Lecture 6. Goals: Determine the optimal threshold, filter, signals for a binary communications problem VI-1
|
|
- Ολυμπία Κόρακας
- 6 χρόνια πριν
- Προβολές:
Transcript
1 Lecue 6 Goals: Deemine e opimal esold, file, signals fo a binay communicaions poblem VI-
2 Minimum Aveage Eo Pobabiliy Poblem: Find e opimum file, esold and signals o minimize e aveage eo pobabiliy. s s Z γ dec s γ dec s n P e P e π Peos ansmied Peos ansmied Pobabiliy s ansmied VI-
3 π π π Pobabiliy s ansmied e aveage pobabiliy of eo is P ep e π Pe π () Le ŝ ŝ s s d d oupu due o s alone oupu due o s alone Since we assume a e eceive will decide s if e oupu of e file is lage an a esold and s if i is smalle, we need o assume a. ŝ ŝ P e P Z γ s ansmied VI-3
4 If s is ansmied en Z akes e fom Z ŝ η wee η is a Gaussian andom vaiable wi mean and vaiance σ N ; σ N N d N H f d f us P e P P ŝ η γ Q ŝ η γ ŝ γ () P e P P Z ŝ γ s ansmied η γ s ansmied P η γ ŝ VI-4
5 s Q γ ŝ (3) Subsiuing () and (3) ino () yields P e γ s π Q ŝ γ π Q γ ŝ (4) e poblem is o minimize e eo pobabiliy ove all coices of γ. s s and VI-5
6 π π Sep : Minimize P e ove γ Facs used: Q Q x x x e x π π e u du Meod: Se e deivaive of P e wi espec o γ equal o. d P e dγ exp ŝ π γ exp γ ŝ π VI-6
7 π exp ŝ π exp ŝ γ γ ŝ exp γ ŝ ŝ γ σn π π γ γŝ ŝ ŝ γŝ γ σ N ln π π γ ŝ ŝ σ N ln π π ŝ ŝ γ σ N ln π π ŝ ŝ ŝ ŝ γ op ŝ σ N ln π π ŝ ŝ (5) Special Case: If π π en γ op ŝ ŝ VI-7
8 Wa is P e fo e opimal esold? ŝ γ op ŝ ŝ σ N ln π π ŝ ŝ ŝ ŝ ŝ ŝ σ N ŝ ln π π ŝ γ op ŝ ŝ ŝ ŝ ln π (6) π Definiion: ŝ f g s ŝ s f g s s d s d s s d VI-8
9 us fom (6) Similaly Remembe a σ N Le λ ŝ γ op ŝ γ op N N s s s s s s ln π (7) π ln π (8) π d(9) d () s s s s s d s s d VI-9
10 Ē s s s s d E E s s Ē Ē E E s Ē s Ē () Combining (7), (8), (9), (), and () ŝ γ op s N N s ln π π s s Ē N 4Ē N s s ln π π VI-
11 Le λ s s en ŝ γ op λ β λ Ē N β 4Ē N ln π π Similaly γ op ŝ λ β λ VI-
12 Summay of Sep : γ op ŝ σ N ln π π ŝ ŝ πq λ ŝ P e γ op s s πq λ β λ β λ λ s s Ē N β 4Ē N ln π π Ē E E s s Ē VI-
13 πq λ λ Find e opimal file Sep : o minimize e aveage pobabiliy of eo Meod: Fis sow a P e is an deceasing funcion of λ by sowing e deivaive is negaive. en find e a maximizes λ (us minimizing P e ). P e s s Pe γ op πqλ s s β λ β λ P e λ π e π λ π β λ β λ β λ π exp exp β λ λ λ π β β e β β λ λ β λ π π π β λ VI-3
14 exp λ β λ π β λ e β π β λ e β β Ē N N 4Ē ln π π ln π ln π π π e β π π e β π π π e β ππ π e βπ π d P e dλ e λ π e β λ λ ππ β λ ππ β λ β λ VI-4
15 op s Since d P e dλ so a Pe is minimized by maximizing λ. λ s s Fom Scwaz s inequaliy s s s us s s λ wi equaliy if s. Coose λ ). Fo opimal esold and opimal file s P e γ op s s πq β πq β s o e signals. s is called e maced file because i is maced γ op op s E E N ln π π VI-5
16 Fo e opimal file e oupu due o signal alone ae ŝ E Ē ŝ Ē E If π π en P eq Q Ē N VI-6
17 Find e opimal signals s eo. Sep 3: and s Meod: P e depends on e signal only oug Ē and. Ē E E o minimize e aveage pobabiliy of s I is obvious a we could jus incease e enegy o infiniy and ge eo pobabiliy. Insead we will fix Ē and vay e signals o vay. Again we sow a P e is an inceasing funcion of and en coose e signals o minimize. s Ē P e πq β πq β Ē N β 4Ē N ln π π VI-7
