THE SYSTEMATIC AND THE RANDOM. ERRORS  DUE TO ELEMENT TOLERANCES OF ELECTRICAL NETWORKS


 Toby Morton
 9 months ago
 Views:
Transcription
1 R775 Philips Res. Repts 26, , 1971' THE SYSTEMATIC AND THE RANDOM. ERRORS  DUE TO ELEMENT TOLERANCES OF ELECTRICAL NETWORKS by H. W. HANNEMAN Abstract Usig the law of propagatio of errors, approximated expressios are derived for the systematic ad the radom error i etwork fuctios due to elemet toleraces. A relatio is derived betwee the radom error ad the summedsquared sesitivity provided that the elemets have equal probability distributios. These expressios have bee foud useful i the aalysis of filter etworks. 1. Itroductio The elemets used i electrical etworks deviate from their face value. That is why it is ot possible to get the specified etwork fuctio but oly a approximatio. It is importat to have a measure of how large the deviatio may be expected to be. Schoeffler 1) has chose the sum of the magitudessquared of the firstorder sesitivity as a measure for the sesitivity of the etwork fuctio (e.g. voltagetrasfer fuctio, iput impedace) due to toleraces of the compoets. We shall use a statistical approach 2). Compoets ca be cosidered as stochastical variables with a mea (equal to the omial value of the compoet) ad a stadard deviatio. I geeral, there will be a systematic error ad a radom oe (the systematic error caused by e.g. temperature deviatios or agig will ot be take ito accout there). A approximated expressio is derived for both the systematic error of a etwork fuctio ad the radom error (sec. 3). The systematic error is show to be proportioal to the secodorder sesitivity while the radom error is proportioal to the firstorder sesitivity. Therefore, the systematic error is a order of magitude smaller tha the radom error if the toleraces are small. The ifluece of the probability distributio of the elemets upo both errors is ivestigated. A relatio is derived betwee the radom error ad the sum of the magitudessquared of the sesitivities of a etwork characteristic to chages i each of the elemets (sec. 4). This aalysis also reveals the idividual cotributio of each elemet to the summed error. The tolerace of a specific elemet ca be chose i terms of the apparet sesitivity of that elemet.. The results are applied to a RCactive filter realizig a Chebyshev characteristic (sec. 5). 2. Defiitios Let y deote the etwork fuctio of iterest. The values of the compoets
2 ERRORS DUE TO ELEMENT TOLERANCES OF ELECTRICAL NETWORKS 415 of the etwork which realizes this fuctio are deoted as XI for i = 1 up, to. We ca write ' y mayalso be a fuctio of frequecy, time, etc. The fuctio y represets e.g. the modulus of the voltagetrasfer ratio of the etwork cofiguratio. The compoets Xl do ot have a fixed value, they are supposed to be stochastical variables. The omial value of compoet Xl which is assumed to be idetical with the mea is deoted as!ll. We assume that the distributio of the compoet values will be symmetrical aroud Xl =!ll. We ow expad y i a dimesioal Taylor series aroud the value (Pl>!l2'...,!lh...,!l): " All derivatives are take at the poit (PI'!l2'...,!lh...,!lIl). F,urther, we assume that the stochastical variables Xl> X2', X are idepedet. (A etwork cosists i geeralof differet types of compoets. Elemet values that represet the same kid of compoets ca have equal elemet values or differet oes. All these facts ifluece the justificatio of the assumptio that the compoets are statistically idepedet.) The, we have " (1) Equatio (1) will be the startig poit for our derivatios. 3. The systematic ad the radom error 3.1. The systematic error The systematic error is defied as the differece of the mea value of the
3 416 H. W. HANNEMAN etwork fuctio 4) ad the exact value of the etwork fuctio /(!11,!12,...,!1,,)' Let E(y) qeote the expected value of y. Usig eq. (1), the expectatio of y ca the be foud as follows: " E(y) = E(f(!1t.!12,...,!1I>...,!1,,)) + E (2: (XI fli) ::,) + " C deotes the sum of the terms cotaiig covariaces. For symmetrical distributios it ca easily be show that all terms cotaiig odd powers of (XI !11). do ot cotribute to the sum 2). Thus, 11 For the systematic error s defied as the differece betwee the mea of the etwork fuctio (E(y)) ad the etwork fuctio itself we obtai i a first approximatio 11 where s = ~"\' a? b2~, 2.f.. ; öx, (2) 3.2. The radom error The calculatio of the radom error is essetially the calculatio of the variace. As i sec. 2 we assume that the compoets are stochastical idepedet variables which are. symmetrically distributed aroud the omial value. We apply the operator variace to eq. (1). This leads to the result
4 ERRORS DUE TO ELEMENT TOLERANCES OF ELECTRICAL NETWORKS 417 which equals where C' deotes the sum of the terms cotaiig covariaces ad al 2 = var (XI)' Neglectig terms of a order higher tha oe yields for the stadard deviatio of y, ay, (3) The stadard deviatio of each compoet al ca be calculated from the probability distributio fuctio of the particular compoet XI' Most likely, the probability fuctio will be gaussia. Because of fiite toleraces the p.d.f. will be trucated at both tolerace limits (example 2). A drastically trucated gaussia distributio is the uiform distributio 3) which is used i example 1. Example 1 The compoet value XI is uiformdistributed betwee the tolerace limits ± a% (fig. I). The value of XI ca be represeted as 2 a III XI = III + (R 0,5), 100 p(xi) (1a)/J; f!.i (I+a)f!.i Xi Fig. 1. Statistical distributio of the compoet values of elemet XI'
5 418 H. W. HANNEMAN where R deotes a umber betwee 0 ad 1, chose at radom (pseudoradom). The distributio fuctio P(X,) ca be deoted as ad X, > (l + a)!ti or X, < (1  a) Il" Now, ad (1 +a)~1 E(x, 2 ) = f P(X,) x? dx, = t (3 + a 2 ). (la)~1 Thus, the variace of the uiform distributio will be (4) If all compoets x, of the etwork fuctio f have a uiform distributio the systematic error ca be expressed as (5) while the variace of the etwork fuctio ay 2 becomes (:~,Y a/ = t'al L!t? (6) Example 2 The probability desity fuctio of compoet value x, is a trucated ormal ~e. The degree of trucatio is 0: (the percetage of compoets with a compoet value less tha the lower tolerace limit (1  a)!ti or larger tha the
6 ERRORS DUE TO ELEMENT TOLERANCE!! OF ELECTRICAL NETWORKS 419 Fig. 2. Statistical distributio of the compoet values of elemet X,; P(XI): ormal distributio, h(x,): trucated distributio. upper tolerace limit (1 + a) #1) (fig. 2). The ormal distributio P(X,) is give by 1 {I (X' fl,)2} p(x,) = a,(2)1/2 exp  2' a, . The doubly trucated distributio fuctio hex,) is equal to p(x,) Now, 1 a E(x,) = #,. The variace of the trucated ormal distributio is calculated as The fuctio cp(a) is give i table J, for some values of the degree of trucatio. TABLE I (7) cp(a) It ca be see from table I (as expected from example 1) that the value of cp(a) approaches t for large degrees of trucatio. 4. Relatio to the sesitivity The firstorder sesitivity is writte as #1 "öf S(/'#I) = , f "ö#1 accordig to Bode's defiitio.
