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JOURNAL OF ELECTROMAGNETIC ENGINEERING AND SCIENCE, VOL. 18, NO. 4, 231~241, OCT. 2018

https://doi.org/10.26866/jees.2018.18.4.231

ISSN 2234-8395 (Online) ∙ ISSN 2234-8409 (Print)

231

I. INTRODUCTION

Ballistic missile defense systems (BMDSs) were developed to

protect territories from enemy ballistic missiles (BMs) in the

1950s. Deployment of a BMDS allows a state to intercept any

attacking missiles before they reach their intended targets [1]. It

is possible to guide multiple interceptors by dedicated radars,

stationary or otherwise, located on the ground, and to intercept

BMs at different altitudes. The radar can begin tracking after

the target has been launched and then communicate with con-

trollers and launchers to release the interceptors and guide them,

based on specific guidance settings [2, 3]. It is naturally desirable

for BMDSs to be able to intercept enemy BMs as rapidly as

possible. This requires estimation accuracy during the tracking

process, which is directly affected by the operating frequency at

the radar station [4].

Frequency selection is difficult for a variety of reasons. Low

frequency may be capable of covering long-range targets because

the signal travels a longer distance, but the radar ranging resolu-

tion is large, owing to limited transmission bandwidth. Also,

the dimension of phased-array radar increases significantly be-

cause of the large distance from antennae elements. High fre-

quency provides more scalable bandwidth resources and optimal

performance at short range, thanks to its small beamwidth [5].

Its drawbacks appear in the case of long-range targets, on ac-

count of free-space or atmospheric losses. The cost of radio fre-

quency components also influences the frequency selection: the

higher the frequency, the more expensive are the components.

Therefore, no frequency can satisfy all conditions, and the radar

only operates at the frequency that is subject to the least number

of constraints. An appropriate frequency must be selected to

provide optimal interception performance.

The purpose of the present study is to compare the intercep-

tion performance at various frequencies for a non-maneuvering

Analysis of the Optimal Frequency Band for a

Ballistic Missile Defense Radar System

Dang-An Nguyen1 · Byoungho Cho1 · Chulhun Seo1,* · Jeongho Park2 · Dong-Hui Lee2

A

bstract

In this paper, we consider the anti-attack procedure of a ballistic missile defense system (BMDS) at different operating frequencies at its

phased-array radar station. The interception performance is measured in terms of lateral divert (LD), which denotes the minimum accel-

eration amount available in an interceptor to compensate for prediction error for a successful intercept. Dependence of the frequency on

estimation accuracy that leads directly to prediction error is taken into account, in terms of angular measurement noises. The estimation

extraction is performed by means of an extended Kalman filter (EKF), considering two typical re-entry trajectories of a non-maneuvering

ballistic missile (BM). The simulation results show better performance at higher frequency for both tracking and intercepting aspects.

Key Words: Intercepting Prediction, Kalman Filter, Midway Guidance, Terminal Guidance, Tracking Radar.

Manuscript received February 23, 2018 ; Accepted May 5, 2018 ; Accepted June 19, 2018. (ID No. 20180223-024J)

1Department of Information Communication, Materials, and Chemistry Convergence Technology, Soongsil University, Seoul, Korea.

2LIG Nex1 Company, Seongnam, Korea.

*Corresponding Author: Chulhun Seo (e-mail: chulhun@ssu.ac.kr)

This is an Open-Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/4.0) which permits

unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited.

ⓒ Copyright The Korean Institute of Electromagnetic Engineering and Science. All Rights Reserved.

JOURNAL OF ELECTROMAGNETIC ENGINEERING AND SCIENCE, VOL. 18, NO. 4, OCT. 2018

232

re-entry enemy BM. An allowable estimation error throughout

the tracking process is determined, based on the acceleration

capability of an interceptor to ensure successful destruction at

the intended altitude. The non-linear motion and measurement

models are discussed mathematically; these reflect the challenge

of tracking precisely within the terminal phase of the BM. The

extended Kalman filter (EKF) is chosen for estimation extrac-

tion, based on two typical trajectories in 3D Cartesian coordi-

nate systems. The effectiveness of the EKF is measured by

means of root-sum-squared position error, denoting the devia-

tion between actual and estimated BM locations. The effect of

frequency on the position error is discussed in terms of radar

measurement noise, which results in a reduction in performance.

The accuracy of estimations leads to precision in predicting the

intercepting point where BM termination will happen. The lat-

eral divert, known as the least amount of acceleration that an

interceptor must attain for a successful intercept, is investigated

from the viewpoint of a zero-lag terminal guidance system.

The arrangement of our material is as follows. Section II pro-

vides an overview of the basic functional principles of a funda-

mental BMDS. In Section III, problems and techniques used

for the tracking process are thoroughly investigated in subsec-

tions on various coordinate systems, motion and measurement

models, and EKF. Section IV discusses the basic terminal guid-

ance system and related issues. The simulation results are shown

in Section V. Conclusions are presented in Section VI.

II.

B

ALLISTIC

M

ISSILE

D

EFENSE

S

YSTEM

The BMDS is commonly a composite system of various

components with different functions. The intercepting proce-

dure of a BMDS is illustrated simply in Fig. 1.

