ARL-TR-7895 ● NOV 2016

US Army Research Laboratory

Numeric Computation of the Radar Cross 

Section of In-flight Projectiles

by Traian Dogaru and Christopher Kenyon

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ARL-TR-7895 ● NOV 2016

US Army Research Laboratory

Numeric Computation of the Radar Cross 

Section of In-flight Projectiles

by Traian Dogaru and Christopher Kenyon

Sensors and Electron Devices Directorate, ARL

Approved for public release; distribution unlimited.
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November 2016

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Oct 2015–Sep 2016

4. TITLE AND SUBTITLE

Numeric Computation of the Radar Cross Section of In-flight Projectiles

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6. AUTHOR(S)

Traian Dogaru and Christopher Kenyon

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US Army Research Laboratory

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ARL-TR-7895

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13. SUPPLEMENTARY NOTES

14. ABSTRACT

In this report, we evaluate the radar cross section (RCS) of in-flight ballistic projectiles (rockets, artillery rounds, and mortars) 

by taking into account the relative position and orientation of the targets with respect to a ground-based radar. The evaluation 

is based on computer models of both the ballistic trajectory and the radar scattering from the targets. Extensive theoretical 

derivations are developed, followed by numerical examples. We consider 2 types of projectiles: a 155-mm artillery round and 

a 120-mm mortar, as well as 4 radar frequency bands: L, S, C, and X. Using simulation tools entirely developed at the US 

Army Research Laboratory, we analyze several sample scenarios and discuss the phenomenological findings associated with 

them. The in-flight projectile RCS calculations will be used in future studies of radar system performance evaluation.

15. SUBJECT TERMS

computational electromagnetics, radar signature, ballistic trajectory, radar cross section, RCS

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ABSTRACT

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19a. NAME OF RESPONSIBLE PERSON

Traian Dogaru

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iii

Contents

List of Figures iv

List of Tables v

1. Introduction 1

2. Calculation of In-Flight RCS of Projectiles 2

2.1 General Methodology 2

2.2 Finding the Radar-Target Relative Orientation 5

2.3 Polarization Transformations 10

2.4 Propagation and Antenna Effects and the Radar Received Power 15

3. Simulation of Ballistics Trajectories 18

3.1 The Modified Point Mass Model 18

3.2 Epicyclic Motion of Flying Projectiles 26

4. Numeric Examples 32

4.1 Dynamic RCS of an In-Flight 155-mm Artillery Round 32

4.2 Dynamic RCS of an In-Flight 120-mm Mortar 39

5. Conclusions 43

6. References 45

List of Symbols, Abbreviations, and Acronyms 47

Distribution List 48
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iv

List of Figures

Fig. 1 The geometric configuration involved in the radar sensing of an inflight ballistic projectile as seen in the ground reference frame x-y-z 

(or G)......................................................................................................2

Fig. 2 The radar targets in the i-j-k (or P) frame, showing the 155-mm 

artillery round (left) and the 120-mm mortar (right). When no spin, 

pitch, and yaw motions are accounted for, these pictures describe the 

radar-projectile relative orientation in the AFDTD radar signature 

models. The origin of the frame coincides with the projectile’s center 

of mass. ..................................................................................................3

Fig. 3 Flowchart describing the methodology of the dynamic RCS calculation 

for in-flight ballistic projectiles .............................................................5

Fig. 4 The in-flight 155-mm artillery round exhibiting the SPY motion 

components, shown here in the i-j-k (or P) frame (left) and the i1-j1-k1

(or P1) frame (right). In the left-hand picture, the k axis coincides with 

the projectile’s velocity vector and the tangent to the trajectory. The 

picture on the right describes the final radar-projectile relative 

orientation in the AFDTD radar signature models. ...............................9

Fig. 5 Diagrams showing the angles and unit vectors relevant to the 

polarization transformation calculation: a) view in the ground (G) 

frame and b) view in the projectile (P1) frame ....................................12

Fig. 6 The geometrical configuration involved in the calculation of the 

ballistic trajectory. The projectile axis coincides with the k1 unit 

vector, while the velocity vector V has the same direction as the k unit 

vector. In this figure, the angle θB is negative. ....................................20

Fig. 7 Various ways of describing the simulated 155-mm artillery round 

ballistic trajectory and its parameter variation vs. time: a) 3-D 

trajectory curve, b) height vs. range curve, c) velocity components vs. 

time, and d) projectile orientation angle φB vs. time. Figure 7a shows 

the approximate radar location in the simulations in Section 4.1........25

Fig. 8 Various ways of describing the simulated 120-mm mortar ballistic 

trajectory and its parameter variation vs. time: a) 3-D trajectory curve, 

b) height vs. range curve, c) velocity components vs. time, and d) 

projectile orientation angle φB vs. time. Figure 8a shows the 

approximate radar location for the simulations in Section 4.2. ...........26

Fig. 9 Trajectory described by the 155-mm projectile tip in the i-j plane over 

a 1-s interval. a) 3-D view and b) view in the i-j plane. The pitch and 

yaw angles in Fig. 9a are exaggerated (not to scale). In Fig. 9b, the 

observer looks along the k axis, in the positive direction....................31

Fig. 10 Relative angles between the radar, projectile orientation, and ground 

obtained in the trajectory simulations of the 155-mm artillery round, 

showing: a) the φB, θ and γ angles, and b) the polarization rotation 

angle η .................................................................................................33
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v

Fig. 11 Dynamic RCS vs. time curves obtained for the simulated trajectory of 

the 155-mm artillery round that does not account for the SPY motion 

components: a) L-band, b) S-band, c) C-band, and d) X-band...........34

Fig. 12 Dynamic RCS vs. time curves obtained for the simulated trajectory of 

the 155-mm artillery round that includes the SPY motion components: 

a) L-band, b) S-band, c) C-band, and d) X-band .................................35

Fig. 13 Received radar power vs. time for the in-flight 155-mm artillery round, 

assuming the fixed gain model in all 4 frequency bands, showing a) VV polarization and b) H-H polarization ...............................................37

Fig. 14 Received radar power vs. time for the in-flight 155-mm artillery round, 

assuming the fixed aperture model in all 4 frequency bands, showing 

a) V-V polarization and b) H-H polarization.......................................37

Fig. 15 Propagation factor F4 during the first 3 s of the 155-mm round ballistic 

flight, for all 4 frequency bands and a) V-V polarization and b) H-H 

polarization ..........................................................................................38

Fig. 16 Received radar power during the first 3 s of the 155-mm artillery 

round ballistic flight, assuming the fixed gain model in all 4 frequency 

bands, showing a) V-V polarization and b) H-H polarization.............38

Fig. 17 Relative angles between the radar, projectile orientation, and ground 

obtained in the trajectory simulations of the 155-mm artillery round, 

showing a) the φB, θ, and γ angles and b) the φ angle for 2 values of 

χ0 ..........................................................................................................40

Fig. 18 Dynamic RCS vs. time curves obtained for the simulated trajectory of 

the 120-mm mortar, comparing various ways to account for the initial 

roll angle χ0, for V-V polarization and a) L-band, b) S-band, c) Cband, and d) X-band. Note: The curves obtained for L- and S-bands 

are on top of one another. ....................................................................41

Fig. 19 Dynamic RCS vs. time curves obtained for the simulated trajectory of 

the 120-mm mortar, using the average over all possible angles χ0, for 

a) L-band, b) S-band, c) C-band, and d) X-band .................................42

Fig. 20 Received radar power vs. time for the in-flight 120-mm mortar, 

assuming the fixed gain model in all 4 frequency bands, showing a) VV polarization and b) H-H polarization ...............................................43

Fig. 21 Received radar power during the first 3 s of the 120-mm mortar 

ballistic flight, assuming the fixed gain model in all 4 frequency bands, 

showing a) V-V polarization and b) H-H polarization .......................43

List of Tables

Table 1 Physical parameters and initial conditions for the trajectory simulation 

examples in this section, involving a 155-mm artillery round and a 

120-mm mortar ....................................................................................25
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vi

INTENTIONALLY LEFT BLANK.
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1

1. Introduction

The US Army is actively pursuing the development of radar technology for the 

detection and tracking of artillery rounds grouped under the generic category of 

rockets, artillery, and mortar (RAM). The electromagnetic (EM) modeling team at 

the US Army Research Laboratory (ARL) is supporting these programs by 

modeling the radar signature of targets relevant to this application, which is a 

critical part of any radar system performance prediction study. This report 

continues previous work done at ARL related to RAM radar targets by considering 

their dynamic signature during the ballistic flight.

Very little work has been published on this topic in the past, particularly in the 

public domain. Otero et al.1 provide a useful analysis of the RAM target radar cross 

section (RCS), including its variations for in-flight projectiles. However, no details 

are offered in that report related to the EM modeling codes used for radar signature 

prediction or the general methodology employed in computing the dynamic RCS 

of flying projectiles.

Our own previous investigations2,3 analyzed the radar signature of RAM targets 

placed in a fixed position, using the AFDTD software developed at ARL.

4 To 

predict the signature of these targets during their ballistic flight, we need to develop 

models of their trajectories, as well as the geometric transformations that link the 

coordinate frames of the ground-based radar and the AFDTD signature simulations. 

While both these problems have been extensively investigated by other research 

communities (particularly those involved with ballistics and computer graphics, 

respectively), we are not aware of other documented work that brings these 2 

engineering fields together with the radar signature analysis into one 

comprehensive study. Note that all the computer simulation codes employed in this 

work were developed entirely at ARL.

This report is organized as follows. In Section 2, we discuss the general 

methodology of in-flight projectile radar signature calculation, with emphasis on 

the geometrical transformations between the radar reference frame and the 

projectile coordinate frame. Section 3 discusses modeling ballistic trajectories, 

presenting the phenomenological aspects relevant to radar sensing of in-flight

ballistic projectiles. In Section 4, we perform the numerical analysis of the dynamic 

radar signature for 2 types of projectiles: a 155-mm artillery round and a 120-mm 

mortar. We finish with conclusions in Section 5.
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2

2. Calculation of In-Flight RCS of Projectiles

2.1 General Methodology

The radar sensing scenario under consideration in this report is described in Fig. 1. 