7 420 H. W. HANNEMAN The sesitivity of a etwork with compoets /hl' /h2'..., /hl>..., /h" is defied by Schoeffler 1) as the sum of the magitudessquared of the firstorder sesitivities, Thus, Comparig ". 1 ~,( "DJ)2 ~ IS (f, /h,)1 2 = J2 L..._; /h/ "D/hl. (8), this result with the expressio we foud for the variace of y (eq. (3» we observe that i the two cases previously discussed (examples 1 ad 2) ~ ISI 2 is proportioal to ay 2 Thus we obtai i case of a trucated ormal distributio for the quotiet of var (y) ad ~ ISI 2 the importat result ((J( ct) a 2 / 2 (/hl' /h2'..., /hl>..., /h). A similar relatio ca be derived from the compariso sesitivity which is equal to of the secodorder ad the systematic error. 5. Applicatio to filter etworks We have derived expressios for the summed systematic ad radom errors. The values of the compoets ca be see as weight fuctios for both errors. It is importat, however, to kow also the value of each of the derivatives (firstorder ad secodorder) of the etwork fuctio. This eables the desiger to choose the tolerace of a specific elemet accordig to the relative importace of the first ad secodorder derivative ad the compoet value itself. Observe that the systematic error is proportioal to the square of the tolerace (a), while the stadard deviatio is proportioal to a. This meas that for small tolerace values the systematic error will be less importat compared with the, I stadard deviatio tha it will be for high tolerace values. Taylor 4) has calculated the compoet tolerace of each elemet so that' the etwork characteristic will remai withi the tolerace limits for each choice of elemet value. This is ofte impractical, however, sice very arrow toleraces o the compoets would be required. A sesitivity aalysis is v~ry importat to the desiger of filter etworks. I order to show the usefuless of the method here preseted we will ow apply the method to a example.
8 ERRORS DUE TO ELEMENT TOLERANCES OF ELECTRICAL NETWORKS 421 Example 3 The secodorder lowpass etwork cofiguratio we shall use is show i fig. 3 ad was first give by Salle ad Key 5). The port voltages are Vi ad Vout' The etwork characteristic of iterest (y) will be the modulus of the trasfer ratio Wout/Vll. With this filter a characteristic of the Chebyshev type (ripple 0 5 db) is realized. Vi Vout Fig. 3. Secodorder lowpass RCactive etwork. The etwork fuctio y is give by We have calculated the systematic error (s) ad the radom error (s.d. of y), deoted by r as a fuctio of frequecy (w). All elemet values have tolerace limits of 5 % ad the probability desity fuctio is uiform. This case is deoted as A i table Il. The systematic error is a order of magitude smaller tha the radom error. For differet toleraces, e.g. a %, table Il A ca still be used. The systematic error the becomes (a/5)2 times a large, while the radom error has to be multiplied by a/5. The percetages of the radom ad systematic errors are take to the value of IVout/Vll at w = O. We have also calculated the magitudes ofthe first ad secodorder derivatives. By comparig the derivatives of y to K with the derivatives of y to the other compoets it ca be see that the first are relatively large. Thereore we have calculated the systematic ad the radom error for a etwork ith a tolerace limit of 1 % for K ad 5 % for the other compoets. The esults are give i table Il B. A oticeable reductio ca be obtaied. The extiarger derivatives belog to Cl' Agai the systematic ad the radom rrors are calculated for a tolerace limit of 1 % for K ad Cl ad 5 % for the th er compoet values (Rl, R2' R 3 ). The results are show i table Il C. Usig the relatioships here derived a aswer ca be obtaied to the questio f what elemet tolerace has to be chose i order to assure that a give peretage of the circuits has a etwork characteristic which deviates ot more ha le (IF (k > 0) from the expected characteristic (fig. 4).
