CHAPTER NINE. Frequency Response Methods
|
|
- Vivien McCormick
- 5 years ago
- Views:
Transcription
1 CHAPTER NINE 9. Itroductio It as poited earlier that i practice the performace of a feedback cotrol system is more preferably measured by its time - domai respose characteristics. This is i cotrast to the aalysis ad desig of systems i the commuicatio field, here the frequecy respose is of more importace, sice i this case most of the sigals to be processed are either siusoidal or periodic i ature. Hoever, aalytically, the time respose of a cotrol system is usually difficult to determie, especially i the case of high - order systems. I the desig aspects, there are o uified ays of arrivig at a desig system give the time - domai specificatios, such as peak overshoot, rise time, delay time ad settig time. O the other had, there is a ealth of graphical methods available i the frequecy - domai aalysis, all suitable for the aalysis ad desig of liear feedback cotrol system. Oce the aalysis ad desig are carried out i the frequecy - domai, the time - domai behavior of the system ca be iterpreted based o the relatioships that exist betee the time - domai ad the frequecy - domai properties. Therefore, might cosider that the mai purpose of coductig cotrol systems aalysis ad desig i the frequecy domai is merely to use the techiques as a coveiet vehicle toard the same objectives as ith time - domai method. Frequecy respose methods ere developed i 930s ad 940s by Nyquist, Bode Nichols, ad may others. The frequecy respose methods are most poerful i covetioal cotrol theory. They are also idispesable to robust cotrol theory. The priciple of frequecy respose testig is based i comparig the iput ad output of the elemets uder test over a ide rage of frequecies, he subjected to a siusoidal iput sigal. The Nyquist stability criterio eables to ivestigate both the absolute ad relative stabilities of liear closed loop systems from koledge of their ope loop frequecy respose characteristics. A advatage of the frequecy respose approach is that frequecy respose tests are; i geeral, simple ad ca be made accurately by use of readily available siusoidal sigal geerators ad precise measuremet equipmet. Ofte the trasfer fuctios of complicated compoets ca be determied experimetally by frequecy respose tests. I additio, the frequecy respose approach has the advatages that a system may be desiged so that the effects of udesirable oise are egligible ad that such aalysis ad desig ca be exteded to certai oliear cotrol systems. Dr. Fadhil Aula Salahaddi Uiversity /College of Egieerig/ Electrical Egieerig Dept. 06 Chapter 9 -
2 Although the frequecy respose of a cotrol system presets a qualitative picture of the trasiet respose, the correlatio betee frequecy ad trasiet resposes is idirect, except for the case of secod- order systems. I desigig a closed - loop system, e adjust the frequecy respose characteristic of the ope - loop trasfer fuctio by usig several desig criteria i order to obtai acceptable trasiet respose characteristics of the system. 9. Frequecy respose coverage the folloig plots 9.. Polar plot Polar plot is a graph of Im [GH] versus Re [GH] o the [GH(j)] - plae for - < <. = o Figure 9. Polar Plot 9.. Magitude - Phase plot The magitude i decibel versus the phase o rectagular coordiates ith as a variable parameter is called Magitude versus Phase Plot. Figure 9. Magitude versus Phase Dr. Fadhil Aula Salahaddi Uiversity /College of Egieerig/ Electrical Egieerig Dept. 06 Chapter 9 -
3 9..3 Bode plot The plot of the magitude i decibel versus (or log ) i semilog (or rectagular) coordiate called Bode plot (corer plot). Figure 9.3 Bode Plot Frequecy Respose Characteristic For closed loop cotrol system Gs ( ) M T. F. GH ( s ) let s j M ( j ) G ( j ) G ( j ) H ( j ) G ( j ) G ( j ) H ( j ) M ( j ) magitude phase agle M(j) may be regarded as the magificatio of the feedback cotrol system. A typical magificatio curve of a feedback cotrol system is sho i the folloig figure: Dr. Fadhil Aula Salahaddi Uiversity /College of Egieerig/ Electrical Egieerig Dept. 06 Chapter 9-3
4 Mp - Resoace Peak: is the maximum value of the magitude of the closed loop frequecy respose. p - Resoace frequecy: is the frequecy at hich Mp occurs. B.W - Bad Width; is the rage of frequecies (of the iput) over hich the system ill respod satisfactorily. Satisfactorily ill be at value equal to of magificatio. c - Cut off frequecy: it is occurs he the magitude ratio ted to of its value Polar Plots (Nyquist Plot) The polar plot of a siusoidal trasfer fuctio G(j) is a plot of the magitude of G(j) versus the phase agle of G(j) o polar coordiates as is varied from zero to ifiity. Thus, the polar plot is the locus of vectors G ( j ) G ( j ) as is varied from zero to ifiity. Note that i polar plots a positive (egative) phase agle is measured couter clockise (clockise) from positive real axis. The polar plot is ofte called Nyquist plot. A example of such a plot is sho i Figure 9.4. Each poit o the polar plot of G(j) represets the termial poit of a vector at a particular value of. i the polar plot, it is importat to sho the frequecy graduatio of the locus. The projectios of G(j) o the real ad imagiary axes is real ad imagiary compoets. A advatage i usig a polar plot is that it depicts the frequecy respose characteristics of a system over the etire frequecy rage i a sigle plot. Oe disadvatage is that the plot does ot clearly idicate the cotributios of each idividual factor of the ope loop trasfer fuctio. Dr. Fadhil Aula Salahaddi Uiversity /College of Egieerig/ Electrical Egieerig Dept. 06 Chapter 9-4
5 Itegrated ad Derivative Factors (j) ± Figure 9.4 Polar Plot The polar plot of G(j) = / j is the egative imagiary axis sice G ( j ) j 90 j The polar plot of G(j) = j is the positive imagiary axis. First order Factors ( + j T) ± For the siusoidal trasfer fuctio G ( j ) ta j T T T The values of G(j) at = 0 ad = /T are respectively G ( j 0) 0 ad G ( j ) 45 T If approaches ifiity, the magitude of G(j) approaches zero ad the phase agle approaches -90 o. The polar plot of this trasfer fuctio is a semicircle as the frequecy is varied from zero to ifiity. Dr. Fadhil Aula Salahaddi Uiversity /College of Egieerig/ Electrical Egieerig Dept. 06 Chapter 9-5