In general, a BMDS is equipped with a ground radar station

whose antenna can be a dipole, parabolic or phased-array, a

command and control system, and a missile launcher, which can

be integrated or located separately.

For an attacking-defense process, the radar starts tracking

the BM, beginning from point A, to obtain useful estimations of

its position and velocity, then predicts an intercepting point C,

where the BM will be terminated. The predicted intercepting

point can be calculated approximately based on the motion

model of the target [3]; this information is sent to the command

and control section. During the BM’s flight, knowledge about

the potential intercepting point continues to be updated and

improved, and the interceptor is guided based on the midway

guidance law until tracking at radar ends (assumed to be at B).

When the interceptor is close enough or acquires the target (as-

sumed to be at D), the seeker with which the missile is equipped

operates as an active radar and takes over tracking through a

terminal guidance system, before destroying the target in an

Fig. 1. Fundamental ballistic missile defense system.

allowable intercept zone, either by explosion (near-fuze) or colli-

sion (hit-to-kill). In Fig. 1, it is assumed that, when the seeker

acquisition happens, the radar stops tracking. The entire defense

procedure can be summarized in three main actions as:

•

The radar tracks the target and predicts the location of the

intercepting point.

•

The prediction point is updated and the interceptor is guid-

ed under the midway guidance law.

•

Terminal guidance happens when the interceptor is close

enough to the target and radar tracking stops.

The phased-array radar is located on the ground and records

the BM on its trajectory. The radar provides indirect measure-

ments of the target, such as range and angle, i.e., azimuth and

elevation, which are corrupted by radar noise, and useful estima-

tion extraction is performed by a noise filter. The accuracy of the

estimation depends greatly on the way that the radar operates,

and can lead to a prediction error, for which the interceptor

must compensate for a successful intercept. The precision of the

predicted intercepting point directly influences interception per-

formance. For example, if the missile cannot see the target ow-

ing to a large prediction error, this will lead to interception fail-

ure. Also, a failed hit will occur if the interceptor, though able to

approach the target, does not have enough energy or accelera-

tion available to correct the prediction error.

The operating frequency is one of the crucial factors impact-

ing on prediction accuracy and effective operation of the radar

station. Some specific radar frequency bands used for typical

BMDSs can be found in [6].

There is no specific standard for choosing the frequency for

the optimal design of the BMDS. However, it is possible to se-

lect an operating frequency from the point of view of perfor-

mance.

III.

P

HASED

-A

RRAY

R

ADAR FOR

T

RACKING

In this section, we consider how the radar works to obtain es-

NGUYEN et al.: ANALYSIS OF THE OPTIMAL FREQUENCY BAND FOR A BALLISTIC MISSILE DEFENSE RADAR SYSTEM

233

timations for the BMs, including the following issues: Cartesian

coordinate systems, re-entry motion model, frequency-depen-

dent radar measurement model, and EKF.

1. Coordinate Systems

For tracking purposes, the radar estimates the position and

velocity of the target on Cartesian coordinate systems (CSs).

Various CSs are commonly used, including earth-centered iner-

tial (ECI) CS, earth-centered fixed (ECF, ECEF, or ECR) CS,

east-north-up (ENU) CS, and radar face (RF) CS. More details

on the first two CSs can be found in [7]. In the present work, we

have selected ENU and RF CSs to express information on the

target, as illustrated in Fig. 2.

The origin of the ENU CS 𝑂𝑥𝑦𝑧 is located at the radar

station above the reference Earth surface ℎ and its vertical axis

𝑂𝑧 is directed along the local vertical line; 𝑂𝑥 and 𝑂𝑦 axes lie

on the local horizontal plane, with 𝑂𝑥 pointing east and 𝑂𝑦

pointing north. The RF CS 𝑂𝑥𝑦𝑧 is normally used in

phased-array radar systems, rather than the ENU CS. Its origin

is located at the radar face and the 𝑂𝑧 axis is normal to the

radar face; 𝑂𝑥 and 𝑂𝑦 axes lie on the radar face, with 𝑂𝑥

lying along the intersection of the radar face and the local hori-

zontal plane.

The radar face is fixed, and therefore the ENU and RF CSs

can be transformed into each other through a transformation

matrix 𝐓 based on known deviation angles, as follows:

𝑥

𝑦𝑧𝐓𝑥

𝑦

𝑧 (1)

where

𝐓𝑐𝑜𝑠𝜆 𝑐𝑜𝑠𝜙𝑠𝑖𝑛𝜆 𝑠𝑖𝑛𝜙𝑠𝑖𝑛𝜆

𝑠𝑖𝑛𝜆 𝑐𝑜𝑠𝜙𝑐𝑜𝑠𝜆 𝑠𝑖𝑛𝜙𝑐𝑜𝑠𝜆

0 𝑠𝑖𝑛𝜙 𝑐𝑜𝑠𝜙

(2)

Note that 𝐓𝐓 because the transformation matrix 𝐓

Fig. 2. East-north-up and radar-face coordinate systems.

is orthogonal and the two Cartesian CSs coincide as 𝜙𝜆

0. This can reduce computational complexity, because both

measurements and estimations of the target are expressed on the

same CS.