9 422 H. W. HANNEMAN TABLE II The systematic ad the radom errors of the modulus of the voltagetrasfer fuctio of a secodorder active RC filter (Chebyshev) for various tolerace limits. The compoet values are:  R, = Rz = 1 Q; Cl = Cz = F; K = 1'841, givig a ripple of 0 5 db (see fig. 3) elemets uiform oe elemet two elemets distributed with (K) (Cl' K) tolerace 5 % tolerace 1% tolerace 1% A B C freq. s (%) r (%) s (%) r (%) s (%) r (%) '0' ruc _Freq.  with the correct elemet values _ characteristic if a systematic error appears I the stadard deviatio ai some frequecies Fig. 4. The modulus of trasfer fuctio of a Chebyshev filter (fifthorder).
10 ERRORS DUE TO ELEMENT TOLERANCES OF ELECTRICAL NETWORKS Coclusio. The sesitivity of a etwork characteristic to deviatios of the elemet values has bee described i a statistical maer, which has resulted i approximated. expressios for the systematic ad. the radom error. A relatio has bee éstablished betwee the sum of the magitudessquared of the firstorder sesitivities ad the radom error. The method has bee applied to a example to demostrate its usefuless. Ackowledgemet I wish to thak Prof. Dr K. M. Adams of the Techological Uiversity of Delft, Ir H. N. Lisse of the Techological Uiversity of Eidhove ad my colleague Ir J. O. Voorma for may valuable suggestios. Eidhove, July 1971 REFERENCES 1) J. Schoeffler, IEEE Tras. Circuit Theory CTtl, , ) J. Madel, The statistical aalysis of experimetal data, Wiley, New York, 1967, p ) P. W. Broome ad F. J. Youg, IRE Tras. Circuit Theory CT9, 1823, ) N. H. Taylor, Proc. IRE 38, , ) R. P. Salle ad E. L. Key, IRE Tras. Circuit Theory CT2, 7485, 1955.
Chapter 6 Principles of Data Reduction
Chapter 6 for BST 695: Special Topics i Statistical Theory. Kui Zhag, 0 Chapter 6 Priciples of Data Reductio Sectio 6. Itroductio Goal: To summarize or reduce the data X, X,, X to get iformatio about a
More informationIntroducing Sample Proportions
Itroducig Sample Proportios Probability ad statistics Aswers & Notes TINspire Ivestigatio Studet 60 mi 7 8 9 0 Itroductio A 00 survey of attitudes to climate chage, coducted i Australia by the CSIRO,
More informationChapter 3. Strong convergence. 3.1 Definition of almost sure convergence
Chapter 3 Strog covergece As poited out i the Chapter 2, there are multiple ways to defie the otio of covergece of a sequece of radom variables. That chapter defied covergece i probability, covergece i
More informationProbability, Expectation Value and Uncertainty
Chapter 1 Probability, Expectatio Value ad Ucertaity We have see that the physically observable properties of a quatum system are represeted by Hermitea operators (also referred to as observables ) such
More informationNumber of fatalities X Sunday 4 Monday 6 Tuesday 2 Wednesday 0 Thursday 3 Friday 5 Saturday 8 Total 28. Day
LECTURE # 8 Mea Deviatio, Stadard Deviatio ad Variace & Coefficiet of variatio Mea Deviatio Stadard Deviatio ad Variace Coefficiet of variatio First, we will discuss it for the case of raw data, ad the
More informationFrequency Response of FIR Filters
EEL335: DiscreteTime Sigals ad Systems. Itroductio I this set of otes, we itroduce the idea of the frequecy respose of LTI systems, ad focus specifically o the frequecy respose of FIR filters.. Steadystate
More informationLecture 1 Probability and Statistics
Wikipedia: Lecture 1 Probability ad Statistics Bejami Disraeli, British statesma ad literary figure (1804 1881): There are three kids of lies: lies, damed lies, ad statistics. popularized i US by Mark
More informationA sequence of numbers is a function whose domain is the positive integers. We can see that the sequence
Sequeces A sequece of umbers is a fuctio whose domai is the positive itegers. We ca see that the sequece,, 2, 2, 3, 3,... is a fuctio from the positive itegers whe we write the first sequece elemet as
More informationA Risk Comparison of Ordinary Least Squares vs Ridge Regression
Joural of Machie Learig Research 14 (2013) 15051511 Submitted 5/12; Revised 3/13; Published 6/13 A Risk Compariso of Ordiary Least Squares vs Ridge Regressio Paramveer S. Dhillo Departmet of Computer
More informationA goodnessoffit test based on the empirical characteristic function and a comparison of tests for normality
A goodessoffit test based o the empirical characteristic fuctio ad a compariso of tests for ormality J. Marti va Zyl Departmet of Mathematical Statistics ad Actuarial Sciece, Uiversity of the Free State,
More information62. Power series Definition 16. (Power series) Given a sequence {c n }, the series. c n x n = c 0 + c 1 x + c 2 x 2 + c 3 x 3 +
62. Power series Defiitio 16. (Power series) Give a sequece {c }, the series c x = c 0 + c 1 x + c 2 x 2 + c 3 x 3 + is called a power series i the variable x. The umbers c are called the coefficiets of
More informationFormation of A Supergain Array and Its Application in Radar
Formatio of A Supergai Array ad ts Applicatio i Radar Tra Cao Quye, Do Trug Kie ad Bach Gia Duog. Research Ceter for Electroic ad Telecommuicatios, College of Techology (Coltech, Vietam atioal Uiversity,
More informationMonte Carlo Integration
Mote Carlo Itegratio I these otes we first review basic umerical itegratio methods (usig Riema approximatio ad the trapezoidal rule) ad their limitatios for evaluatig multidimesioal itegrals. Next we itroduce
More informationKLMED8004 Medical statistics. Part I, autumn Estimation. We have previously learned: Population and sample. New questions
We have previously leared: KLMED8004 Medical statistics Part I, autum 00 How kow probability distributios (e.g. biomial distributio, ormal distributio) with kow populatio parameters (mea, variace) ca give
More informationJoint Probability Distributions and Random Samples. Jointly Distributed Random Variables. Chapter { }
UCLA STAT A Applied Probability & Statistics for Egieers Istructor: Ivo Diov, Asst. Prof. I Statistics ad Neurology Teachig Assistat: Neda Farziia, UCLA Statistics Uiversity of Califoria, Los Ageles, Sprig
More informationDiscrete Orthogonal Moment Features Using Chebyshev Polynomials
Discrete Orthogoal Momet Features Usig Chebyshev Polyomials R. Mukuda, 1 S.H.Og ad P.A. Lee 3 1 Faculty of Iformatio Sciece ad Techology, Multimedia Uiversity 75450 Malacca, Malaysia. Istitute of Mathematical
More informationMOST PEOPLE WOULD RATHER LIVE WITH A PROBLEM THEY CAN'T SOLVE, THAN ACCEPT A SOLUTION THEY CAN'T UNDERSTAND.
