Waveform Optimization in UWB Antenna systemsbased on Prolate Spheroidal Wave Signal Spaces

Pedro Luis Carro, Jesus de Mingo, Paloma Garcia DucarDepartment of Electronic Engineering and Communications

Aragon Institute Engineering Research (I3A)University of ZaragozaZaragoza, 50018 Spain

Email: plcarro@unizar.es, mingo@unizar.es, paloma@unizar.es

Abstract—Antenna waveform distortion evaluation in Ultra-wideband (UWB) communications has become an importantconcern due to the possible degradation in system performance.Fidelity is the typical measure to take into account distortionfor single waveforms. In this paper, a new method designedfor computing approximately the optimum waveform in a two-antenna system, exploiting an important mathematical concept,the orthogonality, is proposed. The method is based on a matrixdecomposition obtained using Prolate-Spheroidal Wave signalspaces and an evaluation of its eigenvalues. It is showed thatthis method allows to improve distortion. Finally, this algorithmis applied in three real links using computed transfer functionswhen two typical radiating structures are used for building aradio link, showing a good improvement in Fidelity values.

Index Terms—UWB, waveform distortion, Fidelity , Prolatespheroidal functions.

I. INTRODUCTION

The development of Ultrawideband (UWB) antennas hasdeeply evolved in the last years. Since 2002, several antennatopologies have been proposed in the literature, so as to fulfillthe UWB regulations. The initial topologies were concernedto show it is possible to achieve large bandwidths by usingdifferent technologies, namely printed antennas. However,researchers go further today, and nowadays, the goal is tooptimize those initial structures in order to improve the perfor-mances of the communication system. Therefore, two differentproblems arise: one is concerned to what and how we shouldmeasure (analysis stage) the antenna performances, and theother is related to the improvement of those structures, takinginto account different Figures of Merit. This paper deals withthe second problem mentioned above, and focuses in signaldistortion minimization.

UWB antennas behave as linear filters that introduce lineardistortion. In order to characterize the pulse-shaping capabil-ity of an antenna, time-domain parameters as Fidelity andCorrelation Energy Patterns are widely used [1-3], as theymeasure distortion in time and space. Distortion measures areusually computed using a single transmitted signal, evaluatingthe received signal, and finally, making a comparison betweenthem. In order to improve the received signal, two differentsolutions have been proposed. The first solution tries to changethe antenna frequency response, leading to optimum radiatingelements. The second keeps the antennas and tries to optimize

the signal so that, for example and typically, Fidelity is themaximum.

As far as distortion minimization related to signal changes isconcerned, D. Pozar has performed an outstanding theoreticalwork [4]. In these studies, he presents waveform optimizationusing variational calculus, worked out in frequency domainand with different constraints, specially focused in signalenergy . Recently [5], he has used orthogonal signal spacesfor dealing with general antennas and optimizing the receivedfield. Regarding experimental and more practical approaches,it is necessary to remark [6], where some antenna optimizationis developed to optimize Fidelity.

In this work, we propose a method for computing wave-forms which optimize Fidelity. This method will be basedon a matrix approximation of the antenna transfer functiondeveloped in a signal space. This proposal constitutes a newway of focusing waveform optimization, which usually reliesin transfer functions either in time or frequency domain.

The paper is organized as follows: section II reviews theantenna system modelling and typical ultrawideband signalspaces, starting from the time-domain effective transfer func-tion. Section III shows the optimization basis from the antennamatrix approximation. The algorithm is presented in the fol-lowing section, and it is applied to some antenna examples.Finally, a conclusion is provided.

