An expression is derived for the maximum proportion of remaining samples in the sample selection technique for mode-stirred chamber measurements. The limit is independent of the selection algorithm used. The limit is verified in simulations as well as measurement examples.This work was supported in part by the Spanish National R&D Programme
through TEC2008-05811, and by Fundacio´n Se´neca, the
R&D coordinating agency for the Region of Murcia (Spain) under
the 11783/PI/09 and 11610/IV2/09 project

Hallbjörner, Paul

Reyes Davó, Elías de los

Sánchez Heredia, Juan Diego

Sánchez Hernández, David Agapito

Microwave and Optical Technology Letters

English

Repositorio Digital de la Universidad Politécnica de Cartagena

LIMIT FOR THE PROPORTION OFREMAINING SAMPLES IN THE MODE-STIRRED CHAMBER SAMPLESELECTION TECHNIQUEPaul Hallbjörner,1 Juan D. Sánchez-Heredia,2Elias de los Reyes,3 and David A. Sánchez-Hernández41 SP Technical Research Institute of Sweden, Box 857, SE-501 15Borås, Sweden; Corresponding author: paul.hallbjorner@sp.se2 EMITE Ing, Edificio CEEIM, Campus Espinardo, E-30100 Murcia,Spain3Departamento de Comunicaciones, Universidad Politécnica deValencia, Camino de Vera, s/n, E-46022 Valencia, Spain4Departamento de Tecnologı́as de la Información yComunicaciones, Universidad Politécnica de Cartagena, Plaza delHospital, 1, E-30202 Cartagena, SpainReceived 25 January 2011ABSTRACT: An expression is derived for the maximum proportion ofremaining samples in the sample selection technique for mode-stirredchamber measurements. The limit is independent of the selectionalgorithm used. The limit is verified in simulations as well asmeasurement examples. VC 2011 Wiley Periodicals, Inc. Microwave OptTechnol Lett 53:2608–2610, 2011; View this article online atwileyonlinelibrary.com. DOI 10.1002/mop.26336Key words: mode-stirred chamber; reverberation chamber; sampleselection; measurement technique1. INTRODUCTIONMode-stirred chamber (MSC) for antenna measurements is awell established technique, presented in several publications [1–4]. Recently, the sample selection technique [5–7] has beenintroduced for the purpose of improving the versatility of MSCmeasurements. In the sample selection technique, some measure-ment samples are selected while the rest are discarded, so thatthe selected sample set conforms to a predetermined probabilitydensity function (PDF). When applying the sample selectiontechnique, the number of samples in a sequence decreases, andsometimes quite drastically. The number of remaining samplesis one of the most critical parameters, and it should be as highas possible to achieve a good accuracy in the results. This articleinvestigates the limit for the number of remaining samples inthe sample selection technique.2. PROPORTION OF REMAINING SAMPLESThe sample selection process will select a certain proportion ofthe initial (input) samples, that is, the number of remaining sam-ples is linearly dependent on the number of initial samples. Inthe following, the proportion of remaining samples is studied, tomake the results more general and applicable to initial samplesets of any length.A function finitial(x) is used to describe the PDF of the initialdata, and the target PDF after sample selection is defined by ftar-get(x). Vector notation is used for the argument because multi-variate distributions can be used, for instance for complex-valued amplitudes, or in measurements of multiple antennas.These functions have the usual properties of PDFs, that is, theyare always positive and the integral over all arguments is unity.The total number of samples before and after sample selectionis denoted as Ninitial and Ntarget, respectively. The number ofsamples within a small interval Dx at any x value isNinitialfinitial(x)Dx and Ntargetftarget(x)Dx, respectively. Naturally,the number of samples at any x value can only remain the sameor decrease as a result of the sample selection process. Conse-quently we haveNtargetftargetðxÞ  NinitialfinitialðxÞ; (1)where Dx is eliminated on both sides. As ftarget(x) is differentfrom finitial(x) (the case