The radar system is placed in a stationary configuration at coordinates ( ) R R R x , y ,z , 

whereas the projectile describes a time-dependent trajectory with instantaneous 

coordinates ( ) P P P x , y ,z of its center of mass. Additionally, the orientation of the 

projectile axis as a function of time must be specified. Our purpose is to compute 

the RCS of the in-flight projectile at a given sequence of time samples, as measured

by the ground-based, stationary radar system.

Fig. 1 The geometric configuration involved in the radar sensing of an in-flight ballistic 

projectile as seen in the ground reference frame x-y-z (or G)
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The RCS calculation method starts with a characterization of the projectile’s 

trajectory. This can be done either via computer simulations or measurements; in 

this study, we use models to compute the target’s coordinates and orientation over 

the flight interval (see Section 3). Combining this information with the radar 

position, one can compute the relative radar-projectile orientation angles (more 

exactly, the angles between the radar line of sight [LOS] and the projectile 

symmetry axis) at each time sample. The geometrical transformations involved in 

this procedure are described in Section 2.2.

In previous work, we evaluated the radar signature of RAM projectiles in fixed 

positions, assuming the radar is placed at various angular coordinates on a sphere 

surrounding the target (Fig. 2). At each time sample of the projectile’s flight, for 

the current radar-projectile aspect angle, the fixed-orientation signature data are 

used as a lookup table to compute the instantaneous RCS. Note that this procedure 

is independent of the radar signature evaluation method for the fixed position 

projectile: this can be obtained through measurements, numerical computations, or 

analytical approximations. In this study, we use data obtained via numerical 

simulation with the AFDTD software,4 as described in our previous work.2,3

Fig. 2 The radar targets in the i-j-k (or P) frame, showing the 155-mm artillery round (left) 

and the 120-mm mortar (right). When no spin, pitch, and yaw motions are accounted for, 

these pictures describe the radar-projectile relative orientation in the AFDTD radar signature 

models. The origin of the frame coincides with the projectile’s center of mass.

When transitioning from the fixed-orientation projectile to the in-flight projectile 

radar sensing scenarios, an additional complication arises due to the change in 

polarization of the radar waves between the 2 geometries. Typically, the radar 

signature is evaluated for a set of orthogonal polarization combinations (at the 

transmitter and receiver, respectively). In this case, we are interested in the vertical-
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4

vertical (V-V) and horizontal-horizontal (H-H) polarizations. These polarizations 

are defined by the orientation of the electric-field vector in the radar (plane) wave 

with respect to the ground (or horizontal, or x-y) plane. Thus, the wave is 

horizontally polarized when the electric field is parallel to the ground plane and 

vertically polarized when the electric field is contained in the plane defined by the 

vertical axis (z) and the direction of propagation.

5

A quick examination of the geometries in Figs. 1 and 2 reveals that the vertical and 

horizontal polarizations have completely different meanings in the 2 sensing 

scenarios. Consequently, when computing the radar signature of the in-flight target, 

the transformation from one coordinate system to the other must take into account 

not only the relative radar-target orientation, but the change in polarization as well. 

As shown in Section 2.3, the polarization transformations involve the full scattering 

matrix of the target,

6 which comprises the complex scattering parameters for each 

polarization combination, including vertical-horizontal (V-H) and horizontalvertical (H-V).

So far, our discussion has been limited to the evaluation of the target response for 

this sensing scenario. Nevertheless, predicting the radar performance must take into

account many other system parameters and environmental factors via the radar 

equation.

7,8 A full system performance analysis is beyond the scope of this report; 

however, we discuss some phenomenological aspects related to radar wave 

propagation and demonstrate how to calculate the radar received power for a given 

set of system parameters in Section 2.4. The signature analysis methodology 

described in this section is illustrated graphically in Fig. 3.
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5

Fig. 3 Flowchart describing the methodology of the dynamic RCS calculation for in-flight

ballistic projectiles

2.2 Finding the Radar-Target Relative Orientation

In this section, we derive the equations describing the relative orientation between 

the radar and the target, given the absolute coordinates of the 2, as well as the 

orientation of the target axis with respect to the ground. This problem can be solved 

by employing geometrical methods frequently used in computer graphics,9

involving vector rotations in the 3-D space.

First, we need to introduce the notations describing the geometric transformations

used in this section. These operations are usually performed on unit vectors in the 

3-D space, defined by their components in a given Cartesian frame, for instance,

the unit vector [ ]

T

u = a b c , with the constraint 1 2 2 2

a + b + c = . A rotation of 

angle α around an axis defined by its unit vector w is described by a 3 × 3 

orthogonal matrix denoted by (α) Rw . (Note: the angle α is considered positive 

when the rotation is performed counterclockwise as one looks against the direction 

of the w axis.) Thus, when we apply this transformation to an arbitrarily oriented 

unit vector u, the result is the unit vector v given by v = Rw (α)u . The rotation 

matrices have the following properties:
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( α) (α) (α) T Rw − = Rw = Rw

−1 . (1)

One can establish a general expression of the rotation matrix for an arbitrary 

direction of the rotation axis in space. However, in this report, we always attempt 

to reduce the problem to a sequence of rotations around one of the Cartesian axes

(x, y or z). These rotations (of angle α) have the following simple expressions:

( )





















= −

α α

α α α

0 sin cos

0 cos sin

1 0 0

Rx , (2)

( )





















−

=

α α

α α

α

sin 0 cos

0 1 0

cos 0 sin

Ry , (3)

( )



















 −

=

0 0 1

sin cos 0

cos sin 0

α α

α α

Rz α . (4)

Another frequently used concept is that of transformation from one Cartesian frame 

A to another frame B. This is described by an orthogonal matrix with the same 

properties as a rotation matrix and denoted QA→B . Thus, if the same unit vector in 

space u is written as A u in frame A and as B u in frame B, then we have 

A

A B

B u = Q → u . The following important result is used in this study: The Cartesian 

frame transformation that takes the arbitrary unit vector [ ] A T

u = a b c (as 

expressed in frame A) to the z axis direction, [ ] B T

u = 0 0 1 (as expressed in 

frame B) is





































+ +

−

− +

+ +

→ =

a b c

a b

a

a b

b

a b

a b

bc

a b

ac

A B 0

2 2 2 2

2 2

2 2 2 2

Q . (5)

Now let us apply this formula to the 2 Cartesian frames described in Figs. 1 and 2. 

We call the frame in Fig. 1 the “ground” frame, denoted by superscript G, and the 

frame in Fig. 2 the “projectile” frame, denoted by superscript P. Notice that the 
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7

projectile axis direction coincides with the k axis in frame P, meaning that its unit 

vector is [ ] P T

P u = 0 0 1 . Therefore, if the same projectile axis is expressed as 

[ ]

T

x y z

G

P u = u u u in frame G, the matrix describing the transformation between 

the 2 frames is









































+ +

−

− +

+ +

→ =

x y z

x y

x

x y

y

x y

x y

y z

x y

x z

G P

u u u

u u

u

u u

u

u u

u u

u u

u u

u u

0

2 2 2 2

2 2

2 2 2 2

Q . (6)

Let u R the unit vector describing the direction of the radar LOS to the projectile 

position. Given the sets of coordinates( ) R R R x , y ,z and ( ) P P P x , y ,z in Fig. 1, as well 

as the angles θ and φ in Fig. 2, we can express this unit vector in the 2 frames as 

follows:

( ) ( ) ( ) 



















−

−

−

− + − + −

=

R P

R P

R P

R P R P R P

G

R

z z

y y

x x

x x y y z z

2 2 2

1

u , (7)





















=

θ

φ θ

φ θ

cos

sin sin

cos sin

P u R . (8)

We use the matrix in Eq. 6 to derive the transformation from G u R to P u R , as one 

transitions from frame G to frame P:

G

G P R

P

R

c

b

a

u = Q → u





















= . (9)

Once we determine the components a, b, and c of P u R , the angles θ and φ are found 

as

c

1 cos− θ = , (10)
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a

1 b

tan − φ = . (11)

These 2 angles are used as indexes in the RCS lookup table obtained from the 

AFDTD simulations, as discussed in the previous section. Interpolation can be used 

when the values computed in Eqs. 10 and 11 fall in between the angle increments 

used in the AFDTD simulations. So far, the evaluation of the radar LOS angles 

required knowledge of the radar coordinates ( ) R R R x , y ,z (fixed in time), as well as 

the projectile coordinates ( ) P P P x , y ,z and orientation ( ) ux uy uz , , with respect to the 

ground frame, which are variable in time and are specified in a separate file 

describing the projectile trajectory.

As illustrated in Section 3, the ballistic trajectories obtained by computer 

simulations are frequently based on the modified point mass model,

10 in which the 

projectile orientation unit vector closely follows the tangent to the trajectory (or the 

velocity vector direction). In this simplified model, the projectile symmetry axis 

coincides with the velocity vector direction (same as the k axis of the P frame in 

Fig. 2), so ( ) ux uy uz , , represent the velocity vector components normalized to this 

vector’s magnitude. In reality, the projectile motion is more complex and its 

complete characterization requires a 6 degrees-of-freedom (DOF) model 

framework.10 Since we do not have access to the 6 DOF trajectory data, our 

simulation strategy involves separately adding spin, pitch, and yaw (SPY) motion 

components to the trajectory data obtained via the modified point mass model (see 

Section 3 for more details). In the following, we call P the frame where the 

projectile velocity vector (or tangent to the trajectory) is aligned with the vertical

axis, and P1 the frame where the projectile axis after SPY is aligned with the vertical

axis. The difference between these 2 frames is illustrated in Fig. 4.
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Fig. 4 The in-flight 155-mm artillery round exhibiting the SPY motion components, shown 

here in the i-j-k (or P) frame (left) and the i1-j1-k1 (or P1) frame (right). In the left-hand picture, 

the k axis coincides with the projectile’s velocity vector and the tangent to the trajectory. The 

picture on the right describes the final radar-projectile relative orientation in the AFDTD 

radar signature models.

