target location and height estimation via multipath signal

Target Location and HeightEstimation via Multipath Signaland 2D Array for Sky-WaveOver-the-Horizon Radar

ZHONGTAO LUOZISHU HEUniversity of Electronic Science and Technology of ChinaChengdu, Sichuan, China

XUYUAN CHENKUN LUNanjing Research Institute of Electronics TechnologyNanjing, Jiangshu, China

In sky-wave over-the-horizon radar (OTHR), it is quite difficultto handle the issues of target location (ground range) and height(altitude) estimation due to their joint effect on group range. Thiswork addresses the joint estimation of target location and height bythe means of multipath propagation of OTHR signals and structureof a two-dimensional (2D) array. Usually, the multipath signal resultsfrom ionosphere propagation in OTHR and can be produced inmulti-input–multi-output (MIMO) radar by transmitting signals ofvarious carrier frequencies. A 2D array provides the potential ofelevation resolution, which is related to ground and slant ranges. Bythe multi-quasi-parabolic (MQP) ionospheric model, we formulatethe multipath propagation and signal model for the OTHR with atarget at location r and height h. Moreover, the jointmaximum-likelihood estimates (MLEs) of r and h are derived, andthe joint Fisher information matrix (FIM) is calculated. With theso-obtained FIM, the estimability is analyzed; that is: r and h areestimable if and only if either a multipath signal or 2D array isavailable. The joint Cramer-Rao bound (CRB) is computed anddiscussed for accuracy improvement. Additionally, the estimability isalso extended to the joint estimation of target location, height, andvelocity.

Manuscript received January 19, 2014; revised March 4, 2015; releasedfor publication September 29, 2015.

DOI. No. 10.1109/TAES.2015.140046.

Refereeing of this contribution was handled by G. San Antonio.

This work was supported by the National Natural Science Foundation ofChina under Grants No. 61032010, No. 61102142, No. 61301262, andNo. 61371164, and the Chongqing Distinguished Youth FoundationGrant No. CSTC2011jjjq40002.

Authors’ addresses: Z. Luo, Z. He, University of Electronic Science andTechnology of China, Chengdu 611731, Sichuan, China; X. Chen, K. Lu,Nanjing Research Institute of Electronics Technology, Nanjing 210039,Jiangshu, China. Corresponding author is Z. Luo, E-mail:([email protected]).

0018-9251/16/$26.00 C© 2016 IEEE

I. INTRODUCTION

Sky-wave over-the-horizon radar (OTHR)takes advantage of the refractive and reflective natureof high-frequency (5–30 MHz) propagation throughthe ionosphere and so provides a range coverage of up to4000 km [1]. Generally, conventional OTHR is a phased-array radar, which places its transmitting and receivingantennas in one-dimensional (1D) arrays, respectively.Signal processing provides the estimates of target azimuth,group range, and radial velocity [1, 2]. However, as thegroup range is a joint effect of location and height, it is noteasy to convert the estimated range into target location [3].

Basically, conventional OTHR considers targetlocation and height estimation in data processing. On onehand, target localization algorithms are mostly developedby a coordinate registration (CR) process, which infers thegeographic position from measured radar returns byray-tracing techniques [4–8]. However, withoutconsidering target height, target localization would bebiased. On the other hand, height estimation is generallystudied via micro-multipath propagation due to specularreflection of the radar signal from Earth’s surface [9–13].A superresolution technique, by iterating the best linearestimate reconstruction of images, has been proposed toinvestigate micro-multipath separation [9]. However, it hasnot been tried on real radar observations. Matched fieldprocessing (MFP) has been proposed to extract targetground range, altitude, and radial velocity [10]. However,experiments in [11] reveal that the MFP performs muchmore poorly with real data collected from the JindaleeOTHR than with results predicted by simulations. In [12],Anderson proposes generalized MFP considering randomionospheric and target-motion effects, under anassumption of known target location. This is unrealistic inpractice. In one word, height estimation based onmicro-multipath still requires further investigation anddemonstration before practical application.

It can be seen that conventional OTHR has not yetfound an effective way for target location and heightestimation. The difficulty is due to the joint effect of targetlocation and height in the group range of the propagationpath. They cannot be decoupled when conventional OTHRprovides only the group range (time delay) and radialvelocity (Doppler shift) of a single path. Different fromconventional OTHR, this paper takes two novel factorsinto consideration. One is a two-dimensional (2D) array,and the other is the multipath effect. The motivations areas follows.

First, a 2D array provides elevation resolution besidesazimuth information. Elevation information is very usefulsince it is related to the ray path through the ionosphere.The advantage of a 2D array has drawn special attentionfrom OTHR researchers. Considering the French OTHRNostradamus with a 2D array [14], elevation applicationmethods have been proposed to improve radarperformances, such as a target location tracking algorithm[15], improved coordinate registration by locating a

IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 52, NO. 2 APRIL 2016 617

beacon [16], and improvement of the Doppler spectrum ofsea echoes [17]. In this paper, we will study the 2D array’seffects on target location and height estimationperformance.

Second, the multipath effect is beneficial for targetlocalization. Compared with micro-multipath in OTHR,the multipath effect is more widely used for targetlocalization in ordinary radar (non-OTHR), especially inmulti-input–multi-output (MIMO) radar. Multiple timedelays produced by multistation radars can be estimatedfor locating a target [18]. Non-collocated MIMO radarplaces the transmitters and receivers widely separated andso produces spatial diversity gain on target detection andestimation performance [19–22]. Godrich and Heanalyzed the antenna placement for the accuracy of targetlocalization and velocity respectively [23, 24]. Unlikeordinary radar with antennas widely separated, an OTHRsystem is usually bistatic. In OTHR, the multipath effectmay be produced by an ionosphere propagationcharacteristic, which permits a single signal to travelthrough different ionosphere layers to illuminate the samearea. Generally, there are one to four paths for one signal.In conventional OTHR, some multipath associationtracking algorithms have been developed withoutconsidering target altitude [25–29].

Herein, we consider making use of the multipath effectfor estimating target location and height in signalprocessing in fast time. Ray tracing is employed based ona widely used sophisticated ionosphere model—themulti-quasi-parabolic (MQP) model [3, 15, 27]. The MQPmodel combines quasi-parabolic layers and joint layerstogether and provides a profile of ionosphere electrondensity, from which the electron magnetic waves’propagation can be deduced. In addition, the multipatheffect depends on the ionosphere state and range ofinterest. When the multipath effect is not produced by asingle signal, OTHR can employ the MIMO technique andtransmit multiple signals. Each signal bears a uniquecarrier frequency to guarantee and accentuate themultipath effect, as electron magnetic waves at differentcarrier frequencies can reach the same area by variouspaths. It is worth noting that our proposed MIMO-OTHRis different from the OTHR in [30–32], in which multipletransmit signals share one carrier frequency, so that thecorrelations of various signals and paths can be utilized, asthe multipath situation is similar to that of conventionalOTHR. Our work is also different from the MIMOtechnique in [33], where height estimation depends onDoppler signatures generated by micro-multipath effectsin slow time.

This paper analyzes the estimability and estimation oftarget location and height in fast time for sky-wave OTHR,considering the effects of multipath propagation and a 2Darray. Traditional azimuth is omitted in this analysis forsimplification. Estimability denotes the feasibility ofeffective estimation, i.e., whether target location andheight are estimable in a certain situation. We also analyzethe joint Cramer-Rao bound (CRB) and discuss the

approaches on accuracy improvement for an OTHR thatwishes to estimate target altitude effectively.

The remainder of the paper is organized as follows. Insection II, we develop a multipath propagation and receivesignal model when OTHR illuminates a target at a certainaltitude. The joint maximum-likelihood estimate (MLE)and Fisher information matrix (FIM) of the target locationand height are derived in section III. Based on the FIM,the estimability is analyzed in section IV. We discusssufficient conditions for effective estimation and also givesome instructive advice on radar system construction andoperation for accuracy improvement. Section V presentssimulation results. Section VI extends the study to thetarget velocity problem. Finally, conclusions are drawn insection VII.