when they are the same is trivial and notuseful), there will be x values at which ftarget(x) > finitial(x). Tofulfill Eq. (1), Ntarget then has to be smaller than Ninitial. It is themaximum ratio of ftarget(x)/finitial(x) for any x that dictates thenecessary ratio of Ntarget/Ninitial for the condition in Eq. (1) to besatisfied. Defining the proportion of remaining samples q asq ¼ NtargetNinitial(2)and substituting this in Eq. (1) gives the conditionq  finitialðxÞftargetðxÞ : (3)In a specific case, there can be some freedom in choosingftarget(x), because PDFs often have one or several parametersand the requirements on ftarget(x) might not decide all of them.Remaining parameters should then preferably be set so that q isoptimized. The maximum value of q is thusqmax ¼ min finitialðxÞftargetðxÞ (4)for all x, and for all ftarget(x) under the given constraints.Different algorithms exist for the process of selecting sam-ples [7]. The different algorithms have been reported to result indifferent proportions of remaining samples. The limit in Eq. (4)is arrived at without any assumptions regarding the algorithmused for selecting samples, so it is valid for any algorithm.3. APPLICATION EXAMPLESTwo examples are provided in which the presented limit is used.In the first, it is showed that the so-called single-step algorithm[7] reaches qmax. In the second, an expression for qmax isderived for a case of commonly used PDFs.3.1 Single-Step AlgorithmThe single-step algorithm works as follows: to decide whether asample should be kept or not, it uses a threshold function D(x),according toDðxÞ ¼ C ftargetðxÞfinitialðxÞ ; (5)where C is a constant. For each input sample at a value xk, arandom number, uniformly distributed in the interval [0,max(D(x))), is generated. If the random number is less thanD(xk), the sample is kept. If it is greater than or equal to D(xk),the sample is discarded.The proportion of remaining samples after all input samplesare covered is given by the multiple integral over the n-dimen-sional space of non-negative real numbers,q ¼Zx2<nþDðxÞfinitialðxÞdx: (6)2608 MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 53, No. 11, November 2011 DOI 10.1002/mopBy substituting Eq. (5) into Eq. (6), and using the fact thatthe integral of a PDF is unity, it is found thatq ¼ C: (7)By letting C ¼ qmax, the single-step algorithm is optimal interms of the proportion of remaining samples.Because of the randomness by which the decision of select-ing/discarding samples is taken, different realizations will resultin different sets of selected samples, and also different numbersof remaining samples. The proportion of remaining samples inEqs. (4) and (6) is exact for infinite length sequences, and accu-rate in the mean for finite length sequences.3.2 From Rayleigh to RiceAn expression for qmax is derived for the case of having an ini-tial data set with Rayleigh distributed amplitude, and a targetPDF which the Rice distribution. In this example, complex-val-ued data samples, that is, amplitude and phase, are assumed.Each complex-valued data sample z can be viewed as a two-dimensional scalar sample (x,y), related asz ¼ xþ jy: (8)The bivariate normal distribution PDF [8] can be used forthe initial data,finitialðx; yÞ ¼ 12ps2iexp  x2 þ y22s2i : (9)By introducing an offset, it can represent Rice distributedscalar amplitudes, and it can therefore be used also for the targetdata. As phases are relative, and a constant phase offset appliedto all data samples will not change the scalar amplitudes, theoffset can for simplicity be chosen to be real valued. With theoffset at,ftargetðx; yÞ ¼ 12ps2texp ðx atÞ2 þ y22s2t !: (10)Substituting Eqs. (9) and (10) into Eq. (4) results in theexpressionqmax ¼ min s2ts2iexpðx atÞ2 þ y22s2t x2 þ y22s2i ! !: (11)To find the minimum, the derivatives of the argument withrespect to x and y are calculated and set to zero. With the opti-mal values of x and y,qmax ¼ s2ts2iexp  12a2ts2i  s2t  : (12)After this, the values of at and st that maximize qmax have tobe chosen. Assume that a certain K-factor is desired in the