Convention Paper 6764


 Meredith Barton
 3 years ago
 Views:
Transcription
1 Audio Egieerig Society Covetio Paper 6764 Preseted at the 10th Covetio 006 May 0 3 Paris, Frace This covetio paper has bee reproduced from the author's advace mauscript, without editig, correctios, or cosideratio by the Review Board. The AES takes o resposibility for the cotets. Additioal papers may be obtaied by sedig request ad remittace to Audio Egieerig Society, 60 East 4 d Street, New York, New York , USA; also see All rights reserved. Reproductio of this paper, or ay portio thereof, is ot permitted without direct permissio from the Joural of the Audio Egieerig Society. Optimisatio of Cocetred Rigid ad Ope Spherical Microphoe Arrays Abhaya Parthy 1, Craig Ji, ad Adré va Schaik 1 School of Iformatio Techology, The Uiversity of Sydey, NSW, 006, Australia School of Electrical ad Iformatio Egieerig, The Uiversity of Sydey, NSW, 006, Australia {craig, ABSTRACT We preset a ovel microphoe array that cosists of a ope spherical array with a smaller rigid spherical array at its cetre. The distributio of microphoes, which results i the array havig the largest frequecy rage, for a give beamformig order, was obtaied by aalysig microphoe errors. For a fixed umber of microphoes, the results for several examples idicate that the maximum frequecy rage is obtaied whe the microphoes are relatively evely distributed betwee the ope ad rigid spheres. 1. INTRODUCTION May spherical microphoe array cofiguratios, such as that preseted by Meyer ad Elko [1] ad Abhayapala ad Ward [], have a limited useable frequecy rage, typically 34 octaves depedig o the umber of microphoes used. This frequecy rage limitatio is due to spatial aliasig at the high frequecies ad microphoe positioig errors at the low frequecies. For a fixed umber of microphoes o a sphere, spatial aliasig ca be reduced by reducig the spacig betwee microphoes ad by reducig the radius of the sphere. The trade off i reducig the radius of the sphere, however, is that the microphoe positioig error icreases due to the smaller size of the sphere. The spatial aliasig error ad the microphoe positioig error for a give arragemet of microphoes o a spherical microphoe array is depedat o the dimesioless parameter kr, where k is the wave umber ad r is the radius of the sphere. Utilisig multiple arrays of differig radii is a techique which allows a larger frequecy rage to be covered. Gover [3] uses two ope spherical microphoe arrays to capture a larger frequecy rage, a smaller spherical microphoe array for capturig high frequecies ad a larger spherical microphoe array for capturig low frequecies. Multiple ope spherical microphoe arrays ca be cetred at the same poit allowig the soud
2 Broadbad Spherical Microphoe Arrays field to be recorded at oe locatio, however, ope spherical microphoe arrays are disadvataged i that their error rises dramatically for certai values of kr. Rigid spherical microphoe arrays do ot have this problem [4]. It is preferable to use rigid spherical microphoe arrays whe recordig a soud field for this reaso. However, multiple rigid spherical microphoe arrays ca ot be cetred at the same poit, ad usig multiple rigid spherical microphoe arrays for recordig a soud field at locatios close to each other is ot practical, as the rigid spheres will scatter soud ad affect the soud field beig recorded by the other rigid spherical microphoe arrays. We preset a spherical microphoe array cofiguratio, which we have ot see reported previously i the literature, with microphoes mouted o both a ope ad rigid sphere with a commo cetre. The smaller rigid spherical microphoe array is used for recordig high frequecies, while the larger ope spherical microphoe array is used for recordig low frequecies. The soud scattered by the cetral rigid sphere ca be calculated aalytically at ay poit surroudig the sphere, ad thus the soud field at the surface of the ope sphere is kow [1]. This cofiguratio allows soud field recordig at oe locatio while retaiig the advatages of the rigid spherical microphoe array. Buildig a spherical microphoe array with this cofiguratio, usig a fixed umber of microphoes, requires that a umber of microphoes be placed o the rigid spherical microphoe array ad the remaiig microphoes be placed o the ope spherical microphoe array. I additio, the frequecy rages that the two spherical microphoe arrays cover must overlap. We preset a optimisatio algorithm that calculates the umber of microphoes that should be placed o the rigid ad ope spherical microphoe arrays to maximise the frequecy rage of the combied arrays for a give spherical harmoic order of the beamformer.. METHODS The optimisatio program was writte usig the MATLAB software eviromet. The useable kr rage is defied as the kr rage for the spherical microphoe array for which the microphoe positioig error ad the spatial aliasig error remai below a fixed value. The iputs for the optimisatio algorithm are the total umber of microphoes, the maximum tolerable sigal error, due to microphoe positioig error ad spatial aliasig error, i the spherical microphoe array, expressed as a oisetosigal ratio, the rage for the uiform distributio of the radom microphoe placemet error for the rigid ad ope spherical microphoe arrays, ad the spherical harmoic order of the beamformer. The optimisatio algorithm calculates the umber of microphoes that should be placed o the rigid ad ope spheres, the ratio of the ope sphere radius to the rigid sphere radius, ad the useable kr rage for the ope array ad the useable kr rage for the rigid array. Several assumptios were made i the desig of the optimisatio algorithm ad are detailed i the followig paragraphs. Firstly, we assume that the spherical harmoic order for the beamformer remais costat to esure reasoably costat gai across the spherical microphoe array s useable kr rage. The gai of the microphoe array is related to the directivity idex which is defied as the peaktoaverage ratio of the beam patter expressed i decibels [3]. For spherical microphoe arrays processed usig spherical harmoics, the directivity idex is related to the spherical harmoic order of the beamformer ad icreases as the spherical harmoic order of the beamformer is icreased. The directivity idex remais relatively costat for a large rage of kr values. By oly beamformig at oe order o both arrays, the directivity idex will remai approximately costat across the etire useable frequecy rage. A secod assumptio is that the microphoes will be arraged o the ope ad rigid spheres with a early uiform spatial samplig scheme []. Nearly uiform spatial samplig schemes have bee show to be the most efficiet i terms of the umber of microphoes required [4]. I additio, oly spatial samplig schemes that satisfy the discrete spherical harmoic orthoormality criterio (see [4]) are used: m m α Y ( Ω ) Y ( Ω ) = δ δmm + ε mm, (1) AES 10th Covetio, Paris, Frace, 006 May 0 3 Page of 6
