# Escola Federal de Engenharia de Itajubá

Save this PDF as:
Size: px
Start display at page:

## Transcription

1 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 Juliao de Lima Jr. home page: Setembro de 2001

2 MPF04 Aálise de Siais e Aquisção de Dados 2/14 Ídice 1.1 Itroductio Project 1: Basic Sigals Exercise: Basic Sigals - Impulses Exercise: Basic Sigals Siusoids Exercise: Sampled Siusoids Exercise: Basic Sigals-Expoetials Project 2: Complex-Valued Sigals Exercise: Complex Expoetials 12

3 Itroductio 3/14 Trabalho 02 SINAIS E SISTEMAS 1.1 Itroductio The basic sigals used ofte i digital sigal processig are the uit impulse sigal δ [, expoetials of the form u[, sie waves, ad their geeralizatio to complex expoetials. The followig projects are directed at the geeratio ad represetatio of these sigals i MATLAB. Sice the oly umerical data type i MATLAB is the M x N matrix, sigal must be represeted as vectors: either M x 1 matrices if colum vectors, or 1 x N matrices if row vectors. I MATLAB all sigals must be fiite i legth. This cotrasts sharply with aalytical problem solvig, where a mathematical formula ca be used to represet a ifiite-legth sigal (e. g., a decayig expoetial, a u[ ). a A Secod issue is the idexig domai associated with a sigal vector. MATLAB assumes by default that a vector is idexed from 1 to N, the vector legth. I cotrast, a sigal vector is ofte the result of samplig a sigal over some domai where the idexig rus from 0 to N-1; or, perhaps, the samplig start at some arbitrary idex that is egative, e. g., at N. The iformatio about the samplig domai caot be attached to the sigal vector cotaiig the sigal values. Istead, the user is forced to keep track of this iformatio separately. Usually, this is ot a problem util it comes time to plot the sigal, i which case the horizotal axis must be labeled properly.

4 MPF04 Aálise de Siais e Aquisção de Dados 4/14 A fial poit is the use of MATLAB's vector otatio to geerate sigals. A sigificat power of the MATLAB eviromet is its high-level otatio for vector maipulatio; for loops are almost always uecessary. Whe creatig sigals such as a sie wave, it is best to apply the si fuctio to a vector argumet, cosistig of all the time samples. I the followig projects, we treat the commo sigals ecoutered i digital sigal processig: impulses, impulse trais, expoetials, ad siusoids. 1.2 Project 1: Basic Sigals This project cocetrates o the issues ivolved with geeratio of some basic discrete-time sigals i MATLAB. Much of the work ceters o usig iteral MATLAB vector routies for sigal geeratio. I additio, a sample MATLAB fuctio will be implemeted. Hits Plottig discrete-time sigals is doe with the stem fuctio i MATLAB. The followig MATLAB code will create 31 poits of a discrete-time siusoid. = 0:30; % vector of time idices sius = si(/2+1); Notice that the = 0 idex must be referred to as (1), due to MATLAB's idexig scheme; likewise, sius(1) is the first value i the siusoid. Whe plottig the sie wave we would use the stem fuctio, which produces the discrete-time sigal plot commoly see i DSP books (see Figura 1.1): stem(,sius ); The first vector argumet must be give i order to get the correct -axis For compariso, try stem(sius) to see the default labelig.

5 Project 1: Basic Sigals 5/14 1 sie wave Time idex () Figura 1.1 -Pottig a discrete-time sigal with stem Exercise: Basic Sigals - Impulses The simplest sigal is the (shifted) uit impulse sigal: 1, = 0 δ[ 0] = (1.1) 0, 0. To create a impulse i MATLAB, we must decide how much of the sigal is of iterest. If the impulse δ [ is goig to be used to drive a causal LTI system, we might wat to see the L poits from = 0 to =L-1. If we choose L = 31, the followig MATLAB code will create a "impulse": L = 31; = 0:(L-1); imp = zeros(l,1); imp(1)=1; Notice that the = 0 idex must be referred to as imp(1), due to MATLAB's idexig scheme. (a). Geerate ad plot the followig sequeces. I each case the horizotal () axis should exted oly over the rage idicated ad should be labeled accordigly. Each sequece should be displayed as a discrete-time sigal usig stem

