B1. Fourier Analysis of Discrete Time Signals


 Jeremy Holmes
 1 years ago
 Views:
Transcription
1 B. Fourier Aalysis of Discrete Time Sigals Objectives Itroduce discrete time periodic sigals Defie the Discrete Fourier Series (DFS) expasio of periodic sigals Defie the Discrete Fourier Trasform (DFT) of sigals with fiite legth Determie the Discrete Fourier Trasform of a complex expoetial. Itroductio I the previous chapter we defied the cocept of a sigal both i cotiuous time (aalog) ad discrete time (digital). Although the time domai is the most atural, sice everythig (icludig our ow lives) evolves i time, it is ot the oly possible represetatio. I this chapter we itroduce the cocept of expadig a geeric sigal i terms of elemetary sigals, such as complex expoetials ad siusoids. This leads to the frequecy domai represetatio of a sigal i terms of its Fourier Trasform ad the cocept of frequecy spectrum so that we characterize a sigal i terms of its frequecy compoets. First we begi with the itroductio of periodic sigals, which eep repeatig i time. For these sigals it is fairly easy to determie a expasio i terms of siusoids ad complex expoetials, sice these are just particular cases of periodic sigals. This is exteded to sigals of a fiite duratio which becomes the Discrete Fourier Trasform (DFT), oe of the most widely used algorithms i Sigal Processig. The cocepts itroduced i this chapter are at the basis of spectral estimatio of sigals addressed i the ext two chapters.. Periodic Sigals VIDEO: Periodic Sigals (9:45) I this sectio we defie a class of discrete time sigals called Periodic Sigals. There are two reasos why these sigals are importat: A umber of sigals i ature exhibit periodic repetitio. Thi of vibratios, ocea waves, seismic waves or electromagetic waves;
2 It is pretty easy to believe that periodic sigals are made of (periodic) siusoids of differet frequecies, which is the goal of the Fourier aalysis of the rest of the chapter. Defiitio: a discrete time sigal x [ ] is periodic if ad oly if there exists a positive iteger such that x [ ] = x [ + ] for all () I other words, a periodic sigal eeps repeatig itself for all values of the idex from to +. The smallest positive iteger satisfyig () is called the period of the sigal. Example: cosider the sigal Figure : periodic sigal ( π π) x [ ] = cos. +.9 It is periodic with period = sice x [ + ] = cos. π ( + ) +.9π for all. ( ) ( π π π) = cos = x [ ] I geeral a siusoidal sigal of the form x [ ] = Acos + α with, itegers is periodic with period, sice x[ + ] = Acos π ( + ) + α = Acos + α + π = x [ ] for all. () (3) Example: cosider the siusoid This ca be writte as ( π π) x [ ] = 5cos.3. 3 x [ ] = 5cos.π
3 Comparig this with () we ca see that the sigal is periodic with period =. I fact we ca easily write x [ + ] = 5cos.3 π ( + ).π ( ) ( π π π) = 5cos = x [ ] for all. I a very similar way a complex expoetial of the form is periodic with period sice x[ + ] = Ae j x[ ] = Ae (4) j ( + ) j j Ae e x = = Example: cosider the complex expoetial. [ ] ( ) j π x = + je This ca be writte as j x [ ] = ( + je ) ad it is periodic with period =. [ ] 3. Expasio of Periodic Sigals: the Discrete Fourier Series (DFS). VIDEO: Referece Frames (:5) ow the problem we wat to address is to determie a referece frame for discrete time periodic sigals. I other words the issue is to determie a set of periodic sigals which serve as a basis of ay other periodic sigal. The cocept of a referece frame is very importat i a umber of fields, icludig geometry, avigatio ad sigal processig. It exteds simple, ituitive geometric ideas to more complex abstract problems. Goig bac to geometry, recall that, give a vector x we ca represet it i terms of a referece frame defied by: A origi ; Referece vectors e, e,... Figure below shows a example of a vector i the plae.
4 Figure : a vector ad a referece frame i the plae Give this defiitio we ca see that a vector (say) i the plae is always represeted by the compoets alog the directios of the basis vectors. I this case, as show i figure 3, the vector x is represeted as show i figure 3 as x = ae + ae where a is the projectio of the vector x alog e a is the projectio of the vector x alog e Figure 3: vector represetatio i terms of its projectios o the referece frame For example you wat to locate yourself with respect to a referece poit ad you say that you are 3m East ad m orth of your referece. I this case the referece vectors e, e are i the East ad orth directios respectively, of uit legth ( meter) ad the compoets a, a of your positio are 3m ad m respectively. This is show i figure 4. Figure 4: example of a geographical positio as a vector i the Earth referece frame. For sigals we use a very similar cocept. We wat to express ay arbitrary sigal [ ] x as the sum of referece sigals havig appropriate physical properties, sigificat for our problems. I our case ad i a lot of applicatios, the referece sigals are siusoids or complex expoetials, as show i figure 5.