3 Broadbad Spherical Microphoe Arrays where Ω = ( θ, ϕ ) are the sample positios o a uit sphere i spherical coordiates, α are the weights for those sample positios, Y m is the spherical harmoic fuctio of order ad mode m, δ is the Kroecker delta fuctio, deotes complex cougatio, ad ε mm deotes the error i the sum for,, mm,. Oly spatial samplig positio lists which satisfy the spherical harmoic orthoormality criterio with 6 ε mm 3 10 for all,, mm, such that, N, where N is the spherical harmoic order of the beamformer, are used for arragig the microphoes o the spherical microphoe arrays. Several spatial samplig positio lists exist that do ot satisfy the orthoormality criterio (1) at a specified order, although there are other samplig positio lists, with fewer positios, that do satisfy the criterio at the same order. Spatial samplig positio lists that do ot satisfy the criterio are replaced with a spatial samplig positio list, with a lower umber of positios, which does satisfy the criterio. For example, at 4th order, we have foud spatial samplig positio lists with 37, 38, 39 ad 41 positios that do ot satisfy the orthoormality criterio, but a spatial samplig positio list with 36 positios that does satisfy the criterio. Fially, we also assume that measuremet oise is idepedet of the cofiguratio of the spherical microphoe array ad do ot iclude it i our sigal error calculatios..1. Optimisatio Algorithm The optimisatio algorithm begis with all microphoes cosidered to be o the rigid spherical microphoe array. For each iteratio of the algorithm, the umber of microphoes o the rigid spherical microphoe array decreases by oe ad the umber of microphoes o the ope spherical microphoe array icreases by oe. This iteratio cotiues util all microphoes are o the ope spherical microphoe array. For each iteratio, the sum of the microphoe positioig error ad the spatial aliasig error, herei referred to as PA error, for both the ope ad rigid spherical microphoe arrays is computed. The PA error is calculated assumig a sigle farfield, plaewave source ad that the beamformer is steered i the directio of the icomig plaewave. The spatial aliasig error, E a, is due to spatial samplig of the soud field o the surface of the spherical microphoe array. Spatial samplig limits the order to which a soud field ca be decomposed o the surface of the sphere ad the spherical harmoic decompositio of the soud field is trucated. The spatial aliasig error [4] is defied as E a = N = 0 = N+ 1 M = 1 b b 4π 4π α P (cos Θ ) P (cos Θ ) y s, () where N is the beamformig order, M is the umber of microphoes, Θ is the agle betwee the icomig plae wave ad the samplig positio Ω, s y is the 4 sigal power, ( N + 1) (4 π ), ad b is defied as where, ( ka) h' ( ka) 4 πi ( ( kr) h ( kr)), (3) h are the spherical Bessel ad Hakel fuctios respectively,, h are their derivatives, i = 1, ad a r is the radius of the cetral rigid sphere. The spatial aliasig error is depedat o the beamformig directio, thus it is calculated for 6 icomig plaewave directios, distributed aroud the sphere as i [6], ad the averaged. It was foud, empirically that after averagig across 6 icomig wave directios the spatial aliasig error varied isigificatly as more icomig wave directios were added ad averaged. The microphoe positioig error, E Ω, is due to errors i the placemet of the microphoes o the spherical microphoe array. The microphoe placemet error, Δ, is the deviatio from the ideal microphoe positio, Ω, to the positio, Ω, give by θ = θ +Δ ad ϕ = ϕ +Δ siθ. (4) The microphoe placemet error, Δ, is assumed to be uiformly distributed such that Δ 0.00 radias. This rage of microphoe placemet error seemed reasoable give the size of the spherical microphoe array. Microphoe positioig error [4] is defied as AES 10th Covetio, Paris, Frace, 006 May 0 3 Page 3 of 6
4 Broadbad Spherical Microphoe Arrays E Ω where = N = 0 = 0 M = 1 b b 4π 4π α P (cos Θ )[ P (cos Θ ) P (cos Θ )] y s,() Θ is the agle betwee the specified microphoe positio ad the icomig wave, ad Θ is the agle betwee the actual microphoe positio ad the icomig wave. The microphoe positioig error is depedat o the directio of beamformig, so a average error is calculated across the sphere for a umber of beamformig directios. The positioig error for the spherical microphoe array is calculated for 100 realisatios of the radom microphoe placemet error for each of 11 icomig plaewave directios, distributed aroud the sphere as i [6], ad the averaged. It was foud empirically that after averagig 100 realisatios of the microphoe placemet error for each of 11 icomig wave directios the error varied isigificatly as more realisatios of the placemet error ad icomig wave directios were added ad averaged. For each iteratio, the PA error with the specified maximum tolerable oisetosigal ratio is used to calculate: firstly, the useable kr rage for the rigid spherical microphoe array, secodly, the largest ratio of the radius of the ope sphere to the radius of the rigid sphere, for which the PA error is less tha the maximum oisetosigal ratio ad such that the highest kr value for the ope spherical microphoe array is idetical to the lowest kr value for the rigid spherical microphoe array, fially, the useable kr rage for the ope spherical microphoe array. It should be oted that the useable kr rage for the ope spherical microphoe array is computed so that the largest value of kr lies before the first local maximum, i the alias error curve, which is greater tha the specified oisetosigal ratio. As show i Fig. 1, the spatial aliasig error curve for the ope sphere cotais umerous rages of kr for which the error rises dramatically. NoisetoSigal Ratio (db) kr Figure 1: Spatial aliasig error curve is show averaged over 6 icomig wave directios, for ope spherical array with 6 microphoes, havig a radius 7 times greater tha the rigid sphere located at its cetre. 3. RESULTS The optimisatio algorithm was executed to fid the optimal distributio for rigidope spherical microphoe array cofiguratios with 96, 64, 3 ad 4 microphoes. All spherical microphoe array cofiguratios were desiged to have a maximum error oisetosigal The first spherical microphoe array cofiguratio cosists of 96 microphoes operatig at 4th order. A miimum of 36 microphoes are required to satisfy the spherical harmoic orthoormality criterio for this cofiguratio. Thus, whe the rigid or ope spherical microphoe array cotais less tha 36 microphoes, that spherical microphoe array caot be used for beamformig. Referrig to Fig., the frequecy (i.e., kr) rage, for this cofiguratio, is at a maximum of.06 octaves whe 46 microphoes are placed o the ope spherical microphoe array ad 0 microphoes are placed o the rigid spherical microphoe array. The ratio of the ope sphere radius to the rigid sphere radius, with this distributio of microphoes, is.66. The frequecy rage curve show i Fig. is ot smooth whe plotted agaist the umber of AES 10th Covetio, Paris, Frace, 006 May 0 3 Page 4 of 6
