The Transfer Function. The Transfer Function. The Transfer Function. The Transfer Function. The Transfer Function. The Transfer Function
|
|
- Emery Curtis
- 6 years ago
- Views:
Transcription
1 A gnraliation of th frquncy rsons function Th convolution sum scrition of an LTI iscrt-tim systm with an imuls rsons h[n] is givn by h y [ n] [ ] x[ n ] Taing th -transforms of both sis w gt n n h n n Y y n x n ( [ ] [ ] [ ] h [ ] x[ n ] n ( l+ h[ ] x[ l] l n Or, Thrfor, Thus, Y( l h x [ ] [l] l Y( H(X( X ( Y( h[ ] X ( H( Hnc, H ( Y( / X ( Th function H(, which is th -transform of th imuls rsons h[n] of th LTI systm, is call th transfr function or th systm function Th invrs -transform of th transfr function H( yils th imuls rsons h[n] 3 5 Consir an LTI iscrt-tim systm charactri by a iffrnc quation x[ n ] y[ n ] Its transfr function is obtain by taing th -transform of both sis of th abov quation Thus H( Or, quivalntly as ( H( An altrnat form of th transfr function is givn by H( ( ξ ( λ
2 7 Or, quivalntly as H ( ( ξ ( ( ξ, ξ,..., ξ ar th finit ros, an λ λ, λ ar th finit ols of H(,,... If >, thr ar aitional ( ros at If <, thr ar aitional ( ols at 8 For a causal IIR igital filtr, th imuls rsons is a causal squnc Th ROC of th causal transfr function H ( ( ξ ( ( is thus xtrior to a circl going through th ol furthst from th origin Thus th ROC is givn by > max λ 9 Examl-Consir th -oint movingavrag FIR filtr with an imuls rsons /, n h[n], othrwis Its transfr function is thn givn by ( n H ( [ n ( ] Th transfr function has ros on th unit circl at j π /, Thr ar ols at an a singl ol at 8 Th ol at.5 xactly cancls th 7 ro at -.5 Th ROC is th ntir - -lan xct Imaginary Part Ral Part Examl-A causal LTI IIR igital filtr is scrib by a constant cofficint iffrnc quation givn by y[ n] x[ n ]. x[ n ] + x[ n 3] +.3 y[ n ]. y[ n ] +. y[ n 3] Its transfr function is thrfor givn by H ( Altrnat forms:. + H ( (. + j.8(. j.8 (.3(.5 + j.7(.5 j.7 ot:pols farthst from hav a magnitu.7 ROC: >.7 Imaginary Part Ral Part
3 3 Frquncy Rsons from Transfr Function If th ROC of th transfr function H( inclus th unit circl, thn th frquncy rsons H ( of th LTI igital filtr can b obtain simly as follows: H ( j H ( ω For a ral cofficint transfr function H( it can b shown that H ( H ( H *( H ( H ( H ( H ( Frquncy Rsons from Transfr Function For a stabl rational transfr function in th form H ( ( ξ ( ( th factor form of th frquncy rsons is givn by ω j H ( ( ( ( 5 Frquncy Rsons from Transfr Function It is convnint to visuali th contributions of th ro factor ( an th ol factor ( from th factor form of th frquncy rsons Th magnitu function is givn by H ( ( Frquncy Rsons from Transfr Function which rucs to H ( Th has rsons for a rational transfr function is of th form arg H ( j ω arg( / + ω( + ω arg( j arg( 7 Frquncy Rsons from Transfr Function Th magnitu-squar function of a ralcofficint transfr function can b comut using H ( ( ( ( ( * * 8 Th factor form of th frquncy rsons ω j H ( ( ( ( is convnint to vlo a gomtric intrrtation of th frquncy rsons comutation from th ol-ro lot as ω varis from to π on th unit circl 3
4 9 Th gomtric intrrtation can b us to obtain a stch of th rsons as a function of th frquncy A tyical factor in th factor form of th frquncy rsons is givn by ( ρ j φ whr ρ jφ is a ro if it is ro factor or is a ol if it is a ol factor As shown blow in th -lan th factor ( ρ j φ rrsnts a vctor starting at th oint ρ jφ an ning on th unit circl at As ω is vari from to π, th ti of th vctor movs countrclocis from th oint tracing th unit circl an bac to th oint As inicat by H ( th magnitu rsons H ( at a scific valu of ω is givn by th rouct of th magnitus of all ro vctors ivi by th rouct of th magnitus of all ol vctors 3 Liwis, from arg H ( j ω arg( / + ω( + arg( arg( w obsrv that th has rsons at a scific valu of ω is obtain by aing th has of th trm / an th linar-has trm ω( to th sum of th angls of th ro vctors minus th angls of th ol vctors Thus, an aroximat lot of th magnitu an has rsonss of th transfr function of an LTI igital filtr can b vlo by xamining th ol an ro locations ow, a ro (ol vctor has th smallst magnitu whn ω φ
5 5 To highly attnuat signal comonnts in a scifi frquncy rang, w n to lac ros vry clos to or on th unit circl in this rang Liwis, to highly mhasi signal comonnts in a scifi frquncy rang, w n to lac ols vry clos to or on th unit circl in this rang A causal LTI igital filtr is BIBO stabl if an only if its imuls rsons h[n] is absolutly summabl, i.., S h [ n] < n W now vlo a stability conition in trms of th ol locations of th transfr function H( Th ROC of th -transform H( of th imuls rsons squnc h[n] is fin n by valus of r for which h[ n] r is absolutly summabl Thus, if th ROC inclus th unit circl, thn th igital filtr is stabl, an vic vrsa In aition, for a stabl an causal igital filtr for which h[n] is a right-si squnc, th ROC will inclu th unit circl an ntir -lan incluing th oint An FIR igital filtr with boun imuls rsons is always stabl 7 8 On th othr han, an IIR filtr may b unstabl if not sign rorly In aition, an originally stabl IIR filtr charactri by infinit rcision cofficints may bcom unstabl whn cofficints gt quanti u to imlmntation Examl-Consir th causal IIR transfr function H ( Th lot of th imuls rsons cofficints is shown on th nxt sli 9 3 5