6 MPF04 Aálise de Siais e Aquisção de Dados 6/14 x [ = 0.9δ[ 5], 1 20, 1 x [ = 0.8δ[, 15 15, 2 x [ = 1.5δ[ 333], , 3 x [ = 4.5δ[ + 7], (b) The shifted impulse, δ ], ca be used to build a weighted [ 0 impulse trai, with period P ad total legth MP: s[ = M 1 l= 0 Alδ[ l P]. (1.2) The weights are AL; if they are all the same. The impulse trai is periodic with period P. Geerate ad plot a periodic impulse trai whose period is P = 5 ad whose legth is 50. Start the sigal at =0. Try to use oe or two vector operatios rather tha a for loop to set the impulse locatios. How may impulses are cotaied withi the fiitelegth sigal? (c). The followig MATLAB code will produce a repetitive sigal i the vector x. x=[0;1;1;0;0;0]*oes(1,7); x=x(:); size(x) % retur the sigal legth Plot x to visualize its form; the give a mathematical formula similar to equatio (1.2) to describe this sigal Exercise: Basic Sigals Siusoids Aother very basic sigal is cosie wave. I geeral, it takes three parameters to describe a real siusoidal sigal completely: amplitude (A), frequecy (Ω0), ad phase (φ). x [ = Acos( Ω0 + φ) (1.3)

7 Project 1: Basic Sigals 7/14 (a). Geerate ad plot each of the followig sequeces. Use MATLAB's vector capability to do this with oe fuctio call by takig the cosie (or sie) of a vector argumet. I each case, the horizotal () axis should exted oly over the rage idicated ad should be labeled accordigly. Each sequece should be displayed as a sequece usig stem. π x1[ = si( ), , π x2[ = si( ), , π x3[ = si(3π + ), , x4[ = cos( π ), Give a simpler formula for x3[ that does ot use trigoometric fuctios. Explai why x4[ is a ot periodic sequece. (b). Write a MATLAB fuctio that will geerate a fiite-legth siusoid. The fuctio will eed a total of five iput argumets: three for the parameters ad two more to specify the first ad last idex of the fiite-legth sigal. The fuctio should retur the result a colum vector that cotais the values of the siusoid. Test this fuctio by plottig the results for various choices of the iput parameters. I particular, show how to geerate the sigal 2si( π /11) for (c). Modificatio: Rewrite the fuctio i part (b) to retur two argumets: a vector of idices over the rager of, ad the values of the sigal Exercise: Sampled Siusoids Ofte a discrete-time sigal is produced by samplig a cotiuous-time sigal such as a costat-frequecy sie wave. The relatioship betwee the cotiuos-time frequecy ad the samplig frequecy is mai poit of

8 MPF04 Aálise de Siais e Aquisção de Dados 8/14 Nyquist-Shao samplig theorem, which requires that samplig frequecy be at least twice the highest frequecy i the sigal for perfect recostructio. I geeral, a cotiuos-time siusoid is give by the followig mathematical formula: s ( t) = Acos(2πf0t + φ) (1.4) where A is the amplitude f0 is the frequecy i Hertz, ad φ is the iitial phase. If a discrete-time sigal is produced by regular samplig of s(t) at a rate of fs=1/t, we get f 0 s[ = s( t) = Acos(2πf T + φ = A π + φ t= T 0 s ) cos 2 (1.5) f s Compariso with formula (1.4) for a discrete-time siusoid, x[=acos(ω0+φ), shows that the ormalized radia frequecy is ow a scaled versio of f0, Ω0=2π(f0Ts). (a). From formula (1.3) for the cotiuous-time siusoid, write a fuctio that will geerate samples of s(t) to create a fiite-legth discrete-time sigal. This fuctio will require six iputs: three for the sigal parameters, two for the start ad stop times, ad oe for samplig rate (i Hertz). It ca call the previously writte MATLAB fuctio for the discrete-time siusoid. To make the MATLAB fuctio correspod to the cotiuous-time sigal defiitio, make the uits of the start ad stop times secods ot idex umber. Use this fuctio to geerate a sample siusoid with the followig defiitio: Sigal freq= 1200 Hz Samplig freq = 8 KHz Iitial Phase= 45 deg Startig Time= 0 s Amplitude = 50 Edig Time = 7 ms

