HW6: MRI Imaging Pulse Sequences (7 Problems for 100 pts)
|
|
- Gary Neal
- 5 years ago
- Views:
Transcription
1 HW6: MRI Imaging Pulse Sequences (7 Problems for 100 ps) GOAL The overall goal of HW6 is o beer undersand pulse sequences for MRI image reconsrucion. OBJECTIVES 1) Design a spin echo pulse sequence o image a poin-like objec (for esing purposes). 2) Image a square-like objec wih differen conras. 3) Image a mysery objec wih differen conras. Your working code will be able o produce T1, T2, or PD weighed images by adjusing he spin echo ime TE and he pulse repeiion inerval TR. DESCRIPTION The basic flow char of he MATLAB-based MRI simulaor is he following: STEP 0: Compue imporan parameers STEP 1: Iniialize he bulk magneizaion. STEP 2: Apply he 90 pulse. STEP 3: Apply phase encode (Gx and Gy) STEP 4: Allow M o evolve unil he 180 pulse. STEP 5: Apply he 180 pulse. STEP 6: Allow M o evolve unil he sar of readou. STEP 7: Apply readou (Gx) STEP 8: Repea Seps 1 o 7 as many imes as necessary o sweep Fourier space. 1
2 Assumpions 1) The saic polarizing field B o = 1.5 Tesla poins along he z-direcion. 2) o 42.57MHz / Tesla 2. 3) The objec is assumed o be a slice. Therefore, you will no use any selecive exciaion or refocusing (G z ). The image size is 64 x 64 pixels, where dx = dy = 1 mm. 4) All magneic fields are specified in he roaing frame x y z. You mus properly se he componens of he effecive magneic field B eff = (B eff,x, B eff,y, B eff,z ) during he 90 pulse, phase encode, 180 pulse, and readou. 5) The ime sep for he simulaion is d = 1/16 ms. 6) The 90 pulse is applied a = 0 for a duraion d (e.g. one ime sep) along he y axis. 7) The phase encode sars immediaely afer he 90 pulse. The duraion of he encode is Tencode = 4 ms. 8) The 180 pulse duraion is applied a = TE/2 for a duraion d (e.g. one ime sep). 9) The readou is Tread = 8 ms. Remember he spin echo ime TE occurs in he middle of readou! RF Gx Gy T ENCODE T READ S() = 0 = TE/2 = TE Fig. 1 Timing diagram for he MRI spin echo pulse sequence. 2
3 Some oher commens: 1) When he load command is used in he program, i will impor hree 64 x 64 elemen marices called PD, T1, and T2. Remember ha he objec is a 2-D marix consising of 64 x 64 pixels. Each objec pixel has is own value of PD, T1, and T2. Therefore, PD is a 64 x 64 elemen marix. Same for T1 and T2. 2) The magneizaion M is a vecor, so i has hree componens M = (M X, M Y, M Z ). T2 T1 PD Remember ha M depends on x and y. Therefore, he x componen of M is a 64 x 64 elemen marix. Same hing for he oher wo componens. In MATLAB, all hree marices are combined ino a single 3-D marix called M. This can be hough of as a book. The firs, second, and hird pages of his book correspond o he M X, M Y, and M Z marices. 3) In STEP 1, he iniial magneizaion is he value of M Z jus before he 90-pulse is applied. M X M Y M Z In class, we discussed ha M Z () = M 0 (1-exp ( - / T1 )). For MRI, we apply he 90-pulse a a regular ime inerval TR. Therefore, he value of M Z jus before he 90-pulse is applied is M 0 (1-exp ( -TR / T1 )). Noe ha M 0 is he magneizaion value afer full recovery, which depends on proon densiy PD. For his assignmen, we will use he simplified definiion M 0 = PD. Iniial M Z Therefore, your MATLAB program should define he iniial value of M Z as PD (1-exp ( -TR / T1 )). Remember ha PD and T1 are 2-D marices! M Z TR TR TR 4) Remember ha your phase encode involves B eff,z = G X X + G Y Y while readou involves only B eff,z = G X X. Keep in mind ha X and Y are 2-D marices conaining he (x,y) coordinaes of each pixel. Also remember ha G Y is a VECTOR, bu you only need ONE value of G Y for each row in Fourier space. 3
4 PROBLEM 1 a) Compue he values for B90 and B180 in unis of Tesla. Show all work. Hin: You should ge B90 = x 10-5 Tesla. b) Specify B90 and B180 in vecor forma. For example, B eff = (B90, 0, 0) applies he B90 along he x -direcion in he roaing frame. c) Make separae skeches of he vecor B eff for he 90 and 180 pulses. Include he vecor orienaion of he magneizaion before and afer boh pulses (in he roaing x y z frame). PROBLEM 2 a) Compue he values for Gx and dgy in unis of Tesla/mm. Hin: You should ge Gx = 2.936x10-6 Tesla/mm. and dgy = 9.176x10-8 Tesla/mm. b) Specify all hree componens of B eff during phase encode. Hin: Remember ha we are working in he roaing frame, so you do NOT include he big B 0 field. c) Do he same for readou. PROBLEM 3 Consider a single voxel (sands for volume pixel ) a (x, y) = (8, 4) mm. You can model his objec as a poin impulse locaed a he appropriae posiion. a) Calculae he 2-D Fourier ransform of he objec. You can use sandard Fourier ransform properies and ables. Hin: You should ge F(u,v) = exp(-j2(8u+4v)). b) Skech he real par of F(u,v) over a 2-D region from u = -0.5 o +0.5 mm -1 and v = -0.5 o +0.5 mm -1. Hin: See page 2.4a of he lecure noes. 4