XI1 (1074) MOST PEOPLE WOULD RATHER LIVE WITH A PROBLEM THEY CAN'T SOLVE, THAN ACCEPT A SOLUTION THEY CAN'T UNDERSTAND. R. E. D. WOOLSEY AND H. S. SWANSON XI2 (1075) STATISTICAL DECISION MAKING Advaced
More informationBinomial Distribution
0.0 0.5 1.0 1.5 2.0 2.5 3.0 0 1 2 3 4 5 6 7 0.0 0.5 1.0 1.5 2.0 2.5 3.0 Overview Example: coi tossed three times Defiitio Formula Recall that a r.v. is discrete if there are either a fiite umber of possible
More informationON POINTWISE BINOMIAL APPROXIMATION
Iteratioal Joural of Pure ad Applied Mathematics Volume 71 No. 1 2011, 5766 ON POINTWISE BINOMIAL APPROXIMATION BY wfunctions K. Teerapabolar 1, P. Wogkasem 2 Departmet of Mathematics Faculty of Sciece
More informationVariance of Discrete Random Variables Class 5, Jeremy Orloff and Jonathan Bloom
Variace of Discrete Radom Variables Class 5, 18.05 Jeremy Orloff ad Joatha Bloom 1 Learig Goals 1. Be able to compute the variace ad stadard deviatio of a radom variable.. Uderstad that stadard deviatio
More informationMASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.436J/15.085J Fall 2008 Lecture 6 9/24/2008 DISCRETE RANDOM VARIABLES AND THEIR EXPECTATIONS
MASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.436J/15.085J Fall 2008 Lecture 6 9/24/2008 DISCRETE RANDOM VARIABLES AND THEIR EXPECTATIONS Cotets 1. A few useful discrete radom variables 2. Joit, margial, ad
More informationIntroduction to Machine Learning DIS10
CS 189 Fall 017 Itroductio to Machie Learig DIS10 1 Fu with Lagrage Multipliers (a) Miimize the fuctio such that f (x,y) = x + y x + y = 3. Solutio: The Lagragia is: L(x,y,λ) = x + y + λ(x + y 3) Takig
More informationSequences of Definite Integrals, Factorials and Double Factorials
47 6 Joural of Iteger Sequeces, Vol. 8 (5), Article 5.4.6 Sequeces of Defiite Itegrals, Factorials ad Double Factorials Thierry DaaPicard Departmet of Applied Mathematics Jerusalem College of Techology
More informationA. Much too slow. C. Basically about right. E. Much too fast
Geeral Questio 1 t this poit, we have bee i this class for about a moth. It seems like this is a good time to take stock of how the class is goig. g I promise ot to look at idividual resposes, so be cadid!
More informationEECE 301 Signals & Systems
EECE 301 Sigals & Systems Prof. Mark Fowler Note Set #8 DT Covolutio: The Tool for Fidig the ZeroState Respose Readig Assigmet: Sectio 2.12.2 of Kame ad Heck 1/14 Course Flow Diagram The arrows here
More informationMechanical Efficiency of Planetary Gear Trains: An Estimate
Mechaical Efficiecy of Plaetary Gear Trais: A Estimate Dr. A. Sriath Professor, Dept. of Mechaical Egieerig K L Uiversity, A.P, Idia Email: sriath_me@klce.ac.i G. Yedukodalu Assistat Professor, Dept.
More information(b) What is the probability that a particle reaches the upper boundary n before the lower boundary m?
MATH 529 The Boudary Problem The drukard s walk (or boudary problem) is oe of the most famous problems i the theory of radom walks. Oe versio of the problem is described as follows: Suppose a particle
More informationBINOMIAL COEFFICIENT AND THE GAUSSIAN
BINOMIAL COEFFICIENT AND THE GAUSSIAN The biomial coefficiet is defied as! k!(! ad ca be writte out i the form of a Pascal Triagle startig at the zeroth row with elemet 0,0) ad followed by the two umbers,
More informationIt should be unbiased, or approximately unbiased. Variance of the variance estimator should be small. That is, the variance estimator is stable.
Chapter 10 Variace Estimatio 10.1 Itroductio Variace estimatio is a importat practical problem i survey samplig. Variace estimates are used i two purposes. Oe is the aalytic purpose such as costructig
More informationSome Properties of the Exact and Score Methods for Binomial Proportion and Sample Size Calculation
Some Properties of the Exact ad Score Methods for Biomial Proportio ad Sample Size Calculatio K. KRISHNAMOORTHY AND JIE PENG Departmet of Mathematics, Uiversity of Louisiaa at Lafayette Lafayette, LA 705041010,
More informationSampling Distributions, ZTests, Power
Samplig Distributios, ZTests, Power We draw ifereces about populatio parameters from sample statistics Sample proportio approximates populatio proportio Sample mea approximates populatio mea Sample variace
More information3/3/2014. CDS M Phil Econometrics. Types of Relationships. Types of Relationships. Types of Relationships. Vijayamohanan Pillai N.