5 Figure 5: a sigal as a sum of referece sigals Give ay periodic sigal with period a very attractive set of referece cadidates is the set of complex expoetials with the same period. These are defied as j e [ ] = e, =,..., (5) It turs out that we ca show the followig Fact: ay discrete time periodic sigal with period ca be writte as for some costats a, a,... a. x [ ] = ae [ ] Example: tae the periodic sigal show i figure 6 below. It is easy to see that it is periodic with period =, sice it eeps repeatig the value,,,, periodically. = Figure 6: periodic sigal with period = Sice the period is = the referece expoetials e [ ] i (5) are give by j e [ ] = e = = j e[ ] = e = ( ) We ca easily verify (try to believe!) that the give periodic sigal i figure 6 ca be writte as x [ ] =.5 e[ ] +.5 e[ ] =.5.5 ( ) I fact this expressio yields x [ ] =.5.5 = for eve ad x [ ] = = for odd.
6 Example: cosider aother example of a periodic sigal with period = show i figure 7. Figure 7: periodic sigal with period = Agai we use the same referece sigals e[ ], =, defied as j e [ ] = e = = j e[ ] = e = ( ) It ca be easily verified that this sigal ca be writte as x [ ] =.5 e[ ] +.8 e[ ] = ( ) 4. Orthogoality of the Referece Sigals VIDEO: Orthogoal Referece Sigals (:5) From the simple geometric examples i the previous sectio (figures,3,4) we see that the vectors i the referece frames are orthogoal to each other. It turs out that this maes it easier i determiig the expasio of the vectors i terms of referece frames. A very similar approach exteds to referece sigals. I particular we ca verify that the j complex expoetials e [ ] = e are all orthogoal to each other i the sese that * if m e[ e ] m[ ] = = if m= (6) This is easy to show just by applyig the defiitio of complex expoetials as follows: m m j π * j π e[ e ] m[ ] = e = e = = = ow we ca apply the geometric sum, defied as (7)
7 j m e π α if α α = α (8) = if α = with α =. The applyig (7) to (8) we obtai the orthogoality coditio i (6). The fact that the complex expoetials e[ ] are orthogoal as i (6) maes the computatio of the expasio coefficiets a a very simple matter. I fact, let x [ ] be a periodic sigal with period ad let s write it agai as x [ ] = ae [ ] The, the coefficiets a, a,..., a of the expasio ca be easily computed from the orthogoality of the complex expoetials as * * * x[ ] e[ ] = amem[ ] e[ ] = am e[ ] em[ ] = a = = m= m= = x [ ] = VIDEO: Discrete Fourier Series (6:8) The rightmost expressio comes from the orthogoality property, for which the summatio is ozero oly whe = m. It is customary to defie X[ ] = a for =,..., ad call it the Discrete Fourier Series (DFS) of the periodic sequece x [ ]. This ca be computed as = j X[ ] = DFS{ x [ ]} = xe [ ], =,..., (9) Coversely, the origial the sigal x [ ] is called the Iverse Discrete Fourier Series (IDFS) ad it ca be writte as j x [ ] = IDFS{ X [ ]} = Xe [ ] () = Example Revisited: cosider the example we have see above (figure 6), with period =. The the Discrete Fourier Series (DFS) of the give sigal is computed as = j X[ ] = xe [ ] = x[] + x[]( ) = + ( ), =, Therefore this yields X [] = 3 ad X [] =. The give sigal ca the be expressed as j x [ ] = IDFS{ X [ ]} = Xe [ ] =.5.5 ( ) = same as i the example above.
8 Example: Cosider the periodic sigal x [ ] show i figure 8 below. Figure 8: a periodic sigal with period =. The we compute its Discrete Fourier Series (DFS) as j e j j if,,...,9 X[ ] = = π xe [ ] = e = j = = e 5 if = This sequece ca be plotted i terms of magitude ad phase as Figure 9: DFS of the sigal i figure 8 Example: Let the sigal x [ ] be defied as x [ ] = cos(.5 π ) It is periodic ad let s see what the period is. Just compute a few values to obtai x[] =, x[] =, x[] =, x[3] =,... This shows that the period is =, which yields its DFS as = j X[ ] = xe [ ] = x[] + x[] ( ), =, Substitutig for x[], x [] we obtai X[] = X[] =.