5 PROBLEM 4 You mus fill in he key pars (i.e. TE, TR, B90, ec.) of he MATLAB emplae program. The following MATLAB files are available on he course websie: 1) A MATLAB emplae for he simulaor called MRI_Imaging_simulaor_emplae.m. 2) A MATLAB funcion MRI_Evolve_M.m ha evolves he magneizaion according o he applied magneic fields for a specific number of ime seps. 3) A MATLAB funcion MRI_Bloch.m ha is used by he MRI_Evolve_M.m funcion. 4) A daa se for a single waer spin MRI_daa_single_spin1.ma conaining T1, T2, gamma, and PD. 5) A daa se for a square objec MRI_daa_square.ma. 6) A daa se for a mysery objec MRI_daa_mysery.ma. a) Reconsruc he image of a single waer spin using TE = 25 ms and TR = 3500 ms. Se case_num=1 in he beginning of your code. Submi he image. You should ge an image of a single whie pixel a (x,y)=(8,4) surrounded by a black background (see Fig. 2). If you do no ge his image, hen your code is no working properly. Image of poin-like objec Real componen of daa Imag componen of daa y (mm) 0 v (1/mm) v (1/mm) x (mm) u (1/mm) u (1/mm) Fig. 2: The reconsruced poin-objec (lef) and raw daa (middle and righ) should look like he images shown above. The middle figure should resemble your skech from Problem 3! b) Submi images of he real and imaginary componens of he daa. Explain he appearance of he wo componens. Are hey idenical? NOTE: Once your code works for his poin-objec, Problems 5 and 6 will go very quickly. 5
6 PROBLEM 5 a) Reconsruc he image of he square objec wih TE = 25 ms and TR = 3500 ms. The square objec is whie brain maer, while he background is waer. Submi he image. Wha ype of weighing is he resuling image? NOTE: The proon densiy, T2, and T1 values of whie maer, gray maer, and cerebrospinal fluid are shown below. Tissue Type Relaive PD T2 (ms) T1 (ms) Whie maer Gray maer Cerebrospinal fluid b) Reconsruc an image of he square objec wih TE = 25 ms and TR = 500 ms. Submi he image. Wha ype of weighing is he resuling image? Explain differences beween he wo images. PROBLEM 6 a) Now implemen hree differen pulse sequences o reconsruc images of he mysery daa. Use he TE and TR values shown o he righ. Submi all hree hree images. Sequence TE (ms) TR (ms) b) Wha ype of weighing is performed by each sequence? For a paricular sequence, idenify he brighes and faines porions of he image. Wha can you conclude abou he issue properies of hese porions? 6
7 PROBLEM 7 v Echo planar imaging (EPI) is an MRI echnique allowing fas image acquisiion. An enire image can be acquired in less han 30 ms, as opposed o he usual duraion of several minues. This is paricularly useful in imaging dynamic evens (i.e. blood flow in he brain). The enire Fourier space (also called k-space) is acquired in a single sho acquisiion. Therefore, daa acquisiion for an enire image involves: (1) A single 90 pulse followed by (2) An iniial phase encode followed by (3) G x and G y waveforms ha produce a zig-zag Fourier rajecory shown o he righ. The following assumpions can be made: 1) B 0 = 3 Tesla and /2 = MHz/Tesla for all issue. Finish Sar u 2) Boh G x and G y are T read /2 in duraion during he iniial phase encode. 3) During he zig-zag pah, he readou ime for each row is T read = 300 s. 4) During he zig-zag pah, each G y pulse is 30 s in duraion. 5) No 180 pulse is used his is a gradien echo EPI pulse sequence. 6) Oupu images are 64 x 64 pixels wih a pixel spacing of x = y = 4 mm. (a) Compue G x and G y during he iniial phase encode (i.e. before he zig-zag readou is performed). Express your answer in Gauss/cm. (remember ha 1 Tesla = 10 4 Gauss). Hin: You should ge G x = G/cm and G y = G/cm. (b) Compue he ampliude of G x and dg y during he zig-zag readou (unis = Gauss/cm). Hin: You should ge dg y = 0.31 G/cm. (c) Skech he RF, G x, G y, and G z fields for your pulse sequence. Obviously, you do no need o skech he enire G x and G y waveform jus skech a porion of he waveforms ha span he firs FOUR rows of he zig-zag pah. You can assume he readou sars immediaely afer he iniial phase encode. 7
SE Sequence: 90º, 180º RF Pulses, Readout Gradient e.g., 256 voxels in a row