Notation: Throughout this paper, we use bolduppercase letters to denote matrices, and bold lowercaseletters to signify vectors. Superscripts {·}H, {·}*, and {·}T

denote the complex conjugate transpose, conjugate, andtranspose of a matrix or vector, respectively, and diag{a}denotes a diagonal matrix with its diagonal given by thevector a. We use E{·} for expectation with respect to allthe random variables within the bracket, and | · | as themodulus for a complex number.

II. MULTIPATH PROPAGATION AND SIGNAL MODEL

Bistatic OTHR system places transmit antennas andreceive antennas separately in two 2D arrays. Owing to thelarge propagation distances compared with the separationbetween transmit and receive arrays, OTHR is consideredas a monostatic system in research. The MIMO techniqueemploys M transmit signals totally. Suppose that there areM rows in the transmit array, and each row emits onesignal. Therefore, there is beamforming in azimuth, andomnidirectional emitting in elevation. We ignore theazimuth analysis and focus on the 2D elevation plane tooffer insight into ionosphere propagation.

Consider monostatic OTHR with M transmit antennasand N receive antennas in the 2D elevation plane (verticalto the horizon), as depicted in Fig. 1, where adjacentreceive antennas have distance d spacing in a uniformlinear array. The mth transmit antenna, for m = 1,. . ., M,emits a narrowband waveform given by

√Emsm(t), where

Em is the transmitted energy, and∫ +∞−∞ |sm(t)|2dt = 1, at

carrier frequency fm. The carrier frequency incrementbetween any two transmitted signals is assumed largeenough, so that the spectrums are non-overlapping to keepsignals mutually orthogonal. A target located at groundrange r and height h is considered. For the convenience ofanalysis, the influence of velocity is not considered here.

The ionosphere is modeled as MQP layers formultipath deducing and ray tracing [3, 15]. Each layer isparameterized by its critical frequency fEi, layer height zi,and layer thickness zhi

, for i = 0,. . ., I, where I denotesthe number of layers in the MQP model. These parametersare assumed to be achieved by the acquisition system ofOTHR [1, 14], and they remain constant during the

618 IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 52, NO. 2 APRIL 2016

Fig. 1. Multipath in OTHR diagram. Assume radar transmits twosignals, s1(t) and s2(t), to illuminate one target with location r and heighth in free space region. Current ionosphere permits one and two one-way

propagations for s1(t) and s2(t), respectively. Thus, five paths areproduced, including one for signal s1(t) and four for s2(t). One-waypropagation length is R

F(B)ml = Gμ(β) − Gh(β, h). Target location

(ground range) is r = gμ(β) − gh(β, h). Slant/group range Rml is sum offorward propagation RF

ml and backward propagation RBml .

observation. Besides, fEi = 0 for i = 0 denotes the freespace region. Under the MQP model, a one-way ray pathstarts from the ground, passes through the ionosphere, andreaches the ground/sea surface beyond the line-of-sighthorizon. With the effect of Earth’s magnetic fieldneglected, the slant range and elevation angle β satisfy thepropagation function [3]

Gμ(β) = 2∫ zβ

z0

zdz√z2μ2(z) − z2

0cos2β

, (1)

in each path, and the ground range satisfies

gμ(β) = 2z20 cos β

∫ zβ

z0

dz

z

√z2μ2(z) − z2

0cos2β

, (2)

where z0 and zβ denote the radius of Earth and the apogeeheight of the ray path (away from Earth’s core),respectively; Gμ (β) and gμ (β) denote the slant range andground range, respectively; and

μ2m(z) = 1 − f 2

Ei

f 2m

[1 ±

(z − zi

zhi

) (zi − zhi

z

)2]

;

z ∈ the ith layer.

(3)

For the positive/negative sign in (3), the negative signapplies to the joining layers, and the positive sign appliesto others. The detailed procedure of layering theionosphere and computing the apogee height zβ can befound in [3]. From altitude 0 to h, the ground and slantranges corresponding to elevation β are

Gh(β, h) =∫ z0+h

z0

zdz√z2μ2(z) − z2

0cos2β

, (4)

and

gh(β, h) = z20 cos β

∫ z0+h

z0

dz

z

√z2μ2(z) − z2

0cos2β

, (5)

respectively.

By the ray-tracing functions (1–5), we can calculatethe multipath along which the transmitted waveform sm(t)reaches the target at (r, h) and returns to the receive array.Suppose the number of multipaths for sm(t) is Lm. Thetime delay corresponding to the lth path of sm(t), for l =1,. . ., Lm, is

τml = (RF

ml + RBml

)/c, (6)

where c denotes the speed of light, and RFml and RB

ml

denote the slant range of the forward propagation (thetransmitted electromagnetic wave travels through theionosphere and reaches the target) and the backwardpropagation (the electromagnetic wave reflected by thetarget travels through the ionosphere and reaches thereceive array) corresponding to the lth path of the mthtransmitted signal. The elevation angles corresponding toRF

ml and RBml , are denoted by βF

ml and βBml , respectively.

Then, the slant range and height h satisfy the followingpropagation equations [3]

RFml = Gμ(βF

ml) − Gh(βFml, h), (7)

RBml = Gμ(βB

ml) − Gh(βBml, h). (8)

Target location r and height h satisfy

r = gμ(βFml) − gh(βF

ml, h), (9)

= gμ(βBml) − gh(βB

ml, h). (10)

By now, the relationship between the time delays andtarget state (r, h) under the multipath effect is establishedby (1–10). Herein we briefly explain their usage. Giventhe ionosphere parameters, transmitting signals, and targetstate (r, h), (9) becomes a function of βF

ml and (10)becomes one of βB

ml . The two functions are solved forpossible solutions of βF

ml and βBml (there may be no

solution, one root, or multiple roots). Then, practicalsolutions are determined, and Lm is evaluated under theconstraints of geography, atmosphere, and radar systemconditions. Substitute the practical βF

ml and βBml in (7) and

(8) for RFml and RB

ml . Finally, τml is obtained by (6). Sincethe aforementioned path parameters vary for carrierfrequencies, multipath diversity is produced as long asOTHR transmits two or more signals accessible to thetarget. The number of paths for all signals isL = ∑M

m=1 Lm. Hence, L is determined by Lm directly,constrained by M indirectly.

Considering the target reflection coefficients, anassumption is made as follows.

ASSUMPTION 1 The reflection coefficient correspondingto the lth path of the mth signal is a zero-mean complexGaussian random variable ξml ∼ CN (0, 2 σ 2

ml), and itremains constant during the observation. For m �= m′ orl �= l′, ξml and ξm‘l ′ are independent.

The background noise level varies with the spectrumin the high-frequency band [1, 14]. When OTHR employsa unique carrier frequency for each signal, the noise levelsat the receivers can be different for various signals.

LUO ET AL.: TARGET LOCATION AND HEIGHT ESTIMATION FOR SKY-WAVE OVER-THE-HORIZON RADAR 619

ASSUMPTION 2 The noise due to the mth signal at the nthantenna is a temporally white, zero-mean complexGaussian random process unm(t) ∼ CN (0, σ 2

um), whereσ 2

um is a constant. The noise is independent for differentreceivers or signals, E{unm(t)u∗

n′m′(t)} = 0,

if n �= n′ or m �= m′.

Under Assumptions 1 and 2, the received signal by thenth receiver due to the mth signal is given by

ynm (t) =Lm∑l=1

ξml

√Emsm (t − τml) ej (n−1)ϕml + unm (t) ,

(11)where

ϕml = 2πd cos βBml · fm/c (12)

denotes the phase interval due to the receive antennaspacing.

Let rnm(t) denote the observed data by the nth antennadue to the mth signal. List the observed data in a columnvector

y (t) = [yT

1 (t) , ..., yTm (t) , ..., yT

M (t)]T

, (13)

where

ym (t) = [y1m (t) , ..., ynm (t) , ..., yNm (t)]T. (14)

We intend to estimate target state (r, h) from theobserved data y(t) and investigate the accuracy ifestimable. These are the main works of the following twosections.