18 d P e d π e β π β π e β π β e β π e β β π e β β e β π π Ē N d P e d Fom Scwaz s inequaliy s s Ē s Ē s VI-8
19 wi equaliy if s Ks K. Fo s Ks E E E E wi equaliy if E E. (Aimeic mean Geomeic mean). wo signals s and s ae said o be anipodal if s s Opimal signals ae anipodal. E N β N 8E ln π π VI-9
20 If π π en P eq Q E N VI-
21 Aside: Scwaz s inequaliy: Fo any f g f f g g Since e polynomial in is neve negaive ee mus be eie no zeos o a double zeo. us e discimanan mus be no be posiive. 4 f g 4 f g f g f g f g Equaliy occus wen f x Kgx. If K is posiive e inequaliy on e ig side becomes equaliy and if K is negaive e inequaliy on e ig side becomes equaliy. is is Scwaz s inequaliy. VI-
22 Aside: Aimeic mean Geomeic mean: Le a and a be eal nonnegaive numbes. a a wi equaliy if a a a aa a a a a a a a a a a a 4aa 4a a a a a a wi equaliy if a a VI-
23 s s ŝ ŝ σ N n Summay γ dec s γ dec s N d s s d d N H f d f VI-3
24 πq λ op P e γ s s π Q ŝ γ π Q γ ŝ Sep : Opimize wi espec o γ. P e γ op s s πq λ β λ β λ Ē N β 4Ē N ln π π λ s s s s s γ op ŝ ŝ ŝ σ N ŝ ln π π Sep : Opimize wi espec o. P e γ op s s πq β πq β VI-4
25 op ˆβ Ē N β 4Ē N ln π π γ op op op s E E s N ln π π e maced file. Sep 3: Opimize wi espec o s and s. P e γ op s op s op πq ˆ ˆβ πq ˆ ˆβ ˆ Ē N N 8Ē ln π π s s op s s s s op op s VI-5
26 γ op op s op s op N ln π π VI-6
27 SPECIAL CASE π P e π Q ŝ γ Q γ ŝ Sep : Opimize wi espec o γ P e Q λ Ē N λ s S Ē E E E s d E s s Ē s d VI-7
28 ˆ γ op ŝ ŝ Sep : Opimize wi espec o P e Q Ē N op s s maced file γ op E E Sep 3: Opimize wi espec o s and s. P e Q ˆ Ē N s s VI-8
29 op s γop VI-9
30 Example: s s Ap s Ap Baseband signals op Ap Assume s ansmied s A A Ap p p Ap p d d d VI-3
31 p e oupu due o signal alone: VI-3
32 ŝ A e oupu due o noise is a Gaussian andom vaiable wi mean zeo and vaiance σ N N 4A Le be e sampling ime. Since e signal ou is a maximum wen and e noise vaiance does no depend on e sample ime e opimum AN VI-3
33 sampling ime is. Equivalen fom of opimal eceive s s γ dec s γ dec s Z d s s s s d VI-33
34 Z s s s d s d If s and s ae ime limied o en Z s s d γ dec s γ dec s s s is is called e Coelaion Receive. VI-34
35 Example: Binay Pase Sif Keying (RF signals) s Acos ω p s s s i i Acos ω p Acos ω iπ p VI-35
36 ω s s γ dec s γ dec s Acos VI-36
37 ω Assume ω oω nπ E i s i d A A A A cos ω d cos sin ω ω d P eq E N Q A N π π γ VI-37
38 ω ω ω Subopimal Receives s i i Acos p ω nπ Z γ dec s γ dec s n cos Claim: σ n N (powe equals /). P e Q ŝ Q ŝ 4. e faco of / is due o muliplying by cos VI-38
39 ωs Poof: σ n E N 4 N 4 N 4 E n n N δ n cos N cos ω cos s s d sin ω cos cos ω ω ω n d s ω ω cos cos cos ωs ωs dds dds dds VI-39
40 povide a ω ŝ i oω nπ. i A i A i Acos ω p low pass cos d ig feq. d w d P e Q ŝ i VI-4
41 Example: Single pole RC file RC e e u RC u d e RC e d VI-4
42 d u d σ N N 4 e u d e e d d e e VI-4
43 e e e ŝ We would like o maximize ŝ ŝ is maximized a andσ N N 4 is consan e opimal sampling ime is is σn. Since VI-43