9 5. Sigals of a Fiite Legth ad the Discrete Fourier Trasform (DFT) VIDEO: the Discrete Fourier Trasform (:) I all data processig experimets we aalyze data of a fiite legth. A typical situatio is show i figure below, where a segmet of data of legth T MAX (i secods, for example) is digitized at a rate F S samples/secod, thus yieldig a vector of = TMAX FS samples. For example we have TMAX = 3msec (ie.3sec ) of data, sampled at FS = Hz, ie, samples/sec. The we obtai a umerical file with legth =.3, = 6 samples. Figure : a sampled sigal of fiite legth As a cosequece we obtai a data vector x [ ], =,... with data poits. I may applicatios we wat the frequecy cotet of this data which allows us to extract relevat iformatio. Thi about a music sigal: the frequecies are related to the otes i the music score. I geeral ay sort of audio sigal, be it speech or from a hydrophoe, has a sigature i frequecy which tells a lot about the ature of the sigal itself. We ca use the frequecy sigature i speech recogitio, for example. I other words it is very importat to determie the spectrum of a sigal umerically directly from the samples. I order to do this, we ca mae use of what we have developed for periodic sigals. I particular we ca loo at a sigal x [ ], =,... of legth samples as oe period of a periodic sigal with period. I this way we ca use exactly the same formulas of the Discrete Fourier Series (DFS) ad defie the Discrete Fourier Trasform (DFT) i exactly the same way: { } = j X[ ] = DFT x [ ] = xe [ ], =,..., j x [ ] = IDFT{ X [ ]} = Xe [ ], =,..., =
10 These expressios are the same as the DFS with the oly differece that the DFS is for periodic sigals, thus x [ ] is periodic ad defied for all samples, while the DFT is for fiite legth sigals, thus x [ ] is defied oly for =,...,. The meaig of this expasio is that ay fiite sequece ca be expaded i terms of complex expoetials with frequecies π /. Example. Cosider the sigal x [ ], =,..,9 of legth = show i figure. Figure : example of a sigal of legth Its DFT ca be computed aalytically usig the geometric series as j e j j if,,...,9 X[ ] = = π xe [ ] = e = j = = e 5 if = This ca be plotted i terms of magitude ad phase as show i figure below. Figure : DFT of the sigal i figure. 6. DFT of Complex Expoetials VIDEO: DFT of Complex Expoetials (:35) Sice we use the DFT to detect spectral compoets, it is particularly useful to see the DFT of a complex expoetial sigal of fiite legth defied as
11 x Ae jω [ ] =, =,... where ω represets the digital frequecy i radias. Applyig the defiitio of its DFT we obtai the followig expressio { } X[ ] = DFT x [ ] = xe [ ] = j j j j ω ω = Ae e = Ae, =,..., = = otice that this expressio has a particular structure ad it ca be writte i the followig way: X [ ] A W π = ω where the fuctio W ( ω ) depeds oly o the data legth ad it is defied as W jω e ( ω) = e = e = jω jω Agai the variable ω idicates digital frequecy ad it is i radias. As we will show below, the same expressio ca be writte i a differet way as ω si j ( )/ W ( ω) e ω = ω si The plot of its magitude for = is show i figure 3. Figure 3: magitude of W ( ω ) otice that the largest values are cocetrated aroud ω =. Also we ca distiguish the mai lobe withi the iterval < ω < ad all the other sidelobes. Most of the eergy is o the mai lobe. Example: cosider the sequece x e j.3π [ ] =, =,...,3
12 I this caseω =.3 π, = 3. The its DFT becomes 3 ( ω π) X [ ] = W.3, =,...,3 π ω= 3 I order to see how this loos lie, let s plot first W ( ω π) 3.3 as i figure 4 The, to see its DFT, we tae samples as This is show i figure 5. Figure 4: plot of W ( ω π) 3.3 { } ( ω π) X [ ] = DFT x [ ] = W.3, =,...,3 π ω= 3 Figure 5: DFT of the example. otice that the maximum correspods to the idex = 5 which yields the digital frequecy ω = 5 π / 3 =.3.3π 7. The Fast Fourier Trasform i Matlab Oe of the most importat reaso (if ot the most importat) reasos why the DFT has become a very popular algorithm is the extremely high efficiecy of its implemetatio. I particular the Fast Fourier Trasform (FFT) ca compute the DFT of a large data set i a very short time. Just to give a idea, if we have data poits, ad is a power of, we ca compute its DFT i a amout of time proportioal to log, usig the FFT, versus by brute force
13 computatio. For example let (say) = = 4. The the time tae by the FFT is 3 proportioal to 4log 4, while the brute force approach would tae a amout of 6 time proportioal to 4. As a cosequece, all computer applicatios, icludig Matlab, use the FFT rather the the DFT. The results of both operatios are idetical, but the FFT computes it i a fractio of the time tae by the brute force DFT. VIDEO: FFT i Matlab (:43) seg4.wmv Agai we wat to determie the DFT of the sequece x e j.3π [ ] = with =,..., 3. The: % Data Legth: =3; %. geerate data =:; x=exp(j.3*pi*); %. tae the FFT X=fft(x); % 3. plot it (choose oe) stem(abs(x)) % if you wat to see the idividual values plot(abs(x)) % to see a cotiuous plot Example: let us tae the same sequece, but with more data poits. Say, let =56. The the same Matlab program yields the graph show i the figure below. Figure 6. DFT of the same sequece, with =56 samples.
