EE 321 Fourier Series Examples Fall 2016

Size: px
Start display at page:

Download "EE 321 Fourier Series Examples Fall 2016"

Transcription

1 EE 31 Fourier Series Examples Fall A periodic sigal is give as xx(tt) = 3 cos(ππ8tt) + 7 si(ππ16tt). Fid the period, fudametal frequecy, ad Fourier series. Sketch the magitude spectrum. TT = ωω 0 = aa 0 = aa = bb =

2 . Fid the Fourier series for the sigal, x(t), show below. Plot the trucated Fourier series represetatio for = 0,1,3, ad 5. Period, T = sec. Fudametal frequecy = ωω 0 = ππ rad/s (ff TT 0 = 1 = 50 Hz). TT x ( t) = a 0 + a cos( ω 0t) + b si( ω0t) = 1 TT/ Trucated Fourier Series: x aa 0 = 1 TT xx(tt)dddd = 1 TT/ bb = 0 π a = si ( a = 0 for eve) π b = 0 (sice x(t) is a eve fuctio) 0 + a cos( ω 0t) + b si( 0t) = 1 ( t) = a ω

3 ow, suppose that the phase of the Fourier series terms are icorrect. Specifically, suppose that for all 1 < i the trucated Fourier series, there phases are off by exactly 180. This is show below, compared to the correct trucated Fourier series, for = 3,5, ad 15.

4 ote that the phase errors cause the recostructed waveforms to look quite differet from the origial square wave.

5 3. Covetioal AM geerated usig a switchig modulator. Problem: Use Fourier Series aalysis to explai how the output sigal i the circuit below, vv 0 (tt), ca be filtered to produce the evelope sigal of the form AA cc (cos(ππff cc tt) + μμμμ(tt)), where 0 μμ 1, ad the message sigal, mm(tt), is assumed scaled so that max tt mm(tt) < 1. Assume that the maximum value of m(t) is much less that A. The the sigal across the resistor is well-modeled as vv 0 (tt) = AA cos(ππff cc tt) + mm(tt) gg(tt) where gg(tt) = 1, iiii cos(ππff cctt) > 0; 0, iiii cos(ππff cc tt) < 0. Hece, gg(tt) is a square wave, as i the previous problem, with period TT = 1, ωω ff 0 = ππ, cc TT bb = 0, ad gg(tt) = 1 + ssssss cos(ωω 0 tt). =1 As a specific example, suppose that the message sigal is a cosie, say mm(tt) = cos (ππff mm tt), where ff mm =,000 Hz, the carrier frequecy is ff mm = 50,000 Hz, ad AA = 100. I AM radio the carrier frequecies are at least 100 times larger tha the highest frequecy i the message (e.g., a carrier frequecy of at least 540 khz ad a maximum message frequecy of 5 khz).

6 4. Fid the Fourier series of the periodic sigal, x (t), show below. Solutio: Write x( t) = xe ( t) + xo ( t) ad fid the Fourier series of the eve part ad odd part of x (t), the combie. xx(tt) + xx( tt) xx(tt) xx( tt) xx ee (tt) =, xx oo (tt) = The Fourier series coefficiets are the computed as aa 0 = 0 (for both xx ee (tt) ad xx oo (tt)). For the eve part, bb = 0 ad aa = 8 si. For the odd part, aa = 0 ad bb = 1 cos. The complete Fourier series for x(t) is the aa 0 = 0, aa = 8 bb = 1 cos. si, ad

7 1. Use Matlab to plot the magitude ad phase spectra.. Determie how may terms are required i a trucated Fourier series to have 10% ormalized MSE. To have 0.5% ormalized MSE. 1 b Magitude: A = a + b, Phase: θ = ta, a fuctio fourier_series1(1,) % Computes the 1-term plot of the magitude ad phase spectra % ad the -term ormalized MSE a0=0; =[1:1:max(1,)]; a=8*si(0.5*pi*)./(pi*); b=-*(1-cos(0.5*pi*))./(pi*); figure(1) subplot(,1,1) stem([0 (1:1)],[a0 sqrt(a(1:1).^ + b(1:1).^)]) xlabel('idex, ') ylabel('a_') title('magitude Spectrum') subplot(,1,) stem([0 (1:1)],[0 ata(b(1:1),a(1:1))*180/pi]) xlabel('idex, ') ylabel('\theta_, Degrees') title('phase Spectrum') ormmse=(4.5-(a0^+0.5*(a(1:)*a(1:)' + b(1:)*b(1:)')))/4.5; Percet_Error = ormmse*100 1 P [ a0 + a + b ] = 1 MSE orm = P 10% MMMMMM = 5 terms 0.5% MMMMMM = 85 terms

