Zanco Journal of Pure and Applied Sciences
Vol.26, No.2, 2014
Theoretical Study of Radar Cross Section - for Various Target Shapes and
at Different Frequency Bands*
(pp. 57-70)
Mudhaffer M. Ameen1 and Hawbash H. Karim2
1-Department of Physics, College of Education Salahaddin University, Erbil - Iraq
2- Department of Physics, Faculty of Science and Health, Koya University
[email protected]
[email protected]
Received: 06/02/2014
Accepted: 03/04/2014
Abstract
In this paper, radar signal was analyzed and processed using MATLAB computer program which
introduced numerous functions to study and evaluated the Radar Cross Section (RCS) for different target’s
shapes; perfectly conducting sphere, ellipsoid, circular flat plate, frustum, cylinder, and complex targets.
RCS for the target’s shapes was studied and analyzed versus aspect angles and frequencies. The frequencies
of the bands are taken as; C-band (4 and 8) GHz, X-band (8 and 12.5) GHz, K-band (18 and 26.5) GHz. The
frequencies are taken for the initial and terminal of the specified bands. All determined parameters for radar were
applied to Baghdad and Erbil International Airports to primary and secondary frequencies at (2.7 and 2.9) GHz
and (1.07 and 1.09) GHz, respectively.
Key words: Radar Cross Section (RCS), Radar target shapes; Radar.
1: Introduction
A
radar is an electromagnetic system for the detection and location of objects. It operates by transmitting a
particular type of waveform, a pulse-modulated sine wave for example, and detects the nature of the
echo signal. Radar is used to extend the capability of one's senses for observing the environment,
especially the sense of vision. Radar can be designed to see through those conditions impervious to normal
human vision, such as darkness, haze, fog, rain, and snow. In addition, radar has the advantage of being able to
measure the distance or range to the object. This is probably its most important attribute [1].
The term “RADAR” was an American acronym created in 1941[2]. The word radar is an abbreviation for
Radio Detection and Ranging [1, 3, 4, and 5]. In general, radar systems use modulated waveforms and
directive antennas to transmit electromagnetic energy into a specific volume in space to search for targets.
Objects (targets) within a search volume will reflect portions of this energy (radar returns or echoes) back to the
radar. These echoes are then processed by the radar receiver to extract target information such as range, velocity,
angular position, and other target identifying characteristics [3]. Early forms of radar carried various names at
different places in the world. However, in modern times the name radar seems to have achieved universal
acceptance [6]
2: Radar Cross Sections (RCS)
The term Radar Cross Section (RCS) was used to describe the amount of scattered power from a target
towards the radar, when the target is illuminated by RF energy [7]. RCS is “a measure of the reflective strength
of a radar target.”[8].
RCS Usually represented by the symbol (σ) and is defined as 4π times the ratio of the power per unit solid
angle scattered in a specified direction to the power per unit area in a plane wave incident on the scattered from a
specified direction [1].

Power reflected toword source / unit solid angle
incident power / 4
(1)
In Eq.(1) the power density of a wave incident on a target located at range R away from the radar is P Di, as
illustrated in Fig. (1) the amount of reflected power from the target is P r
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1
This paper is a part of an M. Sc. thesis written by the second author under supervision of the first.
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Figure 1: Scattering object located at range R [9].
In order to ensure that the radar receiving antenna is in the far field (i.e., scattered waves received by the antenna
are planar), eq. (2) is modified [5]:
  4
 PD r 