The spinning motion component can be described as a rotation of the projectile in 

vertical orientation (as in Fig. 2), around the k axis and of angle χ. The pitch and 

yaw motion components can be described by a sequence of 2 rotations of the 

projectile axis: one around the j axis (of angle α), followed by one around the i axis 

(of angle –β). As a side note, switching the order of the last 2 rotations leads to 

slightly different results, since rotations are noncommutative; however, given the 

fact that α and β are small, both rotation sequences can be approximated by the 

same matrix, so the order does not matter. The time dependence of these 3 motion 

components is discussed in Section 3. Call the unit vector of the new projectile 

orientation (after the 3 SPY rotations) P1

u , coinciding the k1 axis direction. In frame 

P, this is given by

( ) ( ) ( ) P

P

P u P = Ri − β Rj α Rk χ u 1

. (12)

Expressing the unit vector of the projectile axis prior to the 3 additional rotations 

u P , in the new P1 frame, involves performing the inverse of the 3 rotations in 

reverse order:

( ) ( ) ( ) P

P

P

u P = Rk − χ Rj −α Ri β u 1 . (13)

Combining this with previous results, the transformation matrix from the G frame 

to the P1 frame becomes

( ) ( ) ( ) QG→P1 = Rk − χ Rj −α Ri β QG→P (14)
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10

and, finally,

( ) ( ) ( ) G

G P R

P

u R = Rk − χ Rj −α Ri β Q → u 1 . (15)

The calculation of the unit vector P1 u R components allows the evaluation of the radar 

LOS angles θ and φ with respect to the instantaneous projectile axis, after 

accounting for the 3 additional SPY motion components, by using Eqs. 10 and 11.

2.3 Polarization Transformations

Since the EM field is a vector quantity, the full characterization of the target 

response involves the links between all components of the incident and scattered 

electric-field vectors. This is usually expressed by the 2 × 2 scattering matrix of the

target,

6 S, defined by

i

jk r s

r

e E SE

− 0

= (16)

or





















 



 =















 −

i

jk r i

s

s

E

E

S S

S S

r

e

E

E

φ

θ

φθ φφ

θθ θφ

φ

θ 0

. (17)

In these equations, the superscript s indicates scattered field, while the superscript 

i indicates incident field; r is the radar-target range and k0 is the propagation 

constant. Notice that we use the θ and φ components of the electric field instead of 

the traditional vertical and horizontal components, since the latter are typically 

defined with respect to a fixed ground plane. In our analysis, where we use different

reference frames, with different orientations with respect to the ground, these 

traditional definitions are less meaningful.

Our task in this section is to find the transformation of the scattering matrix from a 

hypothetical radar placed in the projectile frame (call this SP) to the ground-based

actual radar frame (call this SR). The former is computed in the AFDTD simulations 

of the target signature, whereas the latter describes the in-flight target response as 

measured by the ground-based radar. All the relevant geometries are illustrated in 

Fig. 5. Notice that in both frames (P1 and G), the far-field electric-field vector of 

the radar waves (both incident and scattered) is perpendicular to the LOS, so only 

2 of its components (θ and φ) need to be considered, as in Eq. 17. The far-field 

scattering model in this study assumes plane wave propagation and linear 

polarization, meaning that the electric-field vectors (incident and scattered) keep 
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11

the same orientation and magnitude along the propagation path. We decompose the 

electric-field vectors in the 2 frames along the 2 components, θ and φ. As shown in 

Fig. 5a, these components have different directions between the 2 frames. The 

relationships between the electric-field vectors in the 2 radar frames can be written 

as















 = =















 = i

R

i

i R

i R

P

i

i P

P E

E

E

E

φ

θ

φ

θ E AE A , (18)















 = =

















= s

R

s

s R

s R

P

s

s P

P E

E

E

E

φ

θ

φ

θ E AE A , (19)

where A is a 2 × 2 orthogonal matrix, with the following elements:





 





− = η η

η η

sin cos

cos sin

A . (20)

Notice that i(s) EP and i(s) ER represent the same vectors in space, although their θ and 

φ components change with the reference frame according to Eqs. 18 and 19.
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12

(a)

(b)

Fig. 5 Diagrams showing the angles and unit vectors relevant to the polarization 

transformation calculation: a) view in the ground (G) frame and b) view in the projectile (P1) 

frame

The angle η can be derived from

( ) R

T

η θP θ cos = e e . (21)
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13

In the last equation, θP e and θR e are the unit vectors of the θ electric-field 

components in the P1 and G frames, respectively (notice that we dropped the i or s

superscript, since the quantities are identical for the incident and scattered fields; 

this is consistent with the monostatic radar assumption and the sign conventions 

used by the AFDTD software). The scattering matrix transformation can be derived 

by writing

i

R R

jk r i

P R

T

jk r i

P P

jk r s T

P

s T

R

r

e

r

e

r

e E A E A S E A S AE S E

− 0 − 0 − 0

= = = = , (22)

which leads to 

S A S PA T

R = . (23)

To perform the scattering matrix transformation in Eq. 23, all we need is the cosη

parameter, defined in Eq. 21. So far, we have not used the superscripts G or P1 to 

denote a specific set of coordinates for any of the vectors introduced in this section. 

In the following, we specifically use these superscripts G and P1 for the θP e and θR e

unit vectors (remember that in the P1 frame the projectile axis is aligned with the 

vertical axis, after taking into account the SPY motion components). When we take 

the scalar product of θP e and θR e in Eq. 21, we must make sure that both unit vectors 

are expressed relative to the same frame.

When the direction of propagation (or radar-target LOS) in the ground frame is 

defined by the elevation angle θG and azimuth angle φG, the θ component of the 

electric field (as expressed in the ground frame) has the unit vector5





















−

−

=

G

G G

G G

G

R

θ

θ φ

θ φ

θ

sin

cos sin

cos cos

e , (24)

where

( ) ( ) ( ) 2 2 2

cos

R P R P R P

R P

G

x x y y z z

z z

− + − + −

− θ = , (25)

( ) ( )

( ) ( ) ( ) 2 2 2

2 2

sin

R P R P R P

R P R P

G

x x y y z z

x x y y

− + − + −

− + − θ = , (26)
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14

( ) ( ) 2 2

cos

R P R P

R P

G

x x y y

x x

− + −

− φ = , (27)

( ) ( ) 2 2

sin

R P R P

R P

G

x x y y

y y

− + −

− φ = . (28)

The signs in Eq. 24 were chosen consistent with the conventions used by the 

AFDTD software. Transforming this unit vector to the projectile frame is done via

G

G P R

P

θR θ e Q e

1

1 = → , (29)

with G→P1

Q given by Eq. 14. By a similar argument, the unit vector P1

θP e in the P1

frame (Fig. 5b) is given by





















−

−

=

θ

θ φ

θ φ

θ

sin

cos sin

cos cos

P1

P e , (30)

where the angles θ and φ are given by Eqs. 10 and 11. It follows that

( ) ( ) G

G P R

T P

P

P

R

T P

η θP θ θ θ e e e Q e

1

1 1 1 cos = = → . (31)

The last equation allows the computation of A and SR via Eqs. 20 and 23, 

respectively.

The entire in-flight target signature calculation procedure presented in the last 2 

sections can be summarized as follows:

• For each time sample, read the projectile coordinates ( ) P P P x , y ,z and 

velocity orientation ( ) ux uy uz , , from the trajectory file. Additionally, 

compute the angles χ, α and β describing the SPY motions.

• Compute the angles θ and φ by Eqs. 15, 10, and 11.

• In the AFDTD target response simulation data, select the scattering matrix 

SP corresponding to the angles θ and φ.

• Use Eqs. 24–28, 30–31, 20, and 23 (in this order) to compute the scattering 

matrix SR characterizing the target response in the ground-based radar 

frame.
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15

• Compute the instantaneous target RCS for each polarization combination as

2 RCS 4 R = π S , where SR is the corresponding scattering matrix element.

2.4 Propagation and Antenna Effects and the Radar Received 

Power

The procedure outlined in the previous 2 sections allows the complete 

characterization of the in-flight target signature as measured by the ground-based 

radar system. However, to predict the radar system performance, many other 

parameters (related to system design, propagation environment, and signal 

processing) must be taken into account. This is done via the radar equation.

7,8

There are many possible forms of the radar equation, involving various levels of 

detail of the system design, and tailored to various radar operational modes or 

applications. In this section, we use a relatively simple form of the equation that 

allows the calculation of the radar received power, based on the target trajectory 

and its radar signature, computed by the procedure described in Section 2.3.

For a ground-based radar system designed to search and track RAM projectiles at 

medium to large ranges, the elevation angles (as measured from the ground plane) 

can be very small, on the order of a few degrees or less. In this case, the ground 

bounce of the radar waves can have a strong effect on the radar received power. 

This is usually accounted for by the propagation factor F4 in the radar equation.

7 If

one includes this factor, the radar received power is computed as

( )

3 4

2 4

4 r

G G F

P P t r r t

π

λ σ = . (32)

In this equation, Pt and Pr are the transmitted and received powers, respectively; Gt

and Gr are the transmitter and receiver antenna gains, respectively; λ is the 

wavelength; σ is the RCS; and ( ) ( ) ( ) 2 2 2

R P R P R P r = x − x + y − y + z − z is the 

radar-target range. The usual assumption is that the radar tracks the target within 

its main antenna beam and the gain is computed at boresight. Another possible form 

that uses the antenna effective aperture A instead of the gain is

2 4

4

4 r

A A F

P P t r

r t πλ

σ = . (33)

For a grazing angle γ, the F factor is given by8
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16

( )

( ) ( ) 



 



 = + Γ − r

z z j G

G F P R

λ

π

γ γ 4

exp 0

2 1 , (34)

where the antenna gain G is computed at boresight and at an angle 2γ away from 

boresight, and Γ(γ ) is the Fresnel reflection coefficient of the ground for a grazing 

angle γ. All the other parameters in Eq. 34 were defined previously. For reference, 

the reflection coefficient is given by7

( )

ε γ ε γ

ε γ ε γ

γ 2

2

sin cos

sin cos

+ −

− − Γ =

r r

r r

V , (35)

for vertical polarization, and

( )

γ ε γ

γ ε γ

γ 2

2

sin cos

sin cos

+ −

− − Γ =

r

r

H , (36)

for horizontal polarization, where εr is the complex dielectric constant of the 

ground, and γ is given by

r

z z P − R = −1 γ sin . (37)

As shown in the numerical examples in Section 4, the propagation factor has an 

important impact (meaning it deviates significantly from 1) only for very small 

values of γ (typically, less than 1°). In practice, this impacts the radar received 

power only during the first few seconds of the projectile’s trajectory. However, its 

effect on the detection performance may be important from a radar system 

operational standpoint, because during that time interval the radar works in search 

mode, when target detection represents its most critical function.

