plane requires no additional processing, this UWB filter would

be simple to implement.

3. IMPLEMENTATION OF THE PROPOSED UWB BPF

Figure 5(a) shows a photograph of the fabricated UWB BPF on

the RT/duroid 5880 substrate. As this filter does not need any

apertures in the ground plane, there was no manufacturing pro-

cess in the ground plane. The fabricated filter is 46.7 mm by 50

mm. The simulated and measured S-parameters and group delay

are shown in Figure 5(b). The proposed UWB BPF has a meas-

ured passband from 3.24 to 11.03 GHz, and the measured return

loss is less than 210 dB at all frequencies in the UWB pass-

band. The measured group delay remains less than 0.5 ns and

flat in the passband. Due to the attenuation poles from the SISs,

sharp rejection skirts are obtained at both the lower and upper

ends of the passband. This UWB BPF provides better perform-

ance in the passband than the conventional filter based on a sin-

gle SISLR, without apertures in the ground plane.

4. CONCLUSION

In this article, a new UWB BPF was proposed, with dual

SISLRs. To eliminate the necessity for apertures in the ground

plane, the proposed UWB BPF was designed with dual SISLRs.

This filter offers the advantages of sharp rejection skirts, good

wideband filtering performance, and low return loss (<210

dB), without a complicated coupling structure or a ground struc-

ture with defects. The proposed UWB BPF has a passband cov-

ering the range of 3.24�11.03 GHz. The group delay is less

than 0.5 ns and flat in the passband. It has a fractional band-

width of 113.73%. It also exhibits sharp attenuations near the

passband. The new filter exhibits high performance of the pass-

band, while retaining the resonant frequencies and sharp rejec-

tion skirt characteristics of the conventional filter.

ACKNOWLEDGMENTS

This work was supported in part by Brain Korea 21 project in

2012 and Basic Science Research Program through the National

Research Foundation of Korea (NRF) funded by the Ministry of

Education, Science and Technology (2012R1A1B3002517)

REREFENCES

1. Federal Communications Commission, Revision of part 15 of the com-

mission’s rule regarding ultra-wideband transmission systems, FCC,

Washington, DC, Tech. Rep. ET-Docket 98-153, FCC02-48, Apr. 2002.

2. L. Zhu, S. Sun, and W. Menzel, Ultra-wideband (UWB) bandpass

filter using multiple-mode resonator, IEEE Microwave Wireless

Compon Lett 15 (2005), 796–798.

3. L. Zhu and H. Wang, Ultra-wideband bandpass filter on aperture-

backed microstrip line, Electron Lett 41 (2005), 1015–1016.

4. H. Wang and L. Zhu, Aperture-backed microstrip line multiple-mode

resonator for design of a novel UWB bandpass filter, In: Asia-Pacific

Microwave Conference Proceedings, vol. 4, Suzhou, China, Dec. 2005.

5. S. Gao, S. Xiao, J. Wang, X. Yang, and B.-Z. Wang, A wideband

microstrip bandpass filter for ultra-wideband wireless communication

application, Microwave Opt Technol Lett 49 (2007), 1975–1976.

6. S.T. Li and Q.X. Chu, A compact UWB bandpass filter with

improved upper-stopband performance, In: ICMMT 2008, vol. 1,

Apr. 2008, pp. 363–365.

7. Z. Li, G.-M. Wang, C.-X. Zhang, and G.-N. Long, An ultra-wide

bandpass filter with good out-of-band performance, Microwave Opt

Technol Lett 50 (2008), 1735–1737.

8. I.-T. Tang, D.-B. Lin, C.-M. Li, and M.-Y. Chiu, Ultra-wideband

bandpass filter using hybrid microstrip-defected-ground structure,

Microwave Opt Technol Lett 50 (2008), 3085–3089.

9. Q.X. Chu and X.K. Tian, Design of UWB bandpass filter using

stepped-impedance stub-loaded resonator, IEEE Microwave Wireless

Compon Lett 20 (2010), 501–503.

VC 2014 Wiley Periodicals, Inc.