6 Quadratic Factors [ ( j / ) ( j / ) ] The lo ad high frequecy portios of the polar plot the folloig siusoidal trasfer fuctio G ( j ) for 0 ( j ) ( j ) are give respectively by lim G ( j ) 0 ad lim G ( j ) 0 80 The polar plot of this siusoidal trasfer fuctio starts at 0 ad eds at 0 80 as icreases from zero to ifiity. Thus, the high frequecy portio of G(j) is taget to the egative real axis Geeral Shapes of Polar Plots The polar plots of a trasfer fuctio of the form K ( jta)( jtb)... G ( j ) ( j ) ( jt)( jt )... Where > m or the degree of the deomiator polyomial is greater tha that of the umerator, ill have the folloig geeral shapes:. Type zero cotrol systems, λ = 0: The startig poit of the polar plot (hich correspod to = 0) is fiite ad is o the positive real axis. The taget to the polar plot at = 0 is perpedicular to the real axis. The termial poit, hich correspods to =, is at the origi, ad the curve is taget to oe of the axis. Example: K G ( j ) ( jt )( jt ) at = 0, G ( j ) K 0, G ( j ) 0 80, ta 90 Each factor of deomiator cotributes a agle of -90 o or 90 o i clockise directio. Dr. Fadhil Aula Salahaddi Uiversity /College of Egieerig/ Electrical Egieerig Dept. 06 Chapter 9-6
7 . Type Cotrol System λ=: The j term i the deomiator cotributes - 90 to the total phase agle of G(j) for 0. At = 0, the magitude of G(j) is ifiity, ad the phase agle becomes -90. At lo frequecies, the polar plot is asymptotic to a lie parallel to the egative imagiary axis. At =, the magitude becomes zero, ad the curve coverges to the origi ad is taget to oe of the axes. Example: K G ( j ) ( j )( jt)( jt )( jt3) at 0, G ( j ) 90, G ( j ) Vx is asymptotes s approaches to zero, G(j) approaches to ifiity alog to asymptotes Vx, ad Vx is foud from: Vx lim Re[ G ( j )] 0 ad is deter mied from x Im [ G ( j )] 0 Dr. Fadhil Aula Salahaddi Uiversity /College of Egieerig/ Electrical Egieerig Dept. 06 Chapter 9-7
8 3. Type Cotrol System λ=: The (j) term i the deomiator cotributes -80 to the total phase agle of G(j) for 0. At = 0, the magitude of G(j) is ifiity, ad the phase agle is equal to At lo frequecies, the polar plot is asymptotic to a lie parallel to the egative real axis. At =, the magitude becomes zero, ad the curve is taget to oe of the axes. K Example: G ( j ) ( j ) ( jt )( jt ) at 0, G ( j ) 80, G ( j ) Dr. Fadhil Aula Salahaddi Uiversity /College of Egieerig/ Electrical Egieerig Dept. 06 Chapter 9-8
9 Rough Sketch of the Polar Plot The sketchig of the polar plot is facilitated by the folloig iformatio:.the behavior of the magitude ad the phase at = 0 ad at =..The poit of itersectio of the polar plot ith the real ad imagiary axes, ad the value of at the itersectios Absolute ad Relative Stability The simplified Nyquist criterio for system stability may be stated as follos: If GH(j) does ot have poles i the right half s - plae, the closed loop system is stable, if ad oly if the - pit lies to the left of the polar plot he movig alog this plot i the directio of icreasig. That is, the polar plot passes o the right side of Gai Margi (GM) GM is defied as the reciprocal of the magitude of the ope loop trasfer fuctio he the phase shift is 80. This is, therefore the factor by hich the gai must be icreased i order to produce istability. Phase Margi (ΦPM) Φ PM is defied as 80 mius the phase agle of the ope loop trasfer fuctio [GH] of the frequecy he the gai is uity. This is therefore the amout of the phase agle, ould have to be icreased to make the system ustable Bode Diagrams (or Logarithmic Plots) A Bode diagram (plot) cosists of to graphs: oce is a plot of the logarithm of the magitude of a siusoidal trasfer fuctio; the other is a plot of the phase agle; both are plotted agaist the frequecy o a logarithmic scale. The stadard represetatio of the logarithmic magitude of G(j) is 0log G(j), here the base of the logarithm is 0. The uit used i this represetatio of the magitude is the decibel, usually abbreviated db. I the logarithmic represetatio, the curves are dra o semi log paper, usig the log scale for frequecy ad the liear scale for either magitude (but i decibels) or phase agle (i degrees). The frequecy rage of iterest determies the umber of logarithmic cycles required o the abscissa. The mai advatage of usig Bode diagram is that multiplicatio of the magitudes ca be coverted ito additio. Further more; a simple method for sketchig a approximate logmagitude curve is available. It is based o asymptotic approximatios. Such approximatio Dr. Fadhil Aula Salahaddi Uiversity /College of Egieerig/ Electrical Egieerig Dept. 06 Chapter 9-9
10 by straight-lie asymptotes is sufficiet if oly rough iformatio o the frequecy - respose characteristics is eeded. Should the exact curve be desired, correctios ca be made easily to these basic asymptotic plots. Expadig the lo - frequecy rage by use of a logarithmic scale for the frequecy is highly advatageous sice characteristics at lo frequecies are most importat impractical systems. Although it is ot possible to plot the curves right do to zero frequecy because of the logarithmic frequecy (log 0 = - ). Note that the experimetal determiatio of a trasfer fuctio ca be made simple if frequecy - respose data are preseted i the form of a Bode diagram. Basic Factors of G(j)H(j) The basic factors that vary frequetly occur i arbitrary trasfer fuctio G(j)H(j) are:. Gai K.. Itegrated ad derivative factors (j) ±. 3. First - order factors ( + j T) ±. 4. Quadratic factors [ ( / ) ( / ) ] j j. Oce e become familiar ith the logarithmic plots of these basic factors, it is possible to utilize them i costructig a composite logarithmic plot for ay geeral form of G(j)H(j) by sketchig for each factor ad addig idividual curves graphically; because addig the logarithms of the gais correspods to multiplyig them together. The Gai K A umber greater tha uity has a positive value i decibels, hile a umber smaller tha uity has a egative value. The log - magitude curve for a costat gai K is a horizotal straight lie at the magitude of 0 log K decibels. The phase agle of the gai K is zero. The effect of varyig the gai K i the trasfer fuctio is that it raises or loers the log - magitude curve of the trasfer fuctio by the correspodig costat amout, but it has o effect o the phase curve. db Φ 0 log K Dr. Fadhil Aula Salahaddi Uiversity /College of Egieerig/ Electrical Egieerig Dept. 06 Chapter 9-0