The propagation factor F4 may depend on various EM phenomena other than the 

ground bounce of radar waves.7 Thus, atmospheric effects tend to distort the 

propagation paths due to variations in the refraction index of the medium. The 

Earth’s curvature can also affect the propagation factor, although its impact is 

significant only at very long ranges (on the order of 100 km or longer). 

Additionally, irregularities and objects on the ground surface (hills/valleys, 

buildings, vegetation, etc.) introduce both distortions in the radar wave propagation 

and direct scattering back to the radar, with a profound impact on the radar detection 

performance. Since all these effects are site-specific, they are not considered in this 
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17

investigation. However, any complete radar performance prediction model would 

have to take them into account.

Another EM analysis aspect that has an effect on the radar equation is the radar 

antenna polarization. This is characterized by the antenna effective length,

11 h, 

defined as

t

i sc jk r e

r

jZ I E h0

2

0 − = λ , (38)

for the transmitter antenna, and

H s Vo = hr E , (39)

for the receiver antenna. In the last 2 equations, Z0 is the free-space impedance, Isc

is the short-circuit current of the transmitter antenna, and Vo is the open-circuit 

voltage of the receiver antenna. Also notice that we used the superscript H (from 

Hermitian) to denote the transpose-conjugate of the hr vector in Eq. 39, since this 

vector may have complex components. The effective lengths ht and hr include 

information on both the antenna gains and polarization vectors, and are functions 

of the elevation and azimuth angles with respect to the antenna’s boresight direction 

(θA and φΑ, respectively). After some manipulations, one can show12 that, to

account for both the antenna polarization and the target polarimetric scattering 

effects, the product GtGrσ in the radar equation must be replaced by

2

2

4 t

H hr Sh

λ

π . (40)

This results is in contrast to the case of either vertically or horizontally polarized 

antennas, described in Section 2.3, where σ was simply replaced by 2 4π S , with S

being a scalar element of the scattering matrix.

This kind of analysis is particularly important in radar systems that use antenna 

arrays (of fixed orientation) and electronic scanning. Although the antenna 

elements are typically designed to transmit and receive either vertically or 

horizontally polarized waves at boresight, the wave polarization becomes mixed 

for a beam squinted away from boresight. In that case, one must use the expression 

in Eq. 40, with ht and hr functions of both θA and φΑ. Additional corrections related 

to the so-called cosine loss7 are also needed for the antenna gain when the beam 

points away from boresight. All these effects are dependent on the radar antenna 

design (both at element and array levels) and are not pursued in the numeric 
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18

examples included in this study. Nevertheless, they are important factors the radar 

designer must take into account when evaluating overall system performance.

3. Simulation of Ballistics Trajectories

3.1 The Modified Point Mass Model

In this section, we discuss the ballistic trajectories of symmetric artillery and mortar 

rounds (notice that projectiles equipped with rocket engines are not treated in this 

report). The exterior ballistics of projectiles is a mature and complex engineering 

subject—its detailed presentation is clearly beyond the scope of this investigation. 

However, an introduction to the motion equations governing ballistic projectiles, as 

well as their solutions (analytic or numeric), reveals interesting phenomenological 

aspects that are relevant to the radar sensing of these targets. Our purpose here is to 

develop a numeric model of the ballistic projectile flying motion that can be used 

in predicting its radar signature. Two types of projectiles are considered in this 

study: spin-stabilized projectiles (such as Howitzer-launched artillery shells) and 

fin-stabilized projectiles (such as mortar rounds). The material in Section 3 is based 

on a selection of earlier studies10,13–15; interestingly, some of the most important 

developments in ballistic science since World War II were made at the Ballistic 

Research Laboratory (BRL), which later became part of the current ARL.

The most complete description of the ballistic projectile motion after it leaves the 

gun barrel and until it hits the target is given by the 6 DOF model.

10,13 This 

represents a system of 6 coupled first-order nonlinear differential equations: 3 for 

the time derivatives of the linear momentum (or velocity) components and 3 for the 

time derivatives of the angular momentum components. The equations can be 

formally written in vector forms as

F g Λ Vm m

dt

d

m = ∑ + + , (41)

= ∑M H

dt

d . (42)

As a general note, in Section 3, we mostly use the symbols and notations in

McCoy,

10 with the exception of the Cartesian frame axis names (both the ground 

and projectile frames). For the latter, we use notations consistent with Section 2 of 

this report. Other variable names in this section may coincide but should not be 
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19

confused with the notations in Section 2. On occasions, we use the dot symbol (e.g., 

dt

dV V = ) to denote the time derivative of a motion variable.

In Eqs. 41 and 42, V is the vector velocity of the center of mass with respect to the 

ground, m is the projectile mass, g is the gravitational acceleration, Λ is the Coriolis 

acceleration,

10 and H is the total vector angular momentum. The quantities on the 

right-hand sides of the 2 equations include the sum of all aerodynamic forces F and 

the sum of all aerodynamic moments M acting on the projectile. Since we do not 

intend to go into a detailed description of the 6 DOF model, we do not explicitly 

write the equations for all forces and moments, but simply enumerate them instead, 

as follows10:

• Forces: drag force, lift force, Magnus force, pitch damping force

• Moments: spin damping moment, rolling moment due to fin cant, pitching 

(overturning) moment, Magnus moment, pitch damping moment

If W is the vector wind velocity (relative to the ground frame), then the relative 

velocity of the projectile with respect to air is v = V − W . In the rest of this study, 

we assume that there is no wind (W = 0), so v and V coincide. For nonzero wind 

speed conditions, the vector V should be replaced by v in the expressions of all 

aerodynamic forces and moments. Note that the direction of V is also the vector 

tangent to the trajectory.

A graphic representation of the various coordinate systems used in this section is 

shown in Fig. 6. The x-y-z frame (or G frame) is the usual Cartesian frame attached 

to the ground, while the frame i-j-k (or P frame) is attached to the velocity vector 

V as follows: k has the same direction as V, j is in the horizontal plane and 

perpendicular to k, while i is perpendicular on both k and j and contained in a 

vertical plane (i = j×k ). Notice that the projectile axis (described by the unit vector 

k1) and the velocity vector V direction are not necessarily identical; the angle 

between the 2, αt, is called the angle of attack. The elevation and azimuth angles 

made by the V vector with the x-y-z frame are φB and θB, as in Fig. 6 (note that 

these angles are defined differently from the elevation and azimuth angles in 

Section 2).
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20

Fig. 6 The geometrical configuration involved in the calculation of the ballistic trajectory. 

The projectile axis coincides with the k1 unit vector, while the velocity vector V has the same 

direction as the k unit vector. In this figure, the angle θB is negative.

The numerical integration of the differential equations Eqs. 41 and 42 involved in 

the 6 DOF model is discussed in McCoy.10 Nevertheless, we chose not to pursue 

this model in our ballistic trajectory study for 2 reasons: 1) we did not have ready 

access to a 6 DOF simulation code and 2) as discussed elsewhere,10,13 the full 6 

DOF model is not really necessary to model the trajectory of symmetric projectiles, 

except in very particular cases, which are not relevant to most practical scenarios. 

Instead, we investigate the modified point-mass trajectory (MPMT) model,10,13–15

combined with the epicyclic motion derived from the linearized aeroballistic (LA) 

model.10,13 As pointed out in McCoy,10 simulations by the MPMT model yield 

trajectories almost identical to the 6 DOF model in most scenarios of interest; 

however, the MPMT model does not completely characterize the pitching and 

yawing (or epicyclic) motions of the projectile. The epicyclic projectile motion, 

which has an important impact on the radar signature, is treated separately by an 

analytic approximation and combined with the other trajectory parameters derived 

from the MPMT model to create a more realistic representation of the projectile 

flight.

In the MPMT model, the linear momentum (or velocity) equations are preserved as 

in the 6 DOF model, but, from the 3 angular momentum equations, only the one 
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21

describing the roll (or spin) motion is preserved as fully coupled with the 3 velocity 

equations. The other 2 types of rotational motion, pitching and yawing, involve a 

transient and a steady-state component—these are described in more detail in 

Section 3.2. The main simplification in the MPMT model consists of neglecting the 

transient pitch-yaw motion (in the assumption that its amplitude is very small) and 

preserving only the steady-state component, described by the vector αR, called the 

yaw of repose. This variable is inserted back in the remaining 4 differential 

equations; however, the calculation of αR as a function of the other motion 

parameters involves a simple algebraic expression, as compared to the additional 

differential equations required by the 6 DOF model.