8 3. Determie the fractio of the sigal power i i) the first harmoic; i the first 3 harmoics. Percet Power i th harmoic = 0.5 (aa + bb ) PP 100% Usig P = 4.5, Percet Power i 1 st harmoic = 0.5 (aa 1 + bb 1 ) 100 PP ππ + ππ = 100 = 76.6% 4.5 Percet Power i first 3 harmoics = 0.5 (aa 1 + bb 1 + aa + bb + aa 3 + bb 3 ) 100% PP = 89.6% 4. Use Matlab to costruct the trucated Fourier series, x 0 + a cos( 0t) + b si( 0t) = 1 ( t) = a ω ω. 5. Also, use Matlab to costruct the trucated Fourier series from the magitude/phase form of the Fourier series coefficiets, Magitude: A = a + b, 1 b Phase: θ = ta, a x 0 + A cos( 0t ) = 1 ( t) = a ω θ. fuctio y=fs_example(,t,time,origial) % Fourier series example a0=0; fs=48000; % Samplig rate, i samples per secod. =[1:]; a=(8/pi)*si(*pi/)./; b=(-/pi)*(1-cos(*pi/))./; t=[-t:1/fs:t+time]; le_=floor(1+*t*fs); y=a0*oes(1,legth(t)); for k=1: y=y+a(k)*cos(k**pi*t/t)+b(k)*si(k**pi*t/t); ed

9 y0=y; figure(1) plot(t(1:le_),y(1:le_)) % Costruct usig magitude/phase form of Fourier series A=sqrt(a.^ + b.^); theta=ata(b,a); y=a0*oes(1,legth(t)); for k=1: y=y+a(k)*cos(k**pi*t/t-theta(k)); ed

10 6. Fially, radomize the phase, θ, ad costruct the trucated Fourier series, x ~ ( t) = a ω θ, 0 + A cos( 0t ) = 1 where ~ θ is θ plus a uiform radom phase variable over [ π, π ]. fuctio y=fs_example(,t,time,origial) % Fourier series example a0=0; fs=48000; % Samplig rate, i samples per secod. =[1:]; a=(8/pi)*si(*pi/)./; b=(-/pi)*(1-cos(*pi/))./; t=[-t:1/fs:t+time]; le_=floor(1+*t*fs); y=a0*oes(1,legth(t)); for k=1: y=y+a(k)*cos(k**pi*t/t)+b(k)*si(k**pi*t/t); ed y0=y; figure(1) plot(t(1:le_),y(1:le_)) % Costruct usig magitude/phase form of Fourier series A=sqrt(a.^ + b.^); theta=ata(b,a); y=a0*oes(1,legth(t)); for k=1: y=y+a(k)*cos(k**pi*t/t-theta(k)); ed figure() plot(t(1:le_),y(1:le_)) %ow radomize the phase theta1=theta+*pi*(rad(1,legth(theta))-0.5); figure(3) y=a0*oes(1,legth(t)); for k=1: y=y+a(k)*cos(k**pi*t/t-theta1(k)); ed plot(t(1:le_),y(1:le_)) if(origial==0) y=y0; ed >> y=fs_example(19,0.004,,0); % Output the trucated Fourier series waveform >> y1=fs_example(19,0.004,,1); % Output the Fourier series with radom phase % Play out the soud (the samplig rate is 48,000 samples per secod) >> soud(y/50,48000) >> soud(y1/50,48000) ote: Each time the program is ru, a differet radom set of phases is geerated, ad a differet recostructed waveform geerated.

11 Each time the program is ru, a differet recostructed radom waveform geerated.

12 Speech Codig Example: Segmet of speech file (top) ad segmet of improved multi-bad excitatio (IMBE) coded speech (bottom). The origial speech file is 1-bit PCM with a samplig rate of 8,000 samples/sec. Toll-quality μ-law PCM uses 8 bits/sample, ad a trasmissio rate of 64 kb/s. The IMBE speech is ecoded at rate 4,800 bits/s.

Trigonometric 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

Trigonometric 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 information

Tagore Engineering College Department of Electrical and Electronics Engineering EC 2314 Digital Signal Processing University Question Paper Part-A

Tagore Engineering College Department of Electrical and Electronics Engineering EC 2314 Digital Signal Processing University Question Paper Part-A Tagore Egieerig College Departmet of Electrical ad Electroics Egieerig EC 34 Digital Sigal Processig Uiversity Questio Paper Part-A Uit-I. Defie samplig theorem?. What is kow as Aliasig? 3. What is LTI

More information

Confidence Intervals

Confidence Intervals Cofidece Itervals Cofidece Itervals are a extesio of the cocept of Margi of Error which we met earlier i this course. Remember we saw: The sample proportio will differ from the populatio proportio by more

More information

1. MATHEMATICAL INDUCTION

1. MATHEMATICAL INDUCTION 1. MATHEMATICAL INDUCTION EXAMPLE 1: Prove that for ay iteger 1. Proof: 1 + 2 + 3 +... + ( + 1 2 (1.1 STEP 1: For 1 (1.1 is true, sice 1 1(1 + 1. 2 STEP 2: Suppose (1.1 is true for some k 1, that is 1

More information

CHAPTER 7: Central Limit Theorem: CLT for Averages (Means)

CHAPTER 7: Central Limit Theorem: CLT for Averages (Means) CHAPTER 7: Cetral Limit Theorem: CLT for Averages (Meas) X = the umber obtaied whe rollig oe six sided die oce. If we roll a six sided die oce, the mea of the probability distributio is X P(X = x) Simulatio:

More information

Study on the application of the software phase-locked loop in tracking and filtering of pulse signal