6 3 Amlitu h[n] Tim inx n As can b sn from th abov lot, th imuls rsons cofficint h[n] cays raily to ro valu as n incrass 3 Th absolut summability conition of h[n] is satisfi Hnc, H( is a stabl transfr function ow, consir th cas whn th transfr function cofficints ar roun to valus with igits aftr th cimal oint: ^ H ( A lot of th imuls rsons of h^[n] is shown blow Amlitu h^[n] In this cas, th imuls rsons cofficint h^[n] incrass raily to a constant valu as n incrass Hnc, th absolut summability conition of is violat Thus, H^ ( is an unstabl transfr function Tim inx n 3 35 Th stability tsting of a IIR transfr function is thrfor an imortant roblm In most cass it is ifficult to comut th infinit sum S n h [ n] < For a causal IIR transfr function, th sum S can b comut aroximatly as K h [ n] S K n 3 Th artial sum is comut for incrasing valus of K until th iffrnc btwn a sris of conscutiv valus of S K is smallr than som arbitrarily chosn small numbr, which is tyically For a transfr function of vry high orr this aroach may not b satisfactory An altrnat, asy-to-tst, stability conition is vlo nxt
7 37 Consir th causal IIR igital filtr with a rational transfr function H( givn by H ( Its imuls rsons {h[n]} is a right-si squnc Th ROC of H( is xtrior to a circl going through th ol furthst from 38 But stability rquirs that {h[n]} b absolutly summabl This in turn imlis that th DTFT of {h[n]} xists ow, if th ROC of th -transform H( inclus th unit circl, thn H ( j H ( ω H ( 39 Conclusion: All ols of a causal stabl transfr function H( must b strictly insi th unit circl Th stability rgion (shown sha in th -lan is shown blow unit circl j Im j j stability rgion R Examl- Th factor form of is H ( H ( (.9 (.93 which has a ral ol at.9 an a ral ol at.93 Sinc both ols ar insi th unit circl, H( is BIBO stabl Examl- Th factor form of is H^ ( H^ ( ( (.85 which has a ral ol on th unit circl at an th othr ol insi th unit circl Sinc both ols ar not insi th unit circl, H( is unstabl 7
the output is Thus, the output lags in phase by θ( ωo) radians Rewriting the above equation we get
Th output y[ of a frquncy-sctiv LTI iscrt-tim systm with a frquncy rspons H ( xhibits som ay rativ to th input caus by th nonro phas rspons θ( ω arg{ H ( } of th systm For an input A cos( ωo n + φ, < n
More informationLinear-Phase FIR Transfer Functions. Functions. Functions. Functions. Functions. Functions. Let
It is impossibl to dsign an IIR transfr function with an xact linar-phas It is always possibl to dsign an FIR transfr function with an xact linar-phas rspons W now dvlop th forms of th linarphas FIR transfr
More information[ ] [ ] DFT: Discrete Fourier Transform ( ) ( ) ( ) ( ) Congruence (Integer modulo m) N-point signal
Congrunc (Intgr modulo m) : Discrt Fourir Transform In this sction, all lttrs stand for intgrs. gcd ( nm, ) th gratst common divisor of n and m Lt d gcd(n,m) All th linar combinations r n+ s m of n and
More information10. The Discrete-Time Fourier Transform (DTFT)
Th Discrt-Tim Fourir Transform (DTFT Dfinition of th discrt-tim Fourir transform Th Fourir rprsntation of signals plays an important rol in both continuous and discrt signal procssing In this sction w
More informationDigital Signal Processing, Fall 2006
Digital Signal Prossing, Fall 006 Ltur 7: Filtr Dsign Zhng-ua an Dpartmnt of Eltroni Systms Aalborg Univrsity, Dnmar t@om.aau. Cours at a glan MM Disrt-tim signals an systms Systm MM Fourir-omain rprsntation
More informationTypes of Transfer Functions. Types of Transfer Functions. Types of Transfer Functions. Ideal Filters. Ideal Filters
Typs of Transfr Typs of Transfr x[n] X( LTI h[n] H( y[n] Y( y [ n] h[ k] x[ n k] k Y ( H ( X ( Th tim-domain classification of an LTI digital transfr function is basd on th lngth of its impuls rspons h[n]:
More informationTypes of Transfer Functions. Types of Transfer Functions. Ideal Filters. Ideal Filters. Ideal Filters
Typs of Transfr Typs of Transfr Th tim-domain classification of an LTI digital transfr function squnc is basd on th lngth of its impuls rspons: - Finit impuls rspons (FIR) transfr function - Infinit impuls
More informationLecture 2: Discrete-Time Signals & Systems. Reza Mohammadkhani, Digital Signal Processing, 2015 University of Kurdistan eng.uok.ac.