Specifically, the equations describing the MPMT model can be written as follows10:

v α (v α ) g Λ V = − + + × R + + Np

R

D L p

m

SC

v

m

SC

v

m

SC

dt

d

2 2 2

ρ ρ α 2 ρ α , (43)

lp

R

pC I

Sd v

dt

dp

2

2 ρ = , (44)

The approximate equation governing the yaw of repose takes different forms for 

spin- and fin-stabilized projectiles. Thus, in the case of spin stabilization,

( )

ρ Mα

x

R Sdv C

I p

4

2 g v

α

× = , (45a)

while for fin stabilization,

( ( ))

Mα

Mq

R

v C

C d

4

v v g

α

× × = . (45b)

In Eq. 43, the accelerations produced by the drag, lift, and Magnus forces were 

written explicitly as the first 3 terms in the right-hand side. In Eqs. 43–45, we 

introduced the following notations: p for the spin rate (or angular velocity), ρ for

the air density, S for the projectile’s cross-sectional area, IR for its axial moment of 

inertia, d for its diameter (or caliber), CD for the drag force coefficient, CLα for the 

lift force coefficient, CNpα for the Magnus force coefficient, Clp for the spin damping 

moment coefficient, CMq for the pitch damping moment coefficient, and CMα for

the pitching (or overturning) moment coefficient. The following forces and 

moments are neglected in this model: the pitch damping force, the rolling moment 

due to fin cant (assuming there is no fin cant), and the Magnus moment. As 
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mentioned earlier, we assume there is no wind, so we can replace v by V in 

Eqs. 43–45.

By explicitly writing the equations for each velocity component, and neglecting the 

Coriolis acceleration, we obtain the following system of 4 scalar differential 

equations:

( ) y Rz z Ry

Np

Rx

L

x D x p V V

m

SC

V

m

SC VV

m

SC V α α

ρ

α ρ ρ α α = − + + − 2 2 2

2  , (46a)

( ) z Rx x Rz

Np

Ry

L

y

D

y p V V

m

SC

V

m

SC VV

m

SC V α α

ρ

α ρ ρ α α = − + + − 2 2 2

2  , (46b)

p(V V ) g

m

SC

V

m

SC VV

m

SC V x Ry y Rx

Np

Rz

L

z D z = − + + α − α − ρ

α ρ ρ α α

2 2 2

2  , (46c)

lp

R

pC I

Sd V

p 2

2 ρ  = , (46d)

with 2 2 2 V = Vx +Vy +Vz .

One phenomenon with significant effect on ballistic trajectories is the variation of 

air pressure with altitude (height). A simple equation accurately describing this 

variation for altitudes z up to 6 km is10

( ) hz z e− ρ = ρ 0 , (47)

where ρ0 = 1.225 kg/m3 is the air pressure at sea level and h = 1.036×10–4 m–1 is a 

constant.

The Earth’s surface curvature, as well as its rotation, can be accounted for in 

ballistic trajectories by corrections to the ground-based Cartesian coordinates and 

the g vector components. All these effects, as well as the Coriolis acceleration, are 

significant only for very long trajectories and are neglected in the remainder of this 

study. Notice that in Eq. 46, we used g = –gz (with g = 9.8 m/s2).

The aerodynamic coefficients mentioned in a previous paragraph have a complex 

dependence on the Mach number (the ratio of the projectile velocity to the speed of 

sound), which cannot be described by a simple closed-form expression. Typically, 

this dependence is captured in lookup tables, based primarily on experimental data. 

Moreover, these coefficients are very sensitive to the projectile’s shape, meaning 

that separate lookup tables must be created for each type of projectile. The 
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23

aerodynamic coefficients typically vary relatively slowly with the Mach number, 

except for the transonic region (Mach number close to 1), where the variation may 

be more abrupt. In the numeric simulations performed in this report, we keep most 

of the aerodynamic coefficients independent of the projectile velocity. However, 

the coefficient associated with the most significant aerodynamic force, namely, the 

drag force, shows the proper variation with the Mach number, according to tables 

available in the literature.

Some of the aerodynamic coefficients also exhibit a variation with the angle of 

attack, which in the MPMT model coincides with αR . Thus, for small angles of 

attack (which are usually encountered in practice),

CD CD CD α R

2

0 2 = + sin , (48a)

CL CL CL α R

2

0 2 sin α = α + α , (48b)

CM CM CM α R

2

0 2 sin α = α + α , (48c)

CMq CMq CMq α R

2

0 2 = + sin . (48d)

Another aspect of integrating the differential system in Eq. 46 is specifying the 

initial conditions. For the velocity vector component, the initial conditions depend 

on the projectile direction as it leaves the muzzle of the gun. Thus, if the initial 

elevation and azimuth angles are φB0 and θB0 (see Fig. 6), and the initial projectile 

speed is V0, then

0 0 0 0 cos cos Vx = V φ B θ B , (49a)

0 0 0 Vy = V sinθ B , (49b)

0 0 0 0 sin cos Vz = V φ B θ B . (49c)

The initial value of the spin rate is

t d

V

p

c

0

0

2π = , (49d)

where the parameter tc is called the rifling (twist) rate and represents the number of 

turns within the gun barrel in which the projectile travels a linear distance equal to 

its caliber (or d).

A MATLAB16 code was developed to implement the numeric integration of Eq. 46

with initial conditions of Eq. 49. The specific algorithm employed in this code is a 
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24

4th-order Runge-Kutta procedure.

17 Once the velocity vector components are found 

at each time step, the ground-based Cartesian coordinates along the trajectory are 

calculated as

( ) ( ) ∫ = +

t

x t x Vx d

0

0 τ τ , (50a)

( ) ( ) ∫ = +

t

y t y Vy d

0

0 τ τ , (50b)

( ) ( ) ∫ = +

t

z t z Vz d

0

0 τ τ , (50c)

while the spin (or roll) angle χ (as in Section 2.2) is calculated as

( ) ( ) ∫ = +

t

t p d

0

0 χ χ τ τ . (50d)

In practice, all the integrals in equations in Eq. 50 are computed as running sums, 

each of them starting with an appropriate initial condition. Also, notice that, at each 

time step, the current motion parameters are used to update the air pressure via 

Eq. 47, the yaw of repose vector via Eq. 45, and the aerodynamic coefficients via 

Eq. 48.

We applied the trajectory simulation code based on the MPMT model to 2 types of 

projectiles: a 155-mm artillery round and a 120-mm mortar (these are the same 

projectiles investigated in our previous work2,3 from a radar signature perspective). 

In the remainder of this section, we present the results obtained from these 

simulations. In all cases, the launching point is placed in the x-y-z coordinate 

system origin (meaning x0 = 0, y0 = 0, z0 = 0).

The 155-mm artillery round is an example of spin-stabilized projectile. To find the 

trajectory, the full system of 4 equations in Eq. 46 was solved by the numerical 

algorithm. All the parameters appearing in the equations (such as mass, moments 

of inertia, aerodynamic coefficients, etc.) are available from public sources, 

primarily McCoy,10 Carlucci and Jacobson,13 and Balon and Komenda.14 These 

parameters, together with the initial conditions, are listed in Table 1. The center of 

mass trajectory obtained by the MPMT model is displayed in Fig. 7. In the same 

figure, we illustrate the variation of the projectile velocity and orientation (elevation 

angle) over time during the ballistic flight.
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25

Table 1 Physical parameters and initial conditions for the trajectory simulation examples 

in this section, involving a 155-mm artillery round and a 120-mm mortar

Projectile type 155-mm round 120-mm mortar

m (kg) 47.5 13.6

d (m) 0.155 0.120

IR (kg m2) 0.161 0.023

IT (kg m2) 2.106 0.232

CD 0.249 (at Mach = 2) 0.119 (at Mach = 0.7)

CLα 2.41 2.00

CNpα 0.51 N/A

Clp –0.024 N/A

CMα 4.65 –2

CMq –26 –22

V0 (m/s) 564 243

p0 (rad/s) 1143 0

φB0 (°) 40 50

θB0 (°) 0 0

(a) (b)

(c) (d)

Fig. 7 Various ways of describing the simulated 155-mm artillery round ballistic trajectory 

and its parameter variation vs. time: a) 3-D trajectory curve, b) height vs. range curve,

c) velocity components vs. time, and d) projectile orientation angle φB vs. time. Figure 7a shows 

the approximate radar location in the simulations in Section 4.1.
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The 120-mm mortar is an example of fin-stabilized projectile. This projectile is not 

intended to spin during flight, and any kind of inadvertent rolling motion is rapidly 

damped. Since p = 0, Eq. 46d can be dropped from the system, leaving only the 

velocity vector components to solve for. The mortar parameters involved in the 

MPMT model equations were also obtained from publicly available sources and are 

listed in Table 1. Graphic representations of the trajectory, as well as the time 

variation of the mortar velocity and orientation, are shown in Fig. 8.

(a) (b)

(c) (d)

Fig. 8 Various ways of describing the simulated 120-mm mortar ballistic trajectory and its 

parameter variation vs. time: a) 3-D trajectory curve, b) height vs. range curve, c) velocity 

components vs. time, and d) projectile orientation angle φB vs. time. Figure 8a shows the 

approximate radar location for the simulations in Section 4.2.

3.2 Epicyclic Motion of Flying Projectiles

As discussed in the previous section, the MPMT model does not account for the 

transient pitching and yawing motion of the projectile during flight. Nevertheless, 

this well-documented phenomenon is consistently encountered in experimental 

ballistic data, as well as predicted by the more rigorous 6 DOF model. Although 

the typical amplitude of these motion components is relatively small, on the order 

of a few degrees, they may have a rather large impact on the time-varying radar 
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27

signature of the in-flight projectile. Therefore, to properly characterize this radar 

signature, our trajectory model must account for the transient pitching and yawing 

motion. The procedure adopted here superposes these 2 motion components as 

obtained from the LA model10 to the MPMT model solution.

The analysis in this section is valid for spin-stabilized projectiles only; a similar 

treatment can be established for the pitching and yawing motions of fin-stabilized 

projectiles, but these are not considered throughout this report. As shown in

McCoy10 using the 6 DOF model, in most cases of practical interest, the 

pitching/yawing motion of a mortar is rapidly damped during the ballistic flight

(over a duration of 2–3 s), making this analysis less important for radar signature 

calculation purposes.

The pitch and yaw angles (α and β, respectively) were introduced in Section 2.2 

(see Fig. 4a). In the assumption that α and β are small, the unit vector of the 

projectile’s symmetry axis, k1, can be written approximately as

k1 ≅ αi + βj + k . (51)

This equation shows that α and β are the coordinates of the k1 unit vector in the i-j

plane. In other words, as the time progresses, the trace described by the projectile 

tip in the i-j plane is similar (in a geometrical sense) to the trace obtained in the 

α−β plane (taking β as the abscissa and α as the ordinate).