Study on the application of the software phase-locked loop in tracking and filtering of pulse signal Advaced Sciece ad Techology Letters, pp.31-35 http://dx.doi.org/10.14257/astl.2014.78.06 Study o the applicatio of the software phase-locked loop i trackig ad filterig of pulse sigal Sog Wei Xia 1 (College

More information

Math C067 Sampling Distributions

Math C067 Sampling Distributions Math C067 Samplig Distributios Sample Mea ad Sample Proportio Richard Beigel Some time betwee April 16, 2007 ad April 16, 2007 Examples of Samplig A pollster may try to estimate the proportio of voters

More information

M06/5/MATME/SP2/ENG/TZ2/XX MATHEMATICS STANDARD LEVEL PAPER 2. Thursday 4 May 2006 (morning) 1 hour 30 minutes INSTRUCTIONS TO CANDIDATES

M06/5/MATME/SP2/ENG/TZ2/XX MATHEMATICS STANDARD LEVEL PAPER 2. Thursday 4 May 2006 (morning) 1 hour 30 minutes INSTRUCTIONS TO CANDIDATES IB MATHEMATICS STANDARD LEVEL PAPER 2 DIPLOMA PROGRAMME PROGRAMME DU DIPLÔME DU BI PROGRAMA DEL DIPLOMA DEL BI 22067304 Thursday 4 May 2006 (morig) 1 hour 30 miutes INSTRUCTIONS TO CANDIDATES Do ot ope

More information

A probabilistic proof of a binomial identity

A 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 information

The Field Q of Rational Numbers

The Field Q of Rational Numbers Chapter 3 The Field Q of Ratioal Numbers I this chapter we are goig to costruct the ratioal umber from the itegers. Historically, the positive ratioal umbers came first: the Babyloias, Egyptias ad Grees

More information

Heat (or Diffusion) equation in 1D*

Heat (or Diffusion) equation in 1D* Heat (or Diffusio) equatio i D* Derivatio of the D heat equatio Separatio of variables (refresher) Worked eamples *Kreysig, 8 th Ed, Sectios.4b Physical assumptios We cosider temperature i a log thi wire

More information

Partial Di erential Equations

Partial 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 information

1 Computing the Standard Deviation of Sample Means

1 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 information

CHAPTER 3 DIGITAL CODING OF SIGNALS

CHAPTER 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 information

I. Chi-squared Distributions

I. Chi-squared Distributions 1 M 358K Supplemet to Chapter 23: CHI-SQUARED DISTRIBUTIONS, T-DISTRIBUTIONS, AND DEGREES OF FREEDOM To uderstad t-distributios, we first eed to look at aother family of distributios, the chi-squared distributios.

More information

Gregory Carey, 1998 Linear Transformations & Composites - 1. Linear Transformations and Linear Composites

Gregory Carey, 1998 Linear Transformations & Composites - 1. Linear Transformations and Linear Composites Gregory Carey, 1998 Liear Trasformatios & Composites - 1 Liear Trasformatios ad Liear Composites I Liear Trasformatios of Variables Meas ad Stadard Deviatios of Liear Trasformatios A liear trasformatio

More information

0.7 0.6 0.2 0 0 96 96.5 97 97.5 98 98.5 99 99.5 100 100.5 96.5 97 97.5 98 98.5 99 99.5 100 100.5

0.7 0.6 0.2 0 0 96 96.5 97 97.5 98 98.5 99 99.5 100 100.5 96.5 97 97.5 98 98.5 99 99.5 100 100.5 Sectio 13 Kolmogorov-Smirov test. Suppose that we have a i.i.d. sample X 1,..., X with some ukow distributio P ad we would like to test the hypothesis that P is equal to a particular distributio P 0, i.e.

More information

Hypothesis testing. Null and alternative hypotheses

Hypothesis 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 information

{{1}, {2, 4}, {3}} {{1, 3, 4}, {2}} {{1}, {2}, {3, 4}} 5.4 Stirling Numbers

{{1}, {2, 4}, {3}} {{1, 3, 4}, {2}} {{1}, {2}, {3, 4}} 5.4 Stirling Numbers . Stirlig Numbers Whe coutig various types of fuctios from., we quicly discovered that eumeratig the umber of oto fuctios was a difficult problem. For a domai of five elemets ad a rage of four elemets,

More information

Learning outcomes. Algorithms and Data Structures. Time Complexity Analysis. Time Complexity Analysis How fast is the algorithm? Prof. Dr.

Learning outcomes. Algorithms and Data Structures. Time Complexity Analysis. Time Complexity Analysis How fast is the algorithm? Prof. Dr. Algorithms ad Data Structures Algorithm efficiecy Learig outcomes Able to carry out simple asymptotic aalysisof algorithms Prof. Dr. Qi Xi 2 Time Complexity Aalysis How fast is the algorithm? Code the

More information

S. Tanny MAT 344 Spring 1999. be the minimum number of moves required.

S. Tanny MAT 344 Spring 1999. be the minimum number of moves required. S. Tay MAT 344 Sprig 999 Recurrece Relatios Tower of Haoi Let T be the miimum umber of moves required. T 0 = 0, T = 7 Iitial Coditios * T = T + $ T is a sequece (f. o itegers). Solve for T? * is a recurrece,

More information

1. C. The formula for the confidence interval for a population mean is: x t, which was

1. C. The formula for the confidence interval for a population mean is: x t, which was s 1. C. The formula for the cofidece iterval for a populatio mea is: x t, which was based o the sample Mea. So, x is guarateed to be i the iterval you form.. D. Use the rule : p-value

More information

WHEN IS THE (CO)SINE OF A RATIONAL ANGLE EQUAL TO A RATIONAL NUMBER?