Lctur 2: Discrt-Tim Signals & Systms Rza Mohammadkhani, Digital Signal Procssing, 2015 Univrsity of Kurdistan ng.uok.ac.ir/mohammadkhani 1 Signal Dfinition and Exampls 2 Signal: any physical quantity that
More informationExercise 1. Sketch the graph of the following function. (x 2
Writtn tst: Fbruary 9th, 06 Exrcis. Sktch th graph of th following function fx = x + x, spcifying: domain, possibl asymptots, monotonicity, continuity, local and global maxima or minima, and non-drivability
More informationFirst order differential equation Linear equation; Method of integrating factors
First orr iffrntial quation Linar quation; Mtho of intgrating factors Exampl 1: Rwrit th lft han si as th rivativ of th prouct of y an som function by prouct rul irctly. Solving th first orr iffrntial
More informationY 0. Standing Wave Interference between the incident & reflected waves Standing wave. A string with one end fixed on a wall
Staning Wav Intrfrnc btwn th incint & rflct wavs Staning wav A string with on n fix on a wall Incint: y, t) Y cos( t ) 1( Y 1 ( ) Y (St th incint wav s phas to b, i.., Y + ral & positiv.) Rflct: y, t)
More information2. Finite Impulse Response Filters (FIR)
.. Mthos for FIR filtrs implmntation. Finit Impuls Rspons Filtrs (FIR. Th winow mtho.. Frquncy charactristic uniform sampling. 3. Maximum rror minimizing. 4. Last-squars rror minimizing.. Mthos for FIR
More informationMathematics 1110H Calculus I: Limits, derivatives, and Integrals Trent University, Summer 2018 Solutions to the Actual Final Examination
Mathmatics H Calculus I: Limits, rivativs, an Intgrals Trnt Univrsity, Summr 8 Solutions to th Actual Final Eamination Tim-spac: 9:-: in FPHL 7. Brought to you by Stfan B lan k. Instructions: Do parts
More informationSchematic of a mixed flow reactor (both advection and dispersion must be accounted for)
Cas stuy 6.1, R: Chapra an Canal, p. 769. Th quation scribin th concntration o any tracr in an lonat ractor is known as th avction-isprsion quation an may b writtn as: Schmatic o a mi low ractor (both
More informationProblem Set 6 Solutions
6.04/18.06J Mathmatics for Computr Scinc March 15, 005 Srini Dvadas and Eric Lhman Problm St 6 Solutions Du: Monday, March 8 at 9 PM in Room 3-044 Problm 1. Sammy th Shark is a financial srvic providr
More informationAdditional Math (4047) Paper 2 (100 marks) y x. 2 d. d d
Aitional Math (07) Prpar b Mr Ang, Nov 07 Fin th valu of th constant k for which is a solution of th quation k. [7] Givn that, Givn that k, Thrfor, k Topic : Papr (00 marks) Tim : hours 0 mins Nam : Aitional
More informationThe Frequency Response of a Quarter-Wave Matching Network
4/1/29 Th Frquncy Rsons o a Quartr 1/9 Th Frquncy Rsons o a Quartr-Wav Matchg Ntwork Q: You hav onc aga rovidd us with conusg and rhas uslss ormation. Th quartr-wav matchg ntwork has an xact SFG o: a Τ
More informationThe Matrix Exponential
Th Matrix Exponntial (with xrciss) by D. Klain Vrsion 207.0.05 Corrctions and commnts ar wlcom. Th Matrix Exponntial For ach n n complx matrix A, dfin th xponntial of A to b th matrix A A k I + A + k!
More informationa 1and x is any real number.
Calcls Nots Eponnts an Logarithms Eponntial Fnction: Has th form y a, whr a 0, a an is any ral nmbr. Graph y, Graph y ln y y Th Natral Bas (Elr s nmbr): An irrational nmbr, symboliz by th lttr, appars
More informationBSc Engineering Sciences A. Y. 2017/18 Written exam of the course Mathematical Analysis 2 August 30, x n, ) n 2
BSc Enginring Scincs A. Y. 27/8 Writtn xam of th cours Mathmatical Analysis 2 August, 28. Givn th powr sris + n + n 2 x n, n n dtrmin its radius of convrgnc r, and study th convrgnc for x ±r. By th root
More informationThe Matrix Exponential
Th Matrix Exponntial (with xrciss) by Dan Klain Vrsion 28928 Corrctions and commnts ar wlcom Th Matrix Exponntial For ach n n complx matrix A, dfin th xponntial of A to b th matrix () A A k I + A + k!