II. THEORETICAL MODELLING AND ULTRAWIDEBAND

SIGNAL SPACES

A system of two antennas, one working in transmittingmode, the other working in receiving mode, as Fig. 1, ismodelled using time domain formalism. The effective transferfunction may be expressed as [7]:

S21(ω, θt, φt, θt, φr) =

= 2jωμ

Z0

�Hr(ω, θt, φt) �Ht(ω, , θr, φr)e−jβr

4πr(1)

where β is the propagation constant, and hr, ht correspondto the realized effective length, that takes into account themismatch between generator, receiver load, and antennas.Thus:

�Ht(ω,Ω) =1 − S11(ω)

2

(r̂ × r̂ ×

∫V

�J(�r′)ejβr̂�r′dV ′)

(2)

978-1-4244-1722-3/08/$25.00 ©2008 IEEE. 1

Fig. 1. UWB Transceiver simple model

This transfer function is used for computing the receivedsignals, vr from a transmitted reference vt as [8]:

vr(t) =μ

Z0

14π

∂

∂t

[�hr(θr, φr, t) ∗ �ht(θt, φt, t− td) ∗ vt(t)

](3)

where ∗ denotes a dot product and time-convolution operationand td is the group delay link. The link under test (LUT)transfer function is identified from eq. 3 as:

�HLUT21 =

μ

Z0

14π

∂

∂t

[�hr(θr, φr, t) ∗ �ht(θt, φt, t− td)

](4)

where HLUT21 should be written HLUT

21 (t, θt, φt, θr, θr) ormore compactly:

�HLUT21 (t,Ωt, ,Ωr) =

μ

Z0

14π

∂

∂t

[�hr(Ωr, t) ∗ �ht(Ωt, t− td)

](5)

Once the received signal is computed, Fidelity is definedas:

F (vt(t), vr(t)) = maxτ

∫ ∞−∞ vt(t)vr(t+ τ)dt[∫ ∞

−∞ ‖vt(t)‖2dt∫ ∞−∞ ‖vr(t)‖2dt

]1/2

(6)Therefore, Fidelity, as defined previously, depends on a

referenced transmitted signal, usually chosen to fulfill theFCC mask. Instead of computing Fidelity for those kind ofsignals, we can study the received signals in case of sending aproper orthogonal family. If a signal belonging to this family istransmitted, the corresponding received waveform will spreadinto other “spectral” forms (the other signals belonging tothe family). Therefore, some measure may be computed bymeans of this energy. Typically, two different signal familiesare employed in ultrawideband:

1) The Hermite-Gauss signal set defines a L2(R) vector

metric subspace S [9]:

S =

⎧⎨⎩ψn(t)n=0...∞ � ψn(t) =

Hern(t/σ)e−t2

4σ2√2nn!

√π

⎫⎬⎭(7)

where Hern(t) are the Hermite polynomials, and theinner product is :

< ψi(t), ψj(t) >=∫ ∞

−∞ψi(t)ψj(t)dt = δij (8)

where δij is the kronecker delta.2) The Prolate Spheroidal Wave Functions (PSWF) [10].

They are the solution of the integral equation:∫ T/2

−T/2

ψ(x)sin Ω(t− x)π(t− x)

dx = λψ(t) (9)

or the differential equation:

d

dt(1 − t2)

dψ

dt+ (ξ − ct2)ψ = 0 (10)

where ψ(t) are the PSWF, λ is the fraction of the energyof ψ(t) that lies in the interval [−1, 1] and ξ is theeigenvalue of ψ(t), The constant c = ΩT

2 denotes thenumber of degrees of freedom, and Ω is the bandwidthand T is the timewidth. The double orthogonality isamong their features, since they verify :∫ ∞

−∞ψm(t)ψn(t)dt = δmn, and

∫ T/2

−T/2

ψm(t)ψn(t)dt = λmδmn (11)

Thus, ψn(t) are complete in S : L2(−T/2, T/2) andare easily normalized to form an orthonormal basis. Thisbasis will be used as a transmitted set of waveforms, inorder to study the distortion by a spectral representationof the antenna system.