MICROSTRIP MULTISTOPBANDFILTER BASED ON PREFRACTALSPACE-FILLING CURVE

Federico CaramanicaDepartment of Information Engineering and Computer Science,University of Trento, Via Sommarive 14, 38050 Trento, Italy;Corresponding author: federico.caramanica@ieee.org

Received 12 June 2013

ABSTRACT: In this letter, a method for the design of miniaturizedmultistopband filters is presented. This methodology exploits multiple

resonance behavior of a small-size prefractal geometries. The consid-ered fractal structures based on space-filling curve geometry and then,after recasting the design problem as an optimization one by defining a

suitable cost function, is optimized by an unsupervised procedure, thegravitational search algorithm, which is a recent and interesting sto-

chastic optimization method. To assess the effectiveness of this approach,a filter with three stopbands has been designed, numerically assessed, andrealized. The obtained results are in good agreement with the numerical

simulations and this confirms the capabilities of the proposed method as amicrowave Computer Aided Design (CAD) tool. VC 2014 Wiley Periodicals,

Inc. Microwave Opt Technol Lett 56:450–455, 2014; View this article

online at wileyonlinelibrary.com. DOI 10.1002/mop.28116

Key words: microstrip filter; notched filter; evolutionary algorithm;

gravitational search algorithm; space filling curve

Figure 5 (a) Photograph of the fabricated UWB BPF with an RT/

duroid 5880 substrate, and (b) Simulated and measured results for the

proposed UWB BPF using dual SISLRs. [Color figure can be viewed in

the online issue, which is available at wileyonlinelibrary.com]

450 MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 56, No. 2, February 2014 DOI 10.1002/mop

1. INTRODUCTION

The development of electronic and communication devices such

as mobile smart phones, laptop, or wireless sensor nodes employ-

ing different frequencies over a wide frequency range has to take

into account the presence of multiple interference signals that can

have a harmful effect on receivers. Eliminate undesired interfer-

ences signals is mandatory to guarantee the correct transmission

of information. In this framework, the objective of this work is to

develop a strategy to design multistopband filters that fulfill the

following specifications: an arbitrary number of stop-bands equal

to R, namely fr; r 5 1; :::;Rf g, and the minimization of the physi-

cal dimensions respect with operating wavelength to favor inte-

gration and portability of mobile devices [1, 2]. The realization of

an efficient and reliable stopband filter that can operate at differ-

ent multiple frequencies, is both a good solution but also a critical

task as the synthesis procedure is not straightforward. Further-

more, adding the necessity of miniaturization, the synthesis of a

multistopband filters becomes an interesting design challenge.

Classical stopband filter based on quarter-wavelength resonator is

not suitable to meet the multiresonances behavior necessary to

make the synthesis of filters [3, 4].

The idea proposed in this work is to modify the structure of

a standard resonating structure considering fractal geometris as

in [5–7] where interesting examples of filters based on resona-

tors built with fractal geometries are presented. H-shape fractals

[5, 6] and Sierpinski’s triangles [7] allow to get multiple

resonances but in this work the attention is shifted on another

kind of fractal geometry, namely the space-filling curve and in

particular the Hilbert shape. However, it has to be pointed out

that Hilbert curve, in the same way of other classical fractal

geometries, usually presents an harmonic frequency response

rather than a multiresonant behavior, as observed in [8], where a

Koch-like shape have been used in antenna synthesis. To over-

come this problem, tune and control the spacing between the

working stopband frequencies of a filter based on a fractal reso-

nator, a possible solution is to perturb the self-similarity by add-

ing some degrees of freedom in the synthesis process [9].

Following this idea, a simple way to perturb the fractal geome-

try is to vary the width of the segments that constitutes the fil-

ter. Instead of a trial-and-error design method, after defining a

suitable cost function, the design problem has been recast as an

optimization one [10–14]. The selection of the widths of the

segments, to located stop-bands at desired frequencies, and the

related optimization problem are controlled by a stochastic opti-

mization algorithm, namely the gravitational search algorithm

(GSA) [15].

The outline of this letter is as follows. Section 2 describes the

geometry of the multifrequency resonator and the details of the

design procedure of the filter. The unsupervised strategy devel-

oped to meet the technical requirements and to minimize the

overall dimensions of the resonator are described in Section 3. In

Section 4, the efficiency and the reliability of the synthesised fil-

ter are assessed with both numerical and experimental results.

Finally, some concluding remarks are given in Section 5.

2. MULTISTOPBAND FILTER DESIGN

Let us consider a basic single-stopband filter, at a certain fre-

quency fstop, realized with quarter-wavelength open-end stub [3,

4] with microstrip technology on a ceramic substrate (Fig. 1).

The considered geometry is planar and printed on the top side

of the dieletric substrate, whereas the bottom side is composed

by a metallic ground plane.

This kind of filter is composed by two main elements. The first

component is the feeding microstrip line with input port P1 and

output port P2, whereas the second is represented by one or more

microstrip resonators. This kind of filter can be arranged into two

different configurations: asymmetric and symmetric as described

in [4]. The asymmetric configuration [Fig. 1(a)] has the ability to

act as a symmetric filter combined with an impedance transformer

[4]. To achieve a wide stopband, the dimensions of the coupled

lined have to be appropriately chosen. The symmetric filter [Fig.