11 Itegral factor j The logarithmic magitude of /j i decibel is 0 log 0 log db j This is a straight lie of slop - 0 db/decade. The phase agle of /j is costat ad equal to - 90 o. Derivative factor j The logarithmic magitude of j i decibel is 0 log j 0 log db This is a straight lie of slop 0 db/decade. The phase agle of j is costat ad equal to 90 o. First order factor a. Factor j T Log magitude is 0 log 0 log T db j T For lo frequecies such that «/ T, the log magitude may be approximated by 0 log T 0 log 0 Thus, the log magitude curve at lo frequecies is the costat 0 db lie. For high frequecies, such that» / T, 0 log T 0 logt db This is a approximate expressio for the high frequecy rage. At = /T, the log magitude equals 0 db; at = 0/T, the log magitude is -0 db, the value of -0 log T Dr. Fadhil Aula Salahaddi Uiversity /College of Egieerig/ Electrical Egieerig Dept. 06 Chapter 9 -
12 db decreases by 0 db for every decade of. For»/T, the log magitude curve is thus a straight lie ith a slope of -0 db/ decade. The frequecy at hich the to asymptotes meet is called the corer frequecy or break frequecy. For the factor /( + j T), the frequecy = /T is the corer frequecy sice at = /T the to asymptotes have the same value. The corer frequecy divides the frequecy respose curve ito to regios; a curve for the lo frequecy regio ad a curve for high frequecy regio. The corer frequecy is very importat i sketchig logarithmic frequecy respose curves. The exact phase agle Φ of the factor /( + j T) is ta T At zero frequecy ( = 0), the phase agle is 0 o. At corer frequecy ( = /T), the phase agle is ta T / T ta 45. At ifiity ( ), the phase agle becomes - 90 o. Sice the phase agle is give by a iverse taget fuctio, the phase agle is ske symmetric about the iflectio poit at Φ = -45 o. b. Factor + j T Log magitude is 0 log j T 0 log T db For lo frequecies such that «/ T, the log magitude may be approximated by 0 log T 0 log 0 Thus, the log magitude curve at lo frequecies is the costat 0 db lie. For high frequecies, such that» / T, 0 log 0 log T T db The exact phase agle Φ of the factor ( + j T) is ta T At zero frequecy ( = 0), the phase agle is 0 o. At corer frequecy ( = /T), the phase agle is ta T / T ta 45. At ifiity ( ), the phase agle becomes 90 o. Sice the phase agle is give by a iverse taget fuctio, the phase agle is ske symmetric about the iflectio poit at Φ = 45 o. Dr. Fadhil Aula Salahaddi Uiversity /College of Egieerig/ Electrical Egieerig Dept. 06 Chapter 9 -
13 Quadratic Factors a. The factor j ( j ) If ζ >, this quadratic factor ca be expressed as a product of to first order factors ith real poles. If 0 < ζ <, this quadratic factor is the product of to complex - cojugate factors. Asymptotic approximatios to the frequecy - respose curves are ot accurate for a factor ith lo values of ζ. This is because the magitude ad phase of the quadratic factor deped o both the corer frequecy ad the dampig ratio ζ. The asymptotic frequecy-respose curve may be obtaied as the follos. Sice 0 log 0 log ( ) ( ) j ( j ) For lo frequecies such that «, the log magitude becomes -0 log = 0 db The lo-frequecy asymptote is thus a horizotal lie at 0 db. For high frequecies such that», the log magitude becomes 0 log 40 log db The equatio for the high-frequecy asymptote is a straight lie havig the slope - 40 db/decade sice 0 40 log log The high frequecy asymptote itersects the lo-frequecy oe at = sice at this frequecy 40 log 40 log 0 db This frequecy,, is the corer frequecy for the quadratic factor cosidered. The phase agle of the quadratic factor j ( j ) ta ( j ) ( j ) ( ) The phase agle is a fuctio of both ad ζ. At = 0, the phase agle equals 0 o. At the corer frequecy =, the phase agle is - 90 o regardless of ζ, sice is Dr. Fadhil Aula Salahaddi Uiversity /College of Egieerig/ Electrical Egieerig Dept. 06 Chapter 9-3
14 ta ( ) ta 90 0 At =, the phase agle becomes - 80 o. The phase agle curve is ske symmetric about the iflectio poit (the poit here Φ = - 90 o ). 0 Bode Diagram j ( j ) Magitude (db) corer frequecy asymptote slope - 40 db/decade Phase (deg) Frequecy (rad/sec) 80 Bode Diagram j ( j ) 60 Magitude (db) corer frequecy asymptote slope 40 db/decade Phase (deg) Frequecy (rad/sec) Dr. Fadhil Aula Salahaddi Uiversity /College of Egieerig/ Electrical Egieerig Dept. 06 Chapter 9-4
15 9.6 The Resoat Frequecy, r, ad the Resoat Peak Value, Mr The magitude of G ( j ) is G ( j ) j ( ) ( ) ( ) j If G(j) has a peak value at some frequecy, this frequecy is called the resoat frequecy. Sice the umerator of G(j) is costat, a peak value of G(j) ill occur he g ( j ) ( ) ( ) is a miimum, ad this equatio ca e ritte as ( ) ( ) 4 ( ) g j is miimum value of g() occur at Thus the resoat frequecy r is r for As the dampig ratio ζ approaches zero, the resoat frequecy approaches. For , the resoat frequecy r, is less tha the damped atural frequecy d, hich is exhibited i the trasiet respose. From the equatio of r above it ca be see that for ζ > 0.707, there is o resoat peak. The magitude G(j) decrease mootoically ith icreasig frequecy. This magitude is less tha 0 db for all values of > 0. The magitude of the resoat peak M, ca be foud by substitutig equatio of r ito equatio of G(j) for , M r G ( j ) max G ( j r) For ζ > M r =. As ζ approaches zero, M r approaches ifiity. This meas that if the udamped system is excited at its atural frequecy, the magitude of G(j) becomes ifiity. The phase agle of G(j) at the frequecy here the resoat peak occurs ca be obtaied by G ( j ) ta 90 si o e Notes: a. If gai crossover frequecy is less tha the phase crossover frequecy, the system is stable. b. If the phase crossover frequecy is less tha the gai crossover frequecy, the system is ustable. Dr. Fadhil Aula Salahaddi Uiversity /College of Egieerig/ Electrical Egieerig Dept. 06 Chapter 9-5
2.004 Dynamics and Control II Spring 2008
MIT OpeCourseWare http://ocw.mit.edu 2.004 Dyamics ad Cotrol II Sprig 2008 For iformatio about citig these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. Massachusetts Istitute of Techology
More informationMechatronics. Time Response & Frequency Response 2 nd -Order Dynamic System 2-Pole, Low-Pass, Active Filter
Time Respose & Frequecy Respose d -Order Dyamic System -Pole, Low-Pass, Active Filter R 4 R 7 C 5 e i R 1 C R 3 - + R 6 - + e out Assigmet: Perform a Complete Dyamic System Ivestigatio of the Two-Pole,
More informationEE Control Systems
Copyright FL Lewis 7 All rights reserved Updated: Moday, November 1, 7 EE 4314 - Cotrol Systems Bode Plot Performace Specificatios The Bode Plot was developed by Hedrik Wade Bode i 1938 while he worked
More informationCourse Outline. Designing Control Systems. Proportional Controller. Amme 3500 : System Dynamics and Control. Root Locus. Dr. Stefan B.