Ouline for Today 1. 2. 3. Inroducion o MRI Quanum NMR and MRI in 0D Magneizaion, m(x,), in a Voxel Proon T1 Spin Relaxaion in a Voxel Proon Densiy MRI in 1D MRI Case Sudy, and Cavea Skech of he MRI Device
More informationRefocusing t. Small Tip Angle Example. Small Tip Angle Example. Bioengineering 280A Principles of Biomedical Imaging. Fall Quarter 2010 MRI Lecture 5
Bioengineering 280A Principles of Biomedical Imaging Fall Quarer 2010 MRI Lecure 5 RF N random seps of lengh d Refocusing ' M xy (,) = jm 0 "B 1 ()exp( jk(,))d %& 100 seps This has he 2D form random of
More informationRelaxation. T1 Values. Longitudinal Relaxation. dm z dt. = " M z T 1. (1" e "t /T 1 ) M z. (t) = M 0
Relaxaion Bioengineering 28A Principles of Biomedical Imaging Fall Quarer 21 MRI Lecure 2 An exciaion pulse roaes he magneiaion vecor away from is equilibrium sae (purely longiudinal). The resuling vecor
More information[ ]e TE /T 2(x,y ) Saturation Recovery Sequence. T1-Weighted Scans. T1-Weighted Scans. I(x, y) ρ(x, y) 1 e TR /T 1
Sauraion Recovery Sequence 90 TE 90 TE 90 Bioengineering 280A Principles of Biomedical Imaging Fall Quarer 2015 MRI Lecure 5 TR Gradien Echo TR [ ]e TE /T 2 * (x,y ) I(x, y) = ρ(x, y) 1 e TR /T 1 (x,y)
More informationKINEMATICS IN ONE DIMENSION
KINEMATICS IN ONE DIMENSION PREVIEW Kinemaics is he sudy of how hings move how far (disance and displacemen), how fas (speed and velociy), and how fas ha how fas changes (acceleraion). We say ha an objec
More informationLet us start with a two dimensional case. We consider a vector ( x,
Roaion marices We consider now roaion marices in wo and hree dimensions. We sar wih wo dimensions since wo dimensions are easier han hree o undersand, and one dimension is a lile oo simple. However, our
More informationk B 2 Radiofrequency pulses and hardware
1 Exra MR Problems DC Medical Imaging course April, 214 he problems below are harder, more ime-consuming, and inended for hose wih a more mahemaical background. hey are enirely opional, bu hopefully will
More informationPhys1112: DC and RC circuits
Name: Group Members: Dae: TA s Name: Phys1112: DC and RC circuis Objecives: 1. To undersand curren and volage characerisics of a DC RC discharging circui. 2. To undersand he effec of he RC ime consan.
More informationSolutions from Chapter 9.1 and 9.2
Soluions from Chaper 9 and 92 Secion 9 Problem # This basically boils down o an exercise in he chain rule from calculus We are looking for soluions of he form: u( x) = f( k x c) where k x R 3 and k is
More informationBEng (Hons) Telecommunications. Examinations for / Semester 2
BEng (Hons) Telecommunicaions Cohor: BTEL/14/FT Examinaions for 2015-2016 / Semeser 2 MODULE: ELECTROMAGNETIC THEORY MODULE CODE: ASE2103 Duraion: 2 ½ Hours Insrucions o Candidaes: 1. Answer ALL 4 (FOUR)
More informationLearning Objectives: Practice designing and simulating digital circuits including flip flops Experience state machine design procedure
Lab 4: Synchronous Sae Machine Design Summary: Design and implemen synchronous sae machine circuis and es hem wih simulaions in Cadence Viruoso. Learning Objecives: Pracice designing and simulaing digial
More informationRF Excitation. Rotating Frame of Reference. RF Excitation. Bioengineering 280A Principles of Biomedical Imaging. Fall Quarter 2012 MRI Lecture 6
RF Exciaion Bioengineering 8A Principles of Biomedical Imaging Fall Quarer 1 MRI Lecure 6 hp://www.drcmr.dk/main/conen/view/13/74/ RF Exciaion Roaing Frame of Reference Reference everyhing o he magneic
More informationLecture 2-1 Kinematics in One Dimension Displacement, Velocity and Acceleration Everything in the world is moving. Nothing stays still.
Lecure - Kinemaics in One Dimension Displacemen, Velociy and Acceleraion Everyhing in he world is moving. Nohing says sill. Moion occurs a all scales of he universe, saring from he moion of elecrons in
More informationTwo Coupled Oscillators / Normal Modes
Lecure 3 Phys 3750 Two Coupled Oscillaors / Normal Modes Overview and Moivaion: Today we ake a small, bu significan, sep owards wave moion. We will no ye observe waves, bu his sep is imporan in is own
More informationLab 10: RC, RL, and RLC Circuits
Lab 10: RC, RL, and RLC Circuis In his experimen, we will invesigae he behavior of circuis conaining combinaions of resisors, capaciors, and inducors. We will sudy he way volages and currens change in
More informationRC, RL and RLC circuits
Name Dae Time o Complee h m Parner Course/ Secion / Grade RC, RL and RLC circuis Inroducion In his experimen we will invesigae he behavior of circuis conaining combinaions of resisors, capaciors, and inducors.