2. Re-entry Motion Model of BM

In an entire trajectory of a BM, several different forces act on

the missile, and not all trajectory regimes are influenced by the

same number of forces. Therefore, it is difficult to portray the

BM’s full motion by employing a single model only. In many

contexts, the BM’s flight is commonly partitioned into three

phases, as shown in Fig. 3.

• Boost: The BM is exposed to forces of thrust, drag, and

gravity, and this phase lasts from the launch to the burnout,

i.e., turn-off thrusters, around 4 minutes. The BM is pow-

ered and accelerated within endo-atmospheric flight.

• Midcourse: During an exo-atmospheric, free-flight motion,

which lasts approximately 20 minutes, only gravity impacts

on the BM.

• Re-entry: The BM re-enters the atmosphere, and the at-

mospheric drag becomes considerable, enduring until

reaching the intended impact point. The drag-induced ac-

celeration depends on the velocity and altitude of the BM

[7].

It is possible and easier to conceive a more precise motion

model of the BM within a particular phase. Because the earth

model can be considered as flat, spherical, or ellipsoidal, the

BM's motion is described in different forms, with a trade-off

between complexity and accuracy. The relevant model is chosen

for an optimal design, according to the point of view of the de-

signer.

In the present work, we look at the re-entry phase only, draw-

ing on the spherical earth model. As mentioned above, there are

two main impacts on the BM during the re-entry phase, i.e.,

gravity and drag; however, in a maneuvering BM, lift force may

be exerted on it, leading to more complicated estimating process.

Fig. 3. Different trajectory phases of a BM.

z

y

x

z

y

x

r

h

Radar Face

Radar Station

Earth

Target

R

O

Local Vertical

Clouds

Top of the Atmosphere

Reentry

Burnout

km

100

15

1000

1600

Launch

JOURNAL OF ELECTROMAGNETIC ENGINEERING AND SCIENCE, VOL. 18, NO. 4, OCT. 2018

234

A non-maneuvering BM is our focus of interest. Also, depend-

ing on the CS used, a re-entry non-maneuvering BM traveling

in the endo-atmosphere is not only subject to atmospheric drag

and the Earth’s gravity but also to the Coriolis and centrifugal

forces [7].

A motion model of a BM is formed and expressed in the

ENU Cartesian CS and with the assumption that the relevant

information on the target includes the position and velocity. For

the sake of convenience, we assume that 𝐩𝑥,𝑦,𝑧 and

𝐯𝑣,𝑣,𝑣 are two vectors denoting the position and the

velocity of the target, respectively. The dynamic model at the re-

entry regime is usually described in a differential form as:

𝐩𝐯

𝐯𝐚𝐚𝐚𝐚 (3)

where 𝐚 is total acceleration; 𝐚, 𝐚 ,𝐚, 𝐚

are acceleration vectors induced by the Earth’s gravity, the drag,

the Coriolis, and the centrifugal force, respectively. Rewriting (3),

we have:

𝐱𝐩𝐯𝐯

𝐚 (4)

where 𝐱𝑥,𝑦,𝑧,𝑣,𝑣,𝑣 represents the state vector of

the target.

The total acceleration for the re-entry BM in the ENU CS is

given specifically in [8], and is as follows:

𝑥𝑦𝑧 2𝜔𝑧𝑐𝑜𝑠𝜑𝑦𝑠𝑖𝑛𝜑

𝜔𝑠𝑖𝑛𝜑𝑟𝑧𝑐𝑜𝑠𝜑𝑦𝑠𝑖𝑛𝜑

𝜔𝑐𝑜𝑠𝜑𝑟𝑧𝑐𝑜𝑠𝜑𝑦𝑠𝑖𝑛𝜑

⎣

⎢

⎢

⎢

⎡

𝜔𝑥

2𝜔𝑥𝑠𝑖𝑛𝜑

2𝜔𝑥𝑐𝑜𝑠𝜑

⎦

⎥

⎥

⎥

⎤

(5)

where

𝑉𝑥𝑦𝑧 = missile velocity (ft/s)

𝛽 = ballistic coefficient

gg𝑟/𝑟

g = gravitational acceleration at sea level (ft/s)

𝜑 = latitude of the radar station

𝑟 = Earth’s radius ( )

𝑟𝑥𝑦𝑧𝑟

distance from the center of the Earth to the missile (ft)

𝜔 = rotation rate of the Earth (rad/s)

𝜌𝜌𝑒 = air density

𝜌,𝐾 = known parameters

ℎ𝑟𝑟 = altitude of the missile.

For the spherical model, the air density is an exponential

function of altitude. The ballistic coefficient is known as the

inverse drag parameter, given by 𝛼𝑆𝑐/𝑚 , wh e re 𝑚 de-

notes target mass, 𝑆 denotes reference area, and 𝑐 is drag

coefficient. The drag parameter is unknown and not constant;

therefore, in the present work, an unknown drag-related para-

meter 𝜌/𝛽 is added to the state vector and estimated online to

enhance performance.