14 The pea ca be verified to be at = 38 so that the estimated frequecy is as expected. ω = 38 =.969π. 3π 56 I the followig example, we loo at the opposite problem. Give a sigal, we wat to determie the frequecy cotet looig at its DFT. Each pea yields a estimate of the domiat frequecies. Cosider a sigal x [ ], =,..., 55 of legth = 56. Its DFT X [] has its magitude as show i the figure below. So the problem is what ca we say about this sigal. Sice it has two distict ad well defied peas, at = 7 ad = 8 the sigal has two frequecies at ω = 7 =. 9π 56 ω = 8 =. 638π 56 Figure 7. Give DFT for the Example. 8. DFT of Siusoidal Sigals. VIDEO: DFT of a Siusoid (8:54) Recall that a siusoid is the sum of two complex expoetials, as
15 A A x[ ] = Acos jα jω jα jω ( ω + α ) = e e + e e I the spectral domai it is represeted as two frequecies as show i the figure below. Figure 8. Spectral Represetatio of a Siusoid ow we wat to see what is its DFT whe we tae a fiite data legth x [ ], =,...,. Defie agai X [ ] = DFT{ x[ ] } ad sice it is the sum of two complex expoetials, its DFT is the sum of the respective DFT s, as A jα jω A jα jω X[ ] = DFT e e + DFT e e The rightmost term, with egative frequecy ω requires some cosideratio sice the frequecies /, =,..., computed by the DFT caot be egative. We ca easily see jω j( ω ) that e = e, with ω so that the above DFT becomes A jα jω A jα j X[ ] = DFT e e + DFT e e The two frequecies ω ad ω are show i the figure below. ( ω ) Figure 9. Frequecy domai represetatio of a siusoidal sigal, usig positive frequecies oly.
16 As a cosequece of this, the DFT of a siusoidal sigal at frequecy < ω < π is goig to show two peas: oe at ω ad oe at ω. Both compoets have the same magitude A / ad phase ± α of opposite sig. Example. Suppose we have a sigal x ] = cos(. 3π + α ) [ with =,..., 3. I terms of complex expoetials, it has two frequecies, at ω =. 3π ad ω =. 3π. The egative frequecy is equivalet to ω = π.3π =. 7π so that its DFT becomes X [ ] = W3 ( ω.3π ) + W3 ( ω.7π ) ω= / 3 The magitude of ( ω.3π ) + W ( ω. 7π ) W is show i the figure below. 3 3 W. Figure. Magitude Plot of ( ω.3π ) + W ( ω. 7π ) 3 3 otice the two peas, as expected, ad the maximum value / = 6 at the peas. The DFT ow is computed as samples of the values at ω = / 3 with =,..., 3. This yields the 3 values show below. DFT of the Siusoid i the example. We ca easily see the two peas at = 5 ad = 3 5 = 7. The correspodig frequecies are ω = 5 / 3. 3π ad ω = 7 / 3. 7π as expected. By the defiitio of the DFT ad the example we see a importat property called Symmetry.