More informationExamples and applications on SSSP and MST
Exampls an applications on SSSP an MST Dan (Doris) H & Junhao Gan ITEE Univrsity of Qunslan COMP3506/7505, Uni of Qunslan Exampls an applications on SSSP an MST Dijkstra s Algorithm Th algorithm solvs
More informationPHYS ,Fall 05, Term Exam #1, Oct., 12, 2005
PHYS1444-,Fall 5, Trm Exam #1, Oct., 1, 5 Nam: Kys 1. circular ring of charg of raius an a total charg Q lis in th x-y plan with its cntr at th origin. small positiv tst charg q is plac at th origin. What
More information2008 AP Calculus BC Multiple Choice Exam
008 AP Multipl Choic Eam Nam 008 AP Calculus BC Multipl Choic Eam Sction No Calculator Activ AP Calculus 008 BC Multipl Choic. At tim t 0, a particl moving in th -plan is th acclration vctor of th particl
More informationEEO 401 Digital Signal Processing Prof. Mark Fowler
EEO 401 Digital Signal Procssing Prof. Mark Fowlr Dtails of th ot St #19 Rading Assignmnt: Sct. 7.1.2, 7.1.3, & 7.2 of Proakis & Manolakis Dfinition of th So Givn signal data points x[n] for n = 0,, -1
More informationy cos x = cos xdx = sin x + c y = tan x + c sec x But, y = 1 when x = 0 giving c = 1. y = tan x + sec x (A1) (C4) OR y cos x = sin x + 1 [8]
DIFF EQ - OPTION. Sol th iffrntial quation tan +, 0
More informationChapter 10. The singular integral Introducing S(n) and J(n)
Chaptr Th singular intgral Our aim in this chaptr is to rplac th functions S (n) and J (n) by mor convnint xprssions; ths will b calld th singular sris S(n) and th singular intgral J(n). This will b don
More informationStability Condition in Terms of the Pole Locations
Stability Condition in Terms of the Pole Locations A causal LTI digital filter is BIBO stable if and only if its impulse response h[n] is absolutely summable, i.e., 1 = S h [ n] < n= We now develop a stability
More informationDISCRETE TIME FOURIER TRANSFORM (DTFT)
DISCRETE TIME FOURIER TRANSFORM (DTFT) Th dicrt-tim Fourir Tranform x x n xn n n Th Invr dicrt-tim Fourir Tranform (IDTFT) x n Not: ( ) i a complx valud continuou function = π f [rad/c] f i th digital
More informationINCOMPLETE KLOOSTERMAN SUMS AND MULTIPLICATIVE INVERSES IN SHORT INTERVALS. xy 1 (mod p), (x, y) I (j)
INCOMPLETE KLOOSTERMAN SUMS AND MULTIPLICATIVE INVERSES IN SHORT INTERVALS T D BROWNING AND A HAYNES Abstract W invstigat th solubility of th congrunc xy (mod ), whr is a rim and x, y ar rstrictd to li
More informationThe second condition says that a node α of the tree has exactly n children if the arity of its label is n.
CS 6110 S14 Hanout 2 Proof of Conflunc 27 January 2014 In this supplmntary lctur w prov that th λ-calculus is conflunt. This is rsult is u to lonzo Church (1903 1995) an J. arkly Rossr (1907 1989) an is
More informationSolution: APPM 1360 Final (150 pts) Spring (60 pts total) The following parts are not related, justify your answers:
APPM 6 Final 5 pts) Spring 4. 6 pts total) Th following parts ar not rlatd, justify your answrs: a) Considr th curv rprsntd by th paramtric quations, t and y t + for t. i) 6 pts) Writ down th corrsponding
More informationThomas Whitham Sixth Form
Thomas Whitham Sith Form Pur Mathmatics Unit C Algbra Trigonomtr Gomtr Calculus Vctor gomtr Pag Algbra Molus functions graphs, quations an inqualitis Graph of f () Draw f () an rflct an part of th curv
More informationDFT: Discrete Fourier Transform
: Discrt Fourir Trasform Cogruc (Itgr modulo m) I this sctio, all lttrs stad for itgrs. gcd m, = th gratst commo divisor of ad m Lt d = gcd(,m) All th liar combiatios r s m of ad m ar multils of d. a b
More information4. (5a + b) 7 & x 1 = (3x 1)log 10 4 = log (M1) [4] d = 3 [4] T 2 = 5 + = 16 or or 16.