III. JOINT MLE AND FIM

It is known from estimation theory that vectorestimation performance is closely related to the joint FIMof unknown vector elements [34]. If the FIM is invertible,the minimum variances of the estimates are given by thediagonal elements of the inverse matrix of the FIM—theso-called CRB matrix. If the FIM is noninvertible, thejoint estimate is ineffective. In this section, we derive thejoint FIM of (r, h) and then analyze the estimationperformance in the following section.

To derive the joint FIM, we calculate the likelihoodfunction �( y(t)|r, h) necessarily. Due to the mutualindependence of the noise in Assumption 2 and reflectioncoefficients in Assumption 1, the log-likelihood functionis calculated as

ln � ( y (t) |r, h) = Cons

+M∑

m=1

Lm∑l=1

2σ 2mlEm

∣∣∣∣ N∑n=1

∞∫−∞

ynm(t)s∗m(t−τml)e−j (n−1)ϕml dt

∣∣∣∣2

σ 2um(σ 2

um + 2NEmσ 2ml)

,

(15)

where Cons is a constant independent of r or h.

From the log-likelihood function in (15), the MLE oftarget location and height is given by

(r , h) = arg maxr,h

M∑m=1

Lm∑l=1

2σ 2mlEm

∣∣∣∣ N∑n=1

∞∫−∞

ynm(t)s∗m(t − τml)e−j (n−1)ϕml dt

∣∣∣∣2

σ 2um(σ 2

um + 2NEmσ 2ml)

. (16)

It can be seen that the coherent processing approach isemployed in processing the received signals by N receiveantennas due to the same path, while the noncoherentprocessing approach is employed in processing the signalsdue to different paths.

Based on the log-likelihood function in (15), the jointFIM of (r, h) is given as

F(r, h) =M∑

m=1

Lm∑l=1

γml

×[

a2mlεml + w2

mlψml amlbmlεml + wmlvmlψml

amlbmlεml + wmlvmlψml b2mlεml + v2

mlψml

],

(17)

where

aml = ∂τml

∂r= 1

c

(∂RF

ml

∂r+ ∂RB

ml

∂r

), (18)

bml = ∂τml

∂h= 1

c

(∂RF

ml

∂h+ ∂RB

ml

∂h

), (19)

wml = ∂βBml

∂r, (20)

vml = ∂βBml

∂h, (21)

and

γml = 32π2N2E2mσ 4

ml

σ 2um

(σ 2

um + 2N Emσ 2ml

) , (22)

εml =∞∫

−∞f 2|Sm (f )|2df −

∣∣∣∣∣∣+∞∫

−∞f |Sm (f )|2df

∣∣∣∣∣∣2

, (23)

ψml = N2 − 1

12

(d sin βB

ml

λm

)2

. (24)

Herein, Sm (f) is the Fourier transform of sm(t), and λm =c/fm denotes the wavelength of the mth signal. (SeeAppendix A for derivation.)

Path parameters ∂RF(B)ml /∂r, ∂R

F(B)ml /∂h, ∂βB

ml/∂r, and∂βB

ml/∂h in (18–21) can be derived from (1–10), thoughthe expressions are too complicated to present here.Actually, they can be easily evaluated by numerical


methods. Generally ∂RF(B)ml /∂r and ∂R

F(B)ml /∂h are positive,

while ∂βBml/∂r and ∂βB

ml/∂h are negative. Hence aml andbml are positive, and wml and vml are negative.

Next, we briefly interpret the meanings of the variablesin (22–24). The variable γ ml shows the influences ofsignal-to-noise ratio (SNR) 2Emσ 2

ml/σ2um and receive

antenna number N. Related to range estimation accuracy,εml is a measure of signal bandwidth, coinciding with thetraditional radar theory [34]. The novel variable ψml isproduced by the receive array in the elevation dimension.It denotes the effect of elevation resolution on rangeestimation, due to the essential relationship between theelevation angle and the slant/ground range in OTHR.

IV. ESTIMABILITY OF LOCATION AND HEIGHT

The MLE in (16) and the joint FIM F(r, h) in (17) bothdepend on propagation paths and the receive array. In thefollowing, we analyze the rank of F(r, h) for various pathsand antennas, as the indication of feasibility of targetlocation and height estimation. If estimable, the CRBs arepresented, and the accuracy is discussed.

The deduction in Appendix B examines the rank ofF(r, h) and concludes

rank [F(r, h)] = min(LN, 2). (25)

Hereby conclusions of estimability are drawn as follows:

1) For multiple paths or antennas LN > 1, the jointFIM F(r, h) is full rank and invertible. Target location andheight (r, h) are estimable.

2) For single path and single antenna LN = 1, the jointFIM F(r, h) is a singular matrix. Target location and height(r, h) are inestimable.

This reveals that target location and height estimabilityor inestimability depends on the number of paths andantennas in OTHR. In the following, we investigate theestimability, accuracy, and inestimability in detail.

A. Estimable for LN > 1

For multiple paths or antennas LN > 1, target locationand height can be estimated by the MLE in (16). Thelower bounds for jointly estimating target location andheight are given by

σ 2min(r) =

∑L�=1 γ�

(b2

�ε� + v2�ψ�

)det [F(r, h)]

, (26)

σ 2min(h) =

∑L�=1 γ�

(a2

�ε� + w2�ψ�

)det [F(r, h)]

, (27)

where

det[F(r, h)] = 1

2

L∑�=1

L∑�′=1

γ�γ�′

×[

ε�ε�′(a�b�′ − a�′b�)2 + ψ�ψ�′(w�v�′ − w�′v�)2

+ε�ψ�′(a�v�′ − w�′b�)2 + ε�′ψ�(w�b�′ − w�′v�)2

],

(28)

and � denotes the serial number of the (m, l) path in themultipath sequence, � = ∑m

m′=1 Lm′−1 + l,for L0 = 0.There are two sufficient conditions for jointly

estimating target location and height. One is multiplepaths, and the other is multiple antennas. Multipath inOTHR provides the diversity gain for target localization,like that in a multistatic radar system and non-collocatedMIMO radar. For a single path, multiple antennas arenecessary for joint estimation. For L = 1, F(r, h) isinvertible only when N > 1 makes ψml > 0 in (24). Inprinciple, it is the elevation resolution brought by theantennas in the elevation dimension that provides theability of decoupling target location and height from thesingle echo.

REMARK There is no clear definition about the thresholdof the differences between paths for claiming “two pathsare different and the multipath effect is produced.” In theformula of F(r, h) in (17), (aml, bml, wml, vml) are the pathparameters, which change continuously and accordinglyas the path varies. There are situations when theparameters for two paths are not equal but very close. Inthe MQP model, a tiny difference between two carrierfrequencies leads to slightly different parameters (aml, bml,wml, vml). Then, F(r, h) is numerically invertible; however,it gives rise to CRBs that are too huge to be practical. Suchsituations can be regarded as the “transition region”between multipath and single path. In the transition region,the CRB exists in theory, but it is unrealistic in practice.

Generally, a reasonable multipath for effectiveestimation is produced under either of the followingconditions. 1) At least two paths are through differentlayers of the ionosphere. 2) If through the same layer,carrier frequencies are different enough to make fardifferent path parameters. We will further simulate this insection V.

B. Discussion on Accuracy Improvement

Next, we investigate the approaches that OTHR cantake to improve the estimation for LN > 1. The jointlocation and height CRBs shed light on the accuracy. Asthe CRB formulas in (26) and (27) are complicated, weturn back to the joint FIM in (17). Before the discussion, auseful lemma is introduced.

LEMMA 1 Given a 2 × 2 symmetric positive definite(SPD) matrix

F =[

F11 F12

F12 F22

], (29)

and a 2 × 2 symmetric nonnegative definite (SND) matrix

F′ =[

F ′11 F ′

12

F ′12 F ′

22

], (30)

for which the diagonal elements are positiveF ′

11 > 0, F ′22 > 0, the diagonal elements of the inverse

matrix F–1 are not less than the corresponding diagonal


elements of (F + F′)–1, i.e.,[F−1]

11 ≥[(F + F′)−1

]11

, (31)

[F−1

]22 ≥

[(F + F′)−1

]22

. (32)

Equation (31) holds if and only if det(F′) = 0, and

F22F′12 = F ′

22F12. (33)

Equation (32) holds if and only if det(F′) = 0, and

F11F′12 = F ′

11F12. (34)

PROOF See Appendix C.