III. DISTORTION OPTIMIZATION

The vector space may be used for characterizing the antennasystem. The procedure is based on computing for every signalψn(t) ∈ S, the received waveform vn

r (t) and project it ontoS. Let ξn(t) these projected signals and their coefficients αnk,conforming a matrix H. As a consequence, the received signalcoefficients γ = {γ0 . . . γn} corresponding to any transmittedsignal ρ = {ρ0 . . . ρn} may be computed by means of:

γ = Hρ (12)

We assume the signal is well approximated in the basis of S.A measure of similarity (σ(ρ, γ)) between the received signaland the transmitted signal will be mathematically quantifiedby the inner product, such that

σ(ρ, γ) = 〈γ, ρ〉 = γT ρ (13)

where T denotes the transpose operator. By applying eq.12:

σ(ρ, γ) = 〈ρ|H|ρ〉 = ρT Hρ (14)

2

This corresponds to a quadratic form, which may be ex-pressed by means of a symmetric matrix, E, as:

σ(ρ, γ) = 〈ρ|E|ρ〉 (15)

where

E =H + HT

2(16)

This transformation to a symmetric matrix is useful, as itwill be showed later. If the distortion is desired to be mini-mized, the goal will be,in general ,to maximize the similarityfunction. We use the normalized similarity, defined as:

σ(γ, ρ) =〈ρ|E|ρ〉ρT ρ

(17)

Applying the usual derivation rules:

∇σ(γ, ρ) = ∇.( 〈ρ|E|ρ〉

ρT ρ

)=

2E(ρT ρ) − (ρT Eρ)2ρρT ρ

(18)

which implies:

Eρ(ρT ρ) = (ρT Eρ)ρ =⇒ Eρ =(ρT EρρT ρ

)ρ (19)

The term in brackets is a scalar number,

λ =ρT EρρT ρ

(20)

as a result, eq. (19) is a eigenvalue equation, such that:

Eρ = λρ (21)

Therefore, this mathematical approach proves that optimumvalues correspond to the eigenvectors of the matrix E. Asthis matrix is symmetric, the eigenvalues and eigenvectors arereal, which assures the optimized signal is real. In addition,the maximum eigenvalue will be the maximum value of thesimilarity.

According to this mathematical derivation, we can obtainan optimum signal by computing the antenna matrix andperforming a Singular Value Decomposition (SVD) of thecorresponding symmetrized matrix. This is summarized in thenext section.

IV. OPTIMIZATION ALGORITHM

Let us consider a UWB antenna system. For simplicity,we will focus in one link direction, although this algorithmmight be carried out in each space direction. The steps of thealgorithm are:

1) Obtain the transfer functions either by simulations ormeasurements.

2) Evaluate the received waveforms for a specific signalbase. This is made by means of

vnr (t,Ωt,Ωt) =

ψ(t) ∗HLUT21 (t,Ωt,Ωr)∫ ∞

−∞ ‖ψ(t) ∗HLUT21 (t,Ωt,Ωr)‖2dt

(22)

3) Evaluate the projection onto S using the projectionoperator, PS :

ξn(t) = PSvnr =

N−1∑i=0

αinψi(t) (23)

As the selected base is orthonormal, the coefficients inthe projection are easily computed by the inner product,since:

< ψn, ξk >=< ψn,N−1∑i=0

αikψi >=

=N−1∑i=0

αik < ψn, ψi > (24)

< ψn, ψi >= δni −→ αnk =< ψn, ξk > (25)

where ψ(t), ξ(t) are vector representations of input andnormalized output waveforms. This projection must bemade for a proper choice of the time delay, td.

4) Symmetrize the antenna matrix H to compute E.5) Perform a Singular Value Decomposition on the matrix

E. This will lead to a maximum eigenvalue |λopt|, witha eigenvector ρopt.

6) Rebuild the actual waveform using

vt(t) =dimS∑n=0

ρnoptψn(t) (26)

7) Compute the actual received signal vr(t) and evaluatethe distortion (by means of the Fidelity, for example),so as to verify if the algorithm has worked.