. 7 7 7... 7 7 (n )0 7 (M) 0(n ) 00 n (A) S ((7) 0(0)) (M) (7 00) 8897 (A). (5a b) 7 7... (5a)... (M) 7 5 5 (a b ) 5 5 a b (M)(A) So th cofficint is 75 (A) (C) [] S (7 7) (M) () 8897 (A) (C) [] 5. x.55
More information5 Transform Analysis of LTI Systems
5 Transform Analysis of LTI Systms ² For an LTI systm with input x [n], output y [n], and impuls rspons h [n]: Fig. 48-F1 ² Nots: 1. y [n] = h [n] x [n]. 2. Y ( jω ) = H ( jω ) X ( jω ). 3. From th Convolution
More information1973 AP Calculus AB: Section I
97 AP Calculus AB: Sction I 9 Minuts No Calculator Not: In this amination, ln dnots th natural logarithm of (that is, logarithm to th bas ).. ( ) d= + C 6 + C + C + C + C. If f ( ) = + + + and ( ), g=
More informationIntroduction to Medical Imaging. Lecture 4: Fourier Theory = = ( ) 2sin(2 ) Introduction
Introduction Introduction to Mdical aging Lctur 4: Fourir Thory Thory dvlopd by Josph Fourir (768-83) Th Fourir transform of a signal s() yilds its frquncy spctrum S(k) Klaus Mullr s() forward transform
More informationWHAT LIES BETWEEN + AND (and beyond)? H.P.Williams
Working Par LSEOR 10-119 ISSN 2041-4668 (Onlin) WHAT LIES BETWEEN + AND (and byond)? HPWilliams London School of Economics hwilliams@lsacuk First ublishd in Grat Britain in 2010 by th Orational Rsarch
More informationHomework #3. 1 x. dx. It therefore follows that a sum of the
Danil Cannon CS 62 / Luan March 5, 2009 Homwork # 1. Th natural logarithm is dfind by ln n = n 1 dx. It thrfor follows that a sum of th 1 x sam addnd ovr th sam intrval should b both asymptotically uppr-
More informationThomas Whitham Sixth Form
Thomas Whitham Sith Form Pur Mathmatics Cor rvision gui Pag Algbra Moulus functions graphs, quations an inqualitis Graph of f () Draw f () an rflct an part of th curv blow th ais in th ais. f () f () f
More informationSinusoidal Response Notes
ECE 30 Sinusoidal Rspons Nots For BIBO Systms AStolp /29/3 Th sinusoidal rspons of a systm is th output whn th input is a sinusoidal (which starts at tim 0) Systm Sinusoidal Rspons stp input H( s) output
More informationECE602 Exam 1 April 5, You must show ALL of your work for full credit.
ECE62 Exam April 5, 27 Nam: Solution Scor: / This xam is closd-book. You must show ALL of your work for full crdit. Plas rad th qustions carfully. Plas chck your answrs carfully. Calculators may NOT b
More informationFloating Point Number System -(1.3)
Floting Point Numbr Sstm -(.3). Floting Point Numbr Sstm: Comutrs rrsnt rl numbrs in loting oint numbr sstm: F,k,m,M 0. 3... k ;0, 0 i, i,...,k, m M. Nottions: th bs 0, k th numbr o igts in th bs xnsion
More information22/ Breakdown of the Born-Oppenheimer approximation. Selection rules for rotational-vibrational transitions. P, R branches.
Subjct Chmistry Papr No and Titl Modul No and Titl Modul Tag 8/ Physical Spctroscopy / Brakdown of th Born-Oppnhimr approximation. Slction ruls for rotational-vibrational transitions. P, R branchs. CHE_P8_M
More informationFloating Point Number System -(1.3)
Floting Point Numbr Sstm -(.3). Floting Point Numbr Sstm: Comutrs rrsnt rl numbrs in loting oint numbr sstm: F,k,m,M 0. 3... k ;0, 0 i, i,...,k, m M. Nottions: th bs 0, k th numbr o igits in th bs xnsion
More informationRecall that by Theorems 10.3 and 10.4 together provide us the estimate o(n2 ), S(q) q 9, q=1
Chaptr 11 Th singular sris Rcall that by Thorms 10 and 104 togthr provid us th stimat 9 4 n 2 111 Rn = SnΓ 2 + on2, whr th singular sris Sn was dfind in Chaptr 10 as Sn = q=1 Sq q 9, with Sq = 1 a q gcda,q=1
More informationANALYSIS IN THE FREQUENCY DOMAIN
ANALYSIS IN THE FREQUENCY DOMAIN SPECTRAL DENSITY Dfinition Th spctral dnsit of a S.S.P. t also calld th spctrum of t is dfind as: + { γ }. jτ γ τ F τ τ In othr words, of th covarianc function. is dfind
More informationSupplementary Materials
6 Supplmntary Matrials APPENDIX A PHYSICAL INTERPRETATION OF FUEL-RATE-SPEED FUNCTION A truck running on a road with grad/slop θ positiv if moving up and ngativ if moving down facs thr rsistancs: arodynamic
More informationChemistry 342 Spring, The Hydrogen Atom.
Th Hyrogn Ato. Th quation. Th first quation w want to sov is φ This quation is of faiiar for; rca that for th fr partic, w ha ψ x for which th soution is Sinc k ψ ψ(x) a cos kx a / k sin kx ± ix cos x
More informationEXST Regression Techniques Page 1
EXST704 - Rgrssion Tchniqus Pag 1 Masurmnt rrors in X W hav assumd that all variation is in Y. Masurmnt rror in this variabl will not ffct th rsults, as long as thy ar uncorrlatd and unbiasd, sinc thy
More informationEEO 401 Digital Signal Processing Prof. Mark Fowler
EEO 401 Digital Signal Procssing Prof. Mark Fowlr ot St #18 Introduction to DFT (via th DTFT) Rading Assignmnt: Sct. 7.1 of Proakis & Manolakis 1/24 Discrt Fourir Transform (DFT) W v sn that th DTFT is
More information5.80 Small-Molecule Spectroscopy and Dynamics
MIT OpnCoursWar http://ocw.mit.du 5.80 Small-Molcul Spctroscopy and Dynamics Fall 008 For information about citing ths matrials or our Trms of Us, visit: http://ocw.mit.du/trms. Lctur # 3 Supplmnt Contnts
More informationWhy is a E&M nature of light not sufficient to explain experiments?