REMARK For the FIM in (17), the matrix inside thesummation is SND (SPD for N > 1). An increase of anyone variable L, εml, γ ml, or ψml produces an equivalenteffect of adding an SND matrix with positive diagonalelements to the original FIM. By Lemma 1, the diagonalelements of the inverse matrix, i.e., the joint CRBs, aredecreased (the equation conditions are not taken intoconsideration due to their rather low probability inpractice).

Next, we discuss the approaches for accuracyimprovement, roughly classified in three aspects.

1) Traditional factors of a radar system, includingtransmit energy Em, signal bandwidth εml, antenna numberN, and interval d. Due to γ ml being an increasing functionof Em and N in (22), and ψml being an increasing functionof N and d in (24), it is concluded that increasing Em, N, ord decreases the CRBs. Hence, traditional approaches arestill effective for better estimation of target location andheight in OTHR, such as increasing transmit energy Em

and bandwidth εml. The accuracy can also be improved byenlarging the array aperture d × N with higher elevationresolution. As the interval d merely affects array aperture,the number N also contributes to noncoherent SNR.

2) Multipath and MIMO technique. Each sub-FIM ofpath (m, l) is SND (SPD for N > 1), so that the CRBs aredecreased by growing L, owing to Lemma 1. There aretwo ways to improve multipath L. The first is to increasethe transmitting signal number M by the MIMO technique.Signals of various carrier frequencies are used to explorethe potential of multipath propagation through theionosphere. The second is to increase the employed pathnumber Lm instead of abandoning multipath signals, at aprice of more complicated signal processing compared tothat of single path in conventional OTHR.

3) Carrier frequency selection. Carrier frequency fmdetermines not only the path number Lm, but also pathparameters (aml, bml, wml, vml), which are closely relatedto the CRBs. In addition, the echo average amplitude σ ml

and noise covariance σ um in each path depend on fm too.As the carrier frequency influences the CRBs throughmultiple factors that are related to the real-timeenvironment state, it is hard to analyze the optimal carrierfrequency theoretically. Herein, we propose to select

carrier frequency via a numeric method for accuracyimprovement. First, the CRBs of various carrierfrequencies are calculated by (26) and (27), based on thereal-time ionosphere state, surveillance data, andinterested area.1 Then, we select the optimal carrierfrequency of the least CRBs, which is supposed to achievethe best accuracy. As the signal number in OTHR is notlarge, a numeric method will be efficient. The calculationand selection can be included in the process of thefrequency management system (FMS) during OTHRoperation.

C. Inestimable for LN = 1

For single path and single antenna LN = 1, F(r, h) isnoninvertible, and the MLE in (16) does not work. Itdenotes the disability of decoupling r and h from the timedelay of a single path without any elevation information.This is the situation that conventional OTHR usually faces.

However, in signal processing, we can take asubstitutive method instead of the MLE in (16). Supposethat by some prior information, the “predicted” targetheight is h0 (e.g., h0 = 0 at sea surface, which may be thetarget’s real height, not for sure). Then, the substitutivelocation estimate is modified as

r0 = arg maxr

2σ 211E1 · |

∞∫−∞

y11(t)s∗1 (t − τ11)dt |2

σ 2u1(σ 2

u1 + 2E1σ211)

∣∣∣∣∣∣∣∣∣h=h0

= arg maxr

∣∣∣∣∣∣∞∫

−∞y11(t)s∗

1 (t − τ11)dt

∥∥∥∥∥∥h=h0

; (35)

that is, the target location is estimated based on theassumed height h0. The location estimate r0 in (35) andthe predicted height h0 constitute an estimated state(r0, h0), which is expected to achieve the same time delayas the real state (r, h) basically. Specially, if the real heightcoincides with the “predicted height” h0, then r0 isefficient. The corresponding CRB is easy to derive, thoughit is not represented here. Otherwise, r0is biased for h �= h0.

It is not surprising to see a supposed target height inOTHR. For existing phased-array OTHR with a 1D array,conventional signal processing methods estimate targetlocation despite the height factor, or alternately speaking,under zero height by default. This simplistic considerationworks in practice and leads to no degradation of targetdetection performance. Here, our analysis providestheoretical support for the reasonability of the zero-heightassumption in conventional OTHR.

1 Average amplitude σml can refer to the backscattering ionogram fromthe backscatter soundings. A conventional 1D array may valuate σml

approximately, while a 2D array can provide an accurate estimate of σml,since its receive beamforming suppresses other multimode signals [14].Noise covariance σ um can be read by spectral surveillance of theoccupancy of the channels and the level of noise.


V. SIMULATION AND PERFORMANCE ANALYSIS

This section demonstrates the joint MLE of the targetstate and investigates the CRBs by numeric simulations.Based on the set MQP parameters, the rootmean-square-error (RMSE) of the MLE is simulated,compared to the root CRB (RCRB). Then, we illustratethe dependence of the RCRB on the receive array andmultipath, as well as selection of carrier frequencies.

A. The MQP Model and MLE Performance

First of all, ionosphere parameters are set as follows:the number of layers I = 3, where i = 1 denotes the layerE, and fE1 = 3.7 MHz, z1 = z0 + 115 km, and zh1 = 15km; i = 3 denotes the layer F2, and fE3 = 12.7 MHz, z3 =z0 + 310 km, and zh3 = 100 km; i = 2 denotes the jointlayer between the layers E and F2; and z0 = 6743 km forthe beginning of the free space region.

Transmitting waveforms are the most widely usedlinear frequency modulated continuous waveform(LFMCW) modulated on various carrier frequencies

sm(t) = 1√PT

P−1∑p=0

rect

(t − pT

T

)s0(t − pT )

· exp(j2πfmt), (36)

where s0(t) = exp(jπBt2/T) for 0 < t < T, rect(·) denotes arectangular window, and rect(t) = 1 for t ∈ (0, 1) andotherwise rect(t) = 0. In addition, B, T, and P denote thebandwidth, period length, and number, respectively, set asB = 20 kHz, T = 0.02 s, and P = 5 in the followingsimulations. A target is located at ground range r = 1500km and height h = 20 km. The variances of reflectioncoefficients in multipath are assumed to be the same,2σ 2

ml = 1, and the transmit energy of each waveform is Em

= 1. Noise covariance is set uniformly σ 2um = σ 2

u forvarious carrier frequencies. We vary σ 2

u to set the SNR ineach receiver.

The relationship between the slant/ground range andelevation angle for a target with an altitude is calculatedby the MQP model functions (1–10), via numericmethods, e.g., the secant method used here. Fig. 2 depictsthe slant/ground range corresponding to an elevation of0◦–35◦ for a target with an altitude of 0 km (sea surface) or20 km, at carrier frequency 20 MHz or 25 MHz. It can beseen that there is one forward/backward propagation for 25MHz to reach a ground range of 1500 km at elevation 7.6◦

for 0 km and 6.1◦ for 20 km (the rising part is ignored).Contrarily, there are two forward/backward propagationsfor 20 MHz, at elevations about 7◦ and 31.6◦ for 0 km, and5.5◦ and 31.4◦ for 20 km. Thus, 25 MHz produces a singlepath L = 1, and 20 MHz produces a multipath L = 4.

The performance of the location and height MLE in(16) versus SNR is investigated by employing thewaveform in (36) as a transmit signal, at carrier frequency20 MHz or 25 MHz, and M = 1. The RMSE of 1000Monte Carlo simulations is drawn in Fig. 3, comparedwith the RCRB, for N = 200, and d = 10 m. We can see

Fig. 2. Slant and ground ranges versus elevation angle in MQP modelfor carrier frequencies 20 MHz and 25 MHz, considering target is at

altitude 0 km or 20 km.