Ultimately, the complete state vector of the BM is 𝐱

𝑥,𝑦,𝑧,𝑥,𝑦,𝑧,𝜌/𝛽. Let 𝑥𝑥, 𝑥𝑦, 𝑥𝑧, 𝑥𝑥,

𝑥𝑦, 𝑥𝑧, 𝑥𝜌/𝛽. The dynamic motion model is

given by [9], and is as follows:

𝐱⎣

⎢

⎢

⎢

⎢

⎢

⎢

⎡

𝑥𝑦𝑧𝑥𝑦𝑧𝜌

𝛽⎦

⎥

⎥

⎥

⎥

⎥

⎥

⎤

⎣

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎡

𝑥

𝑥

𝑥

2𝜔𝑥𝑐𝑜𝑠𝜑𝑥𝑠𝑖𝑛𝜑

𝜔𝑠𝑖𝑛𝜑𝑟𝑥𝑐𝑜𝑠𝜑𝑥𝑠𝑖𝑛𝜑

𝜔𝑐𝑜𝑠𝜑𝑟𝑥𝑐𝑜𝑠𝜑𝑥𝑠𝑖𝑛𝜑

𝐾𝑥𝑥𝑥𝑥𝑥𝑥𝑟𝑥

𝑟⎦

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎤

⎣

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎡

0

0

0

𝜔𝑥

2𝜔𝑥𝑠𝑖𝑛𝜑

2𝜔𝑥𝑐𝑜𝑠𝜑

0⎦

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎤

(6)

(6) is the non-linear function of state vector 𝐱.

𝐱𝐟𝐱 (7)

where 𝐟𝐱 is the seven-dimensional vector function of 𝐱. The

state vector of the target can be discretized by expanding 𝐱

𝐱𝑡Δ𝑡 by Taylor expansion up to the first order:

𝐱𝑡Δ𝑡𝐱𝑡𝐱𝑡Δ𝑡HOT (8)

where Δ𝑡 denotes the small-time step and HOT denotes high-

er order terms. Defining 𝐱𝐱𝑡 and 𝐱𝐱𝑡Δ𝑡,

(8) can be rewritten as follows:

𝐱𝐱𝐟𝐱Δt𝐪 (9)

where 𝐪 represents the discretization error (including HOT)

and modeling uncertainties in motion, and (9) is the recursive

motion equation of the re-entry BM. It is assumed that the er-

ror 𝐪 is Gaussian, zero-mean, and white:

𝐸𝐪0,𝐸𝐪𝐪

𝐐𝜹 (10)

ft

NGUYEN et al.: ANALYSIS OF THE OPTIMAL FREQUENCY BAND FOR A BALLISTIC MISSILE DEFENSE RADAR SYSTEM

235

where 𝛿1 for 𝑘𝑗 and 𝛿0 for others. Note that

𝐐 is a covariance matrix and is one of the known parameters

for the filtering technique discussed in later sections.

3. Radar Measurement Model

In this section, we present a measurement model for phased-

array radar. As is known, phased-array radar measures the range

and angular information of the BM on a spherical CS, which is

referenced directly to the RF Cartesian CS 𝑂𝑥𝑦𝑧.

Specifically, the phased-array radar used for tracking provides

range 𝑟, which denotes distance between the radar and the tar-

get, and two angular measurements, i.e., azimuth 𝑏 and eleva-

tion 𝑒, as illustrated in Fig. 4.

In the spherical CS, these measurements are generally mod-

eled in the following form of additive noise:

𝑟r𝑤

𝑏b𝑤

𝑒e𝑤 (11)

where r,b,and e, which are in non-italic form, denote true

measurements of the target in the sensor spherical CS, and

𝑤,𝑤,and 𝑤 represent the uncorrelated Gaussian noises

with zero-mean, as:

𝐸𝐰0,𝐑𝐸𝐰𝐰𝑻diagσ,σ

,σ

(12)

where 𝐰𝑤,𝑤,𝑤 is the measurement noise vector and

𝐑 denotes the covariance matrix, which is the known parameter.

Some other measurement models can be found in [10].

Let 𝑥,𝑦,𝑧 be the true position of the BM on the RF

Cartesian CS. The noise-corrupted measurements can be con-

verted into Cartesian coordinates as:

𝑟𝑏𝑒 𝑥𝑦𝑧

tan𝑦/𝑥

tan𝑧/𝑥𝑦𝐠𝐱

(13)

Fig. 4. Radar measurement model.

Clearly, the measurements relate to the state vector 𝐱 in a

non-linear function 𝐠, and, after adding a time index (11), be-

come:

𝐲𝐠𝐱𝐰 (14)

The measurement equation is given by (14), where 𝐲

𝑟,𝑏,𝑒

and 𝐰𝑤,𝑤,𝑤

denote the noise- corrupt-

ed measurement vector and the radar noise vector, respectively,

at time 𝑘, and 𝐠 is the vector function of 𝐱. The measure-

ment noise comes from several different sources, and it is impos-

sible to devise a perfect system without noise. The dependence

of noise on frequency is one of the factors needing to be clarified.

In general, accuracy of each measurement is represented by

standard deviation 𝜎. According to [5], there are three main

noise sources causing range measurement error that is modelled

by range standard deviation 𝜎 as:

𝜎𝜎

𝜎

𝜎

(15)

where

𝜎 = SNR-dependent random range error,

𝜎 = range fixed random error,

𝜎 = range bias error.

