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Horseshoe-Shaped Multiband Antenna
for Wireless Application
Vineet Vishnoi, Praveen Kumar Malik and Manoj Kumar Pal
Abstract This paper presents the usefulness of horseshoe-shaped antenna: mathe-
matically based on baker’s transformation. The proposed antenna has the property
of filling a plane using higher-order iterations and exploited in realization of a multi-
band resonant antenna. The effect of additional iterations resulting in the reduction
of resonant frequency is near-logarithmic pattern. The designed antenna shows mul-
tiple frequency bands ranging from 1.01 to 7.60 GHz. It has been also observed that
the proposed prototype antenna has 75% efficiency, directivity up to 11.5 dBi and
gain of about 10 dB. The antenna characteristics have been studied using IE3D v.14
simulation software based on method of moment (MoM) and also experimentally
verified using VNA network analyzer. Simulation and experimental results are in
good agreement and demonstrate the performance of the design methodology and
the proposed antenna structures.
Keywords Baker’s transformation ·Horseshoe-shaped antenna ·IE3D ·
Multiband antennas
1 Introduction
State-of-the-art wireless telecommunication system needs antenna with wider band-
width, multifunctional and compact in size than conventional antennas. As antenna
length may be order of two or large for efficient radiation, this limits the performance
of other parameters like bandwidth, gain and efficiency [1,2]. This generates a new
family of antenna termed as fractal coined by Mandelbrott [3], and a lot of work
V. Vishnoi (B)
Inderprastha Engineering College, Ghaziabad, Uttar Pradesh, India
e-mail: vishnoivineet@gmail.com
P. K. Malik
Lovely Professional University, Jalandhar, Punjab, India
e-mail: pkmalikmeerut@gmail.com
M. K. Pal
Bharat Sanchar Nigam Limited, Kanpur, Uttar Pradesh, India
e-mail: manoj1976bsnl@gmail.com
© Springer Nature Singapore Pte Ltd. 2020
P. K. Singh et al. (eds.), Proceedings of First International Conference on Computing,
Communications, and Cyber-Security (IC4S 2019), Lecture Notes in Networks
and Systems 121, https://doi.org/10.1007/978-981- 15-3369- 3_3
33

34 V. Vishnoi et al.
has been compiled through various researchers on such class of antenna subfamily
[4–10]. The fractal mathematics is too old, but some of fractal geometries find its
wide application in antenna geometry, and from the last two decades, few of such
fractal geometries were investigated in antenna design and become popular such as
Koch curves, Minkowski curves and Sierpinski carpets which are investigated for
antenna design [11–15] as fractal geometry-based antennas have diverse application
in several fields of science and engineering. The multiband characteristics of fractal
antenna are due to their self-similarity of fractal geometry qualitatively associated
with their multiband characteristics of antenna, along with space-filling properties
resulting to the miniaturization of antenna without degradation in antenna parame-
ters such as gain, bandwidth and efficiency. These antennas resonate at frequencies
in a near-logarithmic interval. The individual bands at these resonant frequencies
are generally small, and allocation of these frequency bands founds arbitrarily in its
frequency spectrum [16–21,22–24].
Apart from this, other important features of this fractal antenna include low pro-
file, low cost, conformability and ease of integration with active devices. There is
limit on bandwidth and gain of such antennas. Fortunately, a lot of communication
systems do not require large bandwidths; therefore, it is not an important problem.
Many techniques have been used to reduce the size of antenna, such as using dielec-
tric substrates with high permittivity [17–21,22–25], applying resistive or reactive
loading [18], increasing the electrical length of antenna by optimizing its shape [19]
and utilizing strategically positioned notches on the patch antenna [20,21,22–26].
As in this regard, the earlier article [21] published on horseshoe-shaped antenna using
multi-L-slots does not represent the true horseshoe shape, but in the present article, we
present a truly horseshoe-shaped pattern which is used here in antenna design. This
consists of baker’s transformation curve patterns (22), which have several important
characteristics hitherto unexplored in antenna engineering.
This paper is organized as follows. Section 2describes the mathematics of horse-
shoe geometry. Sect. 3presents the simulated and measured results of proposed
antenna. Results and discussion is provided in Sect. 4.
2 Horseshoe Fractal Design Methodology
One of the simplest planar dynamical systems with a fractal attractor is the so-called
baker’s transformation [9] because it resembles the process of repeatedly stretching
a piece of dough and folding it in two. Let E=[0, 1] ×[0, 1] be the unit square. For
fixed 0 <λ< 1
2, we define the baker’s transformation f:E→Eby
f(x,y)=(2x,λy), (0≤x≤1/2)
(2x−1,λy+1/2), (1/2≤x≤1)(1)