 PD i 
R lim 
2
R 
(3)
The RCS defined by Eq.(3) is often referred to as either the monocratic RCS, the backscattered RCS, or
simply target RCS.
3: Targets Cross Section Shapes
The calculation of radar cross section is complex, even for targets of simple shape. For target shapes
give exact solutions because of boundary value problems. Even in the few cases where exact solutions are
evaluated by digital computer. Rather, some examples will be given of back scattering cross section for some
simply shaped targets in monostatic radar [6]. The following subsections different shapes are discussed.
3.1: Sphere – RCS – Shape
The perfectly conducting sphere is a good example of a target with symmetry such that the cross polarizing
elements of the scattering matrices are zero. This means that each orthogonal component of the incident wave is
backscattered without cross polarized component from the other orthogonal component: As a result the radar
cross section for scattering from one component of an incident wave is the same for the other (orthogonal)
component and therefore it is adequate to give radar cross section for one component of excitation only, see
Fig.(2) [4]:
Figure 2: Plane wave incident on a sphere [4].
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Therefore the RCS for sphere is defined mathematically as[6]:
  4
/ E(r, , ) /
R lim
/ E 0 /
2
(4)
2
R 
2
After, many mathematical derivations and approximations, the final form of the sphere RCS is given as:
 ( ,0)  a 2 [1  sin c(ka)]2
(5)
Note that, when the frequency becomes very high as (λ→0) the sinc term goes to zero, and eq.(5) simply reduces
to:
 ( 0)  a2
(6)
The backscatter from a sphere as a function of radius(r or a) computed using the physical optics approximation
as shown in Fig. (3) [11].
Figure 3: Radar cross section of a perfectly conducting sphere as a function of the radius – to wavelength ratio
2πr/λ [11]
3.2: Ellipsoid RCS - shape
From analytic geometry, the equation for an ellipsoid centered at (0, 0, 0) is shown in Fig. (4) and is defined
by the following equation [11]:
(7)
Figure 4: Ellipsoid target shape Geometry [11].
One widely accepted approximation for the ellipsoid backscattered RCS as given by [11]:
 
a 2 b 2 c 2
[a 2 (sin  ) 2 (cos  ) 2  b 2 (sin  ) 2 (sin  ) 2  c 2 (cos  ) 2 ]2
(8)
3.3: Circular Flat Plate- RCS-shape
The backscattering cross section of a perfectly conducting flat circular plate that is assumed to be thin
compared to a wavelength. The applicable geometry is shown in fig.(4). The plate is centered on the origin and
lies in the x,y plane with width (2r) in the x and y direction respectively.
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The general RCS formula for circular flat plates is given by [12]:
(9)
Figure 5: Circular flat plate target shape [7].
For normal incidence (θ = 0) , formula RCS for a circular flat plate reduces to [12]:
4 A
2

max


(10)
2
Where A is the surface area of the circular flat plate. Using well – known integral definitions of Bessel functions
which makes eq.(9) to become [6]:

(2kr sin  )
   k r  2 J 1
2kr sin 

2
Where k  2
/
4
2

 cos 


(11)
2
, r is a radius of circular plate and J(B) is the first order spherical Bessel function evaluated at
Band width.
3.4: Truncated Cone (Frustum) – RCS - Shape
Figs. (6) and (7) show the geometry associated with a frustum, the half cone angle α is given by;
(12)
Figure 6: Truncated cone (frustum) [9].
Figure 7: Half cone angle representation [9].
Where H and L are small and large height of cones, respectively, define the aspect angle at normal incidence
(broadside) as θn.
For non-normal incidence, the backscattered RCS due to a linearly polarized incident wave is [9]:
2
(13)
 z tan   sin   cos tan  



8 sin   sin  tan   cos 
Where Z is equal to either Z1 or Z2 depending on whether the RCS contribution is from the small or the large
end of the cone. The RCS for infinite cone is [8]:


16
2
(14)
3.5: Circular Cylinder- RCS – Shape
As another example of backscattering radar cross section for some simple shapes, one discusses a perfectly
conducting, long cylinder of circular cross section centered on the origin, as shown in Fig. (8).
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(15)
Figure 8: (a) Elliptical cylinder: (b) Circular cylinder target – shape (11)
4: The present work
In the present work, expression for RCS and other parameters related to it operating at(C, X and K) bands,
primary and secondary frequency bands at Erbil and Baghdad International airports is coded in MATLAB
computer program to study Radar parameters as given in appendices A and B, respectively. According to the
first program, the value of frequency bands, range, reflection coefficient and wavelengths are used to study and
evaluate the RCS, using equations (7), (9), (10), (12).
Effect of Target Geometry on RCS
This section presents examples of backscattered radar cross section for a number of simple shape and
complex objects. In all cases, except the backscattered cross section, which effected by the different target
shapes and versus dimension of targets.
4.1: Radar Cross Section for Perfectly Conducting Sphere
The normalized backscattered cross sections for perfectly conducting sphere are drawn as a function of its
circumference in the units of wavelength. The plots are shown in figures (9) and (10).. It is obvious from figures
that the backscattered radar cross section for perfectly conducting sphere is almost constant in the optical region.
RCS is independent of frequency whenever operating frequency is sufficiently high that is {λ« Range} and λ«
radius(r)}. Figure (10) shows the RCS versus sphere’s circumferences in the unit of wavelength for different
regions. The optical region (“far field” counterpart) corresponds to
r
≥1.6. In this region, the RCS of a sphere

is independent of frequency. Here, the RCS of a sphere, σ = πr 2. The RCS equation breaks down primarily due to
creeping waves in the area where λ~2πr. This area is known as the Mie or resonance region. Because the
sphere’s cross section is independent of aspect (viewing) angle, it is a most valuable target for calibration and
evaluation the radar system.
Figure 9: Normalized backscattered RCS for a perfectly conducting sphere as a function of the radius -towavelength ratio
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2 r

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Figure10: Normalized backscattered RCS for a perfectly conducting sphere as a function of the radius -towavelength ratio
2 r