targetdata, defined asK ¼ a2t2s2t: (13)Using Eq. (13), Eq. (12) can be rewritten, derived withrespect to st, and set to zero to find the optimal value of st.The result isst ¼ siffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ K2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ K2 21svuut: (14)Substituting this expression into Eq. (12) givesqmax ¼ 1þK2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þK2 21s0@1Aexp Kþ KK2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þK2 21q0B@1CA:(15)4. VERIFICATIONNumerical simulations in Matlab are used to verify Eq. (15).Random samples with Rayleigh distributed amplitude and uni-form phase are generated. A sequence of 20,000 samples isFigure 1 Proportion of remaining samples with the single-step algorithm(circles) and upper limit (solid line), when converting Rayleigh to Rice dis-tribution using sample selection with simulated complex-valued dataFigure 2 Proportion of remaining samples with the single-stepalgorithm (circles) and upper limit (solid line), when convertingRayleigh to Rice distribution using sample selection with measuredcomplex-valued dataDOI 10.1002/mop MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 53, No. 11, November 2011 2609generated, which is enough to make the statistical fluctuationsdue to the randomness of the single-step algorithm negligible.Sample selection is then applied, with the single-step algorithm,for a number of desired K-factors between 0.001 and 100. Fig-ure 1 shows the proportion of remaining samples from the simu-lations, together with the limit according to Eq. (15).Measured samples are also used for verification. The single-step algorithm is here applied to a sequence of 6400 samples,taken in a large MSC with mechanical mode stirring, at 2.6GHz. The test object is a single-element antenna with omnidir-ectional coverage. There is very little unstirred power in thechamber, making the raw data samples Rayleigh distributed.These results, together with Eq. (15), are shown in Figure 2.5. CONCLUSIONSAn expression is derived for the upper limit of the proportion ofremaining samples in the sample selection technique. The limitis independent of the algorithm used for the selection, anddepends only on the PDFs of the initial and target data sets.Examples of how the limit can be used are given. The limit isverified experimentally in both simulations and measurementsfor the case of performing sample selection to achieve a Ricedistributed data set from Rayleigh distributed initial data.Knowledge of the limit in practical situations will help estimat-ing measurement uncertainty as well as enable assessment of theefficiency of sample selection algorithms.ACKNOWLEDGMENTSThis work was supported in part by the Spanish National R&D Pro-gramme through TEC2008-05811, and by Fundación Séneca, theR&D coordinating agency for the Region of Murcia (Spain) underthe 11783/PI/09 and 11610/IV2/09 projects.REFERENCES1. P. Corona, G. Latmiral, E. Paolini, and L. Piccioli, Use of a rever-berating enclosure for measurements of radiated power in themicrowave range, IEEE Trans Electromagn Compat EMC-18(1976), 54–59.2. T. Maeda and T. Morooka, Radiation efficiency measurementmethod for electrically small antennas using radio wave scatterers,IEEE AP-S Int Symp 1 (1988), 324–327.3. J. Krogerus, K. Kiesi, and V. Santomaa, Evaluation of three meth-ods for measuring total radiated power of handset antennas, In:IEEE Instrumentation and Measurement Technology Conference,Budapest, Hungary, May 21–23, 2001, pp. 1005–1010.4. J.F. Valenzuela-Valdés, A.M. Martı́nez-González, and D.A. Sán-chez-Hernández, Emulation of MIMO nonisotropic fading environ-ments with reverberation chambers, IEEE Antennas WirelessPropag Lett 7 (2008), 325–328.5. J.D. Sánchez-Heredia, M. Grudén, J.F. Valenzuela-Valdés, andD.A. Sánchez-Hernández, Sample-selection method for arbitraryfading emulation using mode-stirred chambers, IEEE AntennasWireless Propag Lett 9 (2010), 409–412.6. J.D. Sanchez-Heredia, M.A. Garcia-Fernandez, M. Grudén, P.Hallbjörner, A. Rydberg, and D.A. Sánchez-Hernández, Arbitraryfading