Amme 3500 : System Dyamics ad Cotrol Root Locus Course Outlie Week Date Cotet Assigmet Notes Mar Itroductio 8 Mar Frequecy Domai Modellig 3 5 Mar Trasiet Performace ad the s-plae 4 Mar Block Diagrams Assig
More informationFrequency Response Methods
Frequecy Respose Methods The frequecy respose Nyquist diagram polar plots Bode diagram magitude ad phase Frequecy domai specificatios Frequecy Respose Methods I precedig chapters the respose ad performace
More informationThe z-transform. 7.1 Introduction. 7.2 The z-transform Derivation of the z-transform: x[n] = z n LTI system, h[n] z = re j
The -Trasform 7. Itroductio Geeralie the complex siusoidal represetatio offered by DTFT to a represetatio of complex expoetial sigals. Obtai more geeral characteristics for discrete-time LTI systems. 7.
More informationChapter 7: The z-transform. Chih-Wei Liu
Chapter 7: The -Trasform Chih-Wei Liu Outlie Itroductio The -Trasform Properties of the Regio of Covergece Properties of the -Trasform Iversio of the -Trasform The Trasfer Fuctio Causality ad Stability
More informationChapter 2 Feedback Control Theory Continued
Chapter Feedback Cotrol Theor Cotiued. Itroductio I the previous chapter, the respose characteristic of simple first ad secod order trasfer fuctios were studied. It was show that first order trasfer fuctio,
More informationLecture 13. Graphical representation of the frequency response. Luca Ferrarini - Basic Automatic Control 1
Lecture 3 Graphical represetatio of the frequecy respose Luca Ferrarii - Basic Automatic Cotrol Graphical represetatio of the frequecy respose Polar plot G Bode plot ( j), G Im 3 Re of the magitude G (
More information6.003 Homework #3 Solutions
6.00 Homework # Solutios Problems. Complex umbers a. Evaluate the real ad imagiary parts of j j. π/ Real part = Imagiary part = 0 e Euler s formula says that j = e jπ/, so jπ/ j π/ j j = e = e. Thus the
More informationChapter 7 z-transform
Chapter 7 -Trasform Itroductio Trasform Uilateral Trasform Properties Uilateral Trasform Iversio of Uilateral Trasform Determiig the Frequecy Respose from Poles ad Zeros Itroductio Role i Discrete-Time
More informationCDS 101: Lecture 8.2 Tools for PID & Loop Shaping
CDS : Lecture 8. Tools for PID & Loop Shapig Richard M. Murray 7 November 4 Goals: Show how to use loop shapig to achieve a performace specificatio Itroduce ew tools for loop shapig desig: Ziegler-Nichols,
More information732 Appendix E: Previous EEE480 Exams. Rules: One sheet permitted, calculators permitted. GWC 352,
732 Aedix E: Previous EEE0 Exams EEE0 Exam 2, Srig 2008 A.A. Rodriguez Rules: Oe 8. sheet ermitted, calculators ermitted. GWC 32, 9-372 Problem Aalysis of a Feedback System Cosider the feedback system
More informationSchool of Mechanical Engineering Purdue University. ME375 Frequency Response - 1
Case Study ME375 Frequecy Respose - Case Study SUPPORT POWER WIRE DROPPERS Electric trai derives power through a patograph, which cotacts the power wire, which is suspeded from a cateary. Durig high-speed
More information3. Z Transform. Recall that the Fourier transform (FT) of a DT signal xn [ ] is ( ) [ ] = In order for the FT to exist in the finite magnitude sense,
3. Z Trasform Referece: Etire Chapter 3 of text. Recall that the Fourier trasform (FT) of a DT sigal x [ ] is ω ( ) [ ] X e = j jω k = xe I order for the FT to exist i the fiite magitude sese, S = x [
More informationIntroduction to Signals and Systems, Part V: Lecture Summary
EEL33: Discrete-Time Sigals ad Systems Itroductio to Sigals ad Systems, Part V: Lecture Summary Itroductio to Sigals ad Systems, Part V: Lecture Summary So far we have oly looked at examples of o-recursive
More informationBode Diagrams School of Mechanical Engineering ME375 Frequency Response - 29 Purdue University Example Ex:
ME375 Hadouts Bode Diagrams Recall that if m m bs m + bm s + + bs+ b Gs () as + a s + + as+ a The bm( j z)( j z) ( j zm) G( j ) a ( j p )( j p ) ( j p ) bm( s z)( s z) ( s zm) a ( s p )( s p ) ( s p )
More informationCALCULUS BASIC SUMMER REVIEW
CALCULUS BASIC SUMMER REVIEW NAME rise y y y Slope of a o vertical lie: m ru Poit Slope Equatio: y y m( ) The slope is m ad a poit o your lie is, ). ( y Slope-Itercept Equatio: y m b slope= m y-itercept=
More information1the 1it is said to be overdamped. When 1, the roots of
Homework 3 AERE573 Fall 08 Due 0/8(M) ame PROBLEM (40pts) Cosider a D order uderdamped system trasfer fuctio H( s) s ratio 0 The deomiator is the system characteristic polyomial P( s) s s (a)(5pts) Use
More informationExponential Moving Average Pieter P
Expoetial Movig Average Pieter P Differece equatio The Differece equatio of a expoetial movig average lter is very simple: y[] x[] + (1 )y[ 1] I this equatio, y[] is the curret output, y[ 1] is the previous
More informationTEACHER CERTIFICATION STUDY GUIDE
COMPETENCY 1. ALGEBRA SKILL 1.1 1.1a. ALGEBRAIC STRUCTURES Kow why the real ad complex umbers are each a field, ad that particular rigs are ot fields (e.g., itegers, polyomial rigs, matrix rigs) Algebra
More informationContents Kreatryx. All Rights Reserved.