WHEN IS THE (CO)SINE OF A RATIONAL ANGLE EQUAL TO A RATIONAL NUMBER? WHEN IS THE (CO)SINE OF A RATIONAL ANGLE EQUAL TO A RATIONAL NUMBER? JÖRG JAHNEL 1. My Motivatio Some Sort of a Itroductio Last term I tought Topological Groups at the Göttige Georg August Uiversity. This

More information

MATH 083 Final Exam Review

MATH 083 Final Exam Review MATH 08 Fial Eam Review Completig the problems i this review will greatly prepare you for the fial eam Calculator use is ot required, but you are permitted to use a calculator durig the fial eam period

More information

Lab 4 Sampling, Aliasing, FIR Filtering

Lab 4 Sampling, Aliasing, FIR Filtering 47 Lab 4 Sampling, Aliasing, FIR Filtering This is a software lab. In your report, please include all Matlab code, numerical results, plots, and your explanations of the theoretical questions. The due

More information

Estimating the Mean and Variance of a Normal Distribution

Estimating the Mean and Variance of a Normal Distribution Estimatig the Mea ad Variace of a Normal Distributio Learig Objectives After completig this module, the studet will be able to eplai the value of repeatig eperimets eplai the role of the law of large umbers

More information

Math 113 HW #11 Solutions

Math 113 HW #11 Solutions Math 3 HW # Solutios 5. 4. (a) Estimate the area uder the graph of f(x) = x from x = to x = 4 usig four approximatig rectagles ad right edpoits. Sketch the graph ad the rectagles. Is your estimate a uderestimate

More information

HANDOUT E.17 - EXAMPLES ON BODE PLOTS OF FIRST AND SECOND ORDER SYSTEMS

HANDOUT E.17 - EXAMPLES ON BODE PLOTS OF FIRST AND SECOND ORDER SYSTEMS Lecture 7,8 Augut 8, 00 HANDOUT E7 - EXAMPLES ON BODE PLOTS OF FIRST AND SECOND ORDER SYSTEMS Example Obtai the Bode plot of the ytem give by the trafer fuctio ( We covert the trafer fuctio i the followig

More information

Major Coefficients Recovery: a Compressed Data Gathering Scheme for Wireless Sensor Network

Major Coefficients Recovery: a Compressed Data Gathering Scheme for Wireless Sensor Network This full text paper was peer reviewed at the directio of IEEE Commuicatios Society subject matter experts for publicatio i the IEEE Globecom proceedigs. Major Coefficiets Recovery: a Compressed Data Gatherig

More information

2-3 The Remainder and Factor Theorems

2-3 The Remainder and Factor Theorems - The Remaider ad Factor Theorems Factor each polyomial completely usig the give factor ad log divisio 1 x + x x 60; x + So, x + x x 60 = (x + )(x x 15) Factorig the quadratic expressio yields x + x x

More information

Convexity, Inequalities, and Norms

Convexity, Inequalities, and Norms Covexity, Iequalities, ad Norms Covex Fuctios You are probably familiar with the otio of cocavity of fuctios. Give a twicedifferetiable fuctio ϕ: R R, We say that ϕ is covex (or cocave up) if ϕ (x) 0 for

More information

Divide and Conquer. Maximum/minimum. Integer Multiplication. CS125 Lecture 4 Fall 2015

Divide and Conquer. Maximum/minimum. Integer Multiplication. CS125 Lecture 4 Fall 2015 CS125 Lecture 4 Fall 2015 Divide ad Coquer We have see oe geeral paradigm for fidig algorithms: the greedy approach. We ow cosider aother geeral paradigm, kow as divide ad coquer. We have already see a

More information

Our aim is to show that under reasonable assumptions a given 2π-periodic function f can be represented as convergent series

Our aim is to show that under reasonable assumptions a given 2π-periodic function f can be represented as convergent series 8 Fourier Series Our aim is to show that uder reasoable assumptios a give -periodic fuctio f ca be represeted as coverget series f(x) = a + (a cos x + b si x). (8.) By defiitio, the covergece of the series

More information

Properties of MLE: consistency, asymptotic normality. Fisher information.

Properties of MLE: consistency, asymptotic normality. Fisher information. Lecture 3 Properties of MLE: cosistecy, asymptotic ormality. Fisher iformatio. I this sectio we will try to uderstad why MLEs are good. Let us recall two facts from probability that we be used ofte throughout

More information

Fast Fourier Transform and MATLAB Implementation

Fast Fourier Transform and MATLAB Implementation Fast Fourier Trasform ad MATLAB Implemetatio by aju Huag for Dr. Duca L. MacFarlae Sigals I the fields of commuicatios, sigal processig, ad i electrical egieerig moregeerally, a sigalisay time varyig or

More information

Overview of some probability distributions.