17 VIDEO: Symmetry of the DFT (9:) Symmetry of the DFT. Give a real sigal x [ ], =,...,, its DFT X [ ] = DFT x[ ], =,..., is such that { } Proof. From the defiitio of the DFT replacig the idex with X[ ] = X [ ] = X * [ ] we obtai = x[ ] e j ( ) = = x[ ] e j Sice x [] is real by assumptio, the rightmost expressio is the complex cojugate of X [] which shows the property. The cosequece of this property is that, whe the sigal is real, the whole iformatio of the frequecy spectrum is i the first half as X [ ], =,...,( / ), while the iformatio i the secod half of the DFT is redudat. Example. Referrig to the previous example, sice the sigal is real the whole iformatio of its frequecy spectrum is show i the plot for =,..., 5 as show below. Figure. DFT of a Siusoid: first half of the plot. 9. FFT of a Siusoid i Matlab This sectio (o video oly) shows a implemetatio of the FFT of a siusoid i Matlab. VIDEO: FFT of a Siusoid i Matlab (3:)
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 informationThe second difference is the sequence of differences of the first difference sequence, 2
Differece Equatios I differetial equatios, you look for a fuctio that satisfies ad equatio ivolvig derivatives. I differece equatios, istead of a fuctio of a cotiuous variable (such as time), we look for
More informationSequences 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 informationCooleyTukey. Tukey FFT Algorithms. FFT Algorithms. Cooley
Cooley CooleyTuey Tuey FFT Algorithms FFT Algorithms Cosider a legth sequece x[ with a poit DFT X[ where Represet the idices ad as +, +, Cooley CooleyTuey Tuey FFT Algorithms FFT Algorithms Usig these
More information{{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 informationTrigonometric Form of a Complex Number. The Complex Plane. axis. ( 2, 1) or 2 i FIGURE 6.44. The absolute value of the complex number z a bi is
0_0605.qxd /5/05 0:45 AM Page 470 470 Chapter 6 Additioal Topics i Trigoometry 6.5 Trigoometric Form of a Complex Number What you should lear Plot complex umbers i the complex plae ad fid absolute values
More information7. 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 informationExample 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 informationThe Euler Totient, the Möbius and the Divisor Functions
The Euler Totiet, the Möbius ad the Divisor Fuctios Rosica Dieva July 29, 2005 Mout Holyoke College South Hadley, MA 01075 1 Ackowledgemets This work was supported by the Mout Holyoke College fellowship
More informationModule 4: Mathematical Induction
Module 4: Mathematical Iductio Theme 1: Priciple of Mathematical Iductio Mathematical iductio is used to prove statemets about atural umbers. As studets may remember, we ca write such a statemet as a predicate
More informationCS103X: Discrete Structures Homework 4 Solutions
CS103X: Discrete Structures Homewor 4 Solutios Due February 22, 2008 Exercise 1 10 poits. Silico Valley questios: a How may possible sixfigure salaries i whole dollar amouts are there that cotai at least
More informationTheorems About Power Series
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 oegative umber R, called the radius
More informationEscola Federal de Engenharia de Itajubá
Escola Federal de Egeharia de Itajubá Departameto de Egeharia Mecâica PósGraduação em Egeharia Mecâica MPF04 ANÁLISE DE SINAIS E AQUISÇÃO DE DADOS SINAIS E SISTEMAS Trabalho 02 (MATLAB) Prof. Dr. José
More information8.1 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 informationLecture 13. Lecturer: Jonathan Kelner Scribe: Jonathan Pines (2009)
18.409 A Algorithmist s Toolkit October 27, 2009 Lecture 13 Lecturer: Joatha Keler Scribe: Joatha Pies (2009) 1 Outlie Last time, we proved the BruMikowski iequality for boxes. Today we ll go over the
More informationAsymptotic Growth of Functions
CMPS Itroductio to Aalysis of Algorithms Fall 3 Asymptotic Growth of Fuctios We itroduce several types of asymptotic otatio which are used to compare the performace ad efficiecy of algorithms As we ll
More informationProperties 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 informationSECTION 1.5 : SUMMATION NOTATION + WORK WITH SEQUENCES
SECTION 1.5 : SUMMATION NOTATION + WORK WITH SEQUENCES Read Sectio 1.5 (pages 5 9) Overview I Sectio 1.5 we lear to work with summatio otatio ad formulas. We will also itroduce a brief overview of sequeces,
More information1 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 informationTagore Engineering College Department of Electrical and Electronics Engineering EC 2314 Digital Signal Processing University Question Paper PartA
Tagore Egieerig College Departmet of Electrical ad Electroics Egieerig EC 34 Digital Sigal Processig Uiversity Questio Paper PartA UitI. Defie samplig theorem?. What is kow as Aliasig? 3. What is LTI
More informationMaximum Likelihood Estimators.