Cotets Maual for K-Notes... Basics of Cotrol Systems... 3 Sigal Flow Graphs... 7 Time Respose Aalysis... 0 Cotrol System Stability... 6 Root locus Techique... 8 Frequecy Domai Aalysis... Bode Plots...
More informationEE 205 Dr. A. Zidouri. Electric Circuits II. Frequency Selective Circuits (Filters) Bode Plots: Complex Poles and Zeros.
EE 5 Dr. A. Zidouri Electric Circuits II Frequecy Selective Circuits (Filters) Bode Plots: Complex Poles ad Zeros Lecture #4 - - EE 5 Dr. A. Zidouri The material to be covered i this lecture is as follows:
More information(c) Write, but do not evaluate, an integral expression for the volume of the solid generated when R is
Calculus BC Fial Review Name: Revised 7 EXAM Date: Tuesday, May 9 Remiders:. Put ew batteries i your calculator. Make sure your calculator is i RADIAN mode.. Get a good ight s sleep. Eat breakfast. Brig:
More information(s)h(s) = K( s + 8 ) = 5 and one finite zero is located at z 1
ROOT LOCUS TECHNIQUE 93 should be desiged differetly to eet differet specificatios depedig o its area of applicatio. We have observed i Sectio 6.4 of Chapter 6, how the variatio of a sigle paraeter like
More informationSeptember 2012 C1 Note. C1 Notes (Edexcel) Copyright - For AS, A2 notes and IGCSE / GCSE worksheets 1
September 0 s (Edecel) Copyright www.pgmaths.co.uk - For AS, A otes ad IGCSE / GCSE worksheets September 0 Copyright www.pgmaths.co.uk - For AS, A otes ad IGCSE / GCSE worksheets September 0 Copyright
More informationWe will conclude the chapter with the study a few methods and techniques which are useful
Chapter : Coordiate geometry: I this chapter we will lear about the mai priciples of graphig i a dimesioal (D) Cartesia system of coordiates. We will focus o drawig lies ad the characteristics of the graphs
More informationAnalysis of the No-Load Characteristic of the Moving Coil Linear Compressor
Purdue Uiversity Purdue e-pubs Iteratioal Compressor Egieerig Coferece School of Mechaical Egieerig 008 Aalysis of the No-Load Characteristic of the Movig Coil Liear Compressor Yigbai Xie North Chia Electric
More informationFIR Filter Design: Part II
EEL335: Discrete-Time Sigals ad Systems. Itroductio I this set of otes, we cosider how we might go about desigig FIR filters with arbitrary frequecy resposes, through compositio of multiple sigle-peak
More informationSection 10.3 The Complex Plane; De Moivre's Theorem. abi
Sectio 03 The Complex Plae; De Moivre's Theorem REVIEW OF COMPLEX NUMBERS FROM COLLEGE ALGEBRA You leared about complex umbers of the form a + bi i your college algebra class You should remember that "i"
More informationAPPENDIX F Complex Numbers
APPENDIX F Complex Numbers Operatios with Complex Numbers Complex Solutios of Quadratic Equatios Polar Form of a Complex Number Powers ad Roots of Complex Numbers Operatios with Complex Numbers Some equatios
More informationREVISION SHEET FP1 (MEI) ALGEBRA. Identities In mathematics, an identity is a statement which is true for all values of the variables it contains.
The mai ideas are: Idetities REVISION SHEET FP (MEI) ALGEBRA Before the exam you should kow: If a expressio is a idetity the it is true for all values of the variable it cotais The relatioships betwee
More informationChapter 4 : Laplace Transform
4. Itroductio Laplace trasform is a alterative to solve the differetial equatio by the complex frequecy domai ( s = σ + jω), istead of the usual time domai. The DE ca be easily trasformed ito a algebraic
More informationLab(8) controller design using root locus
Lab(8) cotroller desig usig root locus I this lab we will lear how to desig a cotroller usig root locus but before this we eed to aswer the followig questios: What is root locus? What is the purpose of
More information6.003: Signals and Systems. Feedback, Poles, and Fundamental Modes
6.003: Sigals ad Systems Feedback, Poles, ad Fudametal Modes February 9, 2010 Last Time: Multiple Represetatios of DT Systems Verbal descriptios: preserve the ratioale. To reduce the umber of bits eeded
More informationDynamic Response of Linear Systems
Dyamic Respose of Liear Systems Liear System Respose Superpositio Priciple Resposes to Specific Iputs Dyamic Respose of st Order Systems Characteristic Equatio - Free Respose Stable st Order System Respose
More informationDr. Seeler Department of Mechanical Engineering Fall 2009 Lafayette College ME 479: Control Systems and Mechatronics Design and Analysis
Dr. Seeler Departmet of Mechaical Egieerig Fall 009 Lafayette College ME 479: Cotrol Systems ad Mechatroics Desig ad Aalysis Lab 0: Review of the First ad Secod Order Step Resposes The followig remarks
More information1988 AP Calculus BC: Section I
988 AP Calculus BC: Sectio I 9 Miutes No Calculator Notes: () I this eamiatio, l deotes the atural logarithm of (that is, logarithm to the base e). () Uless otherwise specified, the domai of a fuctio f
More informationChapter 1. Complex Numbers. Dr. Pulak Sahoo
Chapter 1 Complex Numbers BY Dr. Pulak Sahoo Assistat Professor Departmet of Mathematics Uiversity Of Kalyai West Begal, Idia E-mail : sahoopulak1@gmail.com 1 Module-2: Stereographic Projectio 1 Euler
More informationAnswer: 1(A); 2(C); 3(A); 4(D); 5(B); 6(A); 7(C); 8(C); 9(A); 10(A); 11(A); 12(C); 13(C)
Aswer: (A); (C); 3(A); 4(D); 5(B); 6(A); 7(C); 8(C); 9(A); 0(A); (A); (C); 3(C). A two loop positio cotrol system is show below R(s) Y(s) + + s(s +) - - s The gai of the Tacho-geerator iflueces maily the
More informationCONTROL SYSTEMS. Chapter 7 : Bode Plot. 40dB/dec 1.0. db/dec so resultant slope will be 20 db/dec and this is due to the factor s
CONTROL SYSTEMS Chapter 7 : Bode Plot GATE Objective & Numerical Type Solutio Quetio 6 [Practice Book] [GATE EE 999 IIT-Bombay : 5 Mark] The aymptotic Bode plot of the miimum phae ope-loop trafer fuctio
More informationThe z-transform can be used to obtain compact transform-domain representations of signals and systems. It
3 4 5 6 7 8 9 10 CHAPTER 3 11 THE Z-TRANSFORM 31 INTRODUCTION The z-trasform ca be used to obtai compact trasform-domai represetatios of sigals ad systems It provides ituitio particularly i LTI system
More informationDynamic System Response