Overview of some probability distributions. Lecture Overview of some probability distributios. I this lecture we will review several commo distributios that will be used ofte throughtout the class. Each distributio is usually described by its probability

More information

1 The Gaussian channel

1 The Gaussian channel ECE 77 Lecture 0 The Gaussia chael Objective: I this lecture we will lear about commuicatio over a chael of practical iterest, i which the trasmitted sigal is subjected to additive white Gaussia oise.

More information

Printing Letters Correctly

Printing Letters Correctly Printing Letters Correctly The ball and stick method of teaching beginners to print has been proven to be the best. Letters formed this way are easier for small children to print, and this print is similar

More information

Elementary Theory of Russian Roulette

Elementary Theory of Russian Roulette Elemetary Theory of Russia Roulette -iterestig patters of fractios- Satoshi Hashiba Daisuke Miematsu Ryohei Miyadera Itroductio. Today we are goig to study mathematical theory of Russia roulette. If some

More information

7. Sample Covariance and Correlation

7. Sample Covariance and Correlation 1 of 8 7/16/2009 6:06 AM Virtual Laboratories > 6. Radom Samples > 1 2 3 4 5 6 7 7. Sample Covariace ad Correlatio The Bivariate Model Suppose agai that we have a basic radom experimet, ad that X ad Y

More information

TAYLOR SERIES, POWER SERIES

TAYLOR SERIES, POWER SERIES TAYLOR SERIES, POWER SERIES The followig represets a (icomplete) collectio of thigs that we covered o the subject of Taylor series ad power series. Warig. Be prepared to prove ay of these thigs durig the

More information

Mathematical goals. Starting points. Materials required. Time needed

Mathematical goals. Starting points. Materials required. Time needed Level A1 of challege: C A1 Mathematical goals Startig poits Materials required Time eeded Iterpretig algebraic expressios To help learers to: traslate betwee words, symbols, tables, ad area represetatios

More information

1 Introduction to reducing variance in Monte Carlo simulations

1 Introduction to reducing variance in Monte Carlo simulations Copyright c 007 by Karl Sigma 1 Itroductio to reducig variace i Mote Carlo simulatios 11 Review of cofidece itervals for estimatig a mea I statistics, we estimate a uow mea µ = E(X) of a distributio by

More information

Confidence Intervals for One Mean

Confidence 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 information

4.1 Sigma Notation and Riemann Sums

4.1 Sigma Notation and Riemann Sums 0 the itegral. Sigma Notatio ad Riema Sums Oe strategy for calculatig the area of a regio is to cut the regio ito simple shapes, calculate the area of each simple shape, ad the add these smaller areas

More information

, a Wishart distribution with n -1 degrees of freedom and scale matrix.

, a Wishart distribution with n -1 degrees of freedom and scale matrix. UMEÅ UNIVERSITET Matematisk-statistiska istitutioe Multivariat dataaalys D MSTD79 PA TENTAMEN 004-0-9 LÖSNINGSFÖRSLAG TILL TENTAMEN I MATEMATISK STATISTIK Multivariat dataaalys D, 5 poäg.. Assume that

More information

THE UNLIKELY UNION OF PARTITIONS AND DIVISORS

THE UNLIKELY UNION OF PARTITIONS AND DIVISORS THE UNLIKELY UNION OF PARTITIONS AND DIVISORS Abdulkadir Hasse, Thomas J. Osler, Mathematics Departmet ad Tirupathi R. Chadrupatla, Mechaical Egieerig Rowa Uiversity Glassboro, NJ 828 I the multiplicative

More information

8.1 Arithmetic Sequences

8.1 Arithmetic Sequences MCR3U Uit 8: Sequeces & Series Page 1 of 1 8.1 Arithmetic Sequeces Defiitio: A sequece is a comma separated list of ordered terms that follow a patter. Examples: 1, 2, 3, 4, 5 : a sequece of the first

More information

Escola Federal de Engenharia de Itajubá

Escola Federal de Engenharia de Itajubá Escola Federal de Egeharia de Itajubá Departameto de Egeharia Mecâica Pós-Graduação em Egeharia Mecâica MPF04 ANÁLISE DE SINAIS E AQUISÇÃO DE DADOS SINAIS E SISTEMAS Trabalho 02 (MATLAB) Prof. Dr. José

More information

The Relaxation Oscillator

The Relaxation Oscillator Explore More! Points awarded: Name: Net ID: The Relaxation Oscillator Laboratory Outline In Lab 5, we constructed a simple three-element oscillator using a capacitor, resistor, and a Schmitt trigger inverter.

More information

Using Excel to Construct Confidence Intervals

Using Excel to Construct Confidence Intervals OPIM 303 Statistics Ja Stallaert Usig Excel to Costruct Cofidece Itervals This hadout explais how to costruct cofidece itervals i Excel for the followig cases: 1. Cofidece Itervals for the mea of a populatio

More information

Soving Recurrence Relations

Soving 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 information

Infinite Sequences and Series

Infinite Sequences and Series CHAPTER 4 Ifiite Sequeces ad Series 4.1. Sequeces A sequece is a ifiite ordered list of umbers, for example the sequece of odd positive itegers: 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29...