More informationRF Excitation. RF Excitation. Bioengineering 280A Principles of Biomedical Imaging
Bioengineering 280A Principles of Biomedical Imaging Fall Quarer 2010 MRI Lecure 4 Simplified Drawing of Basic Insrumenaion. Body lies on able encompassed by coils for saic field B o, gradien fields (wo
More information' ' ' t. Moving Spins. Phase of Moving Spin. Phase of a Moving Spin. Bioengineering 280A Principles of Biomedical Imaging
Moving Spins Bioengineering 8A Principles of Biomedical Imaging Fall Quarer 1 MRI Lecure 6 So far we have assumed ha he spins are no moving (aside from hermal moion giving rise o relaaion) and conras has
More informationWEEK-3 Recitation PHYS 131. of the projectile s velocity remains constant throughout the motion, since the acceleration a x
WEEK-3 Reciaion PHYS 131 Ch. 3: FOC 1, 3, 4, 6, 14. Problems 9, 37, 41 & 71 and Ch. 4: FOC 1, 3, 5, 8. Problems 3, 5 & 16. Feb 8, 018 Ch. 3: FOC 1, 3, 4, 6, 14. 1. (a) The horizonal componen of he projecile
More information!!"#"$%&#'()!"#&'(*%)+,&',-)./0)1-*23)
"#"$%&#'()"#&'(*%)+,&',-)./)1-*) #$%&'()*+,&',-.%,/)*+,-&1*#$)()5*6$+$%*,7&*-'-&1*(,-&*6&,7.$%$+*&%'(*8$&',-,%'-&1*(,-&*6&,79*(&,%: ;..,*&1$&$.$%&'()*1$$.,'&',-9*(&,%)?%*,('&5
More informationPosition, Velocity, and Acceleration
rev 06/2017 Posiion, Velociy, and Acceleraion Equipmen Qy Equipmen Par Number 1 Dynamic Track ME-9493 1 Car ME-9454 1 Fan Accessory ME-9491 1 Moion Sensor II CI-6742A 1 Track Barrier Purpose The purpose
More information' ' ' t. Moving Spins. Phase of a Moving Spin. Phase of Moving Spin. Bioengineering 280A Principles of Biomedical Imaging
Moving Spins Bioengineering 28A Principles of Biomedical Imaging Fall Quarer 28 MRI Lecure 7 So far we have assumed ha he spins are no moving (aside from hermal moion giving rise o relaxaion), and conras
More informationInventory Analysis and Management. Multi-Period Stochastic Models: Optimality of (s, S) Policy for K-Convex Objective Functions
Muli-Period Sochasic Models: Opimali of (s, S) Polic for -Convex Objecive Funcions Consider a seing similar o he N-sage newsvendor problem excep ha now here is a fixed re-ordering cos (> 0) for each (re-)order.
More informationLie Derivatives operator vector field flow push back Lie derivative of
Lie Derivaives The Lie derivaive is a mehod of compuing he direcional derivaive of a vecor field wih respec o anoher vecor field We already know how o make sense of a direcional derivaive of real valued
More informationFinish reading Chapter 2 of Spivak, rereading earlier sections as necessary. handout and fill in some missing details!
MAT 257, Handou 6: Ocober 7-2, 20. I. Assignmen. Finish reading Chaper 2 of Spiva, rereading earlier secions as necessary. handou and fill in some missing deails! II. Higher derivaives. Also, read his
More information' ' ' t. Moving Spins. Phase of a Moving Spin. Bioengineering 280A Principles of Biomedical Imaging. Fall Quarter 2007 MRI Lecture 6
Moving Spins Bioengineering 28A Principles of Biomedical Imaging Fall Quarer 27 MRI Lecure 6 So far we have assumed ha he spins are no moving (aside from hermal moion giving rise o relaxaion), and conras
More informationPrinciples of MRI. Practical Issues in MRI T2 decay. Tissue is a combination of. Results are complex. Started talking about off-resonance
Projec Principles of MRI Lecure 18 EE225E / BIO265 Insrucor: Miki Lusig UC Berkeley, EECS No eams Oral presenaion (20min) Or, Repor as a wikepedia enry Level of presenaion -- assume exbook level of knowledge
More informationRF Excitation. RF Excitation. Bioengineering 280A Principles of Biomedical Imaging. Fall Quarter 2013 MRI Lecture 4
Bioengineering 80A Principles of Biomedical Imaging Fall Quarer 013 MRI Lecure 4 TT. Liu, BE80A, UCSD Fall 01 Simplified Drawing of Basic Insrumenaion. Body lies on able encompassed by coils for saic field
More informationBiol. 356 Lab 8. Mortality, Recruitment, and Migration Rates
Biol. 356 Lab 8. Moraliy, Recruimen, and Migraion Raes (modified from Cox, 00, General Ecology Lab Manual, McGraw Hill) Las week we esimaed populaion size hrough several mehods. One assumpion of all hese
More informationNMR Spectroscopy: Principles and Applications. Nagarajan Murali 1D - Methods Lecture 5
NMR pecroscop: Principles and Applicaions Nagarajan Murali D - Mehods Lecure 5 D-NMR To full appreciae he workings of D NMR eperimens we need o a leas consider wo coupled spins. omeimes we need o go up
More informationPhysics 235 Chapter 2. Chapter 2 Newtonian Mechanics Single Particle
Chaper 2 Newonian Mechanics Single Paricle In his Chaper we will review wha Newon s laws of mechanics ell us abou he moion of a single paricle. Newon s laws are only valid in suiable reference frames,
More informationKinematics Vocabulary. Kinematics and One Dimensional Motion. Position. Coordinate System in One Dimension. Kinema means movement 8.