using semi log scale.
4.2: Radar Cross Section for Ellipsoid
To compute backscattered RCS for targets with ellipsoidal shapes, equation (7) have been used along with
computer program can be handled for this purpose. The plots of the RCS of an ellipsoid versus the aspect angle
are shown in figures (11) and (12). In these figures, the roll angle (φ)is taken as (φ = 45°) while ellipsoidal
dimensions(a, b, c)are evaluated at (0.1m, 2m and 5m) and (1m ,3m and 9 m) for figures (11) and (12),
respectively. It is obvious from these figures that the ellipsoid roll angle (φ) has no effect on RCS since it is
independent on (φ). For the case a = b= c the ellipsoidal shape will reduce to a spherical one. As a matter of fact,
RCS is independent on frequency, therefore, RCS remains constant during all frequency bands, it depends on
dimensions of the ellipsoid.
Figure 11: Ellipsoid backscattered RCS versus aspect angle, φ = 45°
a = 0.1m, b = 2m and c= 5 m.
Figure 12: Ellipsoid backscattered RCS versus aspect angle, φ = 45°
a= 1m, b = 3m and c = 9 m.
4.3: Radar Cross Section for Circular Flat Plate
To study the backscattered RCS for circular flat plates versus aspect angle, eq.(9) is used and computed. The
results are drawn in figures (13) and (14).The figures illustrate that the plots of the RCS of a perfectly
conducting circular plate with radius (r = 0.2m and 0.3m) as a function of (1800 ≥ θ ≥ 00) for frequency bands Cband:(4 , 8 )GHz , X-band:(8 , 12.5)GHz and K-band:(18 , 26.5)GHz are shown in Figures (13) and (14),
respectively. From these figures, one observes that the RCS for a circular flat plate increases with increasing the
radii of circle from (0.2m) to (0.3m). Moreover, its cross section increases by increasing the frequency of
specific bands.
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(a): f = 4GHz
(c): f = 10.25GHz
(e): f = 18GHz
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(b): f = 8GHz
(d): f = 12.5GHz
(f): f = 26.5GHz
Figure 13: Backscattered RCS versus aspect angle for a circular flat plate at radius(r = 0.2m) at:
(a): f = 4GHz (b): f = 8GHz
(c): f = 10.25GHz (d): f = 12.5GHz (e): f = 18GHz (f): f = 26.5GHz
(a): f = 4GHz
(b): f = 26.5GHz
Figure 14: Backscattered RCS versus aspect angle for a circular flat plate at radius(r = 0.3m) at:
(a): f = 4GHz
(b): f = 26.5GHz
Erbil and Baghdad International Airports (RCS for circular flat plate)
RCS for circular flat plate was evaluated and analyzed for Baghdad and Erbil International Air-ports,. RCS
was evaluated for conducting circular plate with radii, (r = 0.2m and 0.3m), at primary frequency band for Erbil
and Baghdad airport of {2.7GHz,2.9GHz} and secondary frequency bands at{1.07GHz,1.09GHz}. The RCS
versus aspect angle is shown in figures (15) and (16).
(a): f = 2.7GHz
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(b): f = 2.9 GHz
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(c): f = 1.07GHz
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(d): f = 1.09 GHz
Figure 15: Backscattered RCS versus aspect angle for a circular flat plate at radius(r = 0.2m)at: (a): f =
2.7GHz (b): f = 2.9 GHz (c): f = 1.07GHz
(d): f = 1.09 GHz
(a): f = 1.07GHz
(b): f = 2.9 GHz
Figure 16: Backscattered RCS versus aspect angle for a circular flat plate at radius(r = 0.3m) at:
1.07GHz
(b): f = 2.9 GHz
(a): f =
4.4: Radar Cross Section for Frustum (Cone)
The backscattered Radar cross section for frustum target was being studied and analyzed.. The conic
dimensions were given as follows: semi - major axis, r2 = 0.3m and 0.5m, semi – minor axis, r1 = 0.2m and 0.4m
whereas height, h = 0.8m and 0.9m. The RCS for frustum was evaluated for the following frequency bands: Cband: (4, 8) GHz, X-band: (8, 12.5) GHz, K-band: (18, 26.5) GHz. The RCS for frustum versus aspect angle is
shown in figures (17) and (18).
Normal incidence will occur between (800 to 100o) ; Therefore, when viewing the target from large end, the
indicator =1 while for the indicator = 0 when viewing the target from small end. The effect of indicators is
shown in figures (17) and (18) for indicators = 1 and 0, respectively. It is clear from figures that the RCS of a
frustum does depend on the frequency.
(a): f = 4GHz
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(b): f = 8 GHz
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(c): f = 12.5GHz
(d): f = 26.5 GHz
Figure 17: Backscattered RCS versus aspect angle for a frustum for (r1 = 0.2m), (r2 = 0.3m) , (h = 0.8m)
and indicator = 1 at: (a): f = 4GHz (b): f = 8 GHz (c): f = 12.5GHz (d): f = 26.5 GHz