Solutio of Liear, Costat-Coefficiet, Ordiary Differetial Equatios Classical Operator Method Laplace Trasform Method Laplace Trasform Properties 1 st -Order Dyamic System Time ad Frequecy Respose d -Order
More informationEE / EEE SAMPLE STUDY MATERIAL. GATE, IES & PSUs Signal System. Electrical Engineering. Postal Correspondence Course
Sigal-EE Postal Correspodece Course 1 SAMPLE STUDY MATERIAL Electrical Egieerig EE / EEE Postal Correspodece Course GATE, IES & PSUs Sigal System Sigal-EE Postal Correspodece Course CONTENTS 1. SIGNAL
More informationPrinciple Of Superposition
ecture 5: PREIMINRY CONCEP O RUCUR NYI Priciple Of uperpositio Mathematically, the priciple of superpositio is stated as ( a ) G( a ) G( ) G a a or for a liear structural system, the respose at a give
More informationFIR Filter Design: Part I
EEL3: Discrete-Time Sigals ad Systems FIR Filter Desig: Part I. Itroductio FIR Filter Desig: Part I I this set o otes, we cotiue our exploratio o the requecy respose o FIR ilters. First, we cosider some
More informationHigher Course Plan. Calculus and Relationships Expressions and Functions
Higher Course Pla Applicatios Calculus ad Relatioships Expressios ad Fuctios Topic 1: The Straight Lie Fid the gradiet of a lie Colliearity Kow the features of gradiets of: parallel lies perpedicular lies
More informationTime-Domain Representations of LTI Systems
2.1 Itroductio Objectives: 1. Impulse resposes of LTI systems 2. Liear costat-coefficiets differetial or differece equatios of LTI systems 3. Bloc diagram represetatios of LTI systems 4. State-variable
More informationNANYANG TECHNOLOGICAL UNIVERSITY SYLLABUS FOR ENTRANCE EXAMINATION FOR INTERNATIONAL STUDENTS AO-LEVEL MATHEMATICS
NANYANG TECHNOLOGICAL UNIVERSITY SYLLABUS FOR ENTRANCE EXAMINATION FOR INTERNATIONAL STUDENTS AO-LEVEL MATHEMATICS STRUCTURE OF EXAMINATION PAPER. There will be oe 2-hour paper cosistig of 4 questios.
More informationInfinite Sequences and Series
Chapter 6 Ifiite Sequeces ad Series 6.1 Ifiite Sequeces 6.1.1 Elemetary Cocepts Simply speakig, a sequece is a ordered list of umbers writte: {a 1, a 2, a 3,...a, a +1,...} where the elemets a i represet
More informationMTH Assignment 1 : Real Numbers, Sequences
MTH -26 Assigmet : Real Numbers, Sequeces. Fid the supremum of the set { m m+ : N, m Z}. 2. Let A be a o-empty subset of R ad α R. Show that α = supa if ad oly if α is ot a upper boud of A but α + is a
More informationWe are mainly going to be concerned with power series in x, such as. (x)} converges - that is, lims N n
Review of Power Series, Power Series Solutios A power series i x - a is a ifiite series of the form c (x a) =c +c (x a)+(x a) +... We also call this a power series cetered at a. Ex. (x+) is cetered at
More informationCEE 522 Autumn Uncertainty Concepts for Geotechnical Engineering
CEE 5 Autum 005 Ucertaity Cocepts for Geotechical Egieerig Basic Termiology Set A set is a collectio of (mutually exclusive) objects or evets. The sample space is the (collectively exhaustive) collectio
More informationREVISION SHEET FP1 (MEI) ALGEBRA. Identities In mathematics, an identity is a statement which is true for all values of the variables it contains.
the Further Mathematics etwork wwwfmetworkorguk V 07 The mai ideas are: Idetities REVISION SHEET FP (MEI) ALGEBRA Before the exam you should kow: If a expressio is a idetity the it is true for all values
More informationMth 138 College Algebra Review Guide for Exam III
Mth 138 College Algebra Review Guide for Exam III Thomas W. Judso Stephe F. Austi State Uiversity Sprig 2018 Exam III Details Exam III will be o Thursday, April 19 ad will cover material up to Chapter
More informationChapter 9: Numerical Differentiation
178 Chapter 9: Numerical Differetiatio Numerical Differetiatio Formulatio of equatios for physical problems ofte ivolve derivatives (rate-of-chage quatities, such as velocity ad acceleratio). Numerical
More informationCOMM 602: Digital Signal Processing
COMM 60: Digital Sigal Processig Lecture 4 -Properties of LTIS Usig Z-Trasform -Iverse Z-Trasform Properties of LTIS Usig Z-Trasform Properties of LTIS Usig Z-Trasform -ve +ve Properties of LTIS Usig Z-Trasform
More informationAppendix F: Complex Numbers
Appedix F Complex Numbers F1 Appedix F: Complex Numbers Use the imagiary uit i to write complex umbers, ad to add, subtract, ad multiply complex umbers. Fid complex solutios of quadratic equatios. Write
More informationAssignment 1 : Real Numbers, Sequences. for n 1. Show that (x n ) converges. Further, by observing that x n+2 + x n+1
Assigmet : Real Numbers, Sequeces. Let A be a o-empty subset of R ad α R. Show that α = supa if ad oly if α is ot a upper boud of A but α + is a upper boud of A for every N. 2. Let y (, ) ad x (, ). Evaluate
More informationCurve Sketching Handout #5 Topic Interpretation Rational Functions
Curve Sketchig Hadout #5 Topic Iterpretatio Ratioal Fuctios A ratioal fuctio is a fuctio f that is a quotiet of two polyomials. I other words, p ( ) ( ) f is a ratioal fuctio if p ( ) ad q ( ) are polyomials
More informationUniversity of California at Berkeley College of Engineering Department of Electrical Engineering and Computer Sciences
A Uiversity of Califoria at Berkeley College of Egieerig Departmet of Electrical Egieerig ad Computer Scieces U N I V E R S T H E I T Y O F LE T TH E R E B E LI G H T C A L I F O R N 8 6 8 I A EECS : Sigals
More informationHonors Calculus Homework 13 Solutions, due 12/8/5
Hoors Calculus Homework Solutios, due /8/5 Questio Let a regio R i the plae be bouded by the curves y = 5 ad = 5y y. Sketch the regio R. The two curves meet where both equatios hold at oce, so where: y
More informationUnit 4: Polynomial and Rational Functions
48 Uit 4: Polyomial ad Ratioal Fuctios Polyomial Fuctios A polyomial fuctio y px ( ) is a fuctio of the form p( x) ax + a x + a x +... + ax + ax+ a 1 1 1 0 where a, a 1,..., a, a1, a0are real costats ad
More informationADVANCED DIGITAL SIGNAL PROCESSING
ADVANCED DIGITAL SIGNAL PROCESSING PROF. S. C. CHAN (email : sccha@eee.hku.hk, Rm. CYC-702) DISCRETE-TIME SIGNALS AND SYSTEMS MULTI-DIMENSIONAL SIGNALS AND SYSTEMS RANDOM PROCESSES AND APPLICATIONS ADAPTIVE
More informationDefinition of z-transform.