1(b)] can offer a wider bandwidth in addition to a higher stopband

rejection in a comparable size [4]. As stated in Section 1, as this

work is focused on the presentation of the potentialities offered by

a methodology based on prefractal space-filling curve optimized

by evolutionary algorithms, the chosen stopband filter configura-

tion is the simplest one, namely the asymmetric configuration

with only one resonator of [Fig. 1(a)].

To develop a multistopband filter able to eliminate interfer-

ence at arbitrary frequencies, the single quarter-wavelength reso-

nator is not suitable due to the harmonic behavior of this

configuration. In this framework, the proposed solution is to fol-

low the example of [5–7] and make use of fractal geometries. In

particular, in this work, the attention is directed toward the class

of space-filling fractal curve such as Hilbert curve that was suc-

cessfully exploited in different works for multibands antenna

design, for example [16–18]. Hilbert fractal curve at different

stages is depicted in Figure 2.

The first reason to use this kind of fractal curve is that Hil-

bert curve, used for antenna geometry, have shown interesting

multiband behavior [16]. The second reason is that the space-

Figure 1 Quarter wavelength stopband filters in asymmetric (a) and (b) symmetric configurations

DOI 10.1002/mop MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 56, No. 2, February 2014 451

filling configuration allows to design resonant structures whose

physical size is considerably smaller than the longest working

wavelength [16].

These two properties can be exploited in the design problem

considered in this work by substituting the quarter-wavelength

resonator with a microstrip-printed structure whose geometry is

based on Hilbert curve. In more details, the idea is to use the

fractal curve evolved at stage i 5 2 [Fig. 2(b)] and realize a

structure similar to Figure 3.

The microstrip line, with input port P1 and output port P2, is

designed with the procedure described in the following to guar-

antee the matching conditions on the P1 and P2 ports. Given the

dieletric properties of the substrate er , the width of the micro-

strip transmission feeder wfeed, to have the characteristic imped-

ance Zline 550X, can be calculated using the following Eq. (1).

For wfeed =h � 2, the approximate expression in terms of Zline

and er , is

wfeed 52h

p

60p2

Zlineffiffiffiffierp 21

� �2ln 2

60p2

Zlineffiffiffiffierp 21

� �

1er21

2erln

60p2

Zlineffiffiffiffierp 21

� �10:392

0:61

er

� �� (1)

being h the height of the dieletric substrate. After the design of

the feeding microstrip, the problem becomes how to obtain the

desired frequency behavior by means of the Hilbert space-filling

curve. The filter structure with Hilbert fractal curve of Figure 3

has a harmonic behavior. For example, Figures 4(a)–4(c) show

the frequency behavior of a filter with a Hilbert shape resonator

built according to the structure proposed in Figure 3. The prop-

erties of the frequency behavior depend on not only LX and LY,

but also the microstrip width rmicro . Unfortunately, this fre-

quency behavior usually does not match with the required multi-

band frequency response. To solve this problem, the following

practical solution is proposed.

The resonant frequencies can be modified by perturbing self-

similarity of the fractal curve [9]. From a geometrical point of

view, fixed the Hilbert fractal curve at stage i 5 2 as skeleton of

Figure 2 Hilbert self-similar curve at different stages: (a) stage i 5 0,

(b) stage i 5 1, and (c) stage i 5 2

Figure 3 Example of stopband filter structure with resonator based on

Hilbert curve geometry

Figure 4 Frequency response of Hilbert self-similar curve at stage i 5 2

with LX5LY510 mm (a), LX5LY515 mm (b), and LX5LY520 mm (c).

For each plot, different widths rmicro of the microstrip constituting the Hil-

bert resonator are considered. [Color figure can be viewed in the online

issue, which is available at wileyonlinelibrary.com]

452 MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 56, No. 2, February 2014 DOI 10.1002/mop

the resonator shape, the perturbed fractal structure is uniquely

defined by the following vector of parameters

v5 LX;LY ; rj; j51; :::; J� �

(2)

where LX and LY are the horizontal and vertical dimension of

the Hilbert geometry.

Furthermore rj, j51; :::; J, are the widths of the segments

constituting the Hilbert shape as described in the parametric

model of Figure 5. Since the Hilbert curve is considered at stage

i 5 2, the total number of segments is J 5 63.

Summarizing, the total number of geometrical degrees of

freedom added to the basic Hilbert fractal shape, to change the

frequency response, is equal to M5J12565. This choice is

related to the observation that using the fractal curve at stage

i 5 0 or i 5 1 the number of degrees of freedom could be not

sufficient to comply with requirements. Unfortunately M, the

number of unknowns parameters, is too high for a manually

trial-and-error procedure. To overcome this problem, the synthe-

sis procedure is described in the following section.