(a): f = 26.5GHz
(b): f = 4GHz
Figure 18: Backscattered RCS versus aspect angle for a frustum for (r1 = 0.4m),
0.9m) and indicator = 0 at: (a): f = 26.5GHz
(b): f = 4GHz.
(r2 = 0.5m) ,
(h =
Erbil and Baghdad International Airports:
The backscattered Radar cross section for frustum target at Baghdad and Erbil International air- ports was
evaluated and computed. The conic dimensions were taken as follows: semi – major axis, r2 = 0.3m and 0.5m,
semi – minor axis, r1 = 0.2m, 0.4m and height h = 0.8m and 0.9m. The RCS for frustum was evaluated for the
following frequencies: primary frequency bands are (2.7GHz and 2.9GHz) while secondary frequency bands are
(1.07GHz and 1.09GHZ).The results are shown in figures (19) and (20).
(a): f = 2.7GHz
(b): f = 1.07 GHz
Figure 19: Backscattered RCS versus aspect angle for a frustum for (r1 = 0.2m), (r2 = 0.3m) , (h = 0.8m)
and indicator = 1 at: (a): f = 2.7GHz
(b): f = 1.07 GHz
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(a): f = 2.7GHz
(b): f = 1.07GHz
Figure 20: Backscattered RCS versus aspect angle for a frustum for (r1 = 0.4m),
0.9m) and indicator = 0 at: (a): f = 2.7GHz (b): f = 1.07GHz
(r2 = 0.5m) ,
(h =
4.5: Radar Cross Section for Cylindrical shape
The backscattered Radar cross section for cylindrical target was studied and evaluated. The cylinder
dimensions are given as follows: cylinder radii are (0.5m and 0.9m) and the focal length of perfectly conducting
cylinder “height” h = (2m and 4m). The RCS for cylinder was computed for following frequency bands: Cband: (4 and 8) GHz, X-band: (8 and 12.5) GHz and K-band: (18 and 26.5) GHz. The RCS for a cylinder versus
aspect angle is shown in figure (21). It is clear from figures that RCS for cylindrical target has maximum value
when aspect angle is 900, while its value reduces to zero when aspect angle is zero or 180 0. Again, the RCS
increases exponentially by increasing aspect angle up to 90 0 at which resonance occurs then after it decreases
exponentially by increasing aspect angle with range (900 < aspect angle < 1800).
(a): f = 4GHz
(c): f = 10.25GHz
(b): f = 8GHz
(d): f = 12.5GHz
(e): f = 18GHz
(f): f = 26.5GHz
Figure 21: Backscattered RCS versus aspect angle for cylinder for (r = 0.5m and (h = 2m) at: (a): f =
4GHz (b): f = 8GHz (c): f = 10.25GHz
(d): f = 12.5GHz (e): f = 18GHz
(f): f = 26.5GHz
Baghdad and Erbil International Air-ports:
The backscattered Radar cross section for a cylinder target at Baghdad and Erbil International air-ports was
evaluated and analyzed The cylinder dimensions were taken as follows: r = 0.5m and 0.9m, “height” h = (2m and
4m). The RCS for a cylinder for these two air-ports was evaluated for the following frequencies: primary
frequency bands are 2.7GHz and 2.9GHz while secondary frequency bands are 1.07GHz and 1.09GHz. The
results of RCS versus aspect angle are shown in figure (22) and (23).
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(a): f = 1.07GHz
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(b): f = 1.09GHz
(c): f = 2.7GHz
(d): f = 2.9GHz
Figure 22: Backscattered RCS versus aspect angle for cylinder for (r = 0.5m and (h = 2m) at:
1.07GHz (b): f = 1.09GHz
(c): f = 2.7GHz
(d): f = 2.9GHz
(a): f = 1.07GHz
(b): f = 26.5GHz
Figure 23: Backscattered RCS versus aspect angle for cylinder for (r = 0.9m and
(a): f = 1.07GHz (b): f = 26.5GHz
(a): f =
(h = 4m) at:
4.6: Radar Cross Section for Complex Objects
Simple shapes are useful in modeling more complex objects where the cross section of the complex target can
be approximated as the sum of contributions from the scattering of the simple component making up the target.
For example, scattering from a relatively complex target consists of two equal, isotropic objects. Scattering from
a missile with a cylindrically shaped body and two perfectly conducting plates, circular flat plates on both ends
separated by a distance, then the RCS was analyzed and computed using MATLAB programs.
RCS for complex target versus aspect angle is shown in Fig. (24) for specific values of circular cylindrical
height, (h = 3m and 2m) while radius of the circular plates are (a 1 = 0.2m and 0.4m) for previously mentioned
frequency bands. From figures, its clear that RCS for complex target starts to reach its maximum value about 80
degrees for rotation of the circular flat-plate targets. According to the figures, it is interesting to note that the
RCS has significant variations with varying aspect angles.