44 esuls in a signal-o-noise aio of ŝ SNR 3 e A N 4 Maximize w... f e f e e e e Le x en xe x e x x x e x e x VI-44
45 x e x We can numeically solve is o ge x is yeilds a signal-o-noise aio of 56 and so 56 RC 8. SNR A 95 N Loss due o subopimal eceive =.89 db VI-45
Analysis of optimal harvesting of a prey-predator fishery model with the limited sources of prey and presence of toxicity
ES Web of Confeences 7, 68 (8) hps://doiog/5/esconf/8768 ICEIS 8 nalsis of opimal havesing of a pe-pedao fishe model wih he limied souces of pe and pesence of oici Suimin,, Sii Khabibah, and Dia nies Munawwaoh
Διαβάστε περισσότεραLecture 12 Modulation and Sampling
EE 2 spring 2-22 Handou #25 Lecure 2 Modulaion and Sampling The Fourier ransform of he produc of wo signals Modulaion of a signal wih a sinusoid Sampling wih an impulse rain The sampling heorem 2 Convoluion
Διαβάστε περισσότεραExample Sheet 3 Solutions
Example Sheet 3 Solutions. i Regular Sturm-Liouville. ii Singular Sturm-Liouville mixed boundary conditions. iii Not Sturm-Liouville ODE is not in Sturm-Liouville form. iv Regular Sturm-Liouville note
Διαβάστε περισσότεραOscillating dipole system Suppose we have two small spheres separated by a distance s. The charge on one sphere changes with time and is described by
5 Radiation (Chapte 11) 5.1 Electic dipole adiation Oscillating dipole system Suppose we have two small sphees sepaated by a distance s. The chage on one sphee changes with time and is descibed by q(t)
Διαβάστε περισσότερα) 2. δ δ. β β. β β β β. r k k. tll. m n Λ + +
Techical Appedix o Hamig eposis ad Helpig Bowes: The ispaae Impac of Ba Cosolidaio (o o be published bu o be made available upo eques. eails of Poofs of Poposiios 1 ad To deive Poposiio 1 s exac ad sufficie
Διαβάστε περισσότεραExample 1: THE ELECTRIC DIPOLE
Example 1: THE ELECTRIC DIPOLE 1 The Electic Dipole: z + P + θ d _ Φ = Q 4πε + Q = Q 4πε 4πε 1 + 1 2 The Electic Dipole: d + _ z + Law of Cosines: θ A B α C A 2 = B 2 + C 2 2ABcosα P ± = 2 ( + d ) 2 2
Διαβάστε περισσότερα( ) ( t) ( 0) ( ) dw w. = = β. Then the solution of (1.1) is easily found to. wt = t+ t. We generalize this to the following nonlinear differential
Periodic oluion of van der Pol differenial equaion. by A. Arimoo Deparmen of Mahemaic Muahi Iniue of Technology Tokyo Japan in Seminar a Kiami Iniue of Technology January 8 9. Inroducion Le u conider a
Διαβάστε περισσότεραSpace Physics (I) [AP-3044] Lecture 1 by Ling-Hsiao Lyu Oct Lecture 1. Dipole Magnetic Field and Equations of Magnetic Field Lines
Space Physics (I) [AP-344] Lectue by Ling-Hsiao Lyu Oct. 2 Lectue. Dipole Magnetic Field and Equations of Magnetic Field Lines.. Dipole Magnetic Field Since = we can define = A (.) whee A is called the
Διαβάστε περισσότερα21. Stresses Around a Hole (I) 21. Stresses Around a Hole (I) I Main Topics
I Main Topics A Intoducon to stess fields and stess concentaons B An axisymmetic poblem B Stesses in a pola (cylindical) efeence fame C quaons of equilibium D Soluon of bounday value poblem fo a pessuized
Διαβάστε περισσότερα6.003: Signals and Systems. Modulation