More information

Now here is the important step

Now 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 information

Notes on Hypothesis Testing

Notes on Hypothesis Testing Probability & Statistics Grishpa Notes o Hypothesis Testig A radom sample X = X 1,..., X is observed, with joit pmf/pdf f θ x 1,..., x. The values x = x 1,..., x of X lie i some sample space X. The parameter

More information

Multi-server Optimal Bandwidth Monitoring for QoS based Multimedia Delivery Anup Basu, Irene Cheng and Yinzhe Yu

Multi-server Optimal Bandwidth Monitoring for QoS based Multimedia Delivery Anup Basu, Irene Cheng and Yinzhe Yu Multi-server Optimal Badwidth Moitorig for QoS based Multimedia Delivery Aup Basu, Iree Cheg ad Yizhe Yu Departmet of Computig Sciece U. of Alberta Architecture Applicatio Layer Request receptio -coectio

More information

Convention Paper 6764

Convention Paper 6764 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

More information

Chapter 6: Variance, the law of large numbers and the Monte-Carlo method

Chapter 6: Variance, the law of large numbers and the Monte-Carlo method Chapter 6: Variace, the law of large umbers ad the Mote-Carlo 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 information

5.3. Generalized Permutations and Combinations

5.3. Generalized Permutations and Combinations 53 GENERALIZED PERMUTATIONS AND COMBINATIONS 73 53 Geeralized Permutatios ad Combiatios 53 Permutatios with Repeated Elemets Assume that we have a alphabet with letters ad we wat to write all possible

More information

4 n. n 1. You shold think of the Ratio Test as a generalization of the Geometric Series Test. For example, if a n ar n is a geometric sequence then

4 n. n 1. You shold think of the Ratio Test as a generalization of the Geometric Series Test. For example, if a n ar n is a geometric sequence then SECTION 2.6 THE RATIO TEST 79 2.6. THE RATIO TEST We ow kow how to hadle series which we ca itegrate (the Itegral Test), ad series which are similar to geometric or p-series (the Compariso Test), but of

More information

A Theoretical and Experimental Analysis of the Acoustic Guitar. Eric Battenberg ME 173 5-18-09

A Theoretical and Experimental Analysis of the Acoustic Guitar. Eric Battenberg ME 173 5-18-09 A Theoretical ad Experimetal Aalysis of the Acoustic Guitar Eric Batteberg ME 173 5-18-09 1 Itroductio ad Methods The acoustic guitar is a striged musical istrumet frequetly used i popular music. Because

More information

Solving Divide-and-Conquer Recurrences

Solving Divide-and-Conquer Recurrences Solvig Divide-ad-Coquer Recurreces Victor Adamchik A divide-ad-coquer algorithm cosists of three steps: dividig a problem ito smaller subproblems solvig (recursively) each subproblem the combiig solutios

More information

Definition. 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

Definition. 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 information

Vectors and Phasors. A supplement for students taking BTEC National, Unit 5, Electrical and Electronic Principles. Owen Bishop

Vectors and Phasors. A supplement for students taking BTEC National, Unit 5, Electrical and Electronic Principles. Owen Bishop Vectors and phasors Vectors and Phasors A supplement for students taking BTEC National, Unit 5, Electrical and Electronic Principles Owen Bishop Copyrught 2007, Owen Bishop 1 page 1 Electronics Circuits

More information

EE 2274 DIODE OR GATE & CLIPPING CIRCUIT

EE 2274 DIODE OR GATE & CLIPPING CIRCUIT Prelab Part I: Diode OR Gate LTspice use 1N4002 EE 2274 DIODE OR GATE & CLIPPING CIRCUIT 1. Design a diode OR gate, Figure 1 in which the maximum current thru R1 I R1 = 9mA assume Vin = 5Vdc. Design the

More information

.04. This means $1000 is multiplied by 1.02 five times, once for each of the remaining sixmonth

.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 information

Basic Measurement Issues. Sampling Theory and Analog-to-Digital Conversion

Basic Measurement Issues. Sampling Theory and Analog-to-Digital Conversion Theory ad Aalog-to-Digital Coversio Itroductio/Defiitios Aalog-to-digital coversio Rate Frequecy Aalysis Basic Measuremet Issues Reliability the extet to which a measuremet procedure yields the same results

More information

3. Covariance and Correlation

3. Covariance and Correlation Virtual Laboratories > 3. Expected Value > 1 2 3 4 5 6 3. Covariace ad Correlatio Recall that by takig the expected value of various trasformatios of a radom variable, we ca measure may iterestig characteristics

More information

CVE 372 HYDROMECHANICS EXERCISE PROBLEMS OPEN CHANNEL FLOWS

CVE 372 HYDROMECHANICS EXERCISE PROBLEMS OPEN CHANNEL FLOWS CVE 72 HYDROMECHANICS EXERCISE PROBLEMS OPEN CHANNEL FLOWS ) A rectagular irrigatio chael of base width m, is to covey 0.2 m /s discharge at a depth of 0.5 m uder uiform flow coditios. The slope of the

More information

Repeating Decimals are decimal numbers that have number(s) after the decimal point that repeat in a pattern.