5 Broadbad Spherical Microphoe Arrays Frequecy Rage (Octaves) Frequecy Rage (Octaves) Number of Microphoes o Rigid Sphere Figure : The frequecy rage is show for a microphoe array cofiguratio with 96 microphoes ad workig to 4th order with maximum oisetosigal microphoes. This is caused by the spatial aliasig error which is highly oliear across kr ad chages i a oliear fashio as the umber of microphoes are icreased or reduced. The frequecy rage curves show i Figs. 3, 4 ad are ot smooth for the same reaso. The secod spherical microphoe array cofiguratio cosists of 64 microphoes operatig at 3rd order. A miimum of 6 microphoes is required to satisfy the spherical harmoic orthoormality criterio for this cofiguratio. The frequecy rage for this cofiguratio is at a maximum of 6.3 octaves whe 3 microphoes are placed o the ope spherical microphoe array ad 3 microphoes are placed o the rigid spherical microphoe array. The ratio of the ope sphere radius to the rigid sphere radius, with this distributio of microphoes, is 7.3. The third spherical microphoe array cofiguratio cosists of 3 microphoes operatig at d order. A miimum of 1 microphoes is required to satisfy the spherical harmoic orthoormality criterio for this cofiguratio. The frequecy rage, for this cofiguratio, is at a maximum of octaves whe 16 microphoes are placed o the ope spherical microphoe array ad 16 microphoes are placed o the rigid spherical microphoe array. The ratio of the ope sphere radius to the rigid sphere radius, with this distributio of microphoes, is Number of Microphoes o Rigid Sphere Figure 3: The frequecy rage is show for a microphoe array cofiguratio with 64 microphoes, ad workig to 3rd order with maximum oisetosigal Frequecy Rage (Octaves) Number of Microphoes o Rigid Sphere Figure 4: The frequecy rage is show for a microphoe array cofiguratio with 3 microphoes ad workig to d order with maximum oisetosigal The fourth spherical microphoe array cofiguratio cosists of 4 microphoes operatig at d order. A miimum of 1 microphoes is required to satisfy the spherical harmoic orthoormality criterio for this cofiguratio. The frequecy rage, for this cofiguratio, is at a maximum of 9.90 octaves whe 1 microphoes are placed o the ope spherical microphoe array ad 1 microphoes are placed o the rigid spherical microphoe array. The ratio of the ope sphere radius to the rigid sphere radius, with this distributio of microphoes, is 3.7. AES 10th Covetio, Paris, Frace, 006 May 0 3 Page of 6
6 Broadbad Spherical Microphoe Arrays Frequecy Rage (Octaves) Number of Microphoes o Rigid Sphere Figure : The frequecy rage is show for a microphoe array cofiguratio with 4 microphoes ad workig to d order with maximum oisetosigal 3.1. Example Microphoe Array A example spherical microphoe array is discussed to illustrate the practical cosideratios that are required whe costructig a spherical microphoe array. The example spherical microphoe array cofiguratio has 3 microphoes o the ope sphere ad 3 microphoes o the rigid sphere, ad has the highest frequecy rage of all the cofiguratios with 64 microphoes. The kr rage for the rigid microphoe array is , ad the kr rage for the ope microphoe array is First of all, the radius is related to f the frequecy by krc r = π f, (6) where c is the speed of soud. Thus, if we would like the array to work to a maximum frequecy of 14.0 khz, the radius of the rigid sphere would have to be 1.87 cm ad the radius of the ope sphere would the be 14.1cm. With this radius, the ope spherical microphoe array ca work to a low frequecy limit of 17 Hz. However, whe buildig the rigid microphoe array usig DPA type 4060BM microphoes, which have a diameter of 0.4 mm ad a height 1.7 mm, it is ot be possible to place 3 microphoes o a sphere of radius 1.87 cm ad the radius of the rigid sphere has to be icreased to accommodate 3 microphoes. We fid that a sphere with a radius of at least.04 cm has to be used to accommodate 3 microphoes. With this radius for the rigid sphere, the high frequecy limit of the rigid spherical microphoe array becomes 1.8 khz. The radius of the ope sphere the becomes 1.4 cm, ad the low frequecy limit of the ope spherical microphoe array is 160 Hz. 4. CONCLUSION From the results preseted above, it is evidet that the largest useable frequecy rage for a cocetric rigid/ope spherical microphoe array beamformer that operates to a costat order is achieved whe microphoes are placed both o the rigid sphere ad the ope sphere. The results idicate that a relatively eve distributio of microphoes, betwee the ope ad rigid spheres, produces the highest frequecy rage.. REFERENCES [1] J. Meyer ad G. W. Elko, A highly scalable spherical microphoe array based o a orthoormal decompositio of the soudfield, i Proc. ICASSP, vol. II, 00, pp [] T. D. Abhayapala ad D. B. Ward, Theory ad desig of high order soud field microphoes usig spherical microphoe array, i Proc. ICASSP, vol. II, 00, pp [3] B. N. Gover, J. G. Rya, ad M. R. Stiso, Microphoe array measuremet system for aalysis of directioal ad spatial variatios of soud fields, J. Acoust. Soc. Amer., vol. 11, o., pp , 00. [4] B. Rafaely, Aalysis ad desig of spherical microphoe arrays, IEEE Tras. o Speech ad Audio Processig, vol. 13, o. 1, pp , 00. [] R. H. Hardi ad N. J. A. Sloae, McLare s improved sub cube ad other ew spherical desigs i three dimesios, Discrete Computatioal Geometry, vol. 1, pp , 199. [6] J. Fliege ad U. Maier, A twostage approach for computig cubature formulae for the sphere, Ergebisberichte Agewadte Mathematik, No. 139T. Fachbereich Mathematik, Uiversität Dortmud, 441 Dortmud, Germay. September AES 10th Covetio, Paris, Frace, 006 May 0 3 Page 6 of 6
Normal Distribution.