Lecture 2 Maximum Likelihood Estimators. Matlab example. As a motivatio, let us look at oe Matlab example. Let us geerate a radom sample of size 00 from beta distributio Beta(5, 2). We will lear the defiitio
More informationConvexity, 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 informationTHE ARITHMETIC OF INTEGERS.  multiplication, exponentiation, division, addition, and subtraction
THE ARITHMETIC OF INTEGERS  multiplicatio, expoetiatio, divisio, additio, ad subtractio What to do ad what ot to do. THE INTEGERS Recall that a iteger is oe of the whole umbers, which may be either positive,
More informationDivide 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 informationCHAPTER 3 DIGITAL CODING OF SIGNALS
CHAPTER 3 DIGITAL CODING OF SIGNALS Computers are ofte used to automate the recordig of measuremets. The trasducers ad sigal coditioig circuits produce a voltage sigal that is proportioal to a quatity
More informationChapter 5: Inner Product Spaces
Chapter 5: Ier Product Spaces Chapter 5: Ier Product Spaces SECION A Itroductio to Ier Product Spaces By the ed of this sectio you will be able to uderstad what is meat by a ier product space give examples
More informationLecture 4: Cheeger s Inequality
Spectral Graph Theory ad Applicatios WS 0/0 Lecture 4: Cheeger s Iequality Lecturer: Thomas Sauerwald & He Su Statemet of Cheeger s Iequality I this lecture we assume for simplicity that G is a dregular
More informationLearning 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 information3. 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 informationChapter One BASIC MATHEMATICAL TOOLS
Chapter Oe BAIC MATHEMATICAL TOOL As the reader will see, the study of the time value of moey ivolves substatial use of variables ad umbers that are raised to a power. The power to which a variable is
More informationLesson 17 Pearson s Correlation Coefficient
Outlie Measures of Relatioships Pearso s Correlatio Coefficiet (r) types of data scatter plots measure of directio measure of stregth Computatio covariatio of X ad Y uique variatio i X ad Y measurig
More informationINFINITE SERIES KEITH CONRAD
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
More informationCS103A 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 informationSequences 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 informationx(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 informationModified Line Search Method for Global Optimization
Modified Lie Search Method for Global Optimizatio Cria Grosa ad Ajith Abraham Ceter of Excellece for Quatifiable Quality of Service Norwegia Uiversity of Sciece ad Techology Trodheim, Norway {cria, ajith}@q2s.tu.o
More information4 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 pseries (the Compariso Test), but of
More informationThe 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 informationRecursion and Recurrences
Chapter 5 Recursio ad Recurreces 5.1 Growth Rates of Solutios to Recurreces Divide ad Coquer Algorithms Oe of the most basic ad powerful algorithmic techiques is divide ad coquer. Cosider, for example,
More informationA probabilistic proof of a binomial identity
A probabilistic proof of a biomial idetity Joatho Peterso Abstract We give a elemetary probabilistic proof of a biomial idetity. The proof is obtaied by computig the probability of a certai evet i two
More informationEngineering 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
More informationS. 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 informationArithmetic Sequences and Partial Sums. Arithmetic Sequences. Definition of Arithmetic Sequence. Example 1. 7, 11, 15, 19,..., 4n 3,...
3330_090.qxd 1/5/05 11:9 AM Page 653 Sectio 9. Arithmetic Sequeces ad Partial Sums 653 9. Arithmetic Sequeces ad Partial Sums What you should lear Recogize,write, ad fid the th terms of arithmetic sequeces.
More informationDepartment of Computer Science, University of Otago
Departmet of Computer Sciece, Uiversity of Otago Techical Report OUCS200609 Permutatios Cotaiig May Patters Authors: M.H. Albert Departmet of Computer Sciece, Uiversity of Otago Micah Colema, Rya Fly
More informationInfinite 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 informationLesson 15 ANOVA (analysis of variance)
Outlie Variability betwee group variability withi group variability total variability Fratio Computatio sums of squares (betwee/withi/total degrees of freedom (betwee/withi/total mea square (betwee/withi
More informationChapter 7 Methods of Finding Estimators
Chapter 7 for BST 695: Special Topics i Statistical Theory. Kui Zhag, 011 Chapter 7 Methods of Fidig Estimators Sectio 7.1 Itroductio Defiitio 7.1.1 A poit estimator is ay fuctio W( X) W( X1, X,, X ) of
More informationOur aim is to show that under reasonable assumptions a given 2πperiodic 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 informationTaking DCOP to the Real World: Efficient Complete Solutions for Distributed MultiEvent Scheduling
Taig DCOP to the Real World: Efficiet Complete Solutios for Distributed MultiEvet Schedulig Rajiv T. Maheswara, Milid Tambe, Emma Bowrig, Joatha P. Pearce, ad Pradeep araatham Uiversity of Souther Califoria
More information5.3. 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 information4.1 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 informationBasic Measurement Issues. Sampling Theory and AnalogtoDigital Conversion