Next, we introduce the complex variable ξ = α + jβ (also called complex yaw). 

The differential equation characterizing ξ according to the LA model is10

ξ

+ (H − jP)ξ − (M + jPT )ξ = − jPG , (52)

where

( )















 + = −

T

Mq M

L I

md C C

C

m

SV H α

α

ρ  2

2 , (53a)

I V

I pd P

T

R = , (53b)

α

ρ

M

T

C

I

SV d M

2

2

= , (53c)















 = α + α

ρ

Mp

R

L C

I

md C

m

SV T

2

2 , (53d)
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28

2

cos

V

gd G φ B = . (53e)

In these equations, we introduced the following new symbols: IT for the transversal 

moment of inertia and CMα for the pitch moment derivative coefficient. Deriving 

Eq. 51 involves neglecting several higher-order terms—these approximations are 

valid in the assumption of a small angle of attack.

10 Additionally, to obtain a linear 

equation in ξ, only the constant terms are retained in the expressions of the 

aerodynamic coefficients (Eq. 48), while the angle-dependent terms are neglected. 

In Eq. 51, all coefficients (H, M, P, T and G) vary slowly in time, via the parameters 

V, p, and φB.

The general solution to Eq. 52 can be written as

(t) A ( j( (t) )) A ( j( (t) )) j (t) ξ = 1 ψ 1 +ψ 10 + 2 ψ 2 +ψ 20 + β R exp exp , (54)

with the real constants A1, A2, ψ10 and ψ20 determined by the initial conditions. The 

solution in Eq. 54 includes 2 oscillatory (or transient) terms and a slowly varying 

(or steady-state) term. Furthermore,

(t) = + j = [P + jH + P − 4M − H − 2 jP(2T − H )] 2

1 2 2 ψ 1 ω1 λ1  , (55a)

(t) = + j = [P + jH − P − 4M − H − 2 jP(2T − H )] 2

1 2 2 ψ 2 ω2 λ2  , (55b)

( ) M jPT

PG

t R

+

β = . (56)

The validity of the solution given by Eqs. 54–56 hinges on the fact that ω1,2, λ1,2,

and βR are all slow-varying functions of time.

After further manipulations of the expressions in Eq. 56 and neglecting some 

higher-order terms, we obtain

[P P 4M ] 2

1 2 ω1 = + − , (57a)

[P P 4M ] 2

1 2 ω2 = − − , (57b)

( )

















−

− = −

P M

P T H H

4

2

2

1

2

λ1 , (57c)
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29

( )

















−

− = +

P M

P T H H

4

2

2

1

2

λ2 . (57d)

For typical scenarios, we have P >> M 2 and M >> PT ; therefore, the following 

simple expressions are good approximations to Eqs. 58 and 57:

p I

I

P

T

R ω1 ≅ = , (58a)

I p

SdV C

P

M

R

M

2

2

2

ρ α ω ≅ = , (58b)

















+

+

≅ − = −

R

Mp

T

Mq M

I

C

I

SVd C C

H T ρ α α λ 

2

2

1 , (58c)

















≅ = α + α

ρ λ Mp

R

L C

I

md C

m

SV T

2

2 2 , (58d)

( )

ρ α

φ β

M

R B

R SdV C

I pg

M

PG

t 3

2 cos

≅ = . (59)

By separating the real and imaginary parts, we can recast the solution in Eq. 54 as

( ) ( ) ( )

( ) ( ) 















+ 















+ −

















+ 













 = −

∫ ∫

∫ ∫

20

0

2

0

2 2

10

0

1

0

1 1

 exp cos

exp cos

λ τ τ ω τ τ ψ

α λ τ τ ω τ τ ψ

t t

t t

A d d

t A d d

, (60a)

( ) ( ) ( )

A ( )d ( )d (t)

t A d d

R

t t

t t

λ τ τ ω τ τ ψ β

β λ τ τ ω τ τ ψ

+















+ 















+ −

















+ 













 = −

∫ ∫

∫ ∫

20

0

2

0

2 2

10

0

1

0

1 1

 exp sin

exp sin

. (60b)

The physical interpretation of the angular motion described in Eq. 60 is the 

following: both the pitch and yaw angle variations in time are characterized by 2 

transient terms consisting of damped oscillations of variable instantaneous 

frequencies, while the yaw angle has a steady-state component with slow time 

variation, βR, which is the same as the yaw of repose encountered in 
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30

Section 3.1. Notice that the expressions of α R in Eq. 45a and βR in Eq. 59 are 

identical; they imply that, for a spin-stabilized projectile, the yaw of repose vector 

is perpendicular to the velocity vector and has a horizontal (yaw) component only.

Since ω1 > ω2 , the first oscillatory term is called the “fast mode”, whereas the 

second oscillatory term is called the “slow mode”. To avoid dynamic instabilities, 

we must have , 0 λ1 λ2 > .

10

As a numerical example, for a 155-mm artillery round with V = 550 m/s and 

p = 1100 rad/s at ground level (ρ = 1.225 kg/m3), we obtain ω1 = 84.1 rad/s, 

ω2 = 14.2 rad/s, λ1 = 1 s–1, and λ2 = 0.388 s–1. This means that both oscillation 

modes are damped with time constants on the order of a few seconds. However, 

one should keep in mind that the LA model makes a number of approximations by 

neglecting nonlinear aerodynamic effects; consequently, the estimates of the 

damping factors based on this model are not very reliable to use in numerical 

simulations of the projectile motion. The nonlinear effects turn out to have a large 

influence on the time variation of the oscillation amplitudes10

; nevertheless, 

accounting for these effects is beyond the scope of this study. The 6 DOF models, 

as well as experimental data, indicate that, for spin-stabilized artillery shells, the 

fast mode dampens with a typical time constant of 5 to 10 s, whereas the slow 

mode’s amplitude decreases as the projectile approaches the apogee and increases 

again on the way to the ground impact.

10 In the numerical examples in Section 4, 

we choose damping time constants larger (or λ1 and λ2 smaller) than the results of 

calculations by Eqs. 58c and 58d to better emphasize the effect of these oscillations 

on the dynamic target RCS.

The initial conditions (the constants A1, A2, ψ10, and ψ20) are random to a large 

extent and depend on specific asymmetries in the construction of the gun barrel and 

projectile. Here, we follow the guidelines in McCoy10 and choose these parameters 

consistent with typical initial pitch and yaw angles observed in live-fire 

experiments.

Figure 9 illustrates the trajectory described by the projectile tip in a plane 

perpendicular to the velocity vector (the i-j plane in Fig. 6), as we look along the 

positive V vector direction, over the first 1-s time interval. In this example, we

chose ω1 = 84.1 rad/s, ω2 = 14.2 rad/s, λ1 = 0.1 s-1, λ2 = 0.01 s-1, A1 = 1°, A2 = 2°, 

ψ10 = 45°, and ψ20 = 120°. Figure 9a shows the curve described in the 3-D space 

(i-j-k coordinate system), while Fig. 9b shows its projection onto the i-j plane. As 

previously explained, the projectile tip coordinates in this plane are proportional to 

β and α, respectively. Notice that, since the ratio between ω1 and ω2 is not an 

integer (although it is close to 6), the overall motion is not completely periodic, 

meaning that each new cycle is slightly shifted (in angle) as compared to the 
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31

previous one. Also, because of the damping, the motion radius decreases with time 

(although this is hardly noticeable over the short time interval considered here).

There is a small offset along the abscissa (j axis) due to the presence of the yaw of 

repose βR. Numeric evaluations show that this angle is very small, typically a 

fraction of 1°. Consequently, it is not accounted for when we compute the 

orientation of the projectile; however, this angle still needs to be updated via 

Eq. 45a at each time step while solving the system in Eq. 46.

(a) (b)

Fig. 9 Trajectory described by the 155-mm projectile tip in the i-j plane over a 1-s interval. 

a) 3-D view and b) view in the i-j plane. The pitch and yaw angles in Fig. 9a are exaggerated 

(not to scale). In Fig. 9b, the observer looks along the k axis, in the positive direction.

There is a strong analogy between the motion of a spinning ballistic projectile in 

the mobile i-j-k frame and that of a spinning top in the fixed x-y-z ground frame. 

The pitch and yaw motions of the projectile can also be interpreted in terms of 

Euler’s angles9

—in that case, we describe the motion as made of precession and 

nutation components instead of pitch and yaw. Although the epicyclic motion of 

the flying ballistic projectile, as well as the SPY transformations in Section 2.2 can 

be reformulated using the precession and nutation concepts, we preferred the pitchyaw formulation, which is typical for the treatment found in ballistics-related texts.
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32

4. Numeric Examples

4.1 Dynamic RCS of an In-Flight 155-mm Artillery Round

The first numerical example of the dynamic RCS calculation involves a 155-mm

artillery round. This is a spin-stabilized projectile; therefore, its ballistic motion 

model includes both the center of mass trajectory (computed by the MPMT model) 

and the SPY motion components (derived by the LA model). Notice that this 

projectile is rotationally symmetric (with respect to the k axis in Fig. 2b), so its 

radar signature is independent of the azimuth aspect angle φ.

In this section, we use the center of mass trajectories presented in Section 3.1. For 

the epicyclic motion of the 155-mm round, we use the same parameters as in the 

numeric example in Section 3.2 employed in generating the curves in Fig. 9. As 

mentioned in the previous section, the LA epicyclic motion model is not entirely 

realistic—in particular, its amplitude variation over time is too simplistic, since it 

fails to take into account some important nonlinear effects. Nevertheless, the 

purpose of our simulations is to evidence some interesting qualitative phenomena 

regarding the time variation of the dynamic radar signature. From this perspective, 

the trajectory model presented here is adequate.