- Trasforms Frequecy domai represetatios of discretetime sigals ad LTI discrete-time systems are made possible with the use of DTFT. However ot all discrete-time sigals e.g. uit step sequece are guarateed
More informationU8L1: Sec Equations of Lines in R 2
MCVU U8L: Sec. 8.9. Equatios of Lies i R Review of Equatios of a Straight Lie (-D) Cosider the lie passig through A (-,) with slope, as show i the diagram below. I poit slope form, the equatio of the lie
More informationSequences A sequence of numbers is a function whose domain is the positive integers. We can see that the sequence
Sequeces A sequece of umbers is a fuctio whose domai is the positive itegers. We ca see that the sequece 1, 1, 2, 2, 3, 3,... is a fuctio from the positive itegers whe we write the first sequece elemet
More informationmx bx kx F t. dt IR I LI V t, Q LQ RQ V t,
Lecture 5 omplex Variables II (Applicatios i Physics) (See hapter i Boas) To see why complex variables are so useful cosider first the (liear) mechaics of a sigle particle described by Newto s equatio
More informationWarped, Chirp Z-Transform: Radar Signal Processing
arped, Chirp Z-Trasform: Radar Sigal Processig by Garimella Ramamurthy Report o: IIIT/TR// Cetre for Commuicatios Iteratioal Istitute of Iformatio Techology Hyderabad - 5 3, IDIA Jauary ARPED, CHIRP Z
More informationPractical Spectral Anaysis (continue) (from Boaz Porat s book) Frequency Measurement
Practical Spectral Aaysis (cotiue) (from Boaz Porat s book) Frequecy Measuremet Oe of the most importat applicatios of the DFT is the measuremet of frequecies of periodic sigals (eg., siusoidal sigals),
More informationMEM 255 Introduction to Control Systems: Analyzing Dynamic Response
MEM 55 Itroductio to Cotrol Systems: Aalyzig Dyamic Respose Harry G. Kwaty Departmet of Mechaical Egieerig & Mechaics Drexel Uiversity Outlie Time domai ad frequecy domai A secod order system Via partial
More informationPHYSICS 116A Homework 2 Solutions
PHYSICS 6A Homework 2 Solutios I. [optioal] Boas, Ch., 6, Qu. 30 (proof of the ratio test). Just follow the hits. If ρ, the ratio of succcessive terms for is less tha, the hits show that the terms of the
More informationChapter 3. z-transform
Chapter 3 -Trasform 3.0 Itroductio The -Trasform has the same role as that played by the Laplace Trasform i the cotiuous-time theorem. It is a liear operator that is useful for aalyig LTI systems such
More informationTHE SYSTEMATIC AND THE RANDOM. ERRORS - DUE TO ELEMENT TOLERANCES OF ELECTRICAL NETWORKS
R775 Philips Res. Repts 26,414-423, 1971' THE SYSTEMATIC AND THE RANDOM. ERRORS - DUE TO ELEMENT TOLERANCES OF ELECTRICAL NETWORKS by H. W. HANNEMAN Abstract Usig the law of propagatio of errors, approximated
More informationChapter 10: Power Series
Chapter : Power Series 57 Chapter Overview: Power Series The reaso series are part of a Calculus course is that there are fuctios which caot be itegrated. All power series, though, ca be itegrated because
More informationMaximum and Minimum Values
Sec 4.1 Maimum ad Miimum Values A. Absolute Maimum or Miimum / Etreme Values A fuctio Similarly, f has a Absolute Maimum at c if c f f has a Absolute Miimum at c if c f f for every poit i the domai. f
More informationChapter 2 Systems and Signals
Chapter 2 Systems ad Sigals 1 Itroductio Discrete-Time Sigals: Sequeces Discrete-Time Systems Properties of Liear Time-Ivariat Systems Liear Costat-Coefficiet Differece Equatios Frequecy-Domai Represetatio
More information(Figure 2.9), we observe x. and we write. (b) as x, x 1. and we write. We say that the line y 0 is a horizontal asymptote of the graph of f.
The symbol for ifiity ( ) does ot represet a real umber. We use to describe the behavior of a fuctio whe the values i its domai or rage outgrow all fiite bouds. For eample, whe we say the limit of f as
More informationENGI Series Page 6-01
ENGI 3425 6 Series Page 6-01 6. Series Cotets: 6.01 Sequeces; geeral term, limits, covergece 6.02 Series; summatio otatio, covergece, divergece test 6.03 Stadard Series; telescopig series, geometric series,
More informationAlgorithm Analysis. Chapter 3
Data Structures Dr Ahmed Rafat Abas Computer Sciece Dept, Faculty of Computer ad Iformatio, Zagazig Uiversity arabas@zu.edu.eg http://www.arsaliem.faculty.zu.edu.eg/ Algorithm Aalysis Chapter 3 3. Itroductio
More informationMath 105: Review for Final Exam, Part II - SOLUTIONS
Math 5: Review for Fial Exam, Part II - SOLUTIONS. Cosider the fuctio f(x) = x 3 lx o the iterval [/e, e ]. (a) Fid the x- ad y-coordiates of ay ad all local extrema ad classify each as a local maximum
More information62. Power series Definition 16. (Power series) Given a sequence {c n }, the series. c n x n = c 0 + c 1 x + c 2 x 2 + c 3 x 3 +
62. Power series Defiitio 16. (Power series) Give a sequece {c }, the series c x = c 0 + c 1 x + c 2 x 2 + c 3 x 3 + is called a power series i the variable x. The umbers c are called the coefficiets of
More informationDiscrete-Time Systems, LTI Systems, and Discrete-Time Convolution
EEL5: Discrete-Time Sigals ad Systems. Itroductio I this set of otes, we begi our mathematical treatmet of discrete-time s. As show i Figure, a discrete-time operates or trasforms some iput sequece x [
More informationQuadratic Functions. Before we start looking at polynomials, we should know some common terminology.