Kinemaics Vocabulary Kinemaics and One Dimensional Moion 8.1 WD1 Kinema means movemen Mahemaical descripion of moion Posiion Time Inerval Displacemen Velociy; absolue value: speed Acceleraion Averages
More informationConstant Acceleration
Objecive Consan Acceleraion To deermine he acceleraion of objecs moving along a sraigh line wih consan acceleraion. Inroducion The posiion y of a paricle moving along a sraigh line wih a consan acceleraion
More informationUnit 1 Test Review Physics Basics, Movement, and Vectors Chapters 1-3
A.P. Physics B Uni 1 Tes Reiew Physics Basics, Moemen, and Vecors Chapers 1-3 * In sudying for your es, make sure o sudy his reiew shee along wih your quizzes and homework assignmens. Muliple Choice Reiew:
More informationProblem Set #1. i z. the complex propagation constant. For the characteristic impedance:
Problem Se # Problem : a) Using phasor noaion, calculae he volage and curren waves on a ransmission line by solving he wave equaion Assume ha R, L,, G are all non-zero and independen of frequency From
More informationLAB # 2 - Equilibrium (static)
AB # - Equilibrium (saic) Inroducion Isaac Newon's conribuion o physics was o recognize ha despie he seeming compleiy of he Unierse, he moion of is pars is guided by surprisingly simple aws. Newon's inspiraion
More informationLinear Time-invariant systems, Convolution, and Cross-correlation
Linear Time-invarian sysems, Convoluion, and Cross-correlaion (1) Linear Time-invarian (LTI) sysem A sysem akes in an inpu funcion and reurns an oupu funcion. x() T y() Inpu Sysem Oupu y() = T[x()] An
More informationElements of Computer Graphics
CS580: Compuer Graphics Min H. Kim KAIST School of Compuing Elemens of Compuer Graphics Geomery Maerial model Ligh Rendering Virual phoography 2 Foundaions of Compuer Graphics A PINHOLE CAMERA IN 3D 3
More information1 1 + x 2 dx. tan 1 (2) = ] ] x 3. Solution: Recall that the given integral is improper because. x 3. 1 x 3. dx = lim dx.
. Use Simpson s rule wih n 4 o esimae an () +. Soluion: Since we are using 4 seps, 4 Thus we have [ ( ) f() + 4f + f() + 4f 3 [ + 4 4 6 5 + + 4 4 3 + ] 5 [ + 6 6 5 + + 6 3 + ]. 5. Our funcion is f() +.
More information04. Kinetics of a second order reaction
4. Kineics of a second order reacion Imporan conceps Reacion rae, reacion exen, reacion rae equaion, order of a reacion, firs-order reacions, second-order reacions, differenial and inegraed rae laws, Arrhenius
More informationAssignment 6. Tyler Shendruk December 6, 2010
Assignmen 6 Tyler Shendruk December 6, 1 1 Harden Problem 1 Le K be he coupling and h he exernal field in a 1D Ising model. From he lecures hese can be ransformed ino effecive coupling and fields K and
More informationt is a basis for the solution space to this system, then the matrix having these solutions as columns, t x 1 t, x 2 t,... x n t x 2 t...
Mah 228- Fri Mar 24 5.6 Marix exponenials and linear sysems: The analogy beween firs order sysems of linear differenial equaions (Chaper 5) and scalar linear differenial equaions (Chaper ) is much sronger
More information) were both constant and we brought them from under the integral.
YIELD-PER-RECRUIT (coninued The yield-per-recrui model applies o a cohor, bu we saw in he Age Disribuions lecure ha he properies of a cohor do no apply in general o a collecion of cohors, which is wha
More informationMATH 4330/5330, Fourier Analysis Section 6, Proof of Fourier s Theorem for Pointwise Convergence
MATH 433/533, Fourier Analysis Secion 6, Proof of Fourier s Theorem for Poinwise Convergence Firs, some commens abou inegraing periodic funcions. If g is a periodic funcion, g(x + ) g(x) for all real x,
More informationPHYSICS 149: Lecture 9
PHYSICS 149: Lecure 9 Chaper 3 3.2 Velociy and Acceleraion 3.3 Newon s Second Law of Moion 3.4 Applying Newon s Second Law 3.5 Relaive Velociy Lecure 9 Purdue Universiy, Physics 149 1 Velociy (m/s) The
More information6.003 Homework #9 Solutions
6.00 Homework #9 Soluions Problems. Fourier varieies a. Deermine he Fourier series coefficiens of he following signal, which is periodic in 0. x () 0 0 a 0 5 a k sin πk 5 sin πk 5 πk for k 0 a k 0 πk j
More informationA First Course on Kinetics and Reaction Engineering. Class 19 on Unit 18
A Firs ourse on Kineics and Reacion Engineering lass 19 on Uni 18 Par I - hemical Reacions Par II - hemical Reacion Kineics Where We re Going Par III - hemical Reacion Engineering A. Ideal Reacors B. Perfecly
More informationSummary of shear rate kinematics (part 1)
InroToMaFuncions.pdf 4 CM465 To proceed o beer-designed consiuive equaions, we need o know more abou maerial behavior, i.e. we need more maerial funcions o predic, and we need measuremens of hese maerial
More informationBNG/ECE 487 FINAL (W16)
BNG/ECE 487 FINAL (W16) NAME: 4 Problems for 100 pts This exam is closed-everything (no notes, books, etc.). Calculators are permitted. Possibly useful formulas and tables are provided on this page. Fourier
More informationIntroduction to AC Power, RMS RMS. ECE 2210 AC Power p1. Use RMS in power calculations. AC Power P =? DC Power P =. V I = R =. I 2 R. V p.