9 Project 1: Basic Sigals 9/14 Make two plots of the resultig sigal: oe as a fuctio of time t (i millisecod), ad the other as a fuctio of the sample idex used i t=ts. Determie the legth of the resultig discrete-time sigal ad the umber of periods of the siusoid icluded i the vector. (b). Show by mathematical maipulatio that samplig a cosie at the times t = +3/4 T will result i a discrete-time sigal that appears to be a sie wave whe f0 = 1/T. Use the fuctio from part (a) to geerate a discrete-time sie wave by chagig the start ad stop times for the samplig Exercise: Basic Sigals-Expoetials The decayig expoetial is a basic sigal i DSP because it occurs as the solutio to liear costat-coefficiet differece equatios. (a). Study the followig MATLAB fuctio to see how it geerates a discrete-time expoetial sigal. The use the fuctio to plot the expoetial x [ = (0.9) over the rage = 0,1, 2,..., 20. fuctio y = geexp(b,0,l) %GENEXP geerate a expoetial sigal: b^ % usage: y = geexp(b,0,l) % b iput scalar givig ratio betwee terms % 0 startig idex (iteger) % L legth of geerated sigal % y output sigal y(1:l) if(l<= 0) error('genexp: legth ot positive') ed =0+[1:L]'-1; %---vector of idices y=b.^; ed (b). I may derivatios,the expoetial sequece a u[ must be summed over a fiite rage. This sum is kow i closed form: L 1 = 0 a L 1 a = 1 a, for a 1 (1.6)

10 MPF04 Aálise de Siais e Aquisção de Dados 10/14 Use the fuctio from part (a) to geerate a expoetial ad the sum it up; compare the result to formula (1.6). (c). Oe reaso the expoetial sequece occurs so ofte i DSP is that time shiftig does ot chage the character of the sigal. Show that the fiite-legth expoetial sigal satisfies the shiftig relatio: y[ = ay[ 1] over the rage1 L 1 (1.7) By comparig the vectors y(2 :L) ad a *y(1:l-1). Whe shiftig fiitelegth sigals i MATLAB, we must be careful at the edpoits because there is o automatic zero paddig. (d). Aother way to geerate a expoetial sigal is to use a recursive formula give by a differece equatio. The sigal y[=a u[ is the solutio to the followig differece equatio whe the iput, x[, is a impulse: y[ ay[ 1] = x[ iitial coditio : y[ 1] = 0 (1.8) Sice the differece equatio is assumed to recurse i a causal maer (i. e., for icreasig ), the iitial coditios at =-1 is ecessary. I MATLAB the fuctio filter will implemet a differece equatio. Use filter to geerate the same sigal as i part (a) (i. e., a = 0.9). 1.3 Project 2: Complex-Valued Sigals This project ceters o the issues ivolved with represetig ad geeratig complex-valued sigals. Although i the real world, sigals must have real values, it is ofte extremely useful to geerate, process, ad iterpret real-valued sigal pairs as complex-valued sigals. This is doe by combiig the sigals ito a pair, as the real ad imagiary parts of a complex umber, ad processig this pair with other complex-valued sigals

11 Project 2: Complex-Valued Sigals 11/14 usig the rules of complex arithmetic. Use of sigal pairs i this way is a importat part of may sigal processig systems, especially those ivolvig modulatio. Complex expoetials are a class of complex sigals that is extremely importat because the complex amplitude (phasor otatio) provides a cocise way to describe siusoidal sigals. Most electrical egieerig studets are familiar with phasors i coectio with ac circuits or power systems, but their use i radar, wave propagatio, ad Fourier aalysis is just as sigificat (although the term phasor is ot always used). Hits I MATLAB, the fuctios real ad imag will extract the real ad imagiary parts of a complex umber. Whe plottig a complex vector, the defaults for plot ad stem ca be cofusig. If z is complex, the plot(z) will plot the imagiary part versus the real part; ad plot(,z) will plot the real part of z versus. However, stem(z) will just plot the real part. If you wat to view simultaeous plots of the real ad imagiary parts, the subplot(211) ad subplot(212) commads prior to each stem commad will force the two plots to be placed o the same scree, oe above the other. See Figura 1.2, which was created usig the followig code: = 0:25; xx = exp(j*/3); %--- complex expoetial subplot (211) stem(,real(xx)) title('real PART'), xlabel('index ()') subplot (212) stem(,imag(xx)) title('imag PART'), xlabel('index ()')