6.3: Signals and Sysems Modulaion December 6, 2 Subjec Evaluaions Your feedback is imporan o us! Please give feedback o he saff and fuure 6.3 sudens: hp://web.mi.edu/subjecevaluaion Evaluaions are open
Διαβάστε περισσότεραAnalytical Expression for Hessian
Analytical Expession fo Hessian We deive the expession of Hessian fo a binay potential the coesponding expessions wee deived in [] fo a multibody potential. In what follows, we use the convention that
Διαβάστε περισσότεραMatrix Hartree-Fock Equations for a Closed Shell System
atix Hatee-Fock Equations fo a Closed Shell System A single deteminant wavefunction fo a system containing an even numbe of electon N) consists of N/ spatial obitals, each occupied with an α & β spin has
Διαβάστε περισσότεραExercises 10. Find a fundamental matrix of the given system of equations. Also find the fundamental matrix Φ(t) satisfying Φ(0) = I. 1.
Exercises 0 More exercises are available in Elementary Differential Equations. If you have a problem to solve any of them, feel free to come to office hour. Problem Find a fundamental matrix of the given
Διαβάστε περισσότεραTrigonometry 1.TRIGONOMETRIC RATIOS
Trigonometry.TRIGONOMETRIC RATIOS. If a ray OP makes an angle with the positive direction of X-axis then y x i) Sin ii) cos r r iii) tan x y (x 0) iv) cot y x (y 0) y P v) sec x r (x 0) vi) cosec y r (y
Διαβάστε περισσότεραSection 8.3 Trigonometric Equations
99 Section 8. Trigonometric Equations Objective 1: Solve Equations Involving One Trigonometric Function. In this section and the next, we will exple how to solving equations involving trigonometric functions.
Διαβάστε περισσότερα5.4 The Poisson Distribution.
The worst thing you can do about a situation is nothing. Sr. O Shea Jackson 5.4 The Poisson Distribution. Description of the Poisson Distribution Discrete probability distribution. The random variable
Διαβάστε περισσότεραCHAPTER 25 SOLVING EQUATIONS BY ITERATIVE METHODS
CHAPTER 5 SOLVING EQUATIONS BY ITERATIVE METHODS EXERCISE 104 Page 8 1. Find the positive root of the equation x + 3x 5 = 0, correct to 3 significant figures, using the method of bisection. Let f(x) =
Διαβάστε περισσότερα(a,b) Let s review the general definitions of trig functions first. (See back cover of your book) sin θ = b/r cos θ = a/r tan θ = b/a, a 0
TRIGONOMETRIC IDENTITIES (a,b) Let s eview the geneal definitions of tig functions fist. (See back cove of you book) θ b/ θ a/ tan θ b/a, a 0 θ csc θ /b, b 0 sec θ /a, a 0 cot θ a/b, b 0 By doing some
Διαβάστε περισσότερα6.003: Signals and Systems
6.3: Signals and Sysems Modulaion December 6, 2 Communicaions Sysems Signals are no always well mached o he media hrough which we wish o ransmi hem. signal audio video inerne applicaions elephone, radio,
Διαβάστε περισσότεραConcrete Mathematics Exercises from 30 September 2016
Concrete Mathematics Exercises from 30 September 2016 Silvio Capobianco Exercise 1.7 Let H(n) = J(n + 1) J(n). Equation (1.8) tells us that H(2n) = 2, and H(2n+1) = J(2n+2) J(2n+1) = (2J(n+1) 1) (2J(n)+1)
Διαβάστε περισσότεραPARTIAL NOTES for 6.1 Trigonometric Identities