ECE MS I DC Power P I = Inroducion o AC Power, MS I AC Power P =? A Solp //9, // // correced p4 '4 v( ) = p cos( ω ) v( ) p( ) Couldn' we define an "effecive" volage ha would allow us o use he same relaionships
More informationLinear Response Theory: The connection between QFT and experiments
Phys540.nb 39 3 Linear Response Theory: The connecion beween QFT and experimens 3.1. Basic conceps and ideas Q: How do we measure he conduciviy of a meal? A: we firs inroduce a weak elecric field E, and
More informationReading from Young & Freedman: For this topic, read sections 25.4 & 25.5, the introduction to chapter 26 and sections 26.1 to 26.2 & 26.4.
PHY1 Elecriciy Topic 7 (Lecures 1 & 11) Elecric Circuis n his opic, we will cover: 1) Elecromoive Force (EMF) ) Series and parallel resisor combinaions 3) Kirchhoff s rules for circuis 4) Time dependence
More informationMatlab and Python programming: how to get started
Malab and Pyhon programming: how o ge sared Equipping readers he skills o wrie programs o explore complex sysems and discover ineresing paerns from big daa is one of he main goals of his book. In his chaper,
More informationMatrix Versions of Some Refinements of the Arithmetic-Geometric Mean Inequality
Marix Versions of Some Refinemens of he Arihmeic-Geomeric Mean Inequaliy Bao Qi Feng and Andrew Tonge Absrac. We esablish marix versions of refinemens due o Alzer ], Carwrigh and Field 4], and Mercer 5]
More informationSPH3U: Projectiles. Recorder: Manager: Speaker:
SPH3U: Projeciles Now i s ime o use our new skills o analyze he moion of a golf ball ha was ossed hrough he air. Le s find ou wha is special abou he moion of a projecile. Recorder: Manager: Speaker: 0
More informationTHE UNIVERSITY OF TEXAS AT AUSTIN McCombs School of Business
THE UNIVERITY OF TEXA AT AUTIN McCombs chool of Business TA 7.5 Tom hively CLAICAL EAONAL DECOMPOITION - MULTIPLICATIVE MODEL Examples of easonaliy 8000 Quarerly sales for Wal-Mar for quarers a l e s 6000
More informationSimulation-Solving Dynamic Models ABE 5646 Week 2, Spring 2010
Simulaion-Solving Dynamic Models ABE 5646 Week 2, Spring 2010 Week Descripion Reading Maerial 2 Compuer Simulaion of Dynamic Models Finie Difference, coninuous saes, discree ime Simple Mehods Euler Trapezoid
More informationDiebold, Chapter 7. Francis X. Diebold, Elements of Forecasting, 4th Edition (Mason, Ohio: Cengage Learning, 2006). Chapter 7. Characterizing Cycles
Diebold, Chaper 7 Francis X. Diebold, Elemens of Forecasing, 4h Ediion (Mason, Ohio: Cengage Learning, 006). Chaper 7. Characerizing Cycles Afer compleing his reading you should be able o: Define covariance
More informationSolutions of Sample Problems for Third In-Class Exam Math 246, Spring 2011, Professor David Levermore
Soluions of Sample Problems for Third In-Class Exam Mah 6, Spring, Professor David Levermore Compue he Laplace ransform of f e from is definiion Soluion The definiion of he Laplace ransform gives L[f]s
More informationIB Physics Kinematics Worksheet
IB Physics Kinemaics Workshee Wrie full soluions and noes for muliple choice answers. Do no use a calculaor for muliple choice answers. 1. Which of he following is a correc definiion of average acceleraion?
More informationExam 8NC20-8NC29 - Introduction to NMR and MRI
Exam 8NC-8NC9 - Inroducion o NMR and MRI Friday April 5, 8.-. h For his exam you may use an ordinary calculaor (no a graphical one). In oal here are 6 assignmens and a oal of 64 poins can be earned. You
More information2.7. Some common engineering functions. Introduction. Prerequisites. Learning Outcomes
Some common engineering funcions 2.7 Inroducion This secion provides a caalogue of some common funcions ofen used in Science and Engineering. These include polynomials, raional funcions, he modulus funcion
More informationYou must fully interpret your results. There is a relationship doesn t cut it. Use the text and, especially, the SPSS Manual for guidance.
POLI 30D SPRING 2015 LAST ASSIGNMENT TRUMPETS PLEASE!!!!! Due Thursday, December 10 (or sooner), by 7PM hrough TurnIIn I had his all se up in my mind. You would use regression analysis o follow up on your
More informationEE 315 Notes. Gürdal Arslan CLASS 1. (Sections ) What is a signal?