12 MPF04 Aálise de Siais e Aquisção de Dados 12/14 1 REAL PART INDEX () 1 IMAG PART INDEX () Figura Plottig Real ad Imagiary Parts of a Discrete-Time Sigal with subplot Exercise: Complex Expoetials The real expoetial otatio ca be exteded to complex-valued expoetial sigals that embody the sie ad cosie sigals. These sigals form the basis of the Fourier trasform. (a). I MATLAB a complex sigal is a atural extesio of the otatio i Exercise 1.4. Thus the parameters a ca be take as a complex umber to geerate these sigals. Recall Euler's formula for the complex expoetial (i a form that gives a sigal): x[ = ( z 0 ) = e (l Z + j z ) jθ 0 0 = r e = r (cosθ + jsiθ ) (1.9) where zo=re jθ =r θ. Use this relatioship to geerate a complex expoetial with z0= Plot the real ad imagiary parts of x[ over the rage Notice that the agle of z0 cotrols the frequecy of the siusoids. (b). For the sigal i part (a) make a plot of the imagiary part versus the real part. The result should be a spiral. Experimet with differet agles for θ - a smaller value should produce a better picture of a spiral. (c). Equatio (1.9) is ot geeral eough to produce all complex expoetials. What is missig is a complex costat to scale the

13 Project 2: Complex-Valued Sigals 13/14 amplitude ad phase of the siusoids. This is the so-called phasor otatio: jφ jθ j( θ+φ) G. z0 = Ae r e = Ar e = Ar [cos( θ + φ) + jsi( θ + φ)], (1.10) where G = A e jφ = A φ is the complex amplitude of the complex expoetial. Geerate ad plot each of the followig sequeces. Covert the siusoids to complex otatio; the create the sigal vector usig exp. If the sigal is purely real, it should be geerated by takig the real part of a complex sigal. I each plot, the horizotal () axis should exted oly over the rage idicated ad should be labeled accordigly. π π x1[ = 3si( ) + j4cos( ), 0 20, 7 7 π x2[ = si( ), 15 25, 17 π π x3[ = 1.1 si( + ), 0 50, 11 4 π x4[ = 0.9 cos( ), For each sigal, determie the values of amplitude ad phase costats that have to be used i G; also the agle ad magitude of zo. (d). These same complex expoetials ca be geerated by first-order differece equatios (usig filter): y [ = z0 y[ 1] + x[. (1.11) The filter coefficiet, z0=re jθ, is a complex umber. The ratio betwee successive terms i the sequece is easily see to be z0; but the correct amplitude ad phase must be set by choosig a complex amplitude for the impulse which drives the differece equatio (i. e., x[ = Gδ[). Use filter to create the same sigals as i part (c). Verify by plottig both the

### 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

### 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

### 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

### 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

### 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

### FOUNDATIONS OF MATHEMATICS AND PRE-CALCULUS GRADE 10

FOUNDATIONS OF MATHEMATICS AND PRE-CALCULUS GRADE 10 [C] Commuicatio Measuremet A1. Solve problems that ivolve liear measuremet, usig: SI ad imperial uits of measure estimatio strategies measuremet strategies.

### 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...

### 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

### SECTION 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,

### Modified 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

### 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

### 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,

### 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

### 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"

### 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,

### Your organization has a Class B IP address of 166.144.0.0 Before you implement subnetting, the Network ID and Host ID are divided as follows:

Subettig Subettig is used to subdivide a sigle class of etwork i to multiple smaller etworks. Example: Your orgaizatio has a Class B IP address of 166.144.0.0 Before you implemet subettig, the Network

### 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.

### Here are a couple of warnings to my students who may be here to get a copy of what happened on a day that you missed.