1 Th wird world of photons Why is a E&M natur of light not sufficint to xplain xprimnts? Do photons xist? Som quantum proprtis of photons 2 Black body radiation Stfan s law: Enrgy/ ara/ tim = Win s displacmnt
More informationEinstein Equations for Tetrad Fields
Apiron, Vol 13, No, Octobr 006 6 Einstin Equations for Ttrad Filds Ali Rıza ŞAHİN, R T L Istanbul (Turky) Evry mtric tnsor can b xprssd by th innr product of ttrad filds W prov that Einstin quations for
More informationQuasi-Classical States of the Simple Harmonic Oscillator
Quasi-Classical Stats of th Simpl Harmonic Oscillator (Draft Vrsion) Introduction: Why Look for Eignstats of th Annihilation Oprator? Excpt for th ground stat, th corrspondnc btwn th quantum nrgy ignstats
More informationMath 34A. Final Review
Math A Final Rviw 1) Us th graph of y10 to find approimat valus: a) 50 0. b) y (0.65) solution for part a) first writ an quation: 50 0. now tak th logarithm of both sids: log() log(50 0. ) pand th right
More informationHigh Energy Physics. Lecture 5 The Passage of Particles through Matter
High Enrgy Physics Lctur 5 Th Passag of Particls through Mattr 1 Introduction In prvious lcturs w hav sn xampls of tracks lft by chargd particls in passing through mattr. Such tracks provid som of th most
More informationZ-Transform. 清大電機系林嘉文 Original PowerPoint slides prepared by S. K. Mitra 4-1-1
Chapter 6 Z-Transform 清大電機系林嘉文 cwlin@ee.nthu.edu.tw 03-5731152 Original PowerPoint slides prepared by S. K. Mitra 4-1-1 z-transform The DTFT provides a frequency-domain representation of discrete-time
More informationUnit 6: Solving Exponential Equations and More
Habrman MTH 111 Sction II: Eonntial and Logarithmic Functions Unit 6: Solving Eonntial Equations and Mor EXAMPLE: Solv th quation 10 100 for. Obtain an act solution. This quation is so asy to solv that
More informationEECE 301 Signals & Systems Prof. Mark Fowler
EECE 301 Signals & Systms Prof. Mark Fowlr ot St #21 D-T Signals: Rlation btwn DFT, DTFT, & CTFT 1/16 W can us th DFT to implmnt numrical FT procssing This nabls us to numrically analyz a signal to find
More informationUNIT I PARTIAL DIFFERENTIAL EQUATIONS PART B. 3) Form the partial differential equation by eliminating the arbitrary functions
UNIT I PARTIAL DIFFERENTIAL EQUATIONS PART B 1) Form th artial diffrntial quation b liminating th arbitrar functions f and g in z f ( x ) g( x ) ) Form th artial diffrntial quation b liminating th arbitrar
More informationBifurcation Theory. , a stationary point, depends on the value of α. At certain values
Dnamic Macroconomic Thor Prof. Thomas Lux Bifurcation Thor Bifurcation: qualitativ chang in th natur of th solution occurs if a paramtr passs through a critical point bifurcation or branch valu. Local
More informationSCHUR S THEOREM REU SUMMER 2005
SCHUR S THEOREM REU SUMMER 2005 1. Combinatorial aroach Prhas th first rsult in th subjct blongs to I. Schur and dats back to 1916. On of his motivation was to study th local vrsion of th famous quation
More informationSlide 1. Slide 2. Slide 3 DIGITAL SIGNAL PROCESSING CLASSIFICATION OF SIGNALS
Slid DIGITAL SIGAL PROCESSIG UIT I DISCRETE TIME SIGALS AD SYSTEM Slid Rviw of discrt-tim signals & systms Signal:- A signal is dfind as any physical quantity that varis with tim, spac or any othr indpndnt
More informationAddition of angular momentum
Addition of angular momntum April, 0 Oftn w nd to combin diffrnt sourcs of angular momntum to charactriz th total angular momntum of a systm, or to divid th total angular momntum into parts to valuat th
More informationMath 102. Rumbos Spring Solutions to Assignment #8. Solution: The matrix, A, corresponding to the system in (1) is
Math 12. Rumbos Spring 218 1 Solutions to Assignmnt #8 1. Construct a fundamntal matrix for th systm { ẋ 2y ẏ x + y. (1 Solution: Th matrix, A, corrsponding to th systm in (1 is 2 A. (2 1 1 Th charactristic
More informationMATH 1080 Test 2-SOLUTIONS Spring
MATH Tst -SOLUTIONS Spring 5. Considr th curv dfind by x = ln( 3y + 7) on th intrval y. a. (5 points) St up but do not simplify or valuat an intgral rprsnting th lngth of th curv on th givn intrval. =
More informationChapter 6. The Discrete Fourier Transform and The Fast Fourier Transform
Pusan ational Univrsity Chaptr 6. Th Discrt Fourir Transform and Th Fast Fourir Transform 6. Introduction Frquncy rsponss of discrt linar tim invariant systms ar rprsntd by Fourir transform or z-transforms.