The SNR-dependent random range measurement error

dominates the radar range error and is determined as follows:

𝜎 Δ𝑅2SNR

(16)

where Δ𝑅𝑐𝜏/2 is the range resolution, 𝜏 the pulse-width,

𝑐 the light speed; and signal-to-noise ratio (SNR) is known as

the radar sensitivity.

Similarly, accuracy of two angular measurements (azimuth

and elevation angles) is also determined by the root-sum-

squared standard deviation of three main errors, as:

𝜎𝜎

𝜎

𝜎

(17)

where

𝜎 = SNR-dependent random range error,

𝜎 = range fixed random error,

𝜎 = range bias error.

The SNR-dependent random angular measurement error

dominates the radar angular error and is given by:

𝜎𝜃𝑘2SNR

(18)

where 𝑘 is the mono-pulse pattern difference slope and typi-

cally equal to 1.6, and 𝜃 is the half-power broadside beam-

width in the angular coordinate of the measurement. The accu-

racy of the azimuth and elevation angles is identified corre-

sponding to the beamwidth values on the respective plane. For

z

y

x

O

r

JOURNAL OF ELECTROMAGNETIC ENGINEERING AND SCIENCE, VOL. 18, NO. 4, OCT. 2018

236

phased-array radar, the broadside beamwidth on each angular

coordinate can be broadened by a scan angle off-broadside 𝛾,

leading to the angular error increment 𝜎, as:

𝜎𝜃𝑘𝑐𝑜𝑠𝛾2SNR

(19)

The broadside beamwidth relates to wavelength or operating

frequency and antenna size, as follows:

𝜃𝑘𝑐𝑓𝐷

(20)

where 𝑘 is the antenna beamwidth coefficient and nearly uni-

ty, 𝑓 is the operating frequency, and 𝐷 is the dimension of

the antenna on the plane at which antenna patterns are meas-

ured. For example, a rectangular phased-array antenna of size

𝐿𝑊 is capable of steering the beam in two dimensions; the

broadside beamwidths on azimuth and elevation coordinates

can be calculated as:

𝜃𝑘𝑐𝑓𝐿

(21)

𝜃𝑘𝑐𝑓𝑊

(22)

4. Extended Kalman Filter

The Kalman filter (KF) is a highly adaptable iterative algo-

rithm, which can estimate non-measured quantities [11]. In

radar applications, the BM velocity is not provided directly by

radar measurements; therefore, KF is a useful tool for extracting

the entire state of the target. The motion and measurement

equations are known as the recognized knowledge of the KF,

and are constructed in linear forms. The EKF is broadened to

apply to non-linear systems [12]. We restate the motion and

measurement equation of BM mentioned above as:

𝐱𝐱𝐟𝐱𝐪𝒌 (23)

𝐲𝐠𝐱𝐰 (24)

For the sake of convenience, let 𝐱/ denote the estimate of

𝐱 based on measurements up to time 𝑗, and 𝐏/ denote the

error covariance matrix associated with 𝐱/. The whole proce-

dure of EKF can be concisely summarized in the following

equations:

• State prediction equation

𝐱

/𝐱

/𝐟𝐱

/Δ𝑡 (25)

• State correction equation

𝐱

/𝐱

/𝐊𝐲𝐠𝐱

/ (26)

where 𝐊 denotes the filter gain.

𝐊𝐏/𝐆

𝐆𝐏/𝐆

(27)

• Covariance prediction equation

𝐏/ 𝐅𝐏/𝐅𝐐 (28)

𝐅𝐈𝐀𝐱/Δ𝑡 (29)

where 𝐀 is a Jacobian matrix of function 𝐟 and is defined as:

𝐀𝐱/𝜕𝐟∂𝐱

|𝐱/ (30)

each element being calculated as in [9].

• Covariance correction equation

𝐏/𝐈𝐊𝐆𝐏/ (31)

where 𝐆 is a Jacobian matrix of function 𝐠, as:

𝐆𝜕𝐠∂𝐱

|𝐱/ (32)

each element being given in the Appendix.

The detailed flow diagram of EKF for the filtering problem

can be found in [9].

IV. TERMINAL GUIDANCE SYSTEM

Before launching an interceptor, the radar tracks the BM and

predicts an intercepting point in advance. The interceptor is

then guided by the midway guidance law to move to that inter-

cepting point. During the flight of the intercepting missile, the

location of the intercepting point continues to be updated until

seeker acquisition happens, when the interceptor is close enough

and can see the target. If the target’s future location is known

perfectly, a missile guidance system inside the interceptor is not

necessary, because there are no errors to allow for. However, it is

impossible to know the intercepting point precisely; therefore,

the launching interceptor may be flown in the wrong direction,

such an error being the main factor causing fail intercept.