Horseshoe-Shaped Multiband Antenna for Wireless Application 35
Fig. 1 Baker’s transformation. aIts effect on the unit square. bIts attractor
This transformation may be thought of as stretching Einto a 2 ×λrectangle,
cutting it into two 1 ×λrectangles and placing these above each other with a gap
of 1
2−λin between; see Fig. 1. Then, Ek=fk(E)is a decreasing sequence of
sets, with Ekcomprising 2khorizontal strips of height λkseparated by gaps of at
least 1
2−λλ(k−1). Since f(Ek)=Ek+1, the compact limit set F=Eksatisfies
f(F)=F. (Strictly speaking, f(F) does not include part of Fin the left edge of the
square E, a consequence of fbeing discontinuous. However, this has little effect on
our study.) If (x,y)∈E, then fk(x,y)∈Ek,sofk(x,y) lies within distance λkof F.
Thus, all points of Eare attracted to Funder iteration by f. If the initial point (x,y)
has x=0, a1,a2, … in base 2 and x= 1
2, then it is easily checked that
fk(x,y)=(0,ak+1,ak+2, ..., yk)(2)
where ykis some point in the strip of Eknumbered ak,ak−1,…,a1(base 2) counting
from the bottom with the bottom strip numbered 0. Thus, when kis large, the position
of fk(x,y) depends largely on the base 2 digits ai of xwith iclose to k. By choosing
an xwith base 2 expansion containing all finite sequences, we can arrange for fk(x,
y) to be dense in Ffor certain initial (x,y), just as in the case of the tent map. Further
analysis along these lines shows that fhas sensitive dependence on initial conditions
and the periodic points of fare dense in F, so that Fis a chaotic attractor for f.
Certainly, Fis a fractal—it is essentially the product [0, 1]F1, where F1is a Cantor
set that is the IFS attractor of S1(x)=x,S2(x)=1/2 +λx. Since dimHF1=dimBF1
=log2/logλ,dimHF=1+log2/logλ, using Corollary 2. The baker’s transformation
is rather artificial, being piecewise linear and discontinuous. However, it does serve to
illustrate how the ‘stretching and cutting’ procedure results in a fractal attractor. The
closely related process of ‘stretching and folding’ can occur for continuous functions

36 V. Vishnoi et al.
Fig. 2 A horseshoe map. aThe square Eis transformed, by stretching and bending, to the horseshoe
f(E), with a, b, c, d mapped to a,b
,c
,d
, respectively. bThe iterates of Eunder fform a set that
is locally a product of a line segment and a cantor set
on plane regions. Let E=[0, 1] ×[0, 1] and suppose that fmaps Ein a one-to-one
manner onto a horseshoe-shaped region f(E) contained in E. Then, fmay be thought
of as stretching Einto a long thin rectangle which is then bent in the middle. This
figure is repeatedly stretched and bent by f, so that fk(E) consists of an increasing
number of side-by-side strips; see Fig. 2.
We hav e E ⊃f(E)⊃f2(E), …, ⊃fk(E), and the compact set F=∞
k=1attracts
all points of E. Locally, Flooks like the product of a Cantor set and an interval. A
variation on this construction gives a transformation with rather different character-
istics; see Fig. 2.IfDis a plane domain containing the unit square Eand f:D→
Dis such that f(E) is a horseshoe with ends and arch lying in a part of Doutside
Ethat is never iterated back into E, then almost all points of the square E(in the
sense of plane measure) are eventually iterated outside Eby f.Iffk(x,y)∈Efor all
positive k, then (x,y)∈∞
k=1f−1(E). With fsuitably defined, f−1(E) consists of
two horizontal bars across E,so∞
k=1f−1(E)is the product of [0, 1] and a Cantor
set. The set F=∞
k=−∞ fk(E)=∞
k=0fk(E)=∞
k=1f−k(E)is compact and
invariant for fand is the product of two Cantor sets. However, Fis not an attractor,
since points arbitrarily close to Fare iterated outside E.
The sets ∞
k=1f−k(E)and ∞
k=0fk(E)are both products of a Cantor set and a
unit interval. Their intersection Fis an unstable invariant set for f. A specific example