Fig. 3. Joint root CRB and MLE performance for target location andheight versus SNR, by carrier frequency 20 MHz (L = 4) or 25 MHz

(L = 1), for target at location (ground range) 1500 km and height 20 km,with M = 1, N = 200, and d = 10 m.

that the RMSE is close to the RCRB for SNR ≥ 0 dB. Inthe low SNR region, the location RMSE is much greaterthan the location RCRB, and the height RMSE is limitedby the height search range 0–40 km. The advantages ofmultipath are shown in two points. First, it can be seen thatthe location/height RCRB of 20 MHz is less than thecorresponding RCRB of 25 MHz at the same SNR.Second, the RMSE approximates the RCRB at SNR =−10 dB of 20 MHz, i.e., lower than SNR = 0 dB of25 MHz. It is worth noting that the CRBs seem relativelylarge, because the radar system parameters are evaluatedrelatively small, as limited by our simulation platform. Forinstance, the practical coherent integral time may exceed 5s, and bandwidth may be over 100 kHz.

B. Investigation on Receive Array

In the following, we simulate the effects of a receivearray on the location and height RCRB for multipath orsingle path. The effect of antenna number N is investigatedin Fig. 4, for SNR = 0 dB, d = 10 m, and M = 1, at carrier


Fig. 4. Root CRB of target location and height versus number ofreceive antennas N, by carrier frequency 20 MHz or 25 MHz, for M = 1,

d = 10 m, and SNR = 0 dB.

Fig. 5. Root CRB of target location and height versus adjacent receiveantennas interval d, by carrier frequency 20 MHz or 25 MHz, for M = 1,

N = 40, and SNR = 0 dB.

frequency 20 MHz or 25 MHz. We can see that the RCRBdecreases as N grows for both 20 MHz and 25 MHz, atdifferent speeds. When N is small, the RCRB of 20 MHzis much lower than the RCRB of 25 MHz. As N grows, thespeeds of RCRB decline are about 1/

√N for 20 MHz, and

1/(2√

N) for 25 MHz, due to the different significance ofψml in (24) for the location and height CRB in multipathand single path. The RCRB of 25 MHz catches up withthat of 25 MHz as N grows in Fig. 4. However, it is foundthat the two speeds exchange with each other when Nexceeds a threshold 212, a value so huge and unrealisticthat the simulations are not shown here.

The influence of antenna interval d on target locationand height RCRB is depicted in Fig. 5, for SNR = 0 dB, N= 40, M = 1, by the waveform in (36) at carrier frequency20 MHz or 25 MHz. It can be seen that the RCRB isalmost not affected by d for multipath at 20 MHz, while itdeceases obviously for single path at 25 MHz. This showsthe different significance of ψml to the CRB for multipathand singe path. The RCRB is much lower in 20 MHz than

Fig. 6. Root CRB of target location and height versus carrier frequencyf1 employing one signal s1(t) in (36) for M = 1, N = 200, d = 10 m, and

SNR = 0 dB.

25 MHz at small d, and it catches up as d grows, similar toN in Fig. 4.

The simulations demonstrate that the receive arrayplays different roles in estimation accuracy for multipathand single path. In the case of multipath, greater arrayaperture (with fixed N) produces little gain on estimationaccuracy. However, for single path, enlarging d is helpfulto improve the accuracy. In both cases, increasing thenumber of antennas is an effective way to improveaccuracy.

C. Investigation on Carrier Frequency and Multipath

The joint CRB also depends on path parameters thatare determined by the ionosphere and signal carrierfrequencies. Next, we investigate various situations ofcarrier frequencies and paths, for single or multipleantennas.

In the case of multiple antennas, we set N = 200 and d= 10 m, so that target location and height are estimableeven for single path. OTHR transmits only one waveforms1(t) as formulated in (36), under the same ionosphereparameters as set in section VA. The location and heightRCRBs are depicted in Fig. 6 as f1 varies from 5 to 28MHz, for SNR = 0 dB. We can see that the RCRBbehaves differently in three stages. In the lower-frequencyregion (about 5–14 MHz), both the location and heightRCRBs decrease slowly as f1 grows. In thehigher-frequency region (about 22.5–28 MHz) forgrowing f1, the height RCRB decreases still, while thelocation RCRB nearly remains unchanged. There is onlyone path in both regions. The middle-frequency region,where multipath effect is produced, produces the lowestRCRB for the whole band. Note that there are somesingular points where path parameters hop, such as thetwo minimums at 17.5 MHz and 21.5 MHz (18 MHz isregular with single path). The reason is that the partialderivative ∂τml/∂h is extremely large in some areas(which may be due to bugs in MQP model; the minimumsneed further demonstration by real data). Herein, if OTHR


Fig. 7. Joint CRB by employing two signals s1(t) and s2(t) in (36) atcarrier frequency combination (f1, f2), for M = 2, N = 1, and SNR = 0

dB: (a) location RCRB, (b) height RCRB.

needs to illuminate a target at location 1500 km and height20 km by only one signal and expects better accuracy, theoptimal carrier frequency is within 14.5–17 MHz.

In the case of a single antenna N = 1, OTHR canestimate target location and height jointly only when L >

1, as analyzed before. To guarantee the multipath effect,OTHR transmits two signals at carrier frequencies f1 andf2 varying independently in 5–28 MHz. Other parametersare set the same as the previous subsection. The locationand height RCRBs versus combinations of [f1, f2] aredepicted in Fig. 7, for SNR = 0 dB. It is obvious that thelocation and height RCRBs depend on the carrierfrequency combination significantly.

In detail, there are three different regions. The first isthe transition region, where the RCRB does not exist or ishuge and unpractical, mainly when f1 and f2 belong to thesame band 5–14.5 MHz, 17.7–18.4 MHz, and 22.1–28MHz. When f1 ≈ f2 produces one path each, the path

parameters of one signal are the same or close to the otherpath, leading to singular FIM. The second region achievesthe best RCRB and suboptimal RCRB when both signalsor only one signal produces multipath, respectively. In thelast region, where two carrier frequencies differ greatly,and each signal produces single path, the RCRB retains inan available level, larger than that of multipath by about14%–28%. It is worth noting that the minimum heightRCRB appears in 17.3–17.7 MHz and 21.5–21.9 MHz inFig. 7b, corresponding to the two minimums at 17.5 MHzand 21.5 MHz in Fig. 6.

D. Comments on Practical Application

The simulation results demonstrate that OTHR caneffectively estimate target location and height jointly, aslong as it satisfies either of two sufficient conditions. Itmay employ multiple antennas in the elevation dimension,or, alternatively, it may transmit one signal with multiplepaths or multiple signals with quite different carrierfrequencies. The last method shows the significanceof the MIMO technique to produce the multipatheffect.

OTHR can draw lessons from the analysis andinvestigations herein. First, sufficient conditions foreffective estimation provide necessary guides for theOTHR system and array construction. Researchers canevaluate the probability of the multipath effect for a singlesignal in routine operation and then decide whether a 2Darray or the MIMO technique is necessary. Take thesimulated situation for example. If the multipath issatisfied in regular missions (such as location 1500 km andfrequency 20 MHz), then OTHR may not need to employa 2D array or the MIMO technique. Otherwise, OTHR hasto decide whether to build a 2D array or update the radarsystem to use the MIMO technique. The joint CRBs in(26) and (27) reflect the estimation accuracy, givingguidance on the details of the 2D array, MIMO technique,transmit power and bandwidth, etc.

Second, accuracy analysis is instructive for radaroperation. Based on the joint CRBs, OTHR can choose tooperate in an appropriate mode for better accuracy. Afterthe ionosphere parameters and interested area are obtainedin real time, OTHR needs to decide the carrier frequenciesof transmit signals, which are crucial for propagationpaths. Ray paths are predicted by (1–10). The locationCRB σ 2

min(r) and height CRB σ 2min(h) are calculated by

(26) and (27) for various carrier frequencies. Then,optimal carrier frequencies are determined according toOTHR’s overall consideration of σ 2

min(r) and σ 2min(h).

Fig. 7 can be considered as a simple example for M = 2,assuming that both signal amplitude and noise covarianceare uniform for various carrier frequencies.

VI. EXTENSION TO VELOCITY (SPEED ANDHEADING)

In the previous sections, we have studied the targetlocation and height estimation. Here, we extend the study


to velocity, still in the 2D elevation plane. Thevelocity information includes two aspects: speed andheading.