Once the seeker sees the target, the terminal guidance acti-

vates, and the seeker plays a role as active radar, taking over the

tracking throughout the remaining time until intercept. The

intercepting missile supplies an acceleration amount whose di-

rection is perpendicular to its velocity direction, by fuel burn or

removing its control surface [3]. The commanded amount of

acceleration depends on the heading error, and takes the form of

the proportional navigation law, which is given as:

𝑛𝑁𝑉𝛿 (33)

where 𝑛 is the acceleration command (in ft/s), 𝑁 is a

unitless, designer-chosen gain, known as the effective navigation

ratio, and is usually within a range as set out in [3, 5]; 𝑉 is the

missile-target closing velocity (in ft/s), and the line of sight an-

gle 𝛿 (in rad) is the angle between an imaginary line connect-

ing the interceptor and the ballistic target and a fixed reference,

as illustrated in Fig. 5. The over-dot denotes the time derivative

NGUYEN et al.: ANALYSIS OF THE OPTIMAL FREQUENCY BAND FOR A BALLISTIC MISSILE DEFENSE RADAR SYSTEM

237

Fig. 5. Line of sight angle.

of the line of sight angle. More detailed information on the

proportional navigation law can be found in [13].

A diagram of a typical terminal guidance system takes the

form of a control loop, as shown in Fig. 6 [13]. In this diagram,

the interceptor acceleration 𝑛 is subtracted from the target

acceleration to generate a relative acceleration, and then a rela-

tive distance is formed after two integrations; at the end of the

flight, the relative distance, called miss distance, is considered as

a performance parameter. Most missile designers desire zero-

miss distance. The line of sight angle 𝛿 is extracted by head-

ing-error addition. For a zero-lag guidance system (not dynamic)

and a non-maneuvering target, the miss distance will always be

zero if the interceptor has sufficient acceleration to offset head-

ing error throughout the seeker acquisition time.

If a zero-miss distance determines a successful intercept, the

required acceleration to compensate for heading error at an in-

stant time within flight time 𝑡 or the amount of time from

seeker acquisition until intercept is given by:

𝑛𝑉𝐻𝐸𝑁1𝑡 𝑡

⁄ 𝑡

(34)

where 𝑉 is the velocity of the interceptor, 𝐻𝐸 is the angular

heading error, and 𝑡 is instantaneous time.

The prediction error (PE) (in ft) and the heading error have

a relationship according to:

PE𝑉𝐻𝐸𝑡 (35)

Substituting (35) into (34), we have

𝑛PE𝑁1𝑡/𝑡 𝑡

(36)

Fig. 6. Terminal guidance system.

The lateral divert or total acceleration Δ𝑉 required during

the flight time 𝑡 relates to 𝑛 according to:

Δ𝑉|𝑛|𝑑𝑡

PE𝑁𝑁1𝑡

(37)

(37) indicates the minimum amount of lateral divert that

must be available in an interceptor to ensure successful destruc-

tion. It can be seen that the longer the flight time, the smaller

the lateral divert; therefore, techniques increasing the seeker

acquisition range increase the acceleration capability of the in-

terceptor.

V. SIMULATION RESULTS

In simulation, we consider two typical trajectories of the BM

in its re-entry phase [8]. The BM at each trajectory is assumed

to have the same ballistic coefficient and begin its re-entry phase

at different altitudes with (nearly) the same beginning velocity.

The actual initial state 𝐱 includes the following elements:

𝑥338,110 ft, 𝑥338,110 ft, 𝑥199,910 ft, 𝑥

-15,297 ft/s, 𝑥-15,297 ft/s, 𝑥-8,653 ft/s, 𝑥4.395

10 lb/ft for case 1, and 𝑥920,640 ft, 𝑥451,515

ft, 𝑥327,897 ft, 𝑥-18,187 ft/s, 𝑥-11,232 ft/s,

𝑥-7,014 ft/s,𝑥7.674210 lb/ft for case 2.

The effectiveness of EKF is compared across five frequency

bands: L-band (1.3 GHz), S-band (2.5 GHz), C-band (5.5

GHz), X-band (9 GHz), and Ku-band (13.5 GHz), by position

error given by:

𝑃𝑜𝑠𝑖𝑡𝑖𝑜𝑛 𝑒𝑟𝑟𝑜𝑟

𝑥𝑥𝑥𝑥𝑥𝑥 (38)

The position error is averaged over Monte-Carlo simulation

runs. At the radar, the sampling interval is set at Δ𝑡=0.1 s, the

radar sensitivity SNR = 12 dB, and the pulse-width 𝜏1 μs.

The phased-array rectangular antenna has a size of 3 m × 5 m,

the scan angle off-broadside 𝛾30°, the measurement covari-

ance matrix 𝐑 is given by (12), whose range variance is given by

(15), and angular variances are calculated by (19).

Fig. 7 shows the actual altitude of the BM during re-entry

flight time at (nearly) the same beginning velocity (around 23

kft/s). The BM at higher altitude takes a longer interval than

lower-altitude BM to reach the same altitude. For example, the

BM in case 2 flies to an altitude of 100 kft in 40 seconds, and in

just 12 seconds in case 1. Also, the BM in case 1 is decelerated

faster, owing to a higher drag effect at lower altitude, and vice

versa, as shown in Fig. 8.