Repeating Decimals are decimal numbers that have number(s) after the decimal point that repeat in a pattern. 5.5 Fractios ad Decimals Steps for Chagig a Fractio to a Decimal. Simplify the fractio, if possible. 2. Divide the umerator by the deomiator. d d Repeatig Decimals Repeatig Decimals are decimal umbers

More information

USING STATISTICAL FUNCTIONS ON A SCIENTIFIC CALCULATOR

USING STATISTICAL FUNCTIONS ON A SCIENTIFIC CALCULATOR USING STATISTICAL FUNCTIONS ON A SCIENTIFIC CALCULATOR Objective:. Improve calculator skills eeded i a multiple choice statistical eamiatio where the eam allows the studet to use a scietific calculator..

More information

Sampling Distribution And Central Limit Theorem

Sampling Distribution And Central Limit Theorem () Samplig Distributio & Cetral Limit Samplig Distributio Ad Cetral Limit Samplig distributio of the sample mea If we sample a umber of samples (say k samples where k is very large umber) each of size,

More information

Example 2 Find the square root of 0. The only square root of 0 is 0 (since 0 is not positive or negative, so those choices don t exist here).

Example 2 Find the square root of 0. The only square root of 0 is 0 (since 0 is not positive or negative, so those choices don t exist here). BEGINNING ALGEBRA Roots ad Radicals (revised summer, 00 Olso) Packet to Supplemet the Curret Textbook - Part Review of Square Roots & Irratioals (This portio ca be ay time before Part ad should mostly

More information

APPLICATION NOTE 30 DFT or FFT? A Comparison of Fourier Transform Techniques

APPLICATION NOTE 30 DFT or FFT? A Comparison of Fourier Transform Techniques APPLICATION NOTE 30 DFT or FFT? A Compariso of Fourier Trasform Techiques This applicatio ote ivestigates differeces i performace betwee the DFT (Discrete Fourier Trasform) ad the FFT(Fast Fourier Trasform)

More information

x(x 1)(x 2)... (x k + 1) = [x] k n+m 1

x(x 1)(x 2)... (x k + 1) = [x] k n+m 1 1 Coutig mappigs For every real x ad positive iteger k, let [x] k deote the fallig factorial ad x(x 1)(x 2)... (x k + 1) ( ) x = [x] k k k!, ( ) k = 1. 0 I the sequel, X = {x 1,..., x m }, Y = {y 1,...,

More information

Experiment 3: Double Sideband Modulation (DSB)

Experiment 3: Double Sideband Modulation (DSB) Experiment 3: Double Sideband Modulation (DSB) This experiment examines the characteristics of the double-sideband (DSB) linear modulation process. The demodulation is performed coherently and its strict

More information

In nite Sequences. Dr. Philippe B. Laval Kennesaw State University. October 9, 2008

In 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 information

A Combined Continuous/Binary Genetic Algorithm for Microstrip Antenna Design

A 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 16804-0030 haupt@ieee.org Abstract:

More information

3. Greatest Common Divisor - Least Common Multiple

3. 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 information

PSYCHOLOGICAL STATISTICS

PSYCHOLOGICAL 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 information

Frequency Domain Characterization of Signals. Yao Wang Polytechnic University, Brooklyn, NY11201 http: //eeweb.poly.edu/~yao

Frequency Domain Characterization of Signals. Yao Wang Polytechnic University, Brooklyn, NY11201 http: //eeweb.poly.edu/~yao Frequency Domain Characterization of Signals Yao Wang Polytechnic University, Brooklyn, NY1121 http: //eeweb.poly.edu/~yao Signal Representation What is a signal Time-domain description Waveform representation

More information

Sequences II. Chapter 3. 3.1 Convergent Sequences

Sequences II. Chapter 3. 3.1 Convergent Sequences Chapter 3 Sequeces II 3. Coverget Sequeces Plot a graph of the sequece a ) = 2, 3 2, 4 3, 5 + 4,...,,... To what limit do you thik this sequece teds? What ca you say about the sequece a )? For ǫ = 0.,

More information

Cooley-Tukey. Tukey FFT Algorithms. FFT Algorithms. Cooley

Cooley-Tukey. Tukey FFT Algorithms. FFT Algorithms. Cooley Cooley Cooley-Tuey Tuey FFT Algorithms FFT Algorithms Cosider a legth- sequece x[ with a -poit DFT X[ where Represet the idices ad as +, +, Cooley Cooley-Tuey Tuey FFT Algorithms FFT Algorithms Usig these

More information

SAMPLE QUESTIONS FOR FINAL EXAM. (1) (2) (3) (4) Find the following using the definition of the Riemann integral: (2x + 1)dx

SAMPLE QUESTIONS FOR FINAL EXAM. (1) (2) (3) (4) Find the following using the definition of the Riemann integral: (2x + 1)dx SAMPLE QUESTIONS FOR FINAL EXAM REAL ANALYSIS I FALL 006 3 4 Fid the followig usig the defiitio of the Riema itegral: a 0 x + dx 3 Cosider the partitio P x 0 3, x 3 +, x 3 +,......, x 3 3 + 3 of the iterval