A. Signal Model and Joint FIM

Consider a target at location r and height h that ismoving with a constant speed v and heading directionangle α. The relationships of target state and incident andreflection paths are depicted in Fig. 8.

The time delay corresponding to the lth path of sm(t) isgiven by (6), and the Doppler shift is given by

fd,ml = fmv[cos(α − βF

ml) + cos(α − βBml)

]/c. (37)

Under Assumptions 1–2, the received signal by the nthreceiver due to the mth signal is given by

yVnm (t) =

Lm∑l=1

ξml

√Emsm (t − τml) ej2πfd,ml t ej (n−1)ϕml

+ unm(t). (38)

There are four unknown parameters to be estimated,i.e., target location, height, speed, and heading, listed in avector

θ = [r, h, v, α]. (39)

By derivations similar to section III, the joint MLE ofθ is given by

θ = arg maxθ

M∑m=1

Lm∑l=1

2σ 2mlEm

σ 2um(σ 2

um + 2NEmσ 2ml

×∣∣∣∣∣∣

N∑n=1

∞∫−∞

yVnm(t)s∗

m(t − τml)e−j2πfd,ml t e−j (n−1)ϕml dt

∣∣∣∣∣∣2

,

(40)

FV (θ ) =M∑

m=1

Lm∑l=1

γml

×

⎡⎢⎢⎢⎢⎣

[FVml(θ )]11 [FV

ml(θ )]12 pml(amlρml + emlηml) qml(amlρml + emlηml)

[FVml(θ )]12 [FV

ml(θ )]22 pml(bmlρml + kmlηml) qml(bmlρml + kmlηml)

pml(amlρml + emlηml) pml(bmlρml + kmlηml) p2mlηml pmlqmlηml

qml(amlρml + emlηml) qml(bmlρml + kmlηml) pmlqmlηml q2mlηml

⎤⎥⎥⎥⎥⎦ , (41)

[FVml(θ )]11 = a2

mlεml + 2amlemlρml + e2mlηml + w2

mlψml,

(42)

[FVml(θ)]12 = aml(bmlεml+kmlρml)+eml(bmlρml+kmlηml)

+ wmlvmlψml, (43)

Fig. 8. Target velocity, and incident and reflection paths in OTHRdiagram. Target is considered in free space region, moving at speed v,heading direction angle α to horizon. Elevation angles of incident and

reflection paths are considered to be equal to departure and arrival anglesat radar base, respectively.

[FVml(θ )]22 = b2

mlεml + 2bmlkmlρml + k2mlηml + v2

mlψml,

(44)where yV

nm(t) denotes the observed signal by the nthantennas due to the mth signal.

The FIM of θ is a 4 × 4 matrix, formulated as (41),where

eml = fmv

c

[sin(α − βF

ml)∂βF

ml

∂r+ sin(α − βB

ml)∂βB

ml

∂r

],

(45)

kml = fmv

c

[sin(α − βF

ml)∂βF

ml

∂h+ sin(α − βB

ml)∂βB

ml

∂h

],

(46)

pml = fm

c

[cos(α − βF

ml) + cos(α − βBml)

], (47)

qml = −fm

c

[sin(α − βF

ml) + sin(α − βBml)

], (48)

ηml =∞∫

−∞|sm (t − τml)|2t2dt −

⎡⎣ ∞∫

−∞|sm (t − τml)|2tdt

⎤⎦

2

,

(49)

ρml

1

2π

⎡⎣ ∞∫

−∞s∗m (t − τml)

∂sm (t − τml)

∂τml

· t dt

⎤⎦

−∞∫

−∞f |Sm (f )|2df

∞∫−∞

|sm (t − τml)|2tdt, (50)

and {·} denotes the imaginary operator.


B. Investigation on Estimability

Detailed analysis on the rank of FV(θ) is not presentedhere for brevity. The conclusions on estimability aredrawn as follows.

1) For multipath L > 1, the FIM FV(θ) is full rank,[FV(θ)] = 4. Target location r, height h, speed v, andheading α are estimable by the MLE in (40), and theCRBs are given by the diagonal elements of the inversematrix of FV(θ).

2) For single path L = 1, the FIM FV(θ) is singular.Thus, target parameters (r, h, v, α) are inestimable.Specially, rank [FV(θ)] = 3 for N > 1, rank [FV(θ)] = 2for N = 1.

These conclusions indicate that multipath is essentialfor the estimability of target speed and heading. WhenOTHR illuminates a target from various directions bymultiple paths, spatial diversity is brought so that targetspeed and heading are estimable. Otherwise, for the caseof single path, the echo carries the information of aDoppler shift due to the radial velocity as the projection oftarget speed and heading (v, α). However, the real speedand heading cannot be resolved from the echo, even whenelevation information is provided. Elevation information isunhelpful for estimating target speed and heading in singlepath (though useful for estimating target location andheight). The joint MLE in (40) fails for single path, as aseries of combinations of speed and heading evaluationcan maximize the log-likelihood function. Meanwhile,location and height estimating still works, as analyzed insection IVA.

In the worst case for a single path and single antennaLN = 1, all the target parameters are inestimable. Instead,we can take a substitutive method, similar to section IVC.Two parameters are needed to be supposed by priorinformation. For example, consider the supposed height h0

and heading α0, the parameters left to be estimated are(r, v). The substitutive MLE is modified as

(r , v)

= arg maxr,v

∣∣∣∣∣∣∞∫

−∞yV

11(t)s∗1 (t − τ11)e−j2πfd,11t dt

∣∣∣∣∣∣∣∣∣∣∣∣ h = h0

α = α0

.

(51)

The estimate (r , v) in (51) and supposed (h0, α0) formthe estimated state (r0, h0, v0, α0), which is expected toachieve the same time delay and Doppler shift as the realstate (r, h, v, α) basically.

The consideration of assuming height and headingmakes sense in practical operation. For example, for avessel in the sea, the height and heading angle are bothzero. A plane has its frequent cruising altitude andgenerally flies parallel to the horizon, except inmaneuvering stages.

Fig. 9. Joint root CRB (in m for location and height, m/s for speed, andrad for heading) and MLE performance versus SNR, by carrier frequency

20 MHz, M = 1, L = 4, and N = 1.

C. Simulation on the MLE

Numeric simulations are employed to demonstrate thejoint estimation of target location, height, speed, andheading under the multipath effect. The ionosphereparameters are set the same as in section VA, and thetransmit waveform is given in (36) at carrier frequency20 MHz, M = 1, and L = 4. Target parameters are set(r, h, v, α) = (1500 km, 20 km, 100 m/s, 10◦). Receivedata are simulated by the signal model in (38) for variousSNR values. The MLE in (40) is employed for jointlyestimating target state. The RMSE and RCRB for the fourparameters are depicted in Fig. 9, for N = 1. It can be seenthat the RMSE is close to the RCRB for SNR ≥ 12.5 dB.In the low SNR region, the height and heading RMSEvalues are lower than the RCRB, because of the limitedsearch range. It is demonstrated that OTHR is able toestimate target location, height, speed, and heading jointlyunder the multipath effect, and it does not require multipleantennas necessarily.

VII. CONCLUSION

This paper has addressed the joint estimation of targetlocation and height in OTHR by employing the diversityof multipath signal and structure of a 2D array. Bydetermining the joint FIM, it is revealed that an efficientestimation can be obtained provided that the diversity ofmultipath signals or the structure of a 2D array can beemployed. The multipath effect may be produced bytransmitting a signal that experiences multipathpropagation. However, such a favorite carrier frequencymay not exist. As an alternative, the MIMO technique canbe employed to produce the multipath effect bytransmitting multiple signals at the same instant. We alsoanalyze the performance improvement based on the jointCRBs. Several points of advice are proposed for theOTHR construction and operation.