3/3/04 CDS M Phil Old Least Squares (OLS) Vijayamohaa Pillai N CDS M Phil Vijayamoha CDS M Phil Vijayamoha Types of Relatioships Oly oe idepedet variable, Relatioship betwee ad is Liear relatioships Curviliear
More informationMath 2784 (or 2794W) University of Connecticut
ORDERS OF GROWTH PAT SMITH Math 2784 (or 2794W) Uiversity of Coecticut Date: Mar. 2, 22. ORDERS OF GROWTH. Itroductio Gaiig a ituitive feel for the relative growth of fuctios is importat if you really
More informationCentral Limit Theorem the Meaning and the Usage
Cetral Limit Theorem the Meaig ad the Usage Covetio about otatio. N, We are usig otatio X is variable with mea ad stadard deviatio. i lieu of sayig that X is a ormal radom Assume a sample of measuremets
More informationSTATISTICAL INFERENCE
STATISTICAL INFERENCE POPULATION AND SAMPLE Populatio = all elemets of iterest Characterized by a distributio F with some parameter θ Sample = the data X 1,..., X, selected subset of the populatio = sample
More informationThe McClelland approximation and the distribution of electron molecular orbital energy levels
J. Serb. Chem. Soc. 7 (10) 967 973 (007) UDC 54 74+537.87:53.74+539.194 JSCS 369 Origial scietific paper The McClellad approximatio ad the distributio of electro molecular orbital eergy levels IVAN GUTMAN*
More informationAnalysis of Algorithms. Introduction. Contents
Itroductio The focus of this module is mathematical aspects of algorithms. Our mai focus is aalysis of algorithms, which meas evaluatig efficiecy of algorithms by aalytical ad mathematical methods. We
More informationHOMEWORK #10 SOLUTIONS
Math 33  Aalysis I Sprig 29 HOMEWORK # SOLUTIONS () Prove that the fuctio f(x) = x 3 is (Riema) itegrable o [, ] ad show that x 3 dx = 4. (Without usig formulae for itegratio that you leart i previous
More informationRecall the study where we estimated the difference between mean systolic blood pressure levels of users of oral contraceptives and nonusers, x  y.
Testig Statistical Hypotheses Recall the study where we estimated the differece betwee mea systolic blood pressure levels of users of oral cotraceptives ad ousers, x  y. Such studies are sometimes viewed
More informationSINGLECHANNEL QUEUING PROBLEMS APPROACH
SINGLECHANNEL QUEUING ROBLEMS AROACH Abdurrzzag TAMTAM, Doctoral Degree rogramme () Dept. of Telecommuicatios, FEEC, BUT Email: xtamta@stud.feec.vutbr.cz Supervised by: Dr. Karol Molár ABSTRACT The paper
More informationsin(n) + 2 cos(2n) n 3/2 3 sin(n) 2cos(2n) n 3/2 a n =
60. Ratio ad root tests 60.1. Absolutely coverget series. Defiitio 13. (Absolute covergece) A series a is called absolutely coverget if the series of absolute values a is coverget. The absolute covergece
More informationNumerical Methods in Fourier Series Applications
Numerical Methods i Fourier Series Applicatios Recall that the basic relatios i usig the Trigoometric Fourier Series represetatio were give by f ( x) a o ( a x cos b x si ) () where the Fourier coefficiets
More informationElementary Statistics
Elemetary Statistics M. Ghamsary, Ph.D. Sprig 004 Chap 0 Descriptive Statistics Raw Data: Whe data are collected i origial form, they are called raw data. The followig are the scores o the first test of
More informationHOMEWORK 2 SOLUTIONS
HOMEWORK SOLUTIONS CSE 55 RANDOMIZED AND APPROXIMATION ALGORITHMS 1. Questio 1. a) The larger the value of k is, the smaller the expected umber of days util we get all the coupos we eed. I fact if = k
More informationSensitivity Analysis of Daubechies 4 Wavelet Coefficients for Reduction of Reconstructed Image Error
Proceedigs of the 6th WSEAS Iteratioal Coferece o SIGNAL PROCESSING, Dallas, Texas, USA, March 4, 7 67 Sesitivity Aalysis of Daubechies 4 Wavelet Coefficiets for Reductio of Recostructed Image Error DEVINDER
More informationMECHANICAL INTEGRITY DESIGN FOR FLARE SYSTEM ON FLOW INDUCED VIBRATION
MECHANICAL INTEGRITY DESIGN FOR FLARE SYSTEM ON FLOW INDUCED VIBRATION Hisao Izuchi, Pricipal Egieerig Cosultat, Egieerig Solutio Uit, ChAS Project Operatios Masato Nishiguchi, Egieerig Solutio Uit, ChAS
More informationNotes on iteration and Newton s method. Iteration
Notes o iteratio ad Newto s method Iteratio Iteratio meas doig somethig over ad over. I our cotet, a iteratio is a sequece of umbers, vectors, fuctios, etc. geerated by a iteratio rule of the type 1 f
More informationReview Questions, Chapters 8, 9. f(y) = 0, elsewhere. F (y) = f Y(1) = n ( e y/θ) n 1 1 θ e y/θ = n θ e yn
Stat 366 Lab 2 Solutios (September 2, 2006) page TA: Yury Petracheko, CAB 484, yuryp@ualberta.ca, http://www.ualberta.ca/ yuryp/ Review Questios, Chapters 8, 9 8.5 Suppose that Y, Y 2,..., Y deote a radom
More informationSTAT 203 Chapter 18 Sampling Distribution Models
STAT 203 Chapter 18 Samplig Distributio Models Populatio vs. sample, parameter vs. statistic Recall that a populatio cotais the etire collectio of idividuals that oe wats to study, ad a sample is a subset
More informationROTATIONEQUIVALENCE CLASSES OF BINARY VECTORS. 1. Introduction
t m Mathematical Publicatios DOI: 10.1515/tmmp20160033 Tatra Mt. Math. Publ. 67 (2016, 93 98 ROTATIONEQUIVALENCE CLASSES OF BINARY VECTORS Otokar Grošek Viliam Hromada ABSTRACT. I this paper we study
More information11. FINITE FIELDS. Example 1: The following tables define addition and multiplication for a field of order 4.