Horseshoe-Shaped Multiband Antenna for Wireless Application 37
Fig. 3 A horseshoe map. aAn alternative horseshoe map. bThe square Eis transformed, so that
the arch and ends of f(E) lie outside E
of a ‘stretching and folding’ transformation is the Henon map f:R2→R2 (Fig. 3).
f(x,y)=(y+1−ax2,bx)(3)
where aand bare constants. (The values a=1.4 and b=0.3 are usually chosen for
study. For these values, there is a quadrilateral Dfor which f(D)⊂Dto which we
can restrict attention.) This mapping has Jacobian bfor all (x,y), so it contracts area
at a constant rate throughout R2; to within a linear change of coordinates, Fig. 2is
the most general quadratic mapping with this property. The transformation in Fig. 2
may be decomposed into an area-preserving bend, a contraction and a reflection,
the net effect being horseshoe-like; see Fig. 4. This leads us to expect fto have a
fractal attractor, and this is borne out by computer pictures, Fig. 4. Detailed pictures
show banding indicative of a set that is locally the product of a line segment and a
Cantor-like set. The diagrams show the effect of this successive transformation on
a rectangle. Estimates suggest that the attractor has box dimension of about 1.26
when a=1.4 and b=0.3. Detailed analysis of the dynamics of the Henon map
is complicated. In particular, the qualitative changes in behavior (bifurcations) that
occur as aand bvary that are highly intricate. Many other types of stretching and
folding are possible. Transformations can fold several times or even be many-to-one;
for example the ends of a horseshoe might cross. Such transformations often have
fractal attractors, but their analysis tends to be difficult (Fig. 5).
3 Simulation Setup
The resonant properties of the proposed antenna have been predicted and optimized
using a frequency domain three-dimensional full-wave electromagnetic field solver
(IE3D-Zeland), as the proposed horseshoe antenna is designed on the patch size of
3.8 ×5.5 cm, using scale factor one-third on glass–epoxy material having height
1.6 mm and dielectric constant 4.5 as shown in Fig. 6.
As explained above, the base shape for horseshoe geometry is designed on rect-
angular patch of specific dimension as provided above, and for further iterations,

38 V. Vishnoi et al.
Fig. 4 Henon map may be
decomposed into an
area-preserving bend,
followed by a contraction
andfollowedbyareflection
in the line y=x
the geometry is scaled by a factor of one-third keeping outer dimension of the patch
invariant as 3.87 ×5.56 cm for first iteration and second iteration which are generated
in the same manner as its originator base shape as shown in Fig. 7.
After completion of simulation setup, IE3D provides various antenna parameters
through its easily accessible user graphics format for analysis point of view, and here,
all such typical antenna parameters are analyzed viz return loss (S11), directivity,
antenna gain and its radiation/antenna efficiency.
Fig. 8illustrates the comparison in between return loss for base shape of horseshoe
and its iterated geometries, and it is observable that lowering of resonance frequency
form 1.229 to 1.034 Ghz along with incorporation of multiple frequency bands from
1.034 to 7.6 GHz
Similarly, other important antenna parameters such as directivity, gain and effi-
ciency for horseshoe’s base shape and its iterated geometries are compared in Figs. 9,
10 and 11, respectively, from Fig. 5, directivity approaches up to 13 dBi for second
iteration, and similarly, gain approaches up to 21 and 10 dB on an average (Fig. 10).

Horseshoe-Shaped Multiband Antenna for Wireless Application 39
Fig. 5 Iterates of a point under the Henon map showing the form of the attractor. Banding is
apparent in the enlarged portion in the inset
Fig. 6 Proposed prototype antenna with design dimensions
Fig. 7 Design antenna. aFirst iteration. bSecond iteration

40 V. Vishnoi et al.
Fig. 8 Return loss (S11) curve for second iteration of horseshoe antenna
Fig. 9 Directivity versus frequency plot for second iteration of horseshoe antenna
In continuation to this, from Fig. 11, it is also observable that efficiency approaches
up to 75%, so all antenna parameters provide satisfactory optimized values (Fig. 12).
Apart from above-shown parameters, the proposed antenna has the promising
radiation patterns for elevation (φ=0° and φ=90°) and azimuth plane (θ=900°)
in Fig. 13 at frequencies 1.22 GHz, 1.821 GHz and 6.935 GHz, respectively. After
the radiation pattern, experimental results are also superimposed (with red in color)
on simulated radiation pattern for comparison point of view in the respective figures.