Theory ad AalogtoDigital Coversio Itroductio/Defiitios Aalogtodigital coversio Rate Frequecy Aalysis Basic Measuremet Issues Reliability the extet to which a measuremet procedure yields the same results
More informationFast 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 informationAlgebra Vocabulary List (Definitions for Middle School Teachers)
Algebra Vocabulary List (Defiitios for Middle School Teachers) A Absolute Value Fuctio The absolute value of a real umber x, x is xifx 0 x = xifx < 0 http://www.math.tamu.edu/~stecher/171/f02/absolutevaluefuctio.pdf
More informationNUMBERS COMMON TO TWO POLYGONAL SEQUENCES
NUMBERS COMMON TO TWO POLYGONAL SEQUENCES DIANNE SMITH LUCAS Chia Lake, Califoria a iteger, The polygoal sequece (or sequeces of polygoal umbers) of order r (where r is r > 3) may be defied recursively
More informationPermutations, the Parity Theorem, and Determinants
1 Permutatios, the Parity Theorem, ad Determiats Joh A. Guber Departmet of Electrical ad Computer Egieerig Uiversity of Wiscosi Madiso Cotets 1 What is a Permutatio 1 2 Cycles 2 2.1 Traspositios 4 3 Orbits
More informationDiscrete Mathematics and Probability Theory Spring 2014 Anant Sahai Note 13
EECS 70 Discrete Mathematics ad Probability Theory Sprig 2014 Aat Sahai Note 13 Itroductio At this poit, we have see eough examples that it is worth just takig stock of our model of probability ad may
More information0.7 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 KolmogorovSmirov 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 informationSequences, Series and Convergence with the TI 92. Roger G. Brown Monash University
Sequeces, Series ad Covergece with the TI 92. Roger G. Brow Moash Uiversity email: rgbrow@deaki.edu.au Itroductio. Studets erollig i calculus at Moash Uiversity, like may other calculus courses, are itroduced
More informationMathematical 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 informationSoving Recurrence Relations
Sovig Recurrece Relatios Part 1. Homogeeous liear 2d degree relatios with costat coefficiets. Cosider the recurrece relatio ( ) T () + at ( 1) + bt ( 2) = 0 This is called a homogeeous liear 2d degree
More informationCHAPTER 3 THE TIME VALUE OF MONEY
CHAPTER 3 THE TIME VALUE OF MONEY OVERVIEW A dollar i the had today is worth more tha a dollar to be received i the future because, if you had it ow, you could ivest that dollar ad ear iterest. Of all
More informationA 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,
More informationGCE 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 Alevel Further Mathematics 660 Further Pure : Cotets Chapter : Comple umbers 4 Itroductio 5 The geeral comple umber 5
More informationAn antenna is a transducer of electrical and electromagnetic energy. electromagnetic. electrical
Basic Atea Cocepts A atea is a trasducer of electrical ad electromagetic eergy electromagetic electrical electrical Whe We Desig A Atea, We Care About Operatig frequecy ad badwidth Sometimes frequecies
More informationSection 7: Free electron model
Physics 97 Sectio 7: ree electro model A free electro model is the simplest way to represet the electroic structure of metals. Although the free electro model is a great oversimplificatio of the reality,
More informationLecture 5: Span, linear independence, bases, and dimension
Lecture 5: Spa, liear idepedece, bases, ad dimesio Travis Schedler Thurs, Sep 23, 2010 (versio: 9/21 9:55 PM) 1 Motivatio Motivatio To uderstad what it meas that R has dimesio oe, R 2 dimesio 2, etc.;
More informationLinear Algebra II. 4 Determinants. Notes 4 1st November Definition of determinant
MTH6140 Liear Algebra II Notes 4 1st November 2010 4 Determiats The determiat is a fuctio defied o square matrices; its value is a scalar. It has some very importat properties: perhaps most importat is
More informationFOUNDATIONS OF MATHEMATICS AND PRECALCULUS GRADE 10
FOUNDATIONS OF MATHEMATICS AND PRECALCULUS GRADE 10 [C] Commuicatio Measuremet A1. Solve problems that ivolve liear measuremet, usig: SI ad imperial uits of measure estimatio strategies measuremet strategies.
More informationFast 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.
More informationNormal 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 informationRunning 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 stepbystep procedure for solvig a problem i a fiite amout of time. Most algorithms trasform iput objects ito output objects.
More information1.3 Binomial Coefficients
18 CHAPTER 1. COUNTING 1. Biomial Coefficiets I this sectio, we will explore various properties of biomial coefficiets. Pascal s Triagle Table 1 cotais the values of the biomial coefficiets ( ) for 0to
More information5: Introduction to Estimation
5: Itroductio to Estimatio Cotets Acroyms ad symbols... 1 Statistical iferece... Estimatig µ with cofidece... 3 Samplig distributio of the mea... 3 Cofidece Iterval for μ whe σ is kow before had... 4 Sample
More informationDefinition. A variable X that takes on values X 1, X 2, X 3,...X k with respective frequencies f 1, f 2, f 3,...f k has mean
1 Social Studies 201 October 13, 2004 Note: The examples i these otes may be differet tha used i class. However, the examples are similar ad the methods used are idetical to what was preseted i class.