More information

CS103A Handout 23 Winter 2002 February 22, 2002 Solving Recurrence Relations

CS103A Handout 23 Winter 2002 February 22, 2002 Solving Recurrence Relations CS3A Hadout 3 Witer 00 February, 00 Solvig Recurrece Relatios Itroductio A wide variety of recurrece problems occur i models. Some of these recurrece relatios ca be solved usig iteratio or some other ad

More information

Domain 1 - Describe Cisco VoIP Implementations

Domain 1 - Describe Cisco VoIP Implementations Maual ONT (642-8) 1-800-418-6789 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 information

THE ARITHMETIC OF INTEGERS. - multiplication, exponentiation, division, addition, and subtraction

THE 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 information

Section 11.3: The Integral Test

Section 11.3: The Integral Test Sectio.3: The Itegral Test Most of the series we have looked at have either diverged or have coverged ad we have bee able to fid what they coverge to. I geeral however, the problem is much more difficult

More information

Chapter 7 - Sampling Distributions. 1 Introduction. What is statistics? It consist of three major areas:

Chapter 7 - Sampling Distributions. 1 Introduction. What is statistics? It consist of three major areas: Chapter 7 - Samplig Distributios 1 Itroductio What is statistics? It cosist of three major areas: Data Collectio: samplig plas ad experimetal desigs Descriptive Statistics: umerical ad graphical summaries

More information

UC Berkeley Department of Electrical Engineering and Computer Science. EE 126: Probablity and Random Processes. Solutions 9 Spring 2006

UC Berkeley Department of Electrical Engineering and Computer Science. EE 126: Probablity and Random Processes. Solutions 9 Spring 2006 Exam format UC Bereley Departmet of Electrical Egieerig ad Computer Sciece EE 6: Probablity ad Radom Processes Solutios 9 Sprig 006 The secod midterm will be held o Wedesday May 7; CHECK the fial exam

More information

Class Meeting # 16: The Fourier Transform on R n

Class Meeting # 16: The Fourier Transform on R n MATH 18.152 COUSE NOTES - CLASS MEETING # 16 18.152 Itroductio to PDEs, Fall 2011 Professor: Jared Speck Class Meetig # 16: The Fourier Trasform o 1. Itroductio to the Fourier Trasform Earlier i the course,

More information

Normal Distribution.

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 information

bello words how to build them? tt tt

bello words how to build them? tt tt 99 practical stuff Reproducing and using this material in any form without a written permission from Underware is probhibited. p 1/8 bello words how to build them? tt tt tt 11 Rep input the same character

More information

Approximating the Sum of a Convergent Series

Approximating the Sum of a Convergent Series Approximatig the Sum of a Coverget Series Larry Riddle Ages Scott College Decatur, GA 30030 lriddle@agesscott.edu The BC Calculus Course Descriptio metios how techology ca be used to explore covergece

More information

Free Animal Alphabet Match. Halloween Version & Anytime version. October and November Calendar pieces

Free Animal Alphabet Match. Halloween Version & Anytime version. October and November Calendar pieces Free Animal Alphabet Match Halloween Version & Anytime version October and November Calendar pieces Aa alligator Recognize and name all upper- and lowercase letters of the alphabet. Table of Contents ALL

More information

UNIT 3 SUMMARY STATIONS THROUGHOUT THE NEXT 2 DAYS, WE WILL BE SUMMARIZING THE CONCEPT OF EXPONENTIAL FUNCTIONS AND THEIR VARIOUS APPLICATIONS.

UNIT 3 SUMMARY STATIONS THROUGHOUT THE NEXT 2 DAYS, WE WILL BE SUMMARIZING THE CONCEPT OF EXPONENTIAL FUNCTIONS AND THEIR VARIOUS APPLICATIONS. Name: Group Members: UNIT 3 SUMMARY STATIONS THROUGHOUT THE NEXT DAYS, WE WILL BE SUMMARIZING THE CONCEPT OF EXPONENTIAL FUNCTIONS AND THEIR VARIOUS APPLICATIONS. EACH ACTIVITY HAS A COLOR THAT CORRESPONDS

More information

ACCESS - MATH July 2003 Notes on Body Mass Index and actual national data

ACCESS - MATH July 2003 Notes on Body Mass Index and actual national data ACCESS - MATH July 2003 Notes o Body Mass Idex ad actual atioal data What is the Body Mass Idex? If you read ewspapers ad magazies it is likely that oce or twice a year you ru across a article about the

More information

A Resource for Free-standing Mathematics Qualifications Working with %

A Resource for Free-standing Mathematics Qualifications Working with % Ca you aswer these questios? A savigs accout gives % iterest per aum.. If 000 is ivested i this accout, how much will be i the accout at the ed of years? A ew car costs 16 000 ad its value falls by 1%

More information

Sequences and Series

Sequences and Series CHAPTER 9 Sequeces ad Series 9.. Covergece: Defiitio ad Examples Sequeces The purpose of this chapter is to itroduce a particular way of geeratig algorithms for fidig the values of fuctios defied by their

More information

University of California, Los Angeles Department of Statistics. Distributions related to the normal distribution

University 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 Chi-square (χ ) distributio.

More information