Normal Distributio www.icrf.l Normal distributio I probability theory, the ormal or Gaussia distributio, is a cotiuous probability distributio that is ofte used as a first approimatio to describe realvalued
More informationCHAPTER 3 DIGITAL CODING OF SIGNALS
CHAPTER 3 DIGITAL CODING OF SIGNALS Computers are ofte used to automate the recordig of measuremets. The trasducers ad sigal coditioig circuits produce a voltage sigal that is proportioal to a quatity
More informationModified Line Search Method for Global Optimization
Modified Lie Search Method for Global Optimizatio Cria Grosa ad Ajith Abraham Ceter of Excellece for Quatifiable Quality of Service Norwegia Uiversity of Sciece ad Techology Trodheim, Norway {cria, ajith}@q2s.tu.o
More informationIncremental calculation of weighted mean and variance
Icremetal calculatio of weighted mea ad variace Toy Fich faf@cam.ac.uk dot@dotat.at Uiversity of Cambridge Computig Service February 009 Abstract I these otes I eplai how to derive formulae for umerically
More informationResearch Article Sign Data Derivative Recovery
Iteratioal Scholarly Research Network ISRN Applied Mathematics Volume 0, Article ID 63070, 7 pages doi:0.540/0/63070 Research Article Sig Data Derivative Recovery L. M. Housto, G. A. Glass, ad A. D. Dymikov
More informationBasic Measurement Issues. Sampling Theory and AnalogtoDigital Conversion
Theory ad AalogtoDigital Coversio Itroductio/Defiitios Aalogtodigital coversio Rate Frequecy Aalysis Basic Measuremet Issues Reliability the extet to which a measuremet procedure yields the same results
More informationCOMPARISON OF THE EFFICIENCY OF SCONTROL CHART AND EWMAS 2 CONTROL CHART FOR THE CHANGES IN A PROCESS
COMPARISON OF THE EFFICIENCY OF SCONTROL CHART AND EWMAS CONTROL CHART FOR THE CHANGES IN A PROCESS Supraee Lisawadi Departmet of Mathematics ad Statistics, Faculty of Sciece ad Techoology, Thammasat
More informationI. Chisquared Distributions
1 M 358K Supplemet to Chapter 23: CHISQUARED DISTRIBUTIONS, TDISTRIBUTIONS, AND DEGREES OF FREEDOM To uderstad tdistributios, we first eed to look at aother family of distributios, the chisquared distributios.
More informationDomain 1: Designing a SQL Server Instance and a Database Solution
Maual SQL Server 2008 Desig, Optimize ad Maitai (70450) 18004186789 Domai 1: Desigig a SQL Server Istace ad a Database Solutio Desigig for CPU, Memory ad Storage Capacity Requiremets Whe desigig a
More information*The most important feature of MRP as compared with ordinary inventory control analysis is its time phasing feature.
Itegrated Productio ad Ivetory Cotrol System MRP ad MRP II Framework of Maufacturig System Ivetory cotrol, productio schedulig, capacity plaig ad fiacial ad busiess decisios i a productio system are iterrelated.
More informationDefinition. A variable X that takes on values X 1, X 2, X 3,...X k with respective frequencies f 1, f 2, f 3,...f k has mean
1 Social Studies 201 October 13, 2004 Note: The examples i these otes may be differet tha used i class. However, the examples are similar ad the methods used are idetical to what was preseted i class.
More informationThe following example will help us understand The Sampling Distribution of the Mean. C1 C2 C3 C4 C5 50 miles 84 miles 38 miles 120 miles 48 miles
The followig eample will help us uderstad The Samplig Distributio of the Mea Review: The populatio is the etire collectio of all idividuals or objects of iterest The sample is the portio of the populatio
More informationPSYCHOLOGICAL STATISTICS
UNIVERSITY OF CALICUT SCHOOL OF DISTANCE EDUCATION B Sc. Cousellig Psychology (0 Adm.) IV SEMESTER COMPLEMENTARY COURSE PSYCHOLOGICAL STATISTICS QUESTION BANK. Iferetial statistics is the brach of statistics
More informationVolatility of rates of return on the example of wheat futures. Sławomir Juszczyk. Rafał Balina
Overcomig the Crisis: Ecoomic ad Fiacial Developmets i Asia ad Europe Edited by Štefa Bojec, Josef C. Brada, ad Masaaki Kuboiwa http://www.hippocampus.si/isbn/9789616832328/cotets.pdf Volatility of
More information1 Correlation and Regression Analysis
1 Correlatio ad Regressio Aalysis I this sectio we will be ivestigatig the relatioship betwee two cotiuous variable, such as height ad weight, the cocetratio of a ijected drug ad heart rate, or the cosumptio
More informationIn nite Sequences. Dr. Philippe B. Laval Kennesaw State University. October 9, 2008
I ite Sequeces Dr. Philippe B. Laval Keesaw State Uiversity October 9, 2008 Abstract This had out is a itroductio to i ite sequeces. mai de itios ad presets some elemetary results. It gives the I ite Sequeces
More informationAutomatic Tuning for FOREX Trading System Using Fuzzy Time Series
utomatic Tuig for FOREX Tradig System Usig Fuzzy Time Series Kraimo Maeesilp ad Pitihate Soorasa bstract Efficiecy of the automatic currecy tradig system is time depedet due to usig fixed parameters which
More informationStudy on the application of the software phaselocked loop in tracking and filtering of pulse signal
Advaced Sciece ad Techology Letters, pp.3135 http://dx.doi.org/10.14257/astl.2014.78.06 Study o the applicatio of the software phaselocked loop i trackig ad filterig of pulse sigal Sog Wei Xia 1 (College
More informationHypothesis testing. Null and alternative hypotheses
Hypothesis testig Aother importat use of samplig distributios is to test hypotheses about populatio parameters, e.g. mea, proportio, regressio coefficiets, etc. For example, it is possible to stipulate
More informationNATIONAL SENIOR CERTIFICATE GRADE 12
NATIONAL SENIOR CERTIFICATE GRADE MATHEMATICS P EXEMPLAR 04 MARKS: 50 TIME: 3 hours This questio paper cosists of 8 pages ad iformatio sheet. Please tur over Mathematics/P DBE/04 NSC Grade Eemplar INSTRUCTIONS
More informationGCSE STATISTICS. 4) How to calculate the range: The difference between the biggest number and the smallest number.