More informationLecture 4: Cauchy sequences, BolzanoWeierstrass, and the Squeeze theorem
Lecture 4: Cauchy sequeces, BolzaoWeierstrass, 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
More informationNonlife insurance mathematics. Nils F. Haavardsson, University of Oslo and DNB Skadeforsikring
Nolife isurace mathematics Nils F. Haavardsso, Uiversity of Oslo ad DNB Skadeforsikrig Mai issues so far Why does isurace work? How is risk premium defied ad why is it importat? How ca claim frequecy
More informationOverview 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 informationWeek 3 Conditional probabilities, Bayes formula, WEEK 3 page 1 Expected value of a random variable
Week 3 Coditioal probabilities, Bayes formula, WEEK 3 page 1 Expected value of a radom variable We recall our discussio of 5 card poker hads. Example 13 : a) What is the probability of evet A that a 5
More informationIncremental calculation of weighted mean and variance
Icremetal calculatio of weighted mea ad variace Toy Fich faf@cam.ac.uk dot@dotat.at Uiversity of Cambridge Computig Service February 009 Abstract I these otes I eplai how to derive formulae for umerically
More informationChapter 6: Variance, the law of large numbers and the MonteCarlo method
Chapter 6: Variace, the law of large umbers ad the MoteCarlo method Expected value, variace, ad Chebyshev iequality. If X is a radom variable recall that the expected value of X, E[X] is the average value
More informationAPPLICATION 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 informationPartial Di erential Equations
Partial Di eretial Equatios Partial Di eretial Equatios Much of moder sciece, egieerig, ad mathematics is based o the study of partial di eretial equatios, where a partial di eretial equatio is a equatio
More information1 Computing the Standard Deviation of Sample Means
Computig the Stadard Deviatio of Sample Meas Quality cotrol charts are based o sample meas ot o idividual values withi a sample. A sample is a group of items, which are cosidered all together for our aalysis.
More informationHeat (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 informationYour 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
More informationClass 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 informationVladimir N. Burkov, Dmitri A. Novikov MODELS AND METHODS OF MULTIPROJECTS MANAGEMENT
Keywords: project maagemet, resource allocatio, etwork plaig Vladimir N Burkov, Dmitri A Novikov MODELS AND METHODS OF MULTIPROJECTS MANAGEMENT The paper deals with the problems of resource allocatio betwee
More informationProject Deliverables. CS 361, Lecture 28. Outline. Project Deliverables. Administrative. Project Comments
Project Deliverables CS 361, Lecture 28 Jared Saia Uiversity of New Mexico Each Group should tur i oe group project cosistig of: About 612 pages of text (ca be loger with appedix) 612 figures (please
More informationBasic Elements of Arithmetic Sequences and Series
MA40S PRECALCULUS 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
More informationSAMPLE 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 informationMARTINGALES AND A BASIC APPLICATION
MARTINGALES AND A BASIC APPLICATION TURNER SMITH Abstract. This paper will develop the measuretheoretic approach to probability i order to preset the defiitio of martigales. From there we will apply this
More informationWHEN 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 informationStatistical Methods. Chapter 1: Overview and Descriptive Statistics
Geeral Itroductio Statistical Methods Chapter 1: Overview ad Descriptive Statistics Statistics studies data, populatio, ad samples. Descriptive Statistics vs Iferetial Statistics. Descriptive Statistics
More informationConfidence Intervals for One Mean
Chapter 420 Cofidece Itervals for Oe Mea Itroductio This routie calculates the sample size ecessary to achieve a specified distace from the mea to the cofidece limit(s) at a stated cofidece level for a
More information1 Correlation and Regression Analysis
1 Correlatio ad Regressio Aalysis I this sectio we will be ivestigatig the relatioship betwee two cotiuous variable, such as height ad weight, the cocetratio of a ijected drug ad heart rate, or the cosumptio
More informationQuantum Mechanics for Scientists and Engineers. David Miller
Quatum Mechaics for Scietists ad Egieers David Miller Measuremet ad expectatio values Measuremet ad expectatio values Quatummechaical measuremet Probabilities ad expasio coefficiets Suppose we take some
More informationwhere: T = number of years of cash flow in investment's life n = the year in which the cash flow X n i = IRR = the internal rate of return
EVALUATING ALTERNATIVE CAPITAL INVESTMENT PROGRAMS By Ke D. Duft, Extesio Ecoomist I the March 98 issue of this publicatio we reviewed the procedure by which a capital ivestmet project was assessed. The
More informationTHE HEIGHT OF qbinary SEARCH TREES
THE HEIGHT OF qbinary SEARCH TREES MICHAEL DRMOTA AND HELMUT PRODINGER Abstract. q biary search trees are obtaied from words, equipped with the geometric distributio istead of permutatios. The average
More informationPROCEEDINGS OF THE YEREVAN STATE UNIVERSITY AN ALTERNATIVE MODEL FOR BONUSMALUS SYSTEM
PROCEEDINGS OF THE YEREVAN STATE UNIVERSITY Physical ad Mathematical Scieces 2015, 1, p. 15 19 M a t h e m a t i c s AN ALTERNATIVE MODEL FOR BONUSMALUS SYSTEM A. G. GULYAN Chair of Actuarial Mathematics
More information