APPENDIX A. DERIVATION OF THE FIM IN (17)

To derive the joint FIM F(θ), we define anintermediate parameter vector

ϑ = (τ11, · · · , τMLM

, βB11, · · · , βB

MLM

). (52)

Observe that location r and height h are the uniqueindexes of target state, while the time delays τml andelevation angles βB

ml in ϑ are regarded as the target’sprojections onto multiple paths. Using the chain rule, thejoint FIM is given by

F (r, h) = ∇ϑ · F (ϑ) · [∇ϑ]T, (53)

where

∇ϑ =⎡⎣ ∂τ11

∂r· · · ∂τMLM

∂r

∂βB11

∂r· · · ∂βB

MLM

∂h

∂τ11∂h

· · · ∂τMLM

∂r

∂βB11

∂h· · · ∂βB

MLM

∂h

⎤⎦ , (54)

which represents the differentials of ϑ with respect to (r,h), for which the elements are formulated in (18–21).

By the signal model in (11) under Assumptions 1–2,F(ϑ) is derived after lengthy calculations, partitioned as

F (ϑ) =[

Fττ Oτβ

OTτβ Fββ

], (55)

where Oτ β denotes an L × L null matrix, and

Fττ = diag [γ11ε11, · · · , γMLMεMLM

], (56)

Fββ = diag [γ11ψ11, · · · , γMLMψMLM

], (57)

with elements listed in (22–24). Plug formulas (54–57)and (18–24) into (53), then F(r, h) is obtained by (17).

APPENDIX B. ANALYSIS ON THE RANK OF THE FIM

The rank of the FIM F(r, h) is analyzed based on theformula (53). The elements of ∇ϑ represent therelationship of the ground/slant ranges to elevation anglesin various paths. Basically, for an arbitrary path (m, l),

vectors [ ∂τml

∂r,

∂βBml

∂r] and [ ∂τml

∂h,

∂βBml

∂h] are not linearly

dependent. Thus, ∇ϑ is full rank, i.e., rank ∇ϑ . In thefollowing, we discuss the rank of F(r, h) in two cases,according to the number of receive antennas.

In the case of multiple receive antennas N > 1, for ψml

> 0 in (24), F(ϑ) in (55) is full rank, [F(ϑ)] = 2L. Sincethe product of full rank matrices is a full rank matrix, it isinferred that F(r, h) is full rank, i.e., rank[F(r, h] = 2.

In the case of a single receive antenna N = 1, due toψml = 0 in (24), the formula of F(r, h) in (53) can befurther simplified as

F (r, h) = ([∇ϑ]2×L

) · Fττ · ([∇ϑ]2×L

)T, (58)

where Fτ τ is full rank, rank[Fτ τ ] = L. For various paths,the differentials of time delays to r or h in [∇ϑ]2 × L arelinearly independent generally, hereby ([∇ϑ])2 × L) = min(L, 2). For multipath L > 1, both [∇ϑ]2 × L and Fτ τ are fullrank, so that F(r, h) is full rank. For single path L = 1, rank[Fτ τ ] = 1, rank ([∇ϑ]2 × L) = 1, and so rank [F(r, h)] = 1.

Summarizing the various situations of paths andantennas, it is concluded that:

rank [F (r, h)] = min(LN, 2). (59)

This implies that F(r, h) is full rank as long as there ismultipath or multiple antennas alternately.

APPENDIX C. PROOF OF LEMMA 1

PROOF The sum matrix is represented as

F′′ = F + F′

=[

F11 + F ′11 F12 + F ′

12

F12 + F ′12 F22 + F ′

22

]. (60)

Since the sum of an SPD matrix and an SND matrix isan SPD matrix, F′′ is SPD and invertible. Then, wecompare the diagonal elements of inverse matrices F–1 andF′′–1. Without loss of generality, the differences of the(1,1)th elements are computed

D11 = [F−1

]11 − [

F′′−1]11 (61)

= F22 · det(F′′) − (F22 + F ′22) · det(F)

det(F) · det(F′′). (62)

We are interested in the sign of D11. Due to theproperty of an SPD matrix, we have det(F) > 0, anddet(F

′ ′) > 0, so the denominator of D11 is positive. By F22

> 0 and F ′22 > 0, the numerator of D11 is rewritten as

F22 · det (F′′) − (F22 + F ′22) · det(F)

= F 222F

′11 − 2F22F12F

′12 + F ′

22F212 + F22 · det (F′)

= 1

F ′22

(F 222F

′11F

′22 − 2F22F12F

′12F

′22 + F ′2

22F212)

+F22 · det (F′)

≥ 1

F ′22

(F 222F

′212 − 2F22F12F

′12F

′22 + F ′2

22F212) (63)

= 1

F ′22

(F22F′12 − F ′

22F12)2

≥ 0, (64)

where (63) holds for det(F′) = F ′22F

′11 − F ′2

12 = 0, and(64) holds for F22F

′12 − F ′

22F12 = 0.

Since the denominator of D11 is positive and thenumerator is nonnegative, we have D11 ≥ 0; i.e., [F–1]11 ≥[F′′–1]11, where equality holds if and only if det(F′) = 0,and

F22F′12 = F ′

22F12. (65)

Similarly, the difference of the (2,2)th elements of F–1

and F′′–1 is computed, with D22 = [F–1]22 – [F′′–1]22 ≥ 0,i.e., [F–1]22 ≥ [(F + F′)–1]22, where the equality holds ifand only if det(F′) = 0, and

F11F′12 = F ′

11F12. (66)

Hereby the proof is concluded. �


ACKNOWLEDGMENT

The authors would like to thank Prof. Lei Huang fromShenzhen University for assisting in the writing andediting of this paper. We also appreciate the anonymousreviewers, whose constructive comments have notablycontributed to improve the quality and the readability ofthis paper.

REFERENCES

[1] Fabrizio, G. A.High Frequency Over-the-Horizon Radar: FundamentalPrinciples, Signal Processing, and Practical Applications.New York: McGraw-Hill, 2013.

[2] Headrick, J. M., and Thomason, J. F.Applications of high-frequency radar.Radio Science, 33, 4 (1998), 1045–1054.

[3] Papazoglou, M.Matched-field altitude estimation for over-the horizon radar.Doctoral dissertation, Duke University, Durham, NC, 1998.

[4] Wheadon, N. S., Whitehouse, J. C., Milsom, J. D., and Herring,R. N.Ionospheric modeling and target coordinate registration forHF sky-wave radars.In IET Sixth International Conference on HF Radio Systemsand Techniques, York, UK, Jul. 1994, 258–266.

[5] Krolik, J. L., and Anderson, R. H.Maximum likelihood coordinate registration forover-the-horizon radar.IEEE Transaction on Signal Processing, 45, 4 (Apr. 1997),945–959.

[6] Anderson, R. H., and Krolik, J. L.Over-the-horizon radar target localization using a hiddenMarkov model estimated from ionosonde data.Radio Science, 33, 4 (July-Aug. 1998), 1199–1213.

[7] Anderson, R. H., and Krolik, J. L.Track association for over-the-horizon radar with a statisticalionospheric model.IEEE Transactions on Signal Processing, 50, 11 (2002),2632–2643.

[8] Cacciamano, A., Capria, A., Olivadese, D., Berizzi, F., Mese,E. D., and Cuccoli, F.A coordinate registration technique for OTH sky-wave radarsbased on 3D ray-tracing and sea-land transitions.PIERS Proceedings, Kuala Lumpur, Malaysia, Mar. 2012,146–150.

[9] Anderson, C. W., Green, S. D., and Kingsley, S. P.HF skywave radar: Estimating aircraft heights usingsuper-resolution in range.IEE Proceedings–Radar, Sonar & Navigation, 143, 4 (1996),281–285.

[10] Papazoglou, M., and Krolik, J. L.Matched-field estimation of aircraft altitude from multipleover-the-horizon radar revisits.IEEE Transaction on Signal Processing, 47, 4 (1999),966–976.

[11] Praschifka, J., Durbridge L. J., and Lane J.Investigation of target altitude estimation in skywave OTHradar using a high-resolution ionospheric sounder.In IEEE International Radar Conference—Surveillance for aSafer World, Bordeaux, France, Oct. 2009, 1–6.

[12] Anderson, R. H., Kraut, S., and Krolik, J. L.Robust altitude estimation for over-the-horizon radar using astate-space multipath fading model.IEEE Transaction on Aerospace and Electronic Systems, 39, 1(2003), 192–201.