GCSE STATISTICS You should kow: 1) How to draw a frequecy diagram: e.g. NUMBER TALLY FREQUENCY 1 3 5 ) How to draw a bar chart, a pictogram, ad a pie chart. 3) How to use averages: a) Mea  add up all
More informationSystems Design Project: Indoor Location of Wireless Devices
Systems Desig Project: Idoor Locatio of Wireless Devices Prepared By: Bria Murphy Seior Systems Sciece ad Egieerig Washigto Uiversity i St. Louis Phoe: (805) 6985295 Email: bcm1@cec.wustl.edu Supervised
More informationSECTION 1.5 : SUMMATION NOTATION + WORK WITH SEQUENCES
SECTION 1.5 : SUMMATION NOTATION + WORK WITH SEQUENCES Read Sectio 1.5 (pages 5 9) Overview I Sectio 1.5 we lear to work with summatio otatio ad formulas. We will also itroduce a brief overview of sequeces,
More informationAnalyzing Longitudinal Data from Complex Surveys Using SUDAAN
Aalyzig Logitudial Data from Complex Surveys Usig SUDAAN Darryl Creel Statistics ad Epidemiology, RTI Iteratioal, 312 Trotter Farm Drive, Rockville, MD, 20850 Abstract SUDAAN: Software for the Statistical
More informationPROCEEDINGS OF THE YEREVAN STATE UNIVERSITY AN ALTERNATIVE MODEL FOR BONUSMALUS SYSTEM
PROCEEDINGS OF THE YEREVAN STATE UNIVERSITY Physical ad Mathematical Scieces 2015, 1, p. 15 19 M a t h e m a t i c s AN ALTERNATIVE MODEL FOR BONUSMALUS SYSTEM A. G. GULYAN Chair of Actuarial Mathematics
More informationA Combined Continuous/Binary Genetic Algorithm for Microstrip Antenna Design
A Combied Cotiuous/Biary Geetic Algorithm for Microstrip Atea Desig Rady L. Haupt The Pesylvaia State Uiversity Applied Research Laboratory P. O. Box 30 State College, PA 168040030 haupt@ieee.org Abstract:
More informationZTEST / ZSTATISTIC: used to test hypotheses about. µ when the population standard deviation is unknown
ZTEST / ZSTATISTIC: used to test hypotheses about µ whe the populatio stadard deviatio is kow ad populatio distributio is ormal or sample size is large TTEST / TSTATISTIC: used to test hypotheses about
More informationNonlife insurance mathematics. Nils F. Haavardsson, University of Oslo and DNB Skadeforsikring
Nolife isurace mathematics Nils F. Haavardsso, Uiversity of Oslo ad DNB Skadeforsikrig Mai issues so far Why does isurace work? How is risk premium defied ad why is it importat? How ca claim frequecy
More informationChapter 6: Variance, the law of large numbers and the MonteCarlo method
Chapter 6: Variace, the law of large umbers ad the MoteCarlo method Expected value, variace, ad Chebyshev iequality. If X is a radom variable recall that the expected value of X, E[X] is the average value
More informationChapter 7 Methods of Finding Estimators
Chapter 7 for BST 695: Special Topics i Statistical Theory. Kui Zhag, 011 Chapter 7 Methods of Fidig Estimators Sectio 7.1 Itroductio Defiitio 7.1.1 A poit estimator is ay fuctio W( X) W( X1, X,, X ) of
More information5: Introduction to Estimation
5: Itroductio to Estimatio Cotets Acroyms ad symbols... 1 Statistical iferece... Estimatig µ with cofidece... 3 Samplig distributio of the mea... 3 Cofidece Iterval for μ whe σ is kow before had... 4 Sample
More informationwhere: T = number of years of cash flow in investment's life n = the year in which the cash flow X n i = IRR = the internal rate of return
EVALUATING ALTERNATIVE CAPITAL INVESTMENT PROGRAMS By Ke D. Duft, Extesio Ecoomist I the March 98 issue of this publicatio we reviewed the procedure by which a capital ivestmet project was assessed. The
More informationThe Stable Marriage Problem
The Stable Marriage Problem William Hut Lae Departmet of Computer Sciece ad Electrical Egieerig, West Virgiia Uiversity, Morgatow, WV William.Hut@mail.wvu.edu 1 Itroductio Imagie you are a matchmaker,
More informationCase Study. Normal and t Distributions. Density Plot. Normal Distributions
Case Study Normal ad t Distributios Bret Halo ad Bret Larget Departmet of Statistics Uiversity of Wiscosi Madiso October 11 13, 2011 Case Study Body temperature varies withi idividuals over time (it ca
More information.04. This means $1000 is multiplied by 1.02 five times, once for each of the remaining sixmonth
Questio 1: What is a ordiary auity? Let s look at a ordiary auity that is certai ad simple. By this, we mea a auity over a fixed term whose paymet period matches the iterest coversio period. Additioally,
More informationMeasures of Spread and Boxplots Discrete Math, Section 9.4
Measures of Spread ad Boxplots Discrete Math, Sectio 9.4 We start with a example: Example 1: Comparig Mea ad Media Compute the mea ad media of each data set: S 1 = {4, 6, 8, 10, 1, 14, 16} S = {4, 7, 9,
More informationBiology 171L Environment and Ecology Lab Lab 2: Descriptive Statistics, Presenting Data and Graphing Relationships
Biology 171L Eviromet ad Ecology Lab Lab : Descriptive Statistics, Presetig Data ad Graphig Relatioships Itroductio Log lists of data are ofte ot very useful for idetifyig geeral treds i the data or the
More informationUniversity of California, Los Angeles Department of Statistics. Distributions related to the normal distribution
Uiversity of Califoria, Los Ageles Departmet of Statistics Statistics 100B Istructor: Nicolas Christou Three importat distributios: Distributios related to the ormal distributio Chisquare (χ ) distributio.