It is necessary to note that the RCS is mainly dominated at aspect angles close to 0 degree as well as 180
degrees. Its domination is mainly due to the circular plate. While at aspect angles close to normal incidence,
then the RCS has been almost all dominated by the cylindrical broad side specular return.
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(a): f = 4GHz
(c): f = 10.25GHz
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(b): f = 8GHz
(d): f = 12.5GHz
(e): f = 18GHz
(f): f = 26.5GHz
Figure 24: Backscattered RCS versus aspect angle for a cylinder with flat plates for (a1 = 0.2m) and (h =
2m) at : (a): f =4GHz (b): f = 8GHz (c): f = 10.25GHz (d): f = 12.5GHz (e): f = 18 GHz (f): f = 26.5GHz
Baghdad and Erbil International Air-ports:
The backscattered RCS for complex target was evaluated and analyzed for Baghdad and Erbil International
Airports at mentioned primary and secondary frequency bands. The RCS versus aspect angle is shown in figures
(25)and (26) for a cylindrical shape with flat- plates of dimensions (h = 2m and 3m) and the circular plates with
radii (a1 = 0.2m and 0.4m).
(a): f = 1.07GHz
8:
(b): f = 1.09GHz
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(c): f = 2.7GHz
(d): f = 2.9GHz
Figure 25: Backscattered RCS versus aspect angle for a cylinder with flat plates at (a1 = 0.2m) and
2m). (for Baghdad and Erbil International Air-ports)
(h =
(a): f = 1.07GHz
(b): f = 26.5GHz
Figure 26: Backscattered RCS versus aspect angle for a cylinder with flat plates for (a1 = 0.4m) and (h =
3m) at: (a): f = 1.07GHz
(b): f = 26.5GHz
5: Conclusions
According to the above present analysis procedures and the computed results of Radar- systems, the
following important conclusions can be noticed.
1- The values of backscattered cross section for perfectly conducting sphere were independent on
frequency as shown in Figs. (9) and (10).
2- The backscattered cross section of the ellipsoidal target versus aspect angle as shown in Figs.(11) and
(12) indicated that RCS is independent on the polar angle (φ) and frequency for azimuthally angle, φ =
00 and 450 . Generally speaking, RCS for ellipsoidal shape depends upon the ellipsoidal dimensions and
aspect angle. Therefore, the RCS appeared to be maximum when aspect angle was equal to 900.
Whenever the dimensions of the ellipsoid increased from (0.1, 2, 5) m to (1, 3, 9) m the value of RCS
increased from (-4dBsm to 20 dBsm), respectively.
3- The backscattered cross section of a circular flat plate increased as radius of plate increased from (0.2)
m to (0.3) m as well as it increase as result of increasing frequency as shown in Figs.(13) to (16). Its
value also reduced to minimum when aspect angle equals to 90 0.
4- The backscattered cross section of frustum versus aspect angle increases as a result of increasing
frustum’s dimensions. Moreover, the RCS had minimum and maximum values (-60 dBsm and 20
dBsm) at aspect angles (50 and near 1000), respectively for indicator = 1. While, its values occurred at
minimum and maximum values at aspect angles (180 0 and 800), respectively for indicator = 0 which
resulted upon increasing its dimensions.
5- The backscattered cross section of cylindrical shape versus aspect angle was depends upon aspect angle,
cylinder’s dimensions and operating frequency. Its maximum and minimum values appeared at aspect
angles 900 and 1800, respectively. The RCS values were increased slowly by increasing the dimensions
of cylinder as well as by increasing operating frequency as shown in figs. (21) to (23).
6- The backscattered cross section of complex target versus aspect angle was shown in Figs.(24) to (26).
The RCS was dominated by circular plate for complex target within aspect angles (0 0 and 1800) while it
was dominated by the cylindrical broad at aspect angle nearly 900. Generally, the RCS for complex
targets depends on frequency (wavelength), aspect angle and the geometry of the target. The maximum
value of RCS occurred at aspect angle equals 900.
6: References
1- Merrill I Skolnik, “Introduction to Radar Systems”, 1991 Mac Graw - -Hill Book Company, 2nd
Edition,
2- Eakasit W., "The Effect of Windmill Farms on Military Readiness" 2006, Report to the
Congressional Defense Committees.
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3- Chieh-Ping L., “Through Wall Surveillance Using Ultra Wideband Random” Noise Radar”, 2007
PhD. Thesis, the Pennsylvania State University.
4- Rowsell D., “Application of coherent Radar Using Stepped Frequency Modulation: An Evaluation
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