Figs. 9 and 10 show the position error during the tracking

Reference

Target

Int ercepto r

Geometry Seeker Noise Filter

Guidance

Flight Control

System

Target

Heading

Error Noise

Miss

c

n

L

n

Acceleration

JOURNAL OF ELECTROMAGNETIC ENGINEERING AND SCIENCE, VOL. 18, NO. 4, OCT. 2018

238

Fig. 7. Target's actual altitude during re-entry flight time.

Fig. 8. Target's actual velocity versus re-entry flight time.

Fig. 9. The position error versus re-entry flight time for case 1: 𝛥t

= 0.1, SNR = 12 dB.

time. It can be seen that the position error reduces as the time

increases. The higher frequency yields a smaller position error.

For example, the S-band radar derives a position error of about

3,000 ft, compared with just 800 ft for the C-band at the same

tracking time of 15 seconds in case 1. The performance gap is

Fig. 10. The position error versus re-entry flight time for case 2: 𝛥t

= 0.1, SNR = 12 dB.

negligible after 26 seconds and 40 seconds for case 1 and case 2,

respectively.

Figs. 11 and 12 show the lateral divert for an interceptor to

correct the prediction error at an intercepting point of altitude

100 kft. According to Fig. 7, the time to reach the altitude of

100 kft is 12 seconds for case 1 and 40 seconds for case 2. The

flight time is assumed to be 𝑡3 seconds for both trajectory

cases. Note that the flight time is the time from the point at

which tracking stops at radar until the intercepting time, mean-

ing that the radar stops tracking at t = 9 seconds and t = 37 sec-

onds for case 1 and case 2, respectively. It can be seen that the

lateral divert reduces when the radar operates at a higher fre-

quency, owing to smaller position error. For example, if a fixed

capability of an interceptor is 600 ft/s for case 1, the radar must

operate at a frequency greater than 2.5 GHz for a successful

intercept. Also, the performance gap is negligible at a frequency

larger than 5.5 GHz. When increasing the effective navigation

ratio N′, the missile needs less lateral divert; however, the guid-

Fig. 11. Lateral divert versus frequency for zero-lag guidance sys-

tem for case 1, with intercepting altitude of 100 kft (about

30 km), and flight time of 𝑡3 seconds.

NGUYEN et al.: ANALYSIS OF THE OPTIMAL FREQUENCY BAND FOR A BALLISTIC MISSILE DEFENSE RADAR SYSTEM

239

Fig. 12. Lateral divert versus frequency for zero-lag guidance sys-

tem in case 2, with intercepting altitude of 100 kft (about

30 km), and flight time of 𝑡3 seconds.

Fig. 13. Lateral divert versus frequency for zero-lag guidance sys-

tem in case 2, with intercepting altitude of 230 kft (about

70 km), and flight time of 𝑡3 seconds.

ance noise will increase significantly [14]. Furthermore, the lat-

eral divert required in case 2 is much less than that in case 1,

since the altitude of the BM in case 2 is higher than in case 1.

This causes the BM to travel for a longer time in order to reach

the intercepting point, and the estimation is therefore improved.

This is also obvious when considering the higher altitude of

the intercepting point shown in Fig. 13. The lateral divert in

order to intercept at altitude 70 (km) is much larger than that at

30 (km). For example, at S-band, the interceptor must respond

by an amount of more than 1,500 (ft/s) at an intercepting alti-

tude of 70 km, compared to around 120 (ft/s) at an intercepting

altitude of 30 (km). The lateral divert gap between frequencies is

also broadened.

VI. CONCLUSION

In conclusion, the accuracy of the radar angular measure-

ments is inverse to the frequency. The tracking performance is

therefore improved at high frequency. This increases the inter-

cepting capability of the BMDS, especially at high intercepting

altitude.

REFERENCES

[1] A. Blencowe, "Pursuing peace with the weapons of war: bal-

listic missile defence and international security," 2009;

https://www.e-ir.info/2009/09/05/pursuing-peace-with-the-

weapons-of-war-ballistic-missile-defence-and-international-

security/.

[2] Y. Y. Chen and K. Y. Young, "An intelligent radar predictor

for high-speed moving-target tracking," in Proceedings of

2002 IEEE Region 10 Conference on Computers, Communica-

tions, Control and Power Engineering, Beijing, China, 2002,

pp. 1638–1641.

[3] P. Zarchan, "Ballistic missile defense guidance and control

issues," Science & Global Security, vol. 8, no. 1, pp. 99–124,

1999.

[4] M. A. Richards, J. Sc h e e r, and W. A. Holm, Principles of

Modern Radar: Basic Principles. Raleigh, NC: SciTech Pub-

lishing, 2010.

[5] G. Richard Curry, Radar System Performance Modeling, 2nd

ed. Boston, MA: Artech House, 2005.

[6] IEEE Standard for letter designations for radar-frequency bands

(revision of IEEE 521-1984), IEEE 521-2002, 2002.

[7] X. R. Li and V. P. Jilkov, "Survey of maneuvering target

tracking. Part II: Motion models of ballistic and space tar-

gets," IEEE Transactions on Aerospace and Electronic Sys-

tems, vol. 46, no. 1, pp. 96–119, 2010.

[8] Y. Kashiwagi, Prediction of Ballistic Missile Trajectories. Menlo

Park, CA: Stanford Research Institute, 1968.