More informationThis document contains a collection of formulas and constants useful for SPC chart construction. It assumes you are already familiar with SPC.
SPC Formulas ad Tables 1 This documet cotais a collectio of formulas ad costats useful for SPC chart costructio. It assumes you are already familiar with SPC. Termiology Geerally, a bar draw over a symbol
More informationINVESTMENT PERFORMANCE COUNCIL (IPC)
INVESTMENT PEFOMANCE COUNCIL (IPC) INVITATION TO COMMENT: Global Ivestmet Performace Stadards (GIPS ) Guidace Statemet o Calculatio Methodology The Associatio for Ivestmet Maagemet ad esearch (AIM) seeks
More information5 Boolean Decision Trees (February 11)
5 Boolea Decisio Trees (February 11) 5.1 Graph Coectivity Suppose we are give a udirected graph G, represeted as a boolea adjacecy matrix = (a ij ), where a ij = 1 if ad oly if vertices i ad j are coected
More informationChair for Network Architectures and Services Institute of Informatics TU München Prof. Carle. Network Security. Chapter 2 Basics
Chair for Network Architectures ad Services Istitute of Iformatics TU Müche Prof. Carle Network Security Chapter 2 Basics 2.4 Radom Number Geeratio for Cryptographic Protocols Motivatio It is crucial to
More informationDetermining the sample size
Determiig the sample size Oe of the most commo questios ay statisticia gets asked is How large a sample size do I eed? Researchers are ofte surprised to fid out that the aswer depeds o a umber of factors
More informationOverview on SBox Design Principles
Overview o SBox Desig Priciples Debdeep Mukhopadhyay Assistat Professor Departmet of Computer Sciece ad Egieerig Idia Istitute of Techology Kharagpur INDIA 721302 What is a SBox? SBoxes are Boolea
More informationDomain 1  Describe Cisco VoIP Implementations
Maual ONT (6428) 18004186789 Domai 1  Describe Cisco VoIP Implemetatios Advatages of VoIP Over Traditioal Switches Voice over IP etworks have may advatages over traditioal circuit switched voice etworks.
More informationLesson 17 Pearson s Correlation Coefficient
Outlie Measures of Relatioships Pearso s Correlatio Coefficiet (r) types of data scatter plots measure of directio measure of stregth Computatio covariatio of X ad Y uique variatio i X ad Y measurig
More informationLecture 2: Karger s Min Cut Algorithm
priceto uiv. F 3 cos 5: Advaced Algorithm Desig Lecture : Karger s Mi Cut Algorithm Lecturer: Sajeev Arora Scribe:Sajeev Today s topic is simple but gorgeous: Karger s mi cut algorithm ad its extesio.
More informationCapacity of Wireless Networks with Heterogeneous Traffic
Capacity of Wireless Networks with Heterogeeous Traffic Migyue Ji, Zheg Wag, Hamid R. Sadjadpour, J.J. GarciaLuaAceves Departmet of Electrical Egieerig ad Computer Egieerig Uiversity of Califoria, Sata
More informationDepartment of Computer Science, University of Otago
Departmet of Computer Sciece, Uiversity of Otago Techical Report OUCS200609 Permutatios Cotaiig May Patters Authors: M.H. Albert Departmet of Computer Sciece, Uiversity of Otago Micah Colema, Rya Fly
More informationTrigonometric Form of a Complex Number. The Complex Plane. axis. ( 2, 1) or 2 i FIGURE 6.44. The absolute value of the complex number z a bi is
0_0605.qxd /5/05 0:45 AM Page 470 470 Chapter 6 Additioal Topics i Trigoometry 6.5 Trigoometric Form of a Complex Number What you should lear Plot complex umbers i the complex plae ad fid absolute values
More informationStatistical inference: example 1. Inferential Statistics
Statistical iferece: example 1 Iferetial Statistics POPULATION SAMPLE A clothig store chai regularly buys from a supplier large quatities of a certai piece of clothig. Each item ca be classified either
More informationCHAPTER 3 THE TIME VALUE OF MONEY
CHAPTER 3 THE TIME VALUE OF MONEY OVERVIEW A dollar i the had today is worth more tha a dollar to be received i the future because, if you had it ow, you could ivest that dollar ad ear iterest. Of all
More informationNow here is the important step
LINEST i Excel The Excel spreadsheet fuctio "liest" is a complete liear least squares curve fittig routie that produces ucertaity estimates for the fit values. There are two ways to access the "liest"
More information1 Computing the Standard Deviation of Sample Means
Computig the Stadard Deviatio of Sample Meas Quality cotrol charts are based o sample meas ot o idividual values withi a sample. A sample is a group of items, which are cosidered all together for our aalysis.