emulation using mode-stirred chambers, EuCAP, in press.7. M. Grudén, J.D. Sánchez-Heredia, P. Hallbjörner, J.F. Valenzuela-Valdés, D.A. Sánchez-Hernández, and A. Rydberg, Sample selectionalgorithms for emulation of arbitrary fading distributions using mode-stirred chambers, IEEE Trans Electromagn Compat, submitted.8. L. Råde and B. Westergren, Beta mathematics handbook for sci-ence and engineering, 3rd ed., Studentlitteratur, Lund, Sweden, p.413, ISBN 91–44-25053–3.VC 2011 Wiley Periodicals, Inc.INDUCED ABERRATIONS BYCOMBINATIVE CONVEX/CONCAVEINTERFACES OF REFRACTIVE-INDEX-MISMATCH AND CAPABILITY OFADAPTIVE OPTICS CORRECTIONYiin Kuen Fuh,1,2 Kuo Chan Hsu,1 Jia Ren Fan,1 andMing Xien Lin11 Institute of Opto-Mechatronics Engineering, National CentralUniversity, No. 300, Jhongda Rd., Jhongli City, Taoyuan County32001, Taiwan; Corresponding author: mikefuh@cc.ncu.edu.tw2Department of Mechanical Engineering, National CentralUniversity, No. 300, Jhongda Rd., Jhongli City, Taoyuan County32001, TaiwanReceived 30 January 2011ABSTRACT: Adjustable fluidic adaptive lenses have been extensivelystudied because of the advantage of changing the geometric shape orrefractive index without any mechanical moving parts. However, theinduced aberrations were rarely discussed before. We address this issueby first clarifying the aberrations experimentally due to injected fluidicvolumes. Experimental results show that under the injected fluidicvolume of 0.1 ml, the main aberrations come from Z1 (piston), Z2 (tip),and Z5 (defocus) and Zernike coefficients are 0.97, 0.31, and 1.31 lm,respectively. In what follows, a series of tests specifically designed toexplore the convex/concave interfaces and refractive-index-mismatch(RIM). Furthermore, we demonstrate the capability of adaptive optics(AO) correction on aberrations induced by combinative effects ofmultiple layers with convex/concave interfaces and RIM. A microelectro-mechanical system (MEMS) deformable mirror (DM) with 140 actuatorswas used in conjunction with Shack–Hartmann wavefront sensor torealize the experimentation. In particular, we consider the aberrationintroduced by interfaces of RIM between water/oil and glass. After AOcorrection, we can improve wavefront with root mean square (RMS) of2.17–0.17 lm for an interface between water and glass. As for theinterface between oil and glass, we are capable of improving RMS of0.24–0.10 lm. VC 2011 Wiley Periodicals, Inc. Microwave Opt TechnolLett 53:2610–2615, 2011; View this article online atwileyonlinelibrary.com. DOI 10.1002/mop.26323Key words: adaptive optics; refractive-index-mismatch (RIM);deformable mirror (DM); aberrations; Shack–Hartmann wavefrontsensor1. INTRODUCTIONAdaptive optics (AO) is a widely used concept and technologyin astronomy to compensate the atmospheric turbulence thatconfines the performance of ground-based telescope [1–3]. Inaddition, the biological potentials such as correction of the ocu-lar aberration for the improved in vivo imaging of the retina invision science and ophthalmic applications are of recent thrustareas in research [4–9]. In microscopy, AO has also beenapplied to nonlinear microscopies such as multiphoton micros-copy to image several layers of mismatched and inhomogeneousrefractive index [10]. Moreover, the well-pursuit utilization ofhigh numerical aperture (NA) objectives immersed in a fluidthat is refractive index mismatched will inevitably induce thespherical aberration in the longitudinal direction [11]. AO-assisted imaging system shows great capability of compensatingspherical aberration at a certain depth in a specimen with refrac-tive-index-mismatch (RIM) [12, 13].Adjustable fluidic adaptive lenses have been studied recentlywith one significant advantage such that focal length can beeasily altered by changing the geometric shape or refractive2610 MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 53, No. 11, November 2011 DOI 10.1002/mop

Limit for the proportion of remaining samples in the modestirred chamber sample selection technique

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