[13] Hou, C., Wang Y., and Chen, J.Estimating target heights based on the Earth curvature modeland micromultipath effect in skywave OTH radar.Journal of Applied Mathematics, 2014 (2014), Article ID424191.

[14] Bazin, V., Molinie, J. P., Munoz, J., Dorey, P., Saillant, S.,Auffray, G., Rannou, V., and Lesturgie, M.NOSTRADAMUS: An OTH radar.IEEE Aerospace and Electronic Systems Magazine, 21, 10(2006), 3–11.

[15] Bourgeois, D., Morisseau, C., and Flecheux, M.Over-the-horizon radar target tracking usingmulti-quasi-parabolic ionospheric modelling.IEE Proceedings–Radar, Sonar & Navigation, 153, 5 (2006),409–416.

[16] Saillant, S., Dorey, P., Munoz, J., and Molinie, J. P.Improvement of coordinate registration by locating a beaconby using elevation information measured with a 2-D HFskywave radar.In European Geophysical Society, XXVI General Assembly,Nice, France, March 25–30, 2001.

[17] Kerbiriou, C., and Saillant, S.Use of elevation angle information to improve the Dopplerspectrum of sea echoes in HF radar applications.In IEE 8th International Conference on HF Radio Systemsand Techniques, Guilford, UK, July 10–13, 2000.

[18] Manolakis, D. E.Efficient solution and performance analysis of 3-D positionestimation by trilateration.IEEE Transactions on Aerospace and Electronic Systems, 32,4 (1996), 1239–1248.

[19] Fishler, E., Haimovich, A. M., Blum, R. S., Chizhik, D., Cimini,L., and Valenzuela, R.MIMO radar: An idea whose time has come.In Proceedings of IEEE Radar Conference, Philadelphia, PA,Apr. 2004, 71–78.

[20] Fishler, E., Haimovich, A. M., Blum, R. S., Chizhik, D., Cimini,L., and Valenzuela, R.Spatial diversity in radars—Models and detectionperformance.IEEE Transactions on Signal Processing, 54, 3 (2006),823–838.

[21] Haimovich, A. M., Blum, R. S., and Cimini, L. J.MIMO radar with widely separated antennas.IEEE Signal Processing Magazine, 25, 1 (2008), 116–129.

[22] He, Q., Blum, R. S., and Haimovich, A. M.Noncoherent MIMO radar for location and velocityestimation: More antennas means better performance.IEEE Transactions on Signal Processing, 58, 7 (2010),3661–3680.

[23] Godrich, H., Haimovich, A. M., and Blum R. S.Target localization accuracy gain in MIMO radar-basedsystems.IEEE Transactions on Information Theory, 56, 6 (2010),2783–2803.

[24] He, Q., Blum, R. S., Godrich, H., and Haimovich, A. M.Target velocity estimation and antenna placement for MIMOradar with widely separated antennas.IEEE Journal of Selected Topics in Signal Processing, 4, 1(2010), 79–100.

[25] Rutten, M. G., and Percival, D. J.Joint ionospheric and target state estimation for multipathOTHR track fusion.In Proceedings of the SPIE Conference on Signal and DataProcessing of Small Targets, San Diego, CA, Nov. 2001,118–129.

[26] Bogner, R. E., Bouzerdoum, A., Pope, K. J., and Zhu, J.Association of tracks from over the horizon radar.IEEE Transactions on Aerospace and Electronic Systems


Magazine, 13, 9 (1998), 31–35.[27] Pulford, G. W., and Evans, R. J.

A multipath data association tracker for over-the-horizonradar.IEEE Transactions on Aerospace and Electronic Systems, 34,4 (Oct. 1998), 1165–1183.

[28] Anderson, R. H., and Krolik, J. L.Track association for over-the-horizon radar with a statisticalionospheric model.IEEE Transactions on Signal Processing, 50, 11 (2002),2632–2643.

[29] Subramaniam, M., Tharmarasa, R., McDonald, M., andKirubarajan, T.Multipath-assisted multitarget tracking with reflection pointuncertainty.In Proceedings SPIE 7698, Signal and Data Processing ofSmall Targets 2010, Apr. 2010, 76981A,doi:10.1117/12.851118.

[30] Frazer, G. J., Abramovich, Y. I., and Johnson, B. A.Spatially waveform diverse radar: Perspectives for highfrequency OTHR.

In 2007 IEEE Radar Conference, Boston, MA, Apr. 2007,385–390.

[31] Frazer, G. J., Johnson, B. A., and Abramovich, Y. I.Orthogonal waveform support in MIMO HF OTH radars.In IEEE International Waveform Diversity and DesignConference, Pisa, Italy, Jun. 2007, 423–427.

[32] Frazer, G. J., Meehan, D. H., Abramovich, Y. I., and Johnson,B. A.Mode-selective OTHR: A new cost-effective sensor formaritime domain awareness.In Proceedings of the IEEE Radar Conference, Washington,DC, May 2010, 935–940.

[33] Zhang, Y. D., Amin, M. G., and Himed, B.Altitude estimation of maneuvering targets in MIMOover-the-horizon radar.In IEEE 7th Sensor Array and Multichannel SignalProcessing Workshop (SAM), Hoboken, NJ, Jun. 2012,257–260.

[34] Schonhoff, T. A., and Giordano, A. A.Detection and Estimation Theory and its Applications. NewYork: Pearson College Division, 2006.

Zhongtao Luo received the B.S. degree in electronic engineering and the Ph.D. degreein signal and information processing from University of Electronic Science andTechnology of China (UESTC), Chengdu, China, in 2007 and 2015, respectively. From2007 to 2008, he was with Haier Group, China, where he worked as an electronicsengineer. From 2012 to 2013, he was a visiting scholar in Nanjing Research Institute ofElectronics Technology, China. Since July 2015, he has been with ChongqingUniversity of Posts and Telecommunications (CQUPT), Chongqing, China. Hisresearch interests include statistical signal processing, array signal processing, and theirapplications in radar and communication systems.

Zishu He received the B.S., M.S., and Ph.D. degrees in signal and informationprocessing from University of Electronic Science and Technology of China (UESTC) in1984, 1988, and 2000, respectively. Currently, he is a professor in the School ofElectronic Engineering of UESTC in signal and information processing. His currentresearch interests are involved in array signal processing, digital beamforming, thetheory on MIMO communication and MIMO radar, adaptive signal processing, andchannel estimation. He has finished more than 100 papers, and two books, titled Signalsand Systems, and Modern Digital Signal Processing and its Applications.


Xuyuan Chen received his B.S. degree from Nanjing University of Science andTechnology, China, in 1983, and his M.S. degree from Nanjing Research Institute ofElectronics Technology (NRIET), China, in 1986. Since 1986, he has been with NRIETworking on system design and electronic engineering in radars. Currently, he is a seniorengineer and research supervisor, and the technical leader responsible for the researchprograms of high-frequency (HF) radar in NRIET. His research interests are in radardesign, array processing, and related topics in HF radar.

Kun Lu was born in Guangxi, China, in 1977. He received his B.S. degree in electronicand communication engineering from Harbin Institute of Technology, China, in 1999,and his Ph.D. degree in electronic engineering from Shanghai Jiaotong University,China, in 2004. He is currently a senior engineer in Nanjing Research Institute ofElectronics Technology, China, as the research leader for the high-frequency (HF) radarbranch. His research interests include system design, and signal and informationprocessing in radars, with an emphasis on HF radar.


本文献由“学霸图书馆-文献云下载”收集自网络，仅供学习交流使用。

学霸图书馆（www.xuebalib.com）是一个“整合众多图书馆数据库资源，

提供一站式文献检索和下载服务”的24 小时在线不限IP

图书馆。

图书馆致力于便利、促进学习与科研，提供最强文献下载服务。

图书馆导航：

图书馆首页文献云下载图书馆入口外文数据库大全疑难文献辅助工具

http://www.xuebalib.com/cloud/

http://www.xuebalib.com/

http://www.xuebalib.com/cloud/


http://www.xuebalib.com/vip.html

http://www.xuebalib.com/db.php

http://www.xuebalib.com/zixun/2014-08-15/44.html


target location and height estimation via multipath signal

Documents