PARTIAL NOTES for 6.1 Trigonometric Identities tanθ = sinθ cosθ cotθ = cosθ sinθ BASIC IDENTITIES cscθ = 1 sinθ secθ = 1 cosθ cotθ = 1 tanθ PYTHAGOREAN IDENTITIES sin θ + cos θ =1 tan θ +1= sec θ 1 + cot
Διαβάστε περισσότεραProblem Set 3: Solutions
CMPSCI 69GG Applied Information Theory Fall 006 Problem Set 3: Solutions. [Cover and Thomas 7.] a Define the following notation, C I p xx; Y max X; Y C I p xx; Ỹ max I X; Ỹ We would like to show that C
Διαβάστε περισσότεραPhys460.nb Solution for the t-dependent Schrodinger s equation How did we find the solution? (not required)
Phys460.nb 81 ψ n (t) is still the (same) eigenstate of H But for tdependent H. The answer is NO. 5.5.5. Solution for the tdependent Schrodinger s equation If we assume that at time t 0, the electron starts
Διαβάστε περισσότεραw o = R 1 p. (1) R = p =. = 1
Πανεπιστήµιο Κρήτης - Τµήµα Επιστήµης Υπολογιστών ΗΥ-570: Στατιστική Επεξεργασία Σήµατος 205 ιδάσκων : Α. Μουχτάρης Τριτη Σειρά Ασκήσεων Λύσεις Ασκηση 3. 5.2 (a) From the Wiener-Hopf equation we have:
Διαβάστε περισσότεραApproximate System Reliability Evaluation
Appoximate Sytem Reliability Evaluation Up MTTF Down 0 MTBF MTTR () Time Fo many engineeing ytem component, MTTF MTBF i.e. failue ate, failue fequency, f Fequency, Duation and Pobability Indice: failue
Διαβάστε περισσότεραAppendix. The solution begins with Eq. (2.15) from the text, which we repeat here for 1, (A.1)
Aenix Aenix A: The equaion o he sock rice. The soluion egins wih Eq..5 rom he ex, which we reea here or convenience as Eq.A.: [ [ E E X, A. c α where X u ε, α γ, an c α y AR. Take execaions o Eq. A. as
Διαβάστε περισσότεραThe Simply Typed Lambda Calculus
Type Inference Instead of writing type annotations, can we use an algorithm to infer what the type annotations should be? That depends on the type system. For simple type systems the answer is yes, and
Διαβάστε περισσότερα= e 6t. = t 1 = t. 5 t 8L 1[ 1 = 3L 1 [ 1. L 1 [ π. = 3 π. = L 1 3s = L. = 3L 1 s t. = 3 cos(5t) sin(5t).
Worked Soluion 95 Chaper 25: The Invere Laplace Tranform 25 a From he able: L ] e 6 6 25 c L 2 ] ] L! + 25 e L 5 2 + 25] ] L 5 2 + 5 2 in(5) 252 a L 6 + 2] L 6 ( 2)] 6L ( 2)] 6e 2 252 c L 3 8 4] 3L ] 8L
Διαβάστε περισσότερα6. MAXIMUM LIKELIHOOD ESTIMATION
6 MAXIMUM LIKELIHOOD ESIMAION [1] Maximum Likelihood Estimator (1) Cases in which θ (unknown parameter) is scalar Notational Clarification: From now on, we denote the true value of θ as θ o hen, view θ
Διαβάστε περισσότεραSecond Order RLC Filters
ECEN 60 Circuits/Electronics Spring 007-0-07 P. Mathys Second Order RLC Filters RLC Lowpass Filter A passive RLC lowpass filter (LPF) circuit is shown in the following schematic. R L C v O (t) Using phasor
Διαβάστε περισσότερα3 Frequency Domain Representation of Continuous Signals and Systems
3 Frequency Domain Represenaion of Coninuous Signals and Sysems 3. Fourier Series Represenaion of Periodic Signals............. 2 3.. Exponenial Fourier Series.................... 2 3..2 Discree Fourier
Διαβάστε περισσότεραNumerical Analysis FMN011
Numerical Analysis FMN011 Carmen Arévalo Lund University carmen@maths.lth.se Lecture 12 Periodic data A function g has period P if g(x + P ) = g(x) Model: Trigonometric polynomial of order M T M (x) =