This algorithm will be evaluated in a few antenna examplesin the following section.

V. EXAMPLES

In order to show the performances of the measures, theyhave been evaluated in practical UWB antennas. The analyzedradiating structures (sickle and GA-sickle) correspond to theso called “sickle” geometry [11]. The geometry was simulatedusing IE3D from 3.1 GHz to 10.6 GHz, computing the transferfunction in a hypothetical link, as Fig 2. In order to obtain

Fig. 2. One of the Antenna radio-link analyzed

3

easy and cheap-manufacture antennas, the selected substratewas the FR4, with the following electrical parameters: h =1.6mm, εr = 4.55, tanδ = 0.02. The antennas were placed0.1 meter far away.

Three links were studied following the configurations:sickle-to-sickle (SS), sickle-to-GAsickle (SGS) and GAsickle-to-GAsickle (GSGS), showing results in Fig. 2. Regardingthe PSWF, ten(n=2..11) orthogonal normalized functions wereused for approximating the signal space L2(−T/2, T/2),where T was carefully selected so that the energy lied between3.1 GHz and 10.6 GHz, and c = 15.

4 5 6 7 8 9 10−60

−55

−50

−45

−40

−35

−30

−25

−20Antenna Link Transfer Function

Frequency (GHz)

H(d

B)

SSGSGSSGS

4 5 6 7 8 9 10−35

−30

−25

−20

−15

−10

−5

0Antenna Perfomance

Frequency (GHz)

Ret

urn

Loss

es (

dB)

SickleGA−sickle

Fig. 3. Transmit Antenna Transfer Function.

An example of antenna matrix, H, is showed in Fig. 4.This matrix is symmetrized, and a SVD is carried out. Table Ishows the results obtained in the three links for the maximumeigenvalue and eigenvector. These eigenvectors are used forbuilding the optimum transmitted signal . The results of theFidelity values corresponding to the Prolate Spheroidal Wavesignals are showed in table II. As depicted in Fig. 3, the dif-ferences considering the transfer function are not wide enoughto produce a large difference between the links. However, itis clear the GA-sicke-sickle link produces better performancesthan the others. However these values are far from the theoricalmaximum, which is 1. The Fidelity values corresponding to thewaveforms computed (Fig.5-7) by this new method are 0.93,0.92 and 0.894 (Table III). These values show the improvementregarding distortion when the waveforms are optimized.

1

2

3

4

5

6

7

8

9

10

12

34

56

78

910

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

Spectral Mode

rder

Fig. 4. Matrix Representation of Sickle-Sickle link.

TABLE IEIGENVECTORS OF EVALUATED LINKS

Signal Order ρSS ρGSGS ρSGS

2 -0.432 0.45 -0.4443 0 0 04 0.812 -0.8123 -0.81235 0 0 06 -0,1396 0.1058 - 0.11577 0 0 08 -0.342 0,3271 -0.33389 0 0 010 0.1289 0,1397 -0.135111 0 0 0

VI. CONCLUSION

A new optimization method using Prolate-spheroidal wavesignals has been proposed in order to reduce the distortionintroduced by antenas in UWB radiolinks. This method com-putes the optimum waveform corresponding to an antenna-link by means of signal spaces. The method provides valuableimprovement in terms of Fidelity. Finally, three real examplesare analysed using experimental transfer functions measured inan anechoic chamber or obtained, as here, by electromagneticsimulations, showing the feasibility in practical antennas .

VII. ACKNOWLEDGEMENTS

This work has been funded by the Ministry of Educationand Science and the European funds of Regional Develop-ment (FEDER) under the projects TEC 2007-64536/TCM ,TEC 2004-04529/TCM, the Gobierno de Aragon for WALQAtechnology park and the European Union through the ProgramMarco under the project EUWB (Coexisting Short RangeRadio by advanced Ultrawideband Radio Technology).