This documet was writte ad copyrighted by Paul Dawkis. Use of this documet ad its olie versio is govered by the Terms ad Coditios of Use located at http://tutorial.math.lamar.edu/terms.asp. The olie versio

### 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

### 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

### 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

### CHAPTER 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

### Basic Elements of Arithmetic Sequences and Series

MA40S PRE-CALCULUS UNIT G GEOMETRIC SEQUENCES CLASS NOTES (COMPLETED NO NEED TO COPY NOTES FROM OVERHEAD) Basic Elemets of Arithmetic Sequeces ad Series Objective: To establish basic elemets of arithmetic

### Department of Computer Science, University of Otago

Departmet of Computer Sciece, Uiversity of Otago Techical Report OUCS-2006-09 Permutatios Cotaiig May Patters Authors: M.H. Albert Departmet of Computer Sciece, Uiversity of Otago Micah Colema, Rya Fly

### 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

### 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:

### 1 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

INFINITE SERIES KEITH CONRAD. Itroductio The two basic cocepts of calculus, differetiatio ad itegratio, are defied i terms of limits (Newto quotiets ad Riema sums). I additio to these is a third fudametal

### 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

### Chapter 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

### 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

### Running Time ( 3.1) Analysis of Algorithms. Experimental Studies ( 3.1.1) Limitations of Experiments. Pseudocode ( 3.1.2) Theoretical Analysis

Ruig Time ( 3.) Aalysis of Algorithms Iput Algorithm Output A algorithm is a step-by-step procedure for solvig a problem i a fiite amout of time. Most algorithms trasform iput objects ito output objects.

### 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

### Systems 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) 698-5295 Email: bcm1@cec.wustl.edu Supervised

### Fast Fourier Transform

18.310 lecture otes November 18, 2013 Fast Fourier Trasform Lecturer: Michel Goemas I these otes we defie the Discrete Fourier Trasform, ad give a method for computig it fast: the Fast Fourier Trasform.

### Chapter 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

### 5.4 Amortization. Question 1: How do you find the present value of an annuity? Question 2: How is a loan amortized?

5.4 Amortizatio Questio 1: How do you fid the preset value of a auity? Questio 2: How is a loa amortized? Questio 3: How do you make a amortizatio table? Oe of the most commo fiacial istrumets a perso

### SEQUENCES AND SERIES

Chapter 9 SEQUENCES AND SERIES Natural umbers are the product of huma spirit. DEDEKIND 9.1 Itroductio I mathematics, the word, sequece is used i much the same way as it is i ordiary Eglish. Whe we say

### Taking DCOP to the Real World: Efficient Complete Solutions for Distributed Multi-Event Scheduling

Taig DCOP to the Real World: Efficiet Complete Solutios for Distributed Multi-Evet Schedulig Rajiv T. Maheswara, Milid Tambe, Emma Bowrig, Joatha P. Pearce, ad Pradeep araatham Uiversity of Souther Califoria

### Biology 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

### Measures 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,

Physics 6A Witer 20 Theorems About Power Series Cosider a power series, f(x) = a x, () where the a are real coefficiets ad x is a real variable. There exists a real o-egative umber R, called the radius

### 5 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

### Cantilever 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

### 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

### Engineering 323 Beautiful Homework Set 3 1 of 7 Kuszmar Problem 2.51

Egieerig 33 eautiful Homewor et 3 of 7 Kuszmar roblem.5.5 large departmet store sells sport shirts i three sizes small, medium, ad large, three patters plaid, prit, ad stripe, ad two sleeve legths log

### GCE Further Mathematics (6360) Further Pure Unit 2 (MFP2) Textbook. Version: 1.4

GCE Further Mathematics (660) Further Pure Uit (MFP) Tetbook Versio: 4 MFP Tetbook A-level Further Mathematics 660 Further Pure : Cotets Chapter : Comple umbers 4 Itroductio 5 The geeral comple umber 5

### 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

### Multiple Representations for Pattern Exploration with the Graphing Calculator and Manipulatives

Douglas A. Lapp Multiple Represetatios for Patter Exploratio with the Graphig Calculator ad Maipulatives To teach mathematics as a coected system of cocepts, we must have a shift i emphasis from a curriculum

### Incremental 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

### 7.1 Finding Rational Solutions of Polynomial Equations

4 Locker LESSON 7. Fidig Ratioal Solutios of Polyomial Equatios Name Class Date 7. Fidig Ratioal Solutios of Polyomial Equatios Essetial Questio: How do you fid the ratioal roots of a polyomial equatio?