The radar is placed at coordinates xR = 15 km, yR = 1 km, and zR = 2 m. The 

maximum range of the trajectory is 15.7 km, while the drift is –362 m, so the 

projectile lands past the radar location (in the x direction) and to the side (in the y

direction). The flight time is approximately 56.3 s. This simulation is typical for a 

scenario where the radar is placed in the proximity of the perimeter presumably 

under artillery attack. As one can expect, the dynamic radar signature may change 

significantly if the radar-to-trajectory range and orientation is modified. However, 

the signature computation method remains the same in all these scenarios.

Since some of the most important calculations outlined in Sections 2.2, 2.3, and 2.4 

involve finding relative angles between the radar, projectile, and ground, we think 

it is informative to present the variation of some of these angles during the ballistic 

flight. Thus, in Fig. 10a, we plot on the same graph the projectile-to-ground 

elevation angle φB (the same as in Fig. 7d), the radar-to-projectile elevation angle θ

(given by Eq. 10), and the radar-to-ground grazing angle γ (given by Eq. 37). The 

oscillating nature of the φB and θ angles reflects the epicyclic motion of the 

projectile axis during flight. However, since the radar beam tracks the projectile 

center of mass (whose trajectory is smooth), the grazing angle does not exhibit 

oscillations. In Fig. 10b, we show the polarization rotation angle η, which is defined 

by Eq. 21; this angle also exhibits fast oscillations during the projectile flight.
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(a) (b)

Fig. 10 Relative angles between the radar, projectile orientation, and ground obtained in the 

trajectory simulations of the 155-mm artillery round, showing: a) the φB, θ and γ angles, and

b) the polarization rotation angle η

Next, we present the variation of the RCS with time during the ballistic flight. The 

graphs shown here cover 4 radar bands (L, S, C, and X), with the following 

frequencies: 1.3 GHz for the L-band, 2.4 GHz for the S-band, 5.6 GHz for the Cband, and 9.5 GHz for the X-band, as well as the V-V and H-H polarization 

combinations. We plot separately the RCS obtained by ignoring the SPY motion 

components of the projectile (Fig. 11) and the RCS that includes the SPY 

components (Fig. 12). Note that the radar wave polarizations (V-V or H-H) are 

defined relative to the ground-based radar frame (G).
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34

(a) (b)

(c) (d)

Fig. 11 Dynamic RCS vs. time curves obtained for the simulated trajectory of the 155-mm

artillery round that does not account for the SPY motion components: a) L-band, b) S-band,

c) C-band, and d) X-band
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35

(a) (b)

(c) (d)

Fig. 12 Dynamic RCS vs. time curves obtained for the simulated trajectory of the 155-mm

artillery round that includes the SPY motion components: a) L-band, b) S-band, c) C-band,

and d) X-band

The dynamic RCS plots indicate that, on the average, the projectile’s signature stays 

fairly flat during the ballistic flight, except for the last few seconds of the trajectory. 

The reason for this is that, most of the time, the radar “sees” the projectile close to 

the nose-on aspect, so the RCS changes very little; only at close range and steep 

projectile descent angle (during the last few seconds of the flight) does the aspect 

change to near-broadside. Moreover, the oscillations of the RCS curves in Fig. 11 

are fairly slow for most of the trajectory, but become more rapid at its end—this 

effect can be explained by the fact that the RCS changes more rapidly with the 

aspect angle near broadside than at nose-on orientation. As expected, the RCS

exhibits faster oscillations as the frequency increases (compare, for instance, the Xand L-band curves in Fig. 11).

The graphs in Fig. 12 clearly show the impact of the epicyclic motion of the 

155-mm round on the dynamic RCS. We notice 2 oscillations induced by this 

motion: a fast mode with a frequency of about 13 Hz that lasts for about 10 s and a 

slow mode with a frequency of about 2 Hz that lasts during the entire flight. Notice 
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36

that the frequency of these oscillations in the RCS curves is independent of the 

radar frequency band. It can be shown that the oscillation frequencies are also 

independent of the relative orientation of the radar with respect to the trajectory. 

Possible ways to exploit this signature feature for target discrimination or 

identification will be investigated in a separate study.

In terms of RCS magnitude, the results in Figs. 11 and 12 are consistent with our 

previous findings,

2 which suggest that, on the average, all 4 radar bands offer 

approximately the same signature strength. When we compare the V-V and H-H 

polarizations, we notice that the V-V case consistently exhibits a higher RCS early 

during flight (approximately the first 20 s), after which the 2 signatures become 

comparable. An explanation is the fact that, early during the ascending part of the 

ballistic trajectory, the projectile axis is closely aligned with the electric-field vector 

for the V-V case. Since the projectile axis is also parallel to most of the target’s 

physical boundary, this alignment creates a stronger backscattering response than 

in the H-H case. When the projectile aspect approaches broadside, the V-V and HH signatures tend to become identical.

We further analyzed the radar return from the in-flight projectile by calculating the 

received radar power according to Eqs. 32 or 33. We considered a radar transmitted 

(peak) power of Pt = 8 kW (or 70 dBm). For the computation of the propagation 

factor F4

, we used a ground dielectric constant εr = 8. The radar antenna model that 

we choose is key in comparing the radar performance for different frequency bands. 

Thus, if we choose a fixed antenna gain for all bands, Gt = Gr = 40 dB, the graphs 

in Fig. 13, based on Eq. 32, suggest that the performance is better at lower 

frequencies (largest received power for L-band). However, if our model assumes 

the same antenna aperture for all bands, At = Ar = 4 m2, the graphs in Fig. 14, based 

on Eq. 33, lead us to the opposite conclusion: the performance is better at higher 

frequencies. These differences are clearly dictated by the presence of the factor λ2

in the numerator in Eq. 32 and in the denominator in Eq. 33. As a side note, one 

should consider the fact that, in the X-band, a gain of 40 dB can be achieved with

a 0.8-m2 antenna aperture, whereas, in the L-band, obtaining the same gain requires

a 42-m2 aperture. For completeness, in Eq. 34 we used an antenna gain that varies 

with the squint angle θA as 

( ) ( ) 





























 = +

HB

A

A

G G

θ

θ θ 1 cos π

2

0 . (61)

In this equation, the main antenna beam was represented as a Hanning window with 

respect to the angle θA, where θHB = 5° is the half-beamwidth (to the first zerointercept point). While this is just a generic model of the antenna pattern (does not 
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37

represent any specific antenna design), it is representative enough for the purpose 

of our analysis.

(a) (b)

Fig. 13 Received radar power vs. time for the in-flight 155-mm artillery round, assuming 

the fixed gain model in all 4 frequency bands, showing a) V-V polarization and b) H-H 

polarization

(a) (b)

Fig. 14 Received radar power vs. time for the in-flight 155-mm artillery round, assuming 

the fixed aperture model in all 4 frequency bands, showing a) V-V polarization and b) H-H 

polarization

The increasing trend of the received power with time in Figs. 13 and 14 can be 

largely explained by the reduction in radar-to-projectile range as the projectile gets 

closer to the final impact. More interestingly, we notice a fairly large oscillation of 

the power for the first 1 or 2 s of the flight. This is the effect of the propagation 

factor F4

, which manifests itself only at very low grazing angles. We can 

demonstrate that by separately plotting F4 versus time, for all frequency bands and 

both polarizations, in Fig. 15. This factor is almost identical for V-V and H-H 

polarizations due to the fact that the ground reflection coefficient is approximately 

the same (close to –1) for both wave polarizations at small grazing angles.

7 Notice 

the large oscillations of F4 for the first 1.5 s, involving swings of 20 to 30 dB, which 
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38

can have a dramatic effect on the radar detection performance during that time 

interval. Subsequently, the propagation factor stabilizes to a value of 1, since the 

ground-bounced rays start to propagate outside the antenna main beam, so they are 

mostly suppressed. The oscillations in the propagation factor reappear at the very 

end of the flight, but at that point they are swamped by larger oscillations in the 

target signature that were previously explained. A closer look at the power 

variations for the first few seconds of the flight is presented in Fig. 16, where we 

used the constant gain model in the radar equation. Interestingly, these curves 

clearly capture the 3 types of RCS oscillations discussed in this section: the fast and 

slow modes due to the epicyclic motion, as well as the variations in F4 at very close 

to grazing angles.

(a) (b)

Fig. 15 Propagation factor F4 during the first 3 s of the 155-mm round ballistic flight, for all 

4 frequency bands and a) V-V polarization and b) H-H polarization

(a) (b)

Fig. 16 Received radar power during the first 3 s of the 155-mm artillery round ballistic 

flight, assuming the fixed gain model in all 4 frequency bands, showing a) V-V polarization

and b) H-H polarization
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One should keep in mind that the target signature analysis presented in this report 

is only one aspect of the radar system performance analysis. A complete assessment 

of the latter should include gains/losses induced by various system components and 

signal processing schemes, as well as the accuracy of the radar measurements. 

Other considerations (size, cost, impact of manufacturing errors, maintainability, 

etc.) must also be taken into account when making design decisions.

7

4.2 Dynamic RCS of an In-Flight 120-mm Mortar

The other numeric example of the dynamic RCS considers the 120-mm mortar. 

Since this is a fin-stabilized ballistic projectile, we are not concerned with the SPY 

motions, but only with the motion of the center of mass, as simulated by the MPMT 

model. As with the 155-mm artillery round, we used the same trajectory as that

simulated in Section 3.1. In this case, the radar is placed at coordinates xR = 4.5 km, 

yR = 0.5 km, and zR = 2 m. The flight time is 35.5 s, the maximum range is 4.7 km, 

and there is no trajectory drift in this case.

Although the dynamic RCS analysis of the mortar is simplified as compared to the 

artillery round by the absence of the epicyclic motion, the additional feature that 

we need to take into account is the asymmetry introduced by the mortar’s fins. In 

this case, the radar signature is sensitive to the relative azimuth angle φ. In turn, this 

angle depends on the initial roll angle χ0, which appears in Eq. 50d (notice that, 

since p = 0, the roll angle χ stays constant during the flight). One aspect we want 

to investigate is the impact of the initial roll angle (which is typically unknown in 

advance) on the dynamic RCS curves.