REFERENCES

[1] D.Lamensdorf and L.Susman, “Baseband pulse antenna techniques”,IEEE Antennas Propag. Mag.,vol. 36,pp.20-30, Feb. 1994

[2] J.S. McLean, H. Foltz, and R. Sutton, “Patternn descriptors for UWB an-tennas,” IEEE Trans. Antennas Propag., vol. 51, pp. 3177-3179, Nov.2003

[3] T. Dissanayake and K.P. Esselle, “Correlation Based Pattern StabilityAnalysis and a Figure of Merit ofr UWB Antennas”, IEEE Trans.Antennas Propag., vol. 54, pp. 3184-3191, Nov.2006

[4] D.M. Pozar, “Waveform optimization for ultra wideband radio systems”,IEEE Trans. Antennas Propag., vol. 51, pp. 2335-2345, Sep.2003

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TABLE IIFIDELITY ANALYSIS OF PSWF SETS IN ANTENNA LINKS

Signal Order FSS FGSGS FSGS

2 0,4195 0,4173 0,42593 0,6382 0,6376 0.65714 0,7404 0,0.731 0.7745 0,7319 0,7399 0.77086 0,7066 0,7146 0.77387 0,6611 0,6589 0.76788 0,6834 0,7031 0.70289 0,6479 0,6585 0.638410 0,6576 0,7161 0.635611 0,6879 0,7472 0.6946

TABLE IIIOPTIMIZATION RESULTS

Link F λ = σSS 0,9354 0,6784GSGS 0.9222 0,5829SGS 0.8939 0,6238

[5] D.M. Pozar “Optimal Radiated Waveforms From an Arbitrary UWBAntenna”, IEEE transactions on antennas and propagation, VOL. 55,NO. 12, DEC. 2007

[6] N. Telzhensky and Y. Leviatan, “Novel Method of UWB antennaOptimization for Specified Input Signal Forms by Means of GeneticAlgorithm”, IEEE Trans. Antennas Propag., vol. 54, pp. 2216-2225,Aug.2006

[7] D-H. Kwon, “Effect of Antenna Gain and Group Delay Variations onPulse-Preserving Capabilities of Ultrawideband Antennas”, IEEE Trans.Antennas Propag., vol. 54, pp. 2208-2215, Nov.2003

[8] S. Licul and W.A. Davis, “Unified Frequency and Time-Domain AntennaModeling and Characterization”, IEEE Trans. Antennas Propag., vol. 53,pp. 2882-2888, Sep.2003

[9] C. Abou-Rjeily, N. Daniele and J.C Belfiore, “MIMO UWB Communi-cations using Modified Hermite Pulses”, IEEE Int. Symp. on Personal,Indoor and Mobile Radio Communications (PIMRC’06), Sep.2006

[10] Dilmaghani S.R. ; Ghavami M; Allen B. Aghvami Hamid; “NovelUWB pulse shaping using prolate spheroidal wave functions”,14th IEEEInternational Symposium on Personal, Indoor, and Mobile Radio Com-munication, 2003

[11] Chang H.C.; Yeung H.S.; Chang S.W.; Man F.K.; ”UWB sickle-shapepatch dipolar antenna with stable radiation pattern” , Antennas andPropagation Society International Symposium 2007, IEEE, June 2007

−2 −1.5 −1 −0.5 0 0.5 1 1.5 2−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2

Time(ns)

Optimized Signal Sickle Sickle

OptimumReceived

−2 −1.5 −1 −0.5 0 0.5 1 1.5 2−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2

Time(ns)

Optimized Signal GA−Sickle GA−Sickle

OptimumReceived

−2 −1.5 −1 −0.5 0 0.5 1 1.5 2−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2

Time(ns)

Optimized Signal Sickle GA−Sickle

OptimumReceived

Fig. 5. Computed Optimum Signals in the three antenna-links

5