[9] M. Dressler and W. Ross, Real Time Implementation of the

Kalman Filter for Trajectory Estimation. Me n l o Park, CA:

Stanford Research Institute, 1968.

[10] X. R. Li and V. P. Jilkov, "Survey of maneuvering target

tracking. III. Measurement models," in Signal and Data

Processing of Small Targets 2001. Bellingham , WA: Interna-

tional Society for Optics and Photonics, 2001, pp. 423–447.

[11] R. E. Kalman, "A new approach to linear filtering and pre-

diction problems," Journal of Basic Engineering, vol. 82, no.

1, pp. 35–45, 1960.

[12] M. I. Ribeiro, "Kalman and extended Kalman filters: con-

cept, derivation and properties," Institute for Systems and

Robotics, Lisbon, Portugal, 2004.

[13] P. Zarchan, Tactical and Strategic Missile Guidance, 6th ed.

Washington, DC: American Institute of Aeronautics and

Astronautics Inc., 2012.

[14] M. I. Skolnik, Radar Handbook, 2nd ed. Singapore :

McGraw-Hill, 1991.

JOURNAL OF ELECTROMAGNETIC ENGINEERING AND SCIENCE, VOL. 18, NO. 4, OCT. 2018

240

Dang-An Nguyen

has a degree from the School of Electronics and

Telecommunications at Hanoi University of Science

and Technology, Vietnam. He has three years’ expe-

rience working as a senior member of the Signal

Processing and Radio Communication Laboratory.

He is currently studying for a Master's degree at

Soongsil University, Korea, where his major relates

to microwave signal processing, radar systems, power

amplifiers, and non-Foster circuits.

Byoungho Cho

received his B.S. degree in electrical engineering

from the Republic of Korea Airforce in 1981. He

received his Master’s degree and Ph.D. in electrical

engineering from Florida Institute of Technology in

1987 and 1993, respectively. He worked in the areas

of information and communications in the Republic

of Korea Airforce from 1981 to 2004. Since 2005,

he has worked for the ISR systems department at

LIG Nex1. His current research interests include radar signal processing,

target tracking, and system engineering.

Chulhun Seo (M'97–SM'14)

Please refer to the July 2017 issue (JEES vol. 16, no. 3).

Jeongho Park

received his B.S. degree in electrical engineering

from Yonsei University in 1988 and his Master’s

degree and Ph.D. in electrical engineering from

POSTECH in 1990 and 2001, respectively. Since

1990, he has worked in the development of advanced

radar systems at LIG Nex1. His current research

interests include radar signal processing, target track-

ing, and system engineering.

Dong-Hui Lee

received B.S. and M.S. degrees in telecommunica-

tion engineering from Korea Aerospace University in

2009 and 2011, respectively. Since 2011, he has

worked in the development of advanced radar sys-

tems at LIG Nex1. His current research interests

include radar signal processing, tracking filters, and

system engineering.

NGUYEN et al.: ANALYSIS OF THE OPTIMAL FREQUENCY BAND FOR A BALLISTIC MISSILE DEFENSE RADAR SYSTEM

241

APPENDIX

Calculate Jacobian matrix 𝐆, which is in the following form:

(39)

𝐺

𝑥𝑐𝑜𝑠𝜆𝑥𝑐𝑜𝑠𝜙𝑠𝑖𝑛𝜆𝑥𝑠𝑖𝑛𝜙𝑠𝑖𝑛𝜆

𝐺

𝑥𝑠𝑖𝑛𝜆𝑥𝑐𝑜𝑠𝜙𝑐𝑜𝑠𝜆𝑥𝑠𝑖𝑛𝜙𝑐𝑜𝑠𝜆

𝐺

𝑥𝑠𝑖𝑛𝜙𝑥𝑐𝑜𝑠𝜙

𝐺

𝑥𝑐𝑜𝑠𝜙𝑠𝑖𝑛𝜆𝑥𝑐𝑜𝑠𝜆

𝐺

𝑥𝑐𝑜𝑠𝜙𝑐𝑜𝑠𝜆𝑥𝑠𝑖𝑛𝜆

𝐺

𝑥𝑠𝑖𝑛𝜙

𝐺1

𝑟𝑥𝑥𝑠𝑖𝑛𝜙𝑠𝑖𝑛𝜆𝑥𝑥𝑐𝑜𝑠𝜆

𝑥𝑐𝑜𝑠𝜙𝑠𝑖𝑛𝜆/𝑥𝑥

𝐺

𝑥𝑥𝑠𝑖𝑛𝜙𝑐𝑜𝑠𝜆𝑥𝑥𝑐𝑜𝑠𝜙𝑐𝑜𝑠𝜆

𝑥𝑠𝑖𝑛𝜆/𝑥𝑥

𝐺

𝑥𝑥𝑐𝑜𝑠𝜙𝑥𝑥𝑠𝑖𝑛𝜙/

𝑥𝑥

where

𝑥

𝑥

𝑥𝐓𝑥

𝑥

𝑥 (40)

𝑟𝑥𝑥𝑥 (41)

11 12 13

21 22 23

31 32 33

0000

0000

0000

GGG

GGG

GGG

G