More informationAppendix 2.3 General Solutions for the Step Response of Third- and Fourth-Order Systems (with some unpleasant surprises!)
P.Stariè, E.Margan Appnix 2. A2..1 A2..2 Contnts: Appnix 2. Gnral Solutions for th Stp Rspons of Thir- an Fourth-Orr Systms (with som unplasant surpriss!) Thr is no such thing as instant xprinc! ( Oppnhimr
More informationcycle that does not cross any edges (including its own), then it has at least
W prov th following thorm: Thorm If a K n is drawn in th plan in such a way that it has a hamiltonian cycl that dos not cross any dgs (including its own, thn it has at last n ( 4 48 π + O(n crossings Th
More informationFinite Element Analysis
Finit Elmnt Analysis L4 D Shap Functions, an Gauss Quaratur FEA Formulation Dr. Wiong Wu EGR 54 Finit Elmnt Analysis Roamap for Dvlopmnt of FE Strong form: govrning DE an BCs EGR 54 Finit Elmnt Analysis
More informationMulti-linear Systems and Invariant Theory. in the Context of Computer Vision and Graphics. Class 5: Self Calibration. CS329 Stanford University
Mlti-linar Systms and Invariant hory in th ontt of omtr Vision and Grahics lass 5: Slf alibration S39 Stanford Univrsity Amnon Shasha lass 5 Matrial W Will ovr oday h basic qations and conting argmnts
More informationMATHEMATICS (B) 2 log (D) ( 1) = where z =
MATHEMATICS SECTION- I STRAIGHT OBJECTIVE TYPE This sction contains 9 multipl choic qustions numbrd to 9. Each qustion has choic (A), (B), (C) and (D), out of which ONLY-ONE is corrct. Lt I d + +, J +
More informationCommunication Technologies
Communication Tchnologis. Principls of Digital Transmission. Structur of Data Transmission.2 Spctrum of a Data Signal 2. Digital Modulation 2. Linar Modulation Mthods 2.2 Nonlinar Modulations (CPM-Signals)
More informationModeling with first order equations (Sect. 2.3).
Moling with first orr quations (Sct. 2.3. Main xampl: Salt in a watr tank. Th xprimntal vic. Th main quations. Analysis of th mathmatical mol. Prictions for particular situations. Salt in a watr tank.
More informationSearch sequence databases 3 10/25/2016
Sarch squnc databass 3 10/25/2016 Etrm valu distribution Ø Suppos X is a random variabl with probability dnsity function p(, w sampl a larg numbr S of indpndnt valus of X from this distribution for an
More informationBackground: We have discussed the PIB, HO, and the energy of the RR model. In this chapter, the H-atom, and atomic orbitals.
Chaptr 7 Th Hydrogn Atom Background: W hav discussd th PIB HO and th nrgy of th RR modl. In this chaptr th H-atom and atomic orbitals. * A singl particl moving undr a cntral forc adoptd from Scott Kirby
More informationDiscrete-Time Signal Processing
Discrt-Tim Signal Procssing Hnry D. Pfistr March 3, 07 Th Discrt-Tim Fourir Transform. Dfinition Th discrt-tim Fourir transform DTFT) maps an apriodic discrt-tim signal x[n] to th frquncy-domain function
More informationnd the particular orthogonal trajectory from the family of orthogonal trajectories passing through point (0; 1).
Eamn EDO. Givn th family of curvs y + C nd th particular orthogonal trajctory from th family of orthogonal trajctoris passing through point (0; ). Solution: In th rst plac, lt us calculat th di rntial
More informationMSLC Math 151 WI09 Exam 2 Review Solutions
Eam Rviw Solutions. Comput th following rivativs using th iffrntiation ruls: a.) cot cot cot csc cot cos 5 cos 5 cos 5 cos 5 sin 5 5 b.) c.) sin( ) sin( ) y sin( ) ln( y) ln( ) ln( y) sin( ) ln( ) y y
More informationMinimum Spanning Trees
Yufi Tao ITEE Univrsity of Qunslan In tis lctur, w will stuy anotr classic prolm: finin a minimum spannin tr of an unirct wit rap. Intrstinly, vn tou t prolm appars ratr iffrnt from SSSP (sinl sourc sortst
More informationResponses of Digital Filters Chapter Intended Learning Outcomes:
Responses of Digital Filters Chapter Intended Learning Outcomes: (i) Understanding the relationships between impulse response, frequency response, difference equation and transfer function in characterizing
More information1 Input-Output Stability
Inut-Outut Stability Inut-outut stability analysis allows us to analyz th stability of a givn syst without knowing th intrnal stat x of th syst. Bfor going forward, w hav to introduc so inut-outut athatical
More informationPROBLEM SET Problem 1.