3. UNSUPERVISED OPTIMIZATION OF PREFRACTALHILBERT RESONATOR

In this section, the scheme of the prefractal Hilbert resonator

synthesis, led by GSA, is described in depth and summarized by

the scheme of Figure 6. As a states in the previous Section, we

use an optimization methodology as a selection by hands of LX,

LY, and rj, being j51; :::; J, is not practicable and for this reason

an unsupervised procedure is proposed. In [19–22], particle

swarm optimizer is used to find the geometric configuration of

prefractal shape to obtain monopolar antenna with reduced

dimensions or multiband behavior. In this letter, a similar tech-

nique is applied for the synthesis of the Hilbert prefractal reso-

nator necessary to obtain the desired multistopband filter.

Instead of using particle swarm optimization (PSO) [23–26],

GSA has been chosen to lead the optimization procedure. As

described in [15], the GSA takes inspiration by the laws that

describe behavior of masses: the law of gravity and the law of

motion. This stochastic optimization algorithm shows good per-

formance and seems to converge faster than the other algorithms

such as GA or PSO. The detailed guidelines of GSA are

reported in [15].

The M geometry unknowns constitute the parameter vector

v that is controlled and evolved by GSA at each step of the

iterative optimization procedure. In the scheme of Figure 6,

the GSA starts, at iteration k 5 0, from a set of N initial solu-

tions vkn; n51; :::;N

� �randomly chosen. At each successive

iteration, the optimization algorithm generates a new set of

evolved trial solutions vkn; n51; :::;N

� �, k > 0, according to

the strategy described in [15]. At the k – th iteration of the

GSA, the algorithm controls the prefractal geometry by gener-

ating width of each fractal segment rj, j 5 1; :::; J, and the side

lengths LX and LY. The geometrical solutions proposed by the

GSA algorithm are compared and evaluated according to a

suitable fitness function D vkn

� �. The optimization procedure

ends if one of the stop criteria is met: k 5 K, being K is the

maximum number of iterations, or D vkn

� �� g, being g the

user-defined convergence threshold. The definition of cost

function D is the way to recast the geometrical design problem

as minimization one where, once the optimization algorithm

approaches the global minimum, the filter requirements in the

frequency domain are satisfied. More in details, to link the

geometrical domain with the frequency response domain, it is

mandatory to put in evidence that each trial geometry vkn at

the operative frequency fh 2 fmin ; fmax½ � is simulated with

Method-of-Moments (MoM) full-wave electromagnetic simula-

tor that returns scattering parameters of the considered two-

port microwave device, being S vkn; fh

� �the scattering matrix.

The cost function is defined as

Figure 5 Prefractal Hilbert self-similar curve at stage i 5 2, described

with parametric model

Figure 6 Scheme of the unsupervised synthesis procedure of the reso-

nator geometry

DOI 10.1002/mop MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 56, No. 2, February 2014 453

D vkn

� �5D1 vk

n

� �1D2 vk

n

� �(3)

and it is composed by two terms. The first, D1 vkn

� �, is associated

to the side dimension of the Hilbert resonator

D1 vkn

� �5jLX2Ltarget

X j1jLY2LtargetY j (4)

where LtargetX and Ltarget

Y are the dimension target values. The

second term of (3), namely D2 vkn

� �, is related to the scattering

matrix coefficient and the frequency response of the filter:

D2 vkn

� �5X2

p51

XLmax

l50

jSp1 vkn; fl

� �j2Starget

p1 flð ÞStarget

p1 flð Þ

!224

35 (5)

where fl5l � Df 1fmin is the operative frequency in the interval

fl 2 fmin ; fmax½ �, being Df a suitable sampling frequency interval,

Sp1 vkn; f

� �, being p51; 2, are the values of the scattering matrix

for trial solution vkn, while Starget

p1 fð Þ are the desired target values.

The filter is required to have different response to different fre-

quencies: in particular, in the stop-band high insertion loss and

low impedance matching, while in pass-bands it is highly desira-

ble to have good input impedance matching and very low inser-

tion loss.