Horseshoe-Shaped Multiband Antenna for Wireless Application 41
Fig. 10 Total gain versus frequency plot for second iteration of horseshoe antenna
Fig. 11 Efficiency (antenna and radiation) versus frequency plot for second Iteration of horseshoe
antenna
On comparing in between simulated results and measured results for return loss
as shown in Fig. 12, it is quite promising that for proposed antenna both the curves,
simulated and measured results follow each other with high degree of accuracy, and
the variation in between these two curves can also be anticipated on the basis of
design accuracy.
Since all the three geometries of proposed horseshoe-shaped antenna are analyzed
on IE3D simulation software, in this regard, a combined comparative graph is shown
in Figs. 6and 7, its details are provided in Table 1for mutual comparison of their
return loss (S11) plots for observing the effect of higher-order iteration, and from
the curve and table, it is quite noticeable that nos. of resonating frequency samples

42 V. Vishnoi et al.
Fig. 12 Radiation curve for simulated antenna for second iteration of horseshoe antenna at
frequency 6.935 GHz for elevation aφ=0°, bφ=90°
Fig. 13 Comparison curve in between simulated and measured (in black) return loss (S11) for
second iteration of horseshoe antenna

Horseshoe-Shaped Multiband Antenna for Wireless Application 43
Tabl e 1 Comparison table between the base shape of first and second iterations
S. No. Frequency (GHz) Second iteration First iteration Base shape
11.034 −11.68 −2.73 −4.08
21.229 −30.08 −20.12 −14.06
31.48 −18.18 −5.29 −5.56
41.8 −20.13 −8.36 −4.5
52.19 −10 −6.73 −3.4
62.22 −16 −6.21 −3.46
72.47 −10 −5.06 −3.41
82.6 −20.81 −3.9 −3.3
92.86 −10 −6.17 −2.42
10 2.91 −27.38 −11.54 −2.1
11 2.99 −10 −8.34 −1.48
12 3.36 −14 −5.01 −1.6
13 3.58 −10 −7.8 −4.52
14 3.64 −18.37 −10.02 −6.18
15 3.7 −10 −13.7 −8.07
16 3.95 −10 −12.58 −5.54
17 4−25 −13.85 −5.4
18 4.25 −22 −8.71 −6.9
19 4.34 −10 −7.1 −7.9
20 6.98 −10 0 0
21 7.2 −25 −5.8 0
22 7.6 −10 −8.39 −5.54
go on incrementing while approaching higher-order iteration (as shown in Table 1).
It is also remarkable that about 1.22.
Moreover, from above discussion, it is also noticeable that the proposed antenna
peruses significant directivity from 7 to 11.5 dBi (Fig. 5) for wide range of frequency,
and similarly, Fig. 6depicts the curve of total gain versus frequency plot, as from
the curve, it is noteworthy that total gain approaches up to 10 dB value. The other
important antenna parameters are efficiency of proposed antenna geometry, from
Fig. 7, the curve illustrates antenna and radiation efficiency versus frequency, from
the curve, it is notable that radiation efficiency approaches 100% for a significant
band of freq, also, the antenna efficiency approaches 70% in this frequency range,
the above values are quite promising for a good antenna design, and the proposed
antenna is well suited for wireless applications.

44 V. Vishnoi et al.
4 Conclusion
A novel horseshoe-shaped fractal antenna for wireless application has been proposed,
constructed and tested. The proposed antenna has simple iterative geometry. As the
proposed antenna design provides adoptability of various frequencies ranging from
1.03 up to 7.21 GHz and provides characteristics such as 100% radiation efficiency,
highly permissible gain up to 10 dB and directivity up to 11.5 dBi. These promising
characteristics are found higher than earlier design traditional wireless antennas.
The proposed antenna has good directional-radiation characteristics. Furthermore,
this antenna has many advantages such as easy fabrication, low cost and compact in
size. Therefore, it is found quite suitable for wireless/WLAN applications.
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