solutions of (49) into (44). εp,min of the other conventional filters are similarly calculated, using their RMS indices (see Table 1). We assume that σr and T are normalized to unity and Amax is known. Varying Rrv, cRD, and ΓD, the performance is analyzed to quantitatively investigate the effect of range-Doppler coupling in the RRM system and the performance improvement by the application of the measured range-rate in LFM radar tracking.


## 4.2. Results and Discussion

Figure2 shows the analysis results of the relationship between Rrv and εp,min for cRD = -0.5 (down-chirp), 0 (conventional α-β-η-θ filter without LFM waveforms), and 0.5 (up-chirp); ΓD = 0.1 and 1. As shown in these results, a larger Rrv improves the steady-state tracking accuracy. On comparing the results for cRD= 0.5 and 0, the improvement in the tracking accuracy by use of the up-chirp is confirmed. Although this improvement is known for range-only measured tracking systems [5], these results confirm a similar improvement in the RRM system, as well. As indicated in Figure 2(b), the performance improvement by using the up-chirp becomes considerable, when Rrv is small and ΓD = 1. This establishes that the use of the up-chirp achieves accurate tracking, even when the target maneuvering is relatively large and the measurement accuracy of the range-rate is worse. In contrast, the performance deterioration due to the use of the down-chirp is also confirmed. When Rrv becomes small, indicating that the accuracy of the range-rate measurement is worse, the performance deterioration for cRD = -0.5, compared to the results for cRD= 0, becomes considerable. Moreover, by comparing the results in Figures 2(a) and 2(b), the deterioration of the tracking accuracy for cRD = -0.5 is relatively large for ΓD=1. These results indicate that the down-chirp significantly deteriorates the tracking accuracy, when the range-rate measurement is unreliable, and the accelerating motion of the target is relatively large. In contrast, as indicated by the results for a relatively large Rrv, the tracking accuracy depending on cRD becomes small. This is because the compensation of the bias-error due to range-Doppler coupling is correctly performed in (3) by using a sufficiently accurate vo (sufficiently small σv).Figure 2
Relationship betweenRrv and εp,min for cRD = -0.5, 0, 0.5: (a) ΓD = 0.1. (b) ΓD = 1.
(a) (b)Figure3 shows the analysis results of the relationship between cRD and εp,min for the LFM α-β and RRM-LFM filters with Rrv = 0.1, 1, and 10 and ΓD = 0.1 and 1. The results for Rrv = 1 and 10 clearly indicate the enhancement of the steady-state tracking accuracy by using range-rate measurements. However, the results for Rrv = 0.1 at cRD<0 and ΓD = 0.1 demonstrate that the performance of the RRM-LFM filter is worse than that of the LFM α-β filter. This is because the compensation of the bias error in (3) is not accurate due to the poorer accuracy of the range-rate measurements. The LFM α-β filter compensates such errors using the estimated range-rate acquired in the filtering process, as indicated in (10) and (11). Moreover, errors due to the down-chirp also occur for cRD< 0, and these are also not sufficiently suppressed in the RRM system. Furthermore, when ΓD is small, the effects of a smaller Rrv and cRD for the minimization of the RMS index become considerable because the first term of (38) becomes relatively large. Thus, there are cases where the performance of the RRM-LFM filter is worse than that of the range-only measured LFM α-β filter. This is a notable result because accuracy deterioration due to range-rate measurements does not occur in the conventional α-β-η-θ filter as proved in [18].Figure 3
Relationship betweencRD and εp,min of the LFM α-β and RRM-LFM filters with Rrv = 0.1, 1, and 10: (a) ΓD = 0.1. (b) ΓD = 1.
(a) (b)Figure4 shows the analysis results of relationship between ΓD and εp,min for Rrv = 0.1, 1, and 10 and cRD = -0.5 and 0.5. Although the errors in the LFM α-β filter deteriorate the tracking accuracy with the increase in ΓD, those in the RRM-LFM filter converge because the RRM system effectively compensates the bias error using accurate range-rate information, when Rrv is relatively large. However, we can confirm that the tracking accuracy for Rrv = 0.1 is similar to that of the LFM α-β filter, when cRD = 0.5 because the range-rate measurement accuracy is insufficient for improving the tracking accuracy. On the other hand, the RRM-LFM filter with Rrv = 0.1 and cRD = -0.5 improves the accuracy for a relatively large ΓD because the measured range-rate (or velocity) information is effectively used to compensate the bias errors due to target acceleration. It is relatively difficult for a range-only measured system to compensate the errors due to acceleration because acceleration is the second-derivative of the range. In addition, the performance of the RRM-LFM filter is worse than that of the LFM α-β filter for relatively small ΓD, as discussed in the previous paragraph.Figure 4
Relationship betweenΓD and εp,min of the LFM α-β and RRM-LFM filters with Rrv = 0.1, 1, and 10: (a) cRD= 0.5. (b) cRD= -0.5.
(a) (b)The summary of the analyses results is as follows. All the analysis results prove that the steady-state performance improvement is significant on using an RRM system for tracking with LFM waveforms, when the measurement accuracy of the range-rate is sufficiently large. Even forRrv=1, when the reliabilities of the range and range-rate measurements are the same, the LFM tracking performance is improved. Furthermore, the effects of range-Doppler coupling and its compensation in the RRM system are quantitatively clarified. It is also revealed that there are cases where the performance of the RRM-LFM filter is worse than that of the conventional range-only measured LFM α-β filter, when down-chirp is used (cRD < 0), and Rrv and ΓD are both relatively small. However, for real applications of RRM-LFM radars, this might not be a serious problem due to the following reasons:(i)
For applications, assuming relatively small range measurements, whose order is smaller than 10 m, such as intelligent vehicles, moving robots, and indoor surveillance systems, the sampling interval,T, is less than order of 100 ms. Thus, for such applications, Rrv becomes large, as indicated in (46).(ii)
For applications, assuming relatively large range measurements, whose order is greater than 100 m, such as aircraft tracking, the assumed maximum target acceleration,Amax, and T are large. Thus, ΓD becomes large.Therefore, the representative applications of LFM radars do not consider a parameter setting, whereΓD and Rrv are both small.


## 5. Conclusions

A range tracking filter using LFM waveforms and an RRM system was presented in this paper. The RMS index, which is a performance index that expresses the steady-state range prediction error, was analytically derived for the RRM-LFM filter, and its optimal gain, which minimizes the RMS index, was determined. The optimal steady-state performance of the RRM-LFM filter and its relationship with the measurement accuracy and range-Doppler coupling coefficient were theoretically analyzed. Performance comparison with the LFMα-β filter revealed the effectiveness of range-rate measurement quantitatively. Furthermore, the comparison of the RRM-LFM filter with the conventional α-β-η-θ filter clarified the effect of using LFM waveforms in an RRM system. Thus, it was established that range-rate measurement with sufficient accuracy significantly improves tracking accuracy with LFM waveforms.



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*Source: 9487581-2018-12-04.xml*