More informationCS100: Introduction to Computer Science
Iclass Exercise: CS100: Itroductio to Computer Sciece What is a flipflop? What are the properties of flipflops? Draw a simple flipflop circuit? Lecture 3: Data Storage  Mass storage & represetig
More informationTrading the randomness  Designing an optimal trading strategy under a drifted random walk price model
Tradig the radomess  Desigig a optimal tradig strategy uder a drifted radom walk price model Yuao Wu Math 20 Project Paper Professor Zachary Hamaker Abstract: I this paper the author iteds to explore
More informationOn Formula to Compute Primes. and the n th Prime
Applied Mathematical cieces, Vol., 0, o., 3535 O Formula to Compute Primes ad the th Prime Issam Kaddoura Lebaese Iteratioal Uiversity Faculty of Arts ad cieces, Lebao issam.kaddoura@liu.edu.lb amih AbdulNabi
More informationVladimir N. Burkov, Dmitri A. Novikov MODELS AND METHODS OF MULTIPROJECTS MANAGEMENT
Keywords: project maagemet, resource allocatio, etwork plaig Vladimir N Burkov, Dmitri A Novikov MODELS AND METHODS OF MULTIPROJECTS MANAGEMENT The paper deals with the problems of resource allocatio betwee
More informationA Guide to the Pricing Conventions of SFE Interest Rate Products
A Guide to the Pricig Covetios of SFE Iterest Rate Products SFE 30 Day Iterbak Cash Rate Futures Physical 90 Day Bak Bills SFE 90 Day Bak Bill Futures SFE 90 Day Bak Bill Futures Tick Value Calculatios
More informationChapter 5 Unit 1. IET 350 Engineering Economics. Learning Objectives Chapter 5. Learning Objectives Unit 1. Annual Amount and Gradient Functions
Chapter 5 Uit Aual Amout ad Gradiet Fuctios IET 350 Egieerig Ecoomics Learig Objectives Chapter 5 Upo completio of this chapter you should uderstad: Calculatig future values from aual amouts. Calculatig
More informationPartial Di erential Equations
Partial Di eretial Equatios Partial Di eretial Equatios Much of moder sciece, egieerig, ad mathematics is based o the study of partial di eretial equatios, where a partial di eretial equatio is a equatio
More information3. Greatest Common Divisor  Least Common Multiple
3 Greatest Commo Divisor  Least Commo Multiple Defiitio 31: The greatest commo divisor of two atural umbers a ad b is the largest atural umber c which divides both a ad b We deote the greatest commo gcd
More informationTaking DCOP to the Real World: Efficient Complete Solutions for Distributed MultiEvent Scheduling
Taig DCOP to the Real World: Efficiet Complete Solutios for Distributed MultiEvet Schedulig Rajiv T. Maheswara, Milid Tambe, Emma Bowrig, Joatha P. Pearce, ad Pradeep araatham Uiversity of Souther Califoria
More informationApproximating Area under a curve with rectangles. To find the area under a curve we approximate the area using rectangles and then use limits to find
1.8 Approximatig Area uder a curve with rectagles 1.6 To fid the area uder a curve we approximate the area usig rectagles ad the use limits to fid 1.4 the area. Example 1 Suppose we wat to estimate 1.
More informationMaximum Likelihood Estimators.
Lecture 2 Maximum Likelihood Estimators. Matlab example. As a motivatio, let us look at oe Matlab example. Let us geerate a radom sample of size 00 from beta distributio Beta(5, 2). We will lear the defiitio
More informationEngineering Data Management
BaaERP 5.0c Maufacturig Egieerig Data Maagemet Module Procedure UP128A US Documetiformatio Documet Documet code : UP128A US Documet group : User Documetatio Documet title : Egieerig Data Maagemet Applicatio/Package
More informationSoving Recurrence Relations
Sovig Recurrece Relatios Part 1. Homogeeous liear 2d degree relatios with costat coefficiets. Cosider the recurrece relatio ( ) T () + at ( 1) + bt ( 2) = 0 This is called a homogeeous liear 2d degree
More informationEstimating Probability Distributions by Observing Betting Practices
5th Iteratioal Symposium o Imprecise Probability: Theories ad Applicatios, Prague, Czech Republic, 007 Estimatig Probability Distributios by Observig Bettig Practices Dr C Lych Natioal Uiversity of Irelad,
More informationLECTURE 13: Crossvalidation
LECTURE 3: Crossvalidatio Resampli methods Cross Validatio Bootstrap Bias ad variace estimatio with the Bootstrap Threeway data partitioi Itroductio to Patter Aalysis Ricardo GutierrezOsua Texas A&M
More informationQuadrat Sampling in Population Ecology
Quadrat Samplig i Populatio Ecology Backgroud Estimatig the abudace of orgaisms. Ecology is ofte referred to as the "study of distributio ad abudace". This beig true, we would ofte like to kow how may
More informationConfidence Intervals for One Mean
Chapter 420 Cofidece Itervals for Oe Mea Itroductio This routie calculates the sample size ecessary to achieve a specified distace from the mea to the cofidece limit(s) at a stated cofidece level for a
More informationForecasting techniques
2 Forecastig techiques this chapter covers... I this chapter we will examie some useful forecastig techiques that ca be applied whe budgetig. We start by lookig at the way that samplig ca be used to collect
More informationAn Area Computation Based Method for RAIM Holes Assessment
Joural of Global Positioig Systems (2006) Vol. 5, No. 12:1116 A Area Computatio Based Method for RAIM Holes Assessmet Shaoju Feg, Washigto Y. Ochieg ad Raier Mautz Cetre for Trasport Studies, Departmet
More informationDesigning Incentives for Online Question and Answer Forums
Desigig Icetives for Olie Questio ad Aswer Forums Shaili Jai School of Egieerig ad Applied Scieces Harvard Uiversity Cambridge, MA 0238 USA shailij@eecs.harvard.edu Yilig Che School of Egieerig ad Applied
More informationOutput Analysis (2, Chapters 10 &11 Law)
B. Maddah ENMG 6 Simulatio 05/0/07 Output Aalysis (, Chapters 10 &11 Law) Comparig alterative system cofiguratio Sice the output of a simulatio is radom, the comparig differet systems via simulatio should
More informationChapter XIV: Fundamentals of Probability and Statistics *
Objectives Chapter XIV: Fudametals o Probability ad Statistics * Preset udametal cocepts o probability ad statistics Review measures o cetral tedecy ad dispersio Aalyze methods ad applicatios o descriptive
More informationIrreducible polynomials with consecutive zero coefficients
Irreducible polyomials with cosecutive zero coefficiets Theodoulos Garefalakis Departmet of Mathematics, Uiversity of Crete, 71409 Heraklio, Greece Abstract Let q be a prime power. We cosider the problem
More informationAsymptotic Growth of Functions
CMPS Itroductio to Aalysis of Algorithms Fall 3 Asymptotic Growth of Fuctios We itroduce several types of asymptotic otatio which are used to compare the performace ad efficiecy of algorithms As we ll
More information, a Wishart distribution with n 1 degrees of freedom and scale matrix.