Figure 17a shows the variation of the same angles as in Fig. 10a (namely, φB, θ and 

γ) for the mortar trajectory. These curves are very similar to the ones in Fig. 10a, 

except that they do not exhibit the fast oscillations induced by epicyclic motion. 

Notice that these 3 angles do not depend on χ0. Figure 17b shows the variation of 

the azimuth aspect angle φ, for 2 different values of χ0: 0° and 90°. (Note: Since 

the mortar model considered in this study has 10 fins, the target aspects “seen” by 

the radar for the 2 orientations are different). As expected, there is a direct 

dependency between the 2 angles—note that the difference between the 2 curves 

describing φ stays very close to 90° during the entire trajectory.
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40

(a) (b)

Fig. 17 Relative angles between the radar, projectile orientation, and ground obtained in the 

trajectory simulations of the 155-mm artillery round, showing a) the φB, θ, and γ angles and

b) the φ angle for 2 values of χ0

Next, we look at the RCS variation over time, in the 4 frequency bands, and 

different values of χ0, in Fig. 18 (V-V polarization only). The 3 cases in each graph 

represent χ0 = 0° (blue line), χ0 = 90° (red line), and the average RCS (green line) 

taken over all possible χ0 values (the average is computed in terms of scattering 

matrix elements). For the L- and S-bands, the 3 curves are identical. For the C- and 

X-bands, some differences start to appear between the 3 cases, but they are 

generally small (less than 2 dB in the regions of sizeable RCS). These results, 

consistent with our previous findings,

3 demonstrate that the asymmetry in the 

mortar geometry has very little impact on its radar signature. In other words, there 

is very little dependence of the RCS on the azimuth aspect angle—this dependence 

is more pronounced as the frequency increases. Given this, as well as the fact that 

χ0 is typically unknown, our subsequent results only consider the average RCS over 

all possible χ0 values.
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41

(a) (b)

(c) (d)

Fig. 18 Dynamic RCS vs. time curves obtained for the simulated trajectory of the 120-mm

mortar, comparing various ways to account for the initial roll angle χ0, for V-V polarization 

and a) L-band, b) S-band, c) C-band, and d) X-band. Note: The curves obtained for L- and Sbands are on top of one another.

The RCS versus time graphs in Fig. 19 are equivalent to those in Fig. 11, with the 

mention that we plotted the mean RCS over χ0 in Fig. 19. The conclusions about 

the time variation of the RCS for the mortar are similar to those for the 155-mm

round in the absence of the SPY motion components. However, the differences 

between V-V and H-H polarization, clearly visible in the early part of the 155-mm

round trajectory, are less evident in the case of the mortar. One possible explanation 

is the more rounded shape of the mortar projectile, which makes the target response 

less sensitive to the radar wave polarization. In other words, the argument we made 

in Section 4.1 that the electric-field vector in vertical polarization is aligned with 

the target’s boundary is no longer valid.
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(a) (b)

(c) (d)

Fig. 19 Dynamic RCS vs. time curves obtained for the simulated trajectory of the 120-mm

mortar, using the average over all possible angles χ0, for a) L-band, b) S-band, c) C-band, and

d) X-band

With regard to the received radar power, the results follow the same trends as for 

the 155-mm round. In this section, we only show the received power versus time 

for the fixed gain case (Gt = Gr = 40 dB), with the same radar parameters as in 

Section 4.1, in Fig. 20. Once again, we notice the large oscillations for the first 1 s 

of the trajectory, produced by changes in the propagation factor F4 at close to 

grazing angles. The power received during this part of the trajectory is shown in 

closer detail in Fig. 21. Due to the steeper launch angle φB0 (50° for the mortar vs. 

40° for the 155-mm round), the time it takes for the propagation factor to stabilize 

to unit value is shorter in the mortar case.
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(a) (b)

Fig. 20 Received radar power vs. time for the in-flight 120-mm mortar, assuming the fixed 

gain model in all 4 frequency bands, showing a) V-V polarization and b) H-H polarization

(a) (b)

Fig. 21 Received radar power during the first 3 s of the 120-mm mortar ballistic flight, 

assuming the fixed gain model in all 4 frequency bands, showing a) V-V polarization and

b) H-H polarization

5. Conclusions

In this report, we developed a method of evaluating the radar signature of in-flight

ballistic projectiles based on computer models. The method consists of 3 parts: 

calculating the radar signature of the projectile in a fixed position, creating a model 

of the ballistic trajectory, and evaluating the RCS of the in-flight projectile by 

performing geometrical transformations on the scattering matrix from the groundbased radar frame to the projectile frame. The first part, which employed the 

AFDTD radar signature software, was discussed in detail in our previous work.

2

The second and third parts were presented in Sections 2 and 3 of this report, 

respectively.
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The ballistic trajectory simulations developed in Section 3 were based on the 

MPMT model. Although this model accurately predicts the center of mass 

trajectory, certain approximations are involved in its derivation and some motion 

elements (the pitching and yawing of the projectile axis) are not accounted for. 

Nevertheless, the pitching and yawing motions were added separately based on the 

LA model and turned out to have a significant impact on the radar signature of the 

projectile.

The method developed here allows the evaluation of the full scattering matrix of 

the in-flight projectiles, as well as the RCS for various polarizations. Moreover, we 

used the radar equation to obtain the received radar power; in this formulation, we 

accounted for the propagation factor F4, which typically has a large effect at very 

close to grazing angles. Other issues related to the radar antenna’s impact on the 

system performance were also discussed, although they were not included in the 

numerical examples in Section 4.

The numerical result section analyzes the radar signature of a 155-mm artillery 

round and a 120-mm mortar, for typical trajectories. The results are presented in 4 

radar frequency bands (L, S, C, and X). As already shown in our previous work,

2,3

on the average, there are no significant differences in RCS among the 4 bands; 

nevertheless, we notice more rapid RCS variations as the frequency increases. A 

rigorous statistical analysis of the projectiles radar signature is deferred to a future 

separate study. Another important finding of the simulations in Section 4 is that the 

epicyclic motion of the spin-stabilized projectiles induces relatively fast variations 

in their RCSs during flight. The spectral analysis of these variations and its possible 

exploitation will be the subject of future investigations.

Finally, one should mention that the radar signature evaluation technique developed 

here is general, in the sense that it does not depend on the method of generating the 

trajectory data (whether through models with various degrees of accuracy or 

through measurements). The ballistic trajectory models in Section 3 were mainly 

developed to provide us with data that allowed us to test the radar signature 

calculation algorithms. Application of these algorithms to trajectories generated by 

more rigorous ballistic models would produce even more realistic evaluations of 

the projectile’s radar signature, which is an important component in the overall 

radar system performance prediction. In the future, we hope to acquire experimental 

data collected by this type of radar system, for the purpose of validating the 

computer models.
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6. References

1. Otero M, Deresh B, Teig L, Werth J, Darrah C, Hsu K, Jones D, Holder J, 

Cagle R. RCS modeling of projectiles for counterfire radar performance 

simulations. McLean (VA): MITRE Corporation (US); 2007 Jul. 

2. Kenyon C, Dogaru T. Numerical computation of the radar cross section of 

rockets and artillery rounds. Adelphi (MD): Army Research Laboratory (US);

2015 Sep. Report No.: ARL-TR-7468.

3. Kenyon C, Dogaru T. Numerical computation of the radar cross section of a 

120-mm mortar. Adelphi (MD): Army Research Laboratory (US); 2015 Nov. 

Report No.: ARL-TN-0716.

4. Dogaru T. AFDTD user’s manual. Adelphi (MD): Army Research Laboratory 

(US); 2010 Mar. Report No.: ARL-TR-5145.

5. Balanis C. Advanced engineering electromagnetics. New York (NY): Wiley;

1989.

6. Ruck G, Barrick DE, Stuart WD, Krichbaum CK. Radar cross section 

handbook. New York (NY): Plenum Press; 1970.

7. Richards M, Scheer J, Holm W. Principles of modern radar. Raleigh (NC):

SciTech Publishing; 2010.

8. Skolnik MI. Introduction to radar systems. New York (NY): McGraw Hill;

2001.

9. Foley J D, Van Dam A, Feiner SK, Hughes JF. Computer graphics: principles 

and practice. Boston (MA): Addison-Wesley; 1995.

10. McCoy RL. Modern exterior ballistics. Atglen (PA): Schiffer Publishing; 

2012.

11. Balanis C. Antenna theory – analysis and design. New York (NY): Wiley; 

1997.

12. Dogaru T, Le C. Through the wall radar simulations using polarimetric antenna 

patterns. Adelphi (MD): Army Research Laboratory (US); 2012 Mar. Report 

No.: ARL-TR-5951.

13. Carlucci DE, Jacobson S. Ballistics – theory and design of guns and 

ammunition. Boca Raton (FL): CRC Press; 2014.

14. Balon R, Komenda J. Analysis of the 155-mm ERFB/BB projectile trajectory. 

Advances in Military Technology (Czech Republic). 2001 Jan:91–114.
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15. STANAG 4355, Edition 3. The modified point mass and five degrees of 

freedom trajectory models. Brussels (Belgium): NATO Standardization 

Agency; 2009 Apr.

16. MathWorks web page. Natick (MA): The MathWorks Inc.; 2016 [accessed 

2016 Aug]. http://www.mathworks.com/products/matlab/.

17. Press WH, Vetterling WT, Teukolsky SA, Flannery BP. Numerical recipes in 

Fortran 77. Cambridge (UK): Cambridge University Press; 1992.
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List of Symbols, Abbreviations, and Acronyms

3-D 3-dimensional

ARL US Army Research Laboratory

BRL Ballistic Research Laboratory

DOF degrees of freedom

EM electromagnetic

H-H horizontal-horizontal

H-V horizontal-vertical

LA linear aeroballistic

LOS line of sight

MPMT modified point-mass trajectory

RAM rocket, artillery, and mortar

RCS radar cross section

SPY spinning, pitch, and yaw

V-H vertical-horizontal

V-V vertical-vertical
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