11. FINITE FIELDS 11.1. A Field With 4 Elemets Probably the oly fiite fields which you ll kow about at this stage are the fields of itegers modulo a prime p, deoted by Z p. But there are others. Now although
More informationOBJECTIVES. Chapter 1 INTRODUCTION TO INSTRUMENTATION FUNCTION AND ADVANTAGES INTRODUCTION. At the end of this chapter, students should be able to:
OBJECTIVES Chapter 1 INTRODUCTION TO INSTRUMENTATION At the ed of this chapter, studets should be able to: 1. Explai the static ad dyamic characteristics of a istrumet. 2. Calculate ad aalyze the measuremet
More informationVaranasi , India. Corresponding author
A Geeral Family of Estimators for Estimatig Populatio Mea i Systematic Samplig Usig Auxiliary Iformatio i the Presece of Missig Observatios Maoj K. Chaudhary, Sachi Malik, Jayat Sigh ad Rajesh Sigh Departmet
More informationLecture 9: Diffusion, Electrostatics review, and Capacitors. Context
EECS 5 Sprig 4, Lecture 9 Lecture 9: Diffusio, Electrostatics review, ad Capacitors EECS 5 Sprig 4, Lecture 9 Cotext I the last lecture, we looked at the carriers i a eutral semicoductor, ad drift currets
More informationComplex Numbers Solutions
Complex Numbers Solutios Joseph Zoller February 7, 06 Solutios. (009 AIME I Problem ) There is a complex umber with imagiary part 64 ad a positive iteger such that Fid. [Solutio: 697] 4i + + 4i. 4i 4i
More informationLarge holes in quasirandom graphs
Large holes i quasiradom graphs Joaa Polcy Departmet of Discrete Mathematics Adam Mickiewicz Uiversity Pozań, Polad joaska@amuedupl Submitted: Nov 23, 2006; Accepted: Apr 10, 2008; Published: Apr 18,
More information6.883: Online Methods in Machine Learning Alexander Rakhlin
6.883: Olie Methods i Machie Learig Alexader Rakhli LECTURES 5 AND 6. THE EXPERTS SETTING. EXPONENTIAL WEIGHTS All the algorithms preseted so far halluciate the future values as radom draws ad the perform
More informationAppendix F: Complex Numbers
Appedix F Complex Numbers F1 Appedix F: Complex Numbers Use the imagiary uit i to write complex umbers, ad to add, subtract, ad multiply complex umbers. Fid complex solutios of quadratic equatios. Write
More informationA STATISTICAL SIGNIFICANCE TEST FOR PERSON AUTHENTICATION. IDIAP CP 592, rue du Simplon Martigny, Switzerland
A STATISTICAL SIGFICANCE TEST FOR PERSON AUTHENTICATION Samy Begio Johy Mariéthoz IDIAP CP 592, rue du Simplo 4 1920 Martigy, Switzerlad {begio,marietho}@idiap.ch ABSTRACT Assessig whether two models are
More informationIn number theory we will generally be working with integers, though occasionally fractions and irrationals will come into play.
Number Theory Math 5840 otes. Sectio 1: Axioms. I umber theory we will geerally be workig with itegers, though occasioally fractios ad irratioals will come ito play. Notatio: Z deotes the set of all itegers
More informationREGRESSION (Physics 1210 Notes, Partial Modified Appendix A)
REGRESSION (Physics 0 Notes, Partial Modified Appedix A) HOW TO PERFORM A LINEAR REGRESSION Cosider the followig data poits ad their graph (Table I ad Figure ): X Y 0 3 5 3 7 4 9 5 Table : Example Data
More informationSeunghee Ye Ma 8: Week 5 Oct 28
Week 5 Summary I Sectio, we go over the Mea Value Theorem ad its applicatios. I Sectio 2, we will recap what we have covered so far this term. Topics Page Mea Value Theorem. Applicatios of the Mea Value
More informationSingular Continuous Measures by Michael Pejic 5/14/10
Sigular Cotiuous Measures by Michael Peic 5/4/0 Prelimiaries Give a set X, a σalgebra o X is a collectio of subsets of X that cotais X ad ad is closed uder complemetatio ad coutable uios hece, coutable
More information1. Hilbert s Grand Hotel. The Hilbert s Grand Hotel has infinite many rooms numbered 1, 2, 3, 4
. Hilbert s Grad Hotel The Hilbert s Grad Hotel has ifiite may rooms umbered,,,.. Situatio. The Hotel is full ad a ew guest arrives. Ca the mager accommodate the ew guest?  Yes, he ca. There is a simple
More informationAsymptotic Results for the Linear Regression Model
Asymptotic Results for the Liear Regressio Model C. Fli November 29, 2000 1. Asymptotic Results uder Classical Assumptios The followig results apply to the liear regressio model y = Xβ + ε, where X is
More informationSearching for the Holy Grail of Index Number Theory
Searchig for the Holy Grail of Idex Number Theory Bert M. Balk Rotterdam School of Maagemet Erasmus Uiversity Rotterdam Email bbalk@rsm.l ad Statistics Netherlads Voorburg April 15, 28 Abstract The idex
More informationQBINOMIALS AND THE GREATEST COMMON DIVISOR. Keith R. Slavin 8474 SW Chevy Place, Beaverton, Oregon 97008, USA.
INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 8 2008, #A05 QBINOMIALS AND THE GREATEST COMMON DIVISOR Keith R. Slavi 8474 SW Chevy Place, Beaverto, Orego 97008, USA slavi@dsloly.et Received:
More informationIntroduction to Probability. Ariel Yadin. Lecture 2
Itroductio to Probability Ariel Yadi Lecture 2 1. Discrete Probability Spaces Discrete probability spaces are those for which the sample space is coutable. We have already see that i this case we ca take
More informationPHY4905: NearlyFree Electron Model (NFE)
PHY4905: NearlyFree Electro Model (NFE) D. L. Maslov Departmet of Physics, Uiversity of Florida (Dated: Jauary 12, 2011) 1 I. REMINDER: QUANTUM MECHANICAL PERTURBATION THEORY A. Nodegeerate eigestates
More informationL (x; 1  x)(xi2 x) [(x; x)z +(x;z .xf] 2n i l. 2n i=l. kn i=l j=l
The Itraclass Correlatio: What is it ad why do we care? David J. Pasta, Techology Assessmet Grop, Sa Fracisco, CA Abstract The itraclass correlatio coefficiet (ICC) is a measure of reproducibility that
More informationAvailable online at ScienceDirect. Physics Procedia 57 (2014 ) 48 52
Available olie at www.sciecedirect.com ScieceDirect Physics Procedia 57 (214 ) 48 52 27th Aual CSP Workshops o Recet Developmets i Computer Simulatio Studies i Codesed Matter Physics, CSP 214 Applicatio
More informationTopic 6 Sampling, hypothesis testing, and the central limit theorem
CSE 103: Probability ad statistics Fall 2010 Topic 6 Samplig, hypothesis testig, ad the cetral limit theorem 61 The biomial distributio Let X be the umberofheadswhe acoiofbiaspistossedtimes The distributio
More informationThe Ratio Test. THEOREM 9.17 Ratio Test Let a n be a series with nonzero terms. 1. a. n converges absolutely if lim. n 1
460_0906.qxd //04 :8 PM Page 69 SECTION 9.6 The Ratio ad Root Tests 69 Sectio 9.6 EXPLORATION Writig a Series Oe of the followig coditios guaratees that a series will diverge, two coditios guaratee that
More informationLecture 6 Ecient estimators. RaoCramer bound.
Lecture 6 Eciet estimators. RaoCramer boud. 1 MSE ad Suciecy Let X (X 1,..., X) be a radom sample from distributio f θ. Let θ ˆ δ(x) be a estimator of θ. Let T (X) be a suciet statistic for θ. As we have
More informationSection 5.1 The Basics of Counting
1 Sectio 5.1 The Basics of Coutig Combiatorics, the study of arragemets of objects, is a importat part of discrete mathematics. I this chapter, we will lear basic techiques of coutig which has a lot of
More informationA brief introduction to linear algebra
CHAPTER 6 A brief itroductio to liear algebra 1. Vector spaces ad liear maps I what follows, fix K 2{Q, R, C}. More geerally, K ca be ay field. 1.1. Vector spaces. Motivated by our ituitio of addig ad
More informationThe BornOppenheimer approximation
The BorOppeheimer approximatio 1 Rewritig the Schrödiger equatio We will begi from the full timeidepedet Schrödiger equatio for the eigestates of a molecular system: [ P 2 + ( Pm 2 + e2 1 1 2m 2m m
More informationProbabilistic and Average Linear Widths in L Norm with Respect to rfold Wiener Measure
joural of approximatio theory 84, 3140 (1996) Article No. 0003 Probabilistic ad Average Liear Widths i L Norm with Respect to rfold Wieer Measure V. E. Maiorov Departmet of Mathematics, Techio, Haifa,
More informationHashing and Amortization
Lecture Hashig ad Amortizatio Supplemetal readig i CLRS: Chapter ; Chapter 7 itro; Sectio 7.. Arrays ad Hashig Arrays are very useful. The items i a array are statically addressed, so that isertig, deletig,
More informationFormulas FROM LECTURE 01 TO 22 W X. d n. fx f. Arslan Latif (mt ) & Mohsin Ali (mc ) Mean: Weighted Mean: Mean Deviation: Ungroup Data
1 Formulas FROM LECTURE 01 TO Mea: fx f Weighted Mea: X w W X i i Wi Mea Deviatio: Ugroup Data d M. D Group Data fi di M. D f d ( X X ) Coefficiet of Mea Deviatio: M. D Coefficiet of M. D(for mea) Mea
More informationQuestion 1: Exercise 8.2
Questio 1: Exercise 8. (a) Accordig to the regressio results i colum (1), the house price is expected to icrease by 1% ( 100% 0.0004 500 ) with a additioal 500 square feet ad other factors held costat.