UMEÅ UNIVERSITET Matematiskstatistiska istitutioe Multivariat dataaalys D MSTD79 PA TENTAMEN 00409 LÖSNINGSFÖRSLAG TILL TENTAMEN I MATEMATISK STATISTIK Multivariat dataaalys D, 5 poäg.. Assume that
More informationWeek 3 Conditional probabilities, Bayes formula, WEEK 3 page 1 Expected value of a random variable
Week 3 Coditioal probabilities, Bayes formula, WEEK 3 page 1 Expected value of a radom variable We recall our discussio of 5 card poker hads. Example 13 : a) What is the probability of evet A that a 5
More informationAP Calculus BC 2003 Scoring Guidelines Form B
AP Calculus BC Scorig Guidelies Form B The materials icluded i these files are iteded for use by AP teachers for course ad exam preparatio; permissio for ay other use must be sought from the Advaced Placemet
More information1. Introduction. Scheduling Theory
. Itroductio. Itroductio As a idepedet brach of Operatioal Research, Schedulig Theory appeared i the begiig of the 50s. I additio to computer systems ad maufacturig, schedulig theory ca be applied to may
More informationTruStore: The storage. system that grows with you. Machine Tools / Power Tools Laser Technology / Electronics Medical Technology
TruStore: The storage system that grows with you Machie Tools / Power Tools Laser Techology / Electroics Medical Techology Everythig from a sigle source. Cotets Everythig from a sigle source. 2 TruStore
More informationMTOMTS Production Systems in Supply Chains
NSF GRANT #0092854 NSF PROGRAM NAME: MES/OR MTOMTS Productio Systems i Supply Chais Philip M. Kamisky Uiversity of Califoria, Berkeley Our Kaya Uiversity of Califoria, Berkeley Abstract: Icreasig cost
More informationHow to read A Mutual Fund shareholder report
Ivestor BulletI How to read A Mutual Fud shareholder report The SEC s Office of Ivestor Educatio ad Advocacy is issuig this Ivestor Bulleti to educate idividual ivestors about mutual fud shareholder reports.
More informationLOCATIONAL MARGINAL PRICING FRAMEWORK IN SECURED DISPATCH SCHEDULING UNDER CONTINGENCY CONDITION
IJRET: Iteratioal Joural of Research i Egieerig ad Techology eissn: 23191163 pissn: 23217308 LOCATIONAL MARGINAL PRICING FRAMEWORK IN SECURED DISPATCH SCHEDULING UNDER CONTINGENCY CONDITION R.Maiamda
More informationCooleyTukey. Tukey FFT Algorithms. FFT Algorithms. Cooley
Cooley CooleyTuey Tuey FFT Algorithms FFT Algorithms Cosider a legth sequece x[ with a poit DFT X[ where Represet the idices ad as +, +, Cooley CooleyTuey Tuey FFT Algorithms FFT Algorithms Usig these
More informationDiscrete Mathematics and Probability Theory Spring 2014 Anant Sahai Note 13
EECS 70 Discrete Mathematics ad Probability Theory Sprig 2014 Aat Sahai Note 13 Itroductio At this poit, we have see eough examples that it is worth just takig stock of our model of probability ad may
More informationCantilever Beam Experiment
Mechaical Egieerig Departmet Uiversity of Massachusetts Lowell Catilever Beam Experimet Backgroud A disk drive maufacturer is redesigig several disk drive armature mechaisms. This is the result of evaluatio
More informationA probabilistic proof of a binomial identity
A probabilistic proof of a biomial idetity Joatho Peterso Abstract We give a elemetary probabilistic proof of a biomial idetity. The proof is obtaied by computig the probability of a certai evet i two
More informationThe Binomial Multi Section Transformer
4/15/21 The Bioial Multisectio Matchig Trasforer.doc 1/17 The Bioial Multi Sectio Trasforer Recall that a ultisectio atchig etwork ca be described usig the theory of sall reflectios as: where: Γ ( ω
More informationBaan Service Master Data Management
Baa Service Master Data Maagemet Module Procedure UP069A US Documetiformatio Documet Documet code : UP069A US Documet group : User Documetatio Documet title : Master Data Maagemet Applicatio/Package :
More informationStochastic Online Scheduling with Precedence Constraints
Stochastic Olie Schedulig with Precedece Costraits Nicole Megow Tark Vredeveld July 15, 2008 Abstract We cosider the preemptive ad opreemptive problems of schedulig obs with precedece costraits o parallel
More informationDetecting Voice Mail Fraud. Detecting Voice Mail Fraud  1
Detectig Voice Mail Fraud Detectig Voice Mail Fraud  1 Issue 2 Detectig Voice Mail Fraud Detectig Voice Mail Fraud Several reportig mechaisms ca assist you i determiig voice mail fraud. Call Detail Recordig
More informationListing terms of a finite sequence List all of the terms of each finite sequence. a) a n n 2 for 1 n 5 1 b) a n for 1 n 4 n 2
74 (4 ) Chapter 4 Sequeces ad Series 4. SEQUENCES I this sectio Defiitio Fidig a Formula for the th Term The word sequece is a familiar word. We may speak of a sequece of evets or say that somethig is
More informationTHE ARITHMETIC OF INTEGERS.  multiplication, exponentiation, division, addition, and subtraction
THE ARITHMETIC OF INTEGERS  multiplicatio, expoetiatio, divisio, additio, ad subtractio What to do ad what ot to do. THE INTEGERS Recall that a iteger is oe of the whole umbers, which may be either positive,
More informationCS103X: Discrete Structures Homework 4 Solutions
CS103X: Discrete Structures Homewor 4 Solutios Due February 22, 2008 Exercise 1 10 poits. Silico Valley questios: a How may possible sixfigure salaries i whole dollar amouts are there that cotai at least
More informationSYSTEM INFO. MDK  Multifunctional Digital Communications System. Efficient Solutions for Information and Safety
Commuicatios Systems for Itercom, PA, Emergecy Call ad Telecommuicatios MDK  Multifuctioal Digital Commuicatios System SYSTEM INFO ms NEUMANN ELEKTRONIK GmbH Efficiet Solutios for Iformatio ad Safety
More informationTHE ABRACADABRA PROBLEM
THE ABRACADABRA PROBLEM FRANCESCO CARAVENNA Abstract. We preset a detailed solutio of Exercise E0.6 i [Wil9]: i a radom sequece of letters, draw idepedetly ad uiformly from the Eglish alphabet, the expected
More informationTHE REGRESSION MODEL IN MATRIX FORM. For simple linear regression, meaning one predictor, the model is. for i = 1, 2, 3,, n
We will cosider the liear regressio model i matrix form. For simple liear regressio, meaig oe predictor, the model is i = + x i + ε i for i =,,,, This model icludes the assumptio that the ε i s are a sample
More information