### 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

### Discrete 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

### 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

### 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

### NATIONAL 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

### 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

### A Recursive Formula for Moments of a Binomial Distribution

A Recursive Formula for Momets of a Biomial Distributio Árpád Béyi beyi@mathumassedu, Uiversity of Massachusetts, Amherst, MA 01003 ad Saverio M Maago smmaago@psavymil Naval Postgraduate School, Moterey,

### .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,

### Confidence Intervals. CI for a population mean (σ is known and n > 30 or the variable is normally distributed in the.

Cofidece Itervals A cofidece iterval is a iterval whose purpose is to estimate a parameter (a umber that could, i theory, be calculated from the populatio, if measuremets were available for the whole populatio).

### Lesson 15 ANOVA (analysis of variance)

Outlie Variability -betwee group variability -withi group variability -total variability -F-ratio Computatio -sums of squares (betwee/withi/total -degrees of freedom (betwee/withi/total -mea square (betwee/withi

### Week 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

### INVESTMENT 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

### Research 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

### *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.

### 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

### Approximating 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.

### GCSE 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

### 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

### Overview. Learning Objectives. Point Estimate. Estimation. Estimating the Value of a Parameter Using Confidence Intervals

Overview Estimatig the Value of a Parameter Usig Cofidece Itervals We apply the results about the sample mea the problem of estimatio Estimatio is the process of usig sample data estimate the value of

### BENEFIT-COST ANALYSIS Financial and Economic Appraisal using Spreadsheets

BENEIT-CST ANALYSIS iacial ad Ecoomic Appraisal usig Spreadsheets Ch. 2: Ivestmet Appraisal - Priciples Harry Campbell & Richard Brow School of Ecoomics The Uiversity of Queeslad Review of basic cocepts

### 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.

### Lesson 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

### 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)

### CS103X: 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 six-figure salaries i whole dollar amouts are there that cotai at least

### Exploratory Data Analysis

1 Exploratory Data Aalysis Exploratory data aalysis is ofte the rst step i a statistical aalysis, for it helps uderstadig the mai features of the particular sample that a aalyst is usig. Itelliget descriptios

### 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

### Vladimir 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

### 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,

### 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

### Baan 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 :

### 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

### 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

### Output 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

### 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.

### Complex Numbers. where x represents a root of Equation 1. Note that the ± sign tells us that quadratic equations will have

Comple Numbers I spite of Calvi s discomfiture, imagiar umbers (a subset of the set of comple umbers) eist ad are ivaluable i mathematics, egieerig, ad sciece. I fact, i certai fields, such as electrical

### 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.

### THE 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

### CME 302: NUMERICAL LINEAR ALGEBRA FALL 2005/06 LECTURE 8

CME 30: NUMERICAL LINEAR ALGEBRA FALL 005/06 LECTURE 8 GENE H GOLUB 1 Positive Defiite Matrices A matrix A is positive defiite if x Ax > 0 for all ozero x A positive defiite matrix has real ad positive

### Sequences and Series Using the TI-89 Calculator

RIT Calculator Site Sequeces ad Series Usig the TI-89 Calculator Norecursively Defied Sequeces A orecursively defied sequece is oe i which the formula for the terms of the sequece is give explicitly. For

### How 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.

### Lecture 4: Cauchy sequences, Bolzano-Weierstrass, and the Squeeze theorem

Lecture 4: Cauchy sequeces, Bolzao-Weierstrass, ad the Squeeze theorem The purpose of this lecture is more modest tha the previous oes. It is to state certai coditios uder which we are guarateed that limits

### The 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

### Design of Digital Filters

Chapter 8 Desig of Digital Filters Cotets Overview........................................................ 8. Geeral Cosideratios................................................. 8. Desig of FIR Filters..................................................

### 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

### Chapter 7: Confidence Interval and Sample Size

Chapter 7: Cofidece Iterval ad Sample Size Learig Objectives Upo successful completio of Chapter 7, you will be able to: Fid the cofidece iterval for the mea, proportio, ad variace. Determie the miimum

### 5: 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