PROLEM SET 1 PROFESSOR PETER JOHNSTONE 1. Problm 1. 1.1. Th catgory Mat L. OK, I m not amiliar with th trminology o partially orr sts, so lt s go ovr that irst. Dinition 1.1. partial orr is a binary rlation
More informationOn the irreducibility of some polynomials in two variables
ACTA ARITHMETICA LXXXII.3 (1997) On th irrducibility of som polynomials in two variabls by B. Brindza and Á. Pintér (Dbrcn) To th mmory of Paul Erdős Lt f(x) and g(y ) b polynomials with intgral cofficints
More informationECE 2210 / 00 Phasor Examples
EE 0 / 00 Phasor Exampls. Add th sinusoidal voltags v ( t ) 4.5. cos( t 30. and v ( t ) 3.. cos( t 5. v ( t) using phasor notation, draw a phasor diagram of th thr phasors, thn convrt back to tim domain
More informationIntroduction to the Fourier transform. Computer Vision & Digital Image Processing. The Fourier transform (continued) The Fourier transform (continued)
Introduction to th Fourir transform Computr Vision & Digital Imag Procssing Fourir Transform Lt f(x) b a continuous function of a ral variabl x Th Fourir transform of f(x), dnotd by I {f(x)} is givn by:
More informationWalk Like a Mathematician Learning Task:
Gori Dprtmnt of Euction Wlk Lik Mthmticin Lrnin Tsk: Mtrics llow us to prform mny usful mthmticl tsks which orinrily rquir lr numbr of computtions. Som typs of problms which cn b on fficintly with mtrics
More informationMA 262, Spring 2018, Final exam Version 01 (Green)
MA 262, Spring 218, Final xam Vrsion 1 (Grn) INSTRUCTIONS 1. Switch off your phon upon ntring th xam room. 2. Do not opn th xam booklt until you ar instructd to do so. 3. Bfor you opn th booklt, fill in
More informationFinite element discretization of Laplace and Poisson equations
Finit lmnt discrtization of Laplac and Poisson quations Yashwanth Tummala Tutor: Prof S.Mittal 1 Outlin Finit Elmnt Mthod for 1D Introduction to Poisson s and Laplac s Equations Finit Elmnt Mthod for 2D-Discrtization
More informationEEL3135: Homework #4
EEL335: Homework #4 Problem : For each of the systems below, determine whether or not the system is () linear, () time-invariant, and (3) causal: (a) (b) (c) xn [ ] cos( 04πn) (d) xn [ ] xn [ ] xn [ 5]
More informationCombinatorial Networks Week 1, March 11-12
1 Nots on March 11 Combinatorial Ntwors W 1, March 11-1 11 Th Pigonhol Principl Th Pigonhol Principl If n objcts ar placd in hols, whr n >, thr xists a box with mor than on objcts 11 Thorm Givn a simpl
More informationEE140 Introduction to Communication Systems Lecture 2
EE40 Introduction to Communication Systms Lctur 2 Instructor: Prof. Xiliang Luo ShanghaiTch Univrsity, Spring 208 Architctur of a Digital Communication Systm Transmittr Sourc A/D convrtr Sourc ncodr Channl
More informationON THE NUMBER OF RATIONAL POINTS ON CURVES OVER FINITE FIELDS WITH MANY AUTOMORPHISMS
ON THE NUMBER OF RATIONAL POINTS ON CURVES OVER FINITE FIELDS WITH MANY AUTOMORPHISMS ANTONIO ROJAS-LEÓN Abstract. Using Wil scnt, w giv bouns for th numbr of rational points on two familis of curvs ovr
More informationInstructions for Section 1
Instructions for Sction 1 Choos th rspons tht is corrct for th qustion. A corrct nswr scors 1, n incorrct nswr scors 0. Mrks will not b dductd for incorrct nswrs. You should ttmpt vry qustion. No mrks
More informationCh. 24 Molecular Reaction Dynamics 1. Collision Theory
Ch. 4 Molcular Raction Dynamics 1. Collision Thory Lctur 16. Diffusion-Controlld Raction 3. Th Matrial Balanc Equation 4. Transition Stat Thory: Th Eyring Equation 5. Transition Stat Thory: Thrmodynamic
More information7' The growth of yeast, a microscopic fungus used to make bread, in a test tube can be
N Sction A: Pur Mathmatics 55 marks] / Th rgion R is boundd by th curv y, th -ais, and th lins = V - +7 and = m, whr m >. Find th volum gnratd whn R is rotatd through right angls about th -ais, laving
More informationMultiple Short Term Infusion Homework # 5 PHA 5127
Multipl Short rm Infusion Homwork # 5 PHA 527 A rug is aministr as a short trm infusion. h avrag pharmacokintic paramtrs for this rug ar: k 0.40 hr - V 28 L his rug follows a on-compartmnt boy mol. A 300
More informationu x v x dx u x v x v x u x dx d u x v x u x v x dx u x v x dx Integration by Parts Formula
7. Intgration by Parts Each drivativ formula givs ris to a corrsponding intgral formula, as w v sn many tims. Th drivativ product rul yilds a vry usful intgration tchniqu calld intgration by parts. Starting
More information