Quadratic Fuctios I this sectio we begi the study of fuctios defied by polyomial expressios. Polyomial ad ratioal fuctios are the most commo fuctios used to model data, ad are used extesively i mathematical
More informationRun-length & Entropy Coding. Redundancy Removal. Sampling. Quantization. Perform inverse operations at the receiver EEE
Geeral e Image Coder Structure Motio Video (s 1,s 2,t) or (s 1,s 2 ) Natural Image Samplig A form of data compressio; usually lossless, but ca be lossy Redudacy Removal Lossless compressio: predictive
More informationFAILURE CRITERIA: MOHR S CIRCLE AND PRINCIPAL STRESSES
LECTURE Third Editio FAILURE CRITERIA: MOHR S CIRCLE AND PRINCIPAL STRESSES A. J. Clark School of Egieerig Departmet of Civil ad Evirometal Egieerig Chapter 7.4 b Dr. Ibrahim A. Assakkaf SPRING 3 ENES
More informationAppendix: The Laplace Transform
Appedix: The Laplace Trasform The Laplace trasform is a powerful method that ca be used to solve differetial equatio, ad other mathematical problems. Its stregth lies i the fact that it allows the trasformatio
More informationFrequency Domain Filtering
Frequecy Domai Filterig Raga Rodrigo October 19, 2010 Outlie Cotets 1 Itroductio 1 2 Fourier Represetatio of Fiite-Duratio Sequeces: The Discrete Fourier Trasform 1 3 The 2-D Discrete Fourier Trasform
More informationSignal Processing. Lecture 02: Discrete Time Signals and Systems. Ahmet Taha Koru, Ph. D. Yildiz Technical University.
Sigal Processig Lecture 02: Discrete Time Sigals ad Systems Ahmet Taha Koru, Ph. D. Yildiz Techical Uiversity 2017-2018 Fall ATK (YTU) Sigal Processig 2017-2018 Fall 1 / 51 Discrete Time Sigals Discrete
More informationSignal Processing in Mechatronics. Lecture 3, Convolution, Fourier Series and Fourier Transform
Sigal Processig i Mechatroics Summer semester, 1 Lecture 3, Covolutio, Fourier Series ad Fourier rasform Dr. Zhu K.P. AIS, UM 1 1. Covolutio Covolutio Descriptio of LI Systems he mai premise is that the
More informationChapter 9 - CD companion 1. A Generic Implementation; The Common-Merge Amplifier. 1 τ is. ω ch. τ io
Chapter 9 - CD compaio CHAPTER NINE CD-9.2 CD-9.2. Stages With Voltage ad Curret Gai A Geeric Implemetatio; The Commo-Merge Amplifier The advaced method preseted i the text for approximatig cutoff frequecies
More informationG r a d e 1 1 P r e - C a l c u l u s M a t h e m a t i c s ( 3 0 S )
G r a d e 1 1 P r e - C a l c u l u s M a t h e m a t i c s ( 3 0 S ) Grade 11 Pre-Calculus Mathematics (30S) is desiged for studets who ited to study calculus ad related mathematics as part of post-secodary
More informationCourse Outline. Problem Identification. Engineering as Design. Amme 3500 : System Dynamics and Control. System Response. Dr. Stefan B.
Course Outlie Amme 35 : System Dyamics a Cotrol System Respose Week Date Cotet Assigmet Notes Mar Itrouctio 8 Mar Frequecy Domai Moellig 3 5 Mar Trasiet Performace a the s-plae 4 Mar Block Diagrams Assig
More informationModule 18 Discrete Time Signals and Z-Transforms Objective: Introduction : Description: Discrete Time Signal representation
Module 8 Discrete Time Sigals ad Z-Trasforms Objective:To uderstad represetig discrete time sigals, apply z trasform for aalyzigdiscrete time sigals ad to uderstad the relatio to Fourier trasform Itroductio
More informationECE 422/522 Power System Operations & Planning/Power Systems Analysis II : 6 - Small Signal Stability
ECE 4/5 Power System Operatios & Plaig/Power Systems Aalysis II : 6 - Small Sigal Stability Sprig 014 Istructor: Kai Su 1 Refereces Kudur s Chapter 1 Saadat s Chapter 11.4 EPRI Tutorial s Chapter 8 Power
More informationCh3 Discrete Time Fourier Transform
Ch3 Discrete Time Fourier Trasform 3. Show that the DTFT of [] is give by ( k). e k 3. Determie the DTFT of the two sided sigal y [ ],. 3.3 Determie the DTFT of the causal sequece x[ ] A cos( 0 ) [ ],
More informationThe z transform is the discrete-time counterpart of the Laplace transform. Other description: see page 553, textbook.
The -Trasform 7. Itroductio The trasform is the discrete-time couterpart of the Laplace trasform. Other descriptio: see page 553, textbook. 7. The -trasform Derivatio of the -trasform: x[] re jω LTI system,
More informationAbstract Vector Spaces. Abstract Vector Spaces
Astract Vector Spaces The process of astractio is critical i egieerig! Physical Device Data Storage Vector Space MRI machie Optical receiver 0 0 1 0 1 0 0 1 Icreasig astractio 6.1 Astract Vector Spaces
More informationA sequence of numbers is a function whose domain is the positive integers. We can see that the sequence
Sequeces A sequece of umbers is a fuctio whose domai is the positive itegers. We ca see that the sequece,, 2, 2, 3, 3,... is a fuctio from the positive itegers whe we write the first sequece elemet as
More informationVICTORIA JUNIOR COLLEGE Preliminary Examination. Paper 1 September 2015
VICTORIA JUNIOR COLLEGE Prelimiary Eamiatio MATHEMATICS (Higher ) 70/0 Paper September 05 Additioal Materials: Aswer Paper Graph Paper List of Formulae (MF5) 3 hours READ THESE INSTRUCTIONS FIRST Write
More information