EE 35 Noes Gürdal Arslan CLASS (Secions.-.2) Wha is a signal? In his class, a signal is some funcion of ime and i represens how some physical quaniy changes over some window of ime. Examples: velociy of
More informationKEY. Math 334 Midterm III Winter 2008 section 002 Instructor: Scott Glasgow
KEY Mah 334 Miderm III Winer 008 secion 00 Insrucor: Sco Glasgow Please do NOT wrie on his exam. No credi will be given for such work. Raher wrie in a blue book, or on your own paper, preferably engineering
More informationCHAPTER 12 DIRECT CURRENT CIRCUITS
CHAPTER 12 DIRECT CURRENT CIUITS DIRECT CURRENT CIUITS 257 12.1 RESISTORS IN SERIES AND IN PARALLEL When wo resisors are conneced ogeher as shown in Figure 12.1 we said ha hey are conneced in series. As
More informationLecture 4 Kinetics of a particle Part 3: Impulse and Momentum
MEE Engineering Mechanics II Lecure 4 Lecure 4 Kineics of a paricle Par 3: Impulse and Momenum Linear impulse and momenum Saring from he equaion of moion for a paricle of mass m which is subjeced o an
More informationCourse Notes for EE227C (Spring 2018): Convex Optimization and Approximation
Course Noes for EE7C Spring 018: Convex Opimizaion and Approximaion Insrucor: Moriz Hard Email: hard+ee7c@berkeley.edu Graduae Insrucor: Max Simchowiz Email: msimchow+ee7c@berkeley.edu Ocober 15, 018 3
More information6.003 Homework #9 Solutions
6.003 Homework #9 Soluions Problems. Fourier varieies a. Deermine he Fourier series coefficiens of he following signal, which is periodic in 0. x () 0 3 0 a 0 5 a k a k 0 πk j3 e 0 e j πk 0 jπk πk e 0
More informationSpring Ammar Abu-Hudrouss Islamic University Gaza
Chaper 7 Reed-Solomon Code Spring 9 Ammar Abu-Hudrouss Islamic Universiy Gaza ١ Inroducion A Reed Solomon code is a special case of a BCH code in which he lengh of he code is one less han he size of he
More informationSpin echo. ½πI x -t -πi y -t
y Spin echo ½πI - -πi y - : as needed, no correlaed wih 1/J. Funcions: 1. refocusing; 2. decoupling. Chemical shif evoluion is refocused by he spin-echo. Heeronuclear J-couplings evoluion are refocused
More information( ) ( ) if t = t. It must satisfy the identity. So, bulkiness of the unit impulse (hyper)function is equal to 1. The defining characteristic is
UNIT IMPULSE RESPONSE, UNIT STEP RESPONSE, STABILITY. Uni impulse funcion (Dirac dela funcion, dela funcion) rigorously defined is no sricly a funcion, bu disribuion (or measure), precise reamen requires
More informationBest test practice: Take the past test on the class website
Bes es pracice: Take he pas es on he class websie hp://communiy.wvu.edu/~miholcomb/phys11.hml I have posed he key o he WebAssign pracice es. Newon Previous Tes is Online. Forma will be idenical. You migh
More informationMoving Spins. Phase of a Moving Spin. Phase of Moving Spin. Bioengineering 280A Principles of Biomedical Imaging. Fall Quarter 2009 MRI Lecture 6
Moving Spins Bioengineering 28A Principles of Biomedical Imaging Fall Quarer 29 MRI Lecure 6 So far we have assumed ha he spins are no moving (aside from hermal moion giving rise o relaxaion), and conras
More informationFinal Spring 2007
.615 Final Spring 7 Overview The purpose of he final exam is o calculae he MHD β limi in a high-bea oroidal okamak agains he dangerous n = 1 exernal ballooning-kink mode. Effecively, his corresponds o
More informationMath 116 Practice for Exam 2
Mah 6 Pracice for Exam Generaed Ocober 3, 7 Name: SOLUTIONS Insrucor: Secion Number:. This exam has 5 quesions. Noe ha he problems are no of equal difficuly, so you may wan o skip over and reurn o a problem
More informationWeek 1 Lecture 2 Problems 2, 5. What if something oscillates with no obvious spring? What is ω? (problem set problem)
Week 1 Lecure Problems, 5 Wha if somehing oscillaes wih no obvious spring? Wha is ω? (problem se problem) Sar wih Try and ge o SHM form E. Full beer can in lake, oscillaing F = m & = ge rearrange: F =
More informationMath 105 Second Midterm March 16, 2017
Mah 105 Second Miderm March 16, 2017 UMID: Insrucor: Iniials: Secion: 1. Do no open his exam unil you are old o do so. 2. Do no wrie your name anywhere on his exam. 3. This exam has 9 pages including his
More informationSection 3.5 Nonhomogeneous Equations; Method of Undetermined Coefficients
Secion 3.5 Nonhomogeneous Equaions; Mehod of Undeermined Coefficiens Key Terms/Ideas: Linear Differenial operaor Nonlinear operaor Second order homogeneous DE Second order nonhomogeneous DE Soluion o homogeneous
More informationLab #2: Kinematics in 1-Dimension
Reading Assignmen: Chaper 2, Secions 2-1 hrough 2-8 Lab #2: Kinemaics in 1-Dimension Inroducion: The sudy of moion is broken ino wo main areas of sudy kinemaics and dynamics. Kinemaics is he descripion
More informationTraveling Waves. Chapter Introduction
Chaper 4 Traveling Waves 4.1 Inroducion To dae, we have considered oscillaions, i.e., periodic, ofen harmonic, variaions of a physical characerisic of a sysem. The sysem a one ime is indisinguishable from
More informationSampling in k-space. Aliasing. Aliasing. Bioengineering 280A Principles of Biomedical Imaging. Fall Quarter 2010 MRI Lecture 3. Slower B z (x)=g x x
Sampling in k-space Bioengineering 80A Principles of Biomedical Imaging Fall Quarer 00 MRI Lecure 3 Thomas Liu, BE80A, UCSD, Fall 008 Aliasing Aliasing Slower B z (G Faser Inuiive view of Aliasing FOV
More informationWelcome Back to Physics 215!
Welcome Back o Physics 215! (General Physics I) Thurs. Jan 19 h, 2017 Lecure01-2 1 Las ime: Syllabus Unis and dimensional analysis Today: Displacemen, velociy, acceleraion graphs Nex ime: More acceleraion
More informationRobotics I. April 11, The kinematics of a 3R spatial robot is specified by the Denavit-Hartenberg parameters in Tab. 1.