4. NUMERICAL AND EXPERIMENTAL VALIDATION

In this section, the proposed approach is exploited to synthesize

a three stop-bands filter. The stop-bands are chosen to reduce

the impact of three possible strong interference signals at

fr;R51; :::; 3f g5 1:7; 2:5; 3:6f gGHz . The target values for scat-

tering matrix coefficient in stop-bands are jStarget11 fð Þj520:05 dB

and jStarget21 fð Þj5220:0 dB . Conversely in pass-bands the

requirements become jStarget11 fð Þj5220:0 dB and

jStarget21 fð Þj520:05 dB . Further requirements are the target values

associated to the dimensions of the Hilbert shape resonator that

are LtargetX 51:0 cm and Ltarget

Y 51:0 cm . The ceramic substrate

used both for numerical and experimental validation has a thick-

ness h50:8 mm , a relative permittivity e53:38 and tan d50:002

at f 510GHz . The dielectric properties are considered stable

with good approximation in the considered operational fre-

quency range f 2 fmin ; fmax½ �5 1:0; 4:0½ �GHz . The first step in

the design procedure is to calculate the width of the feeding

microstrip wfeed to obtain an impedance Zline 550 X. By using

(1), wfeed is 1.8 mm.

In the next step, the unsupervised procedure led by GSA is

exploited to synthesize the multistopband filter according to the

desired requirements. The GSA considered in this article adopts

a population of N 5 20 trial solutions, a maximum number of

iterations K 5 200 and a convergence threshold g51022. The

remaining parameters of the GSA have been chosen according

to [15]. The cost function D is plotted in Figure 7. At the end of

the synthesis procedure, the final shape of the Hilbert prefractal

resonator has been obtained.

For the sake of completeness, after the numerical study, an

experimental validation has been carried out. The prefractal fil-

ter prototype has been built by using photolithographic printing

circuit technology on ceramic substrate (Fig. 8).

The side of the prefractal shape is 1.2 cm. The scattering

matrix coefficients jStarget11 j and jStarget

21 j have been measured with

scalar network analyzer. Table 1 and 2 give details about the

comparison among requirements, numerical results, and meas-

urements in stop-bands. In Table 3, there are Q-factors for the

three stopbands. The overall comparison of numerical and

Figure 8 Cost function D vs. the number of iterations k. [Color figure

can be viewed in the online issue, which is available at

wileyonlinelibrary.com]

Figure 7 Photo of the prototype of the optimized Hilbert filter. [Color

figure can be viewed in the online issue, which is available at

wileyonlinelibrary.com]

TABLE 1 Comparison of jS11 frð ÞjdB and jS21 frð ÞjdB Scatter-ing Coefficients Obtained from Numerical and MeasurementEvaluation

Numerical Results Measured Results

Stop Frequency jS11 frð ÞjdB jS21 frð ÞjdB jS11 frð ÞjdB jS21 frð ÞjdB

f1 20.49 214.63 20.03 221.62

f2 20.93 213.88 20.05 219.41

f3 21.13 219.98 20.02 223.37

TABLE 2 The 23dB Stop-Bands Obtained From NumericalAnd Experimental Assessment are Compared withRequirements

Stop Band

Requirements Numerical Results Measured Results

fmin ; fmax½ �Ghz fmin ; fmax½ �Ghz fmin ; fmax½ �

Bstop1 1:55; 1:75½ � 1:52; 1:74½ � 1:52; 1:76½ �Bstop2 2:45; 2:55½ � 2:46; 2:86½ � 2:41; 2:58½ �Bstop3 3:55; 3:65½ � 3:55; 3:66½ � 3:52; 3:71½ �

454 MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 56, No. 2, February 2014 DOI 10.1002/mop

measured scattering coefficients in the working frequency range

is shown in Figure 9. The agreement between project constraints

and results is considered satisfactory.

5. CONCLUSION

In this letter, an synthesis procedure based on space-filling frac-

tal geometries, for the design of multistopbands filters has been

presented. To assess this methodology, a microwave filter based

on Hilbert space-filling curve, capable of eliminating multiple

interference signals, has been designed, fabricated, and experi-

mentally assessed in this letter. The obtained filter shows

reduced dimensions and an acceptable frequency response in all

the considered stop-bands as well as in the working pass-bands.

Numerical simulations have been carried out and compared with

experimental results made on a prototype realized with micro-

strip technology. The results confirm the reliability of the

numerical synthesis, the feasibility, and effectiveness of the

methodology.

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TABLE 3 Comparison of Q-Factors of the R 5 3 Stopbandsfor Numerical and Experimental Results

Stop Band

Q—Factor

Requirements Numerical Results Measured Results

Bstop1 8.0 7.3 6.7

Bstop2 12.5 6.25 14.7

Bstop3 18.0 32.7 18.9

Figure 9 Comparison between simulated and measured jS11j and jS21j.[Color figure can be viewed in the online issue, which is available at

wileyonlinelibrary.com]

DOI 10.1002/mop MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 56, No. 2, February 2014 455