More informationFast Power Flow Methods 1.0 Introduction
Fast ower Flow Methods. Itroductio What we have leared so far is the socalled full ewtoraphso R power flow alorithm. The R alorithm is perhaps the most robust alorithm i the sese that it is most liely
More informationExample 3.3: Rainfall reported at a group of five stations (see Fig. 3.7) is as follows. Kundla. Sabli
3.4.4 Spatial Cosistecy Check Raifall data exhibit some spatial cosistecy ad this forms the basis of ivestigatig the observed raifall values. A estimate of the iterpolated raifall value at a statio is
More informationROLL CUTTING PROBLEMS UNDER STOCHASTIC DEMAND
PacificAsia Joural of Mathematics, Volume 5, No., JauaryJue 20 ROLL CUTTING PROBLEMS UNDER STOCHASTIC DEMAND SHAKEEL JAVAID, Z. H. BAKHSHI & M. M. KHALID ABSTRACT: I this paper, the roll cuttig problem
More informationPAijpam.eu ON TENSOR PRODUCT DECOMPOSITION
Iteratioal Joural of Pure ad Applied Mathematics Volume 103 No 3 2015, 537545 ISSN: 13118080 (prited versio); ISSN: 13143395 (olie versio) url: http://wwwijpameu doi: http://dxdoiorg/1012732/ijpamv103i314
More informationOrthogonal polynomials derived from the tridiagonal representation approach
Orthogoal polyomials derived from the tridiagoal represetatio approach A. D. Alhaidari Saudi Ceter for Theoretical Physics, P.O. Box 374, Jeddah 438, Saudi Arabia Abstract: The tridiagoal represetatio
More informationINFGEO Solutions, Geometrical Optics, Part 1
INFGEO430 20 Solutios, Geometrical Optics, Part Reflectio by a symmetric triagular prism Let be the agle betwee the two faces of a symmetric triagular prism. Let the edge A where the two faces meet be
More informationSequences and Limits
Chapter Sequeces ad Limits Let { a } be a sequece of real or complex umbers A ecessary ad sufficiet coditio for the sequece to coverge is that for ay ɛ > 0 there exists a iteger N > 0 such that a p a q
More informationSolutions to Tutorial 3 (Week 4)
The Uiversity of Sydey School of Mathematics ad Statistics Solutios to Tutorial Week 4 MATH2962: Real ad Complex Aalysis Advaced Semester 1, 2017 Web Page: http://www.maths.usyd.edu.au/u/ug/im/math2962/
More informationA Note on Sums of Independent Random Variables
Cotemorary Mathematics Volume 00 XXXX A Note o Sums of Ideedet Radom Variables Pawe l Hitczeko ad Stehe MotgomerySmith Abstract I this ote a two sided boud o the tail robability of sums of ideedet ad
More informationTesting Statistical Hypotheses for Compare. Means with Vague Data
Iteratioal Mathematical Forum 5 o. 3 656 Testig Statistical Hypotheses for Compare Meas with Vague Data E. Baloui Jamkhaeh ad A. adi Ghara Departmet of Statistics Islamic Azad iversity Ghaemshahr Brach
More informationDefinitions and Theorems. where x are the decision variables. c, b, and a are constant coefficients.
Defiitios ad Theorems Remember the scalar form of the liear programmig problem, Miimize, Subject to, f(x) = c i x i a 1i x i = b 1 a mi x i = b m x i 0 i = 1,2,, where x are the decisio variables. c, b,
More informationIntegrable Functions. { f n } is called a determining sequence for f. If f is integrable with respect to, then f d does exist as a finite real number
MATH 532 Itegrable Fuctios Dr. Neal, WKU We ow shall defie what it meas for a measurable fuctio to be itegrable, show that all itegral properties of simple fuctios still hold, ad the give some coditios
More informationChemical Engineering 160/260 Polymer Science and Engineering. Lecture 7  Statistics of Chain Copolymerization January 31, 2001
Chemical Egieerig 60/60 Polymer Sciece ad Egieerig Lecture 7  Statistics of Chai Copolymerizatio Jauary 3, 00 Objectives! To determie the compositioal relatioships betwee lower ad higher order sequeces
More informationFIXED POINTS OF nvalued MULTIMAPS OF THE CIRCLE
FIXED POINTS OF VALUED MULTIMAPS OF THE CIRCLE Robert F. Brow Departmet of Mathematics Uiversity of Califoria Los Ageles, CA 900951555 email: rfb@math.ucla.edu November 15, 2005 Abstract A multifuctio
More informationINCURSION OF THE GOLDEN RATIO Φ INTO THE SCHRÖDINGER WAVE FUNCTION USING THE Φ RECURSIVE HETERODYNING SET
INCURSION OF THE GOLDEN RATIO Φ INTO THE SCHRÖDINGER WAVE FUNCTION USING THE Φ RECURSIVE HETERODYNING SET S. Giadioto *, R. L. Amoroso & E.A. Rauscher * Advaced Laser Quatum Dyamics Research Istitute (ALQDRI)
More informationarxiv: v1 [math.pr] 13 Oct 2011
A tail iequality for quadratic forms of subgaussia radom vectors Daiel Hsu, Sham M. Kakade,, ad Tog Zhag 3 arxiv:0.84v math.pr] 3 Oct 0 Microsoft Research New Eglad Departmet of Statistics, Wharto School,
More informationAn Efficient LloydMax Quantizer for Matching Pursuit Decompositions
BRAZILVI ITERATIOAL TELECOMMUICATIOS SYMPOSIUM (ITS6), SEPTEMBER 36, 6, FORTALEZACE, BRAZIL A Efficiet LloydMax Quatizer for Matchig Pursuit Decompositios Lisadro Lovisolo, Eduardo A B da Silva ad Paulo
More informationHomework 2. Show that if h is a bounded sesquilinear form on the Hilbert spaces X and Y, then h has the representation
omework 2 1 Let X ad Y be ilbert spaces over C The a sesquiliear form h o X Y is a mappig h : X Y C such that for all x 1, x 2, x X, y 1, y 2, y Y ad all scalars α, β C we have (a) h(x 1 + x 2, y) h(x
More informationA General Family of Estimators for Estimating Population Variance Using Known Value of Some Population Parameter(s)
Rajesh Sigh, Pakaj Chauha, Nirmala Sawa School of Statistics, DAVV, Idore (M.P.), Idia Floreti Smaradache Uiversity of New Meico, USA A Geeral Family of Estimators for Estimatig Populatio Variace Usig
More information