Roboics I April 11, 017 Exercise 1 he kinemaics of a 3R spaial robo is specified by he Denavi-Harenberg parameers in ab 1 i α i d i a i θ i 1 π/ L 1 0 1 0 0 L 3 0 0 L 3 3 able 1: able of DH parameers of
More informationChapters 6 & 7: Trigonometric Functions of Angles and Real Numbers. Divide both Sides by 180
Algebra Chapers & : Trigonomeric Funcions of Angles and Real Numbers Chapers & : Trigonomeric Funcions of Angles and Real Numbers - Angle Measures Radians: - a uni (rad o measure he size of an angle. rad
More informationThis is an example to show you how SMath can calculate the movement of kinematic mechanisms.
Dec :5:6 - Kinemaics model of Simple Arm.sm This file is provided for educaional purposes as guidance for he use of he sofware ool. I is no guaraeed o be free from errors or ommissions. The mehods and
More information1. VELOCITY AND ACCELERATION
1. VELOCITY AND ACCELERATION 1.1 Kinemaics Equaions s = u + 1 a and s = v 1 a s = 1 (u + v) v = u + as 1. Displacemen-Time Graph Gradien = speed 1.3 Velociy-Time Graph Gradien = acceleraion Area under
More informationNEWTON S SECOND LAW OF MOTION
Course and Secion Dae Names NEWTON S SECOND LAW OF MOTION The acceleraion of an objec is defined as he rae of change of elociy. If he elociy changes by an amoun in a ime, hen he aerage acceleraion during
More informationProblem Set 5. Graduate Macro II, Spring 2017 The University of Notre Dame Professor Sims
Problem Se 5 Graduae Macro II, Spring 2017 The Universiy of Nore Dame Professor Sims Insrucions: You may consul wih oher members of he class, bu please make sure o urn in your own work. Where applicable,
More informationADDITIONAL PROBLEMS (a) Find the Fourier transform of the half-cosine pulse shown in Fig. 2.40(a). Additional Problems 91
ddiional Problems 9 n inverse relaionship exiss beween he ime-domain and freuency-domain descripions of a signal. Whenever an operaion is performed on he waveform of a signal in he ime domain, a corresponding
More informationSome Basic Information about M-S-D Systems
Some Basic Informaion abou M-S-D Sysems 1 Inroducion We wan o give some summary of he facs concerning unforced (homogeneous) and forced (non-homogeneous) models for linear oscillaors governed by second-order,
More informationSOLUTIONS TO ECE 3084
SOLUTIONS TO ECE 384 PROBLEM 2.. For each sysem below, specify wheher or no i is: (i) memoryless; (ii) causal; (iii) inverible; (iv) linear; (v) ime invarian; Explain your reasoning. If he propery is no
More information2.1: What is physics? Ch02: Motion along a straight line. 2.2: Motion. 2.3: Position, Displacement, Distance
Ch: Moion along a sraigh line Moion Posiion and Displacemen Average Velociy and Average Speed Insananeous Velociy and Speed Acceleraion Consan Acceleraion: A Special Case Anoher Look a Consan Acceleraion
More information15. Bicycle Wheel. Graph of height y (cm) above the axle against time t (s) over a 6-second interval. 15 bike wheel
15. Biccle Wheel The graph We moun a biccle wheel so ha i is free o roae in a verical plane. In fac, wha works easil is o pu an exension on one of he axles, and ge a suden o sand on one side and hold he
More information= ( ) ) or a system of differential equations with continuous parametrization (T = R
XIII. DIFFERENCE AND DIFFERENTIAL EQUATIONS Ofen funcions, or a sysem of funcion, are paramerized in erms of some variable, usually denoed as and inerpreed as ime. The variable is wrien as a funcion of
More informationCHAPTER 10 VALIDATION OF TEST WITH ARTIFICAL NEURAL NETWORK
175 CHAPTER 10 VALIDATION OF TEST WITH ARTIFICAL NEURAL NETWORK 10.1 INTRODUCTION Amongs he research work performed, he bes resuls of experimenal work are validaed wih Arificial Neural Nework. From he
More informationdt = C exp (3 ln t 4 ). t 4 W = C exp ( ln(4 t) 3) = C(4 t) 3.
Mah Rahman Exam Review Soluions () Consider he IVP: ( 4)y 3y + 4y = ; y(3) = 0, y (3) =. (a) Please deermine he longes inerval for which he IVP is guaraneed o have a unique soluion. Soluion: The disconinuiies
More information2002 November 14 Exam III Physics 191
November 4 Exam III Physics 9 Physical onsans: Earh s free-fall acceleraion = g = 9.8 m/s ircle he leer of he single bes answer. quesion is worh poin Each 3. Four differen objecs wih masses: m = kg, m
More information2001 November 15 Exam III Physics 191
1 November 15 Eam III Physics 191 Physical Consans: Earh s free-fall acceleraion = g = 9.8 m/s 2 Circle he leer of he single bes answer. quesion is worh 1 poin Each 3. Four differen objecs wih masses:
More informationKinematics and kinematic functions
Kinemaics and kinemaic funcions Kinemaics deals wih he sudy of four funcions (called kinemaic funcions or KFs) ha mahemaically ransform join variables ino caresian variables and vice versa Direc Posiion
More information