Διαβάστε περισσότεραMath221: HW# 1 solutions
Math: HW# solutions Andy Royston October, 5 7.5.7, 3 rd Ed. We have a n = b n = a = fxdx = xdx =, x cos nxdx = x sin nx n sin nxdx n = cos nx n = n n, x sin nxdx = x cos nx n + cos nxdx n cos n = + sin
Διαβάστε περισσότεραΕΛΛΗΝΙΚΗ ΔΗΜΟΚΡΑΤΙΑ ΠΑΝΕΠΙΣΤΗΜΙΟ ΚΡΗΤΗΣ. Ψηφιακή Οικονομία. Διάλεξη 10η: Basics of Game Theory part 2 Mαρίνα Μπιτσάκη Τμήμα Επιστήμης Υπολογιστών
ΕΛΛΗΝΙΚΗ ΔΗΜΟΚΡΑΤΙΑ ΠΑΝΕΠΙΣΤΗΜΙΟ ΚΡΗΤΗΣ Ψηφιακή Οικονομία Διάλεξη 0η: Basics of Game Theory part 2 Mαρίνα Μπιτσάκη Τμήμα Επιστήμης Υπολογιστών Best Response Curves Used to solve for equilibria in games
Διαβάστε περισσότεραLifting Entry 2. Basic planar dynamics of motion, again Yet another equilibrium glide Hypersonic phugoid motion MARYLAND U N I V E R S I T Y O F
ifting Entry Basic planar dynamics of motion, again Yet another equilibrium glide Hypersonic phugoid motion MARYAN 1 010 avid. Akin - All rights reserved http://spacecraft.ssl.umd.edu ifting Atmospheric
Διαβάστε περισσότεραVariational Wavefunction for the Helium Atom
Technische Universität Graz Institut für Festkörperphysik Student project Variational Wavefunction for the Helium Atom Molecular and Solid State Physics 53. submitted on: 3. November 9 by: Markus Krammer
Διαβάστε περισσότεραDifferentiation exercise show differential equation
Differentiation exercise show differential equation 1. If y x sin 2x, prove that x d2 y 2 2 + 2y x + 4xy 0 y x sin 2x sin 2x + 2x cos 2x 2 2cos 2x + (2 cos 2x 4x sin 2x) x d2 y 2 2 + 2y x + 4xy (2x cos
Διαβάστε περισσότεραAreas and Lengths in Polar Coordinates
Kiryl Tsishchanka Areas and Lengths in Polar Coordinates In this section we develop the formula for the area of a region whose boundary is given by a polar equation. We need to use the formula for the
Διαβάστε περισσότερα( ) ( ) ( ) Fourier series. ; m is an integer. r(t) is periodic (T>0), r(t+t) = r(t), t Fundamental period T 0 = smallest T. Fundamental frequency ω
Fourier series e jm when m d when m ; m is an ineger. jm jm jm jm e d e e e jm jm jm jm r( is periodi (>, r(+ r(, Fundamenal period smalles Fundamenal frequeny r ( + r ( is periodi hen M M e j M, e j,
Διαβάστε περισσότεραProbabilistic Image Processing by Extended Gauss-Markov Random Fields
Pobablsc mage Pocessng b Eended Gauss-Makov Random Felds Kauuk anaka Munek asuda Ncolas Mon Gaduae School of nfomaon Scences ohoku Unves Japan and D. M. engon Depamen of Sascs Unves of Glasgow UK 3 Sepembe
Διαβάστε περισσότεραHOMEWORK#1. t E(x) = 1 λ = (b) Find the median lifetime of a randomly selected light bulb. Answer:
HOMEWORK# 52258 李亞晟 Eercise 2. The lifetime of light bulbs follows an eponential distribution with a hazard rate of. failures per hour of use (a) Find the mean lifetime of a randomly selected light bulb.
Διαβάστε περισσότεραC.S. 430 Assignment 6, Sample Solutions
C.S. 430 Assignment 6, Sample Solutions Paul Liu November 15, 2007 Note that these are sample solutions only; in many cases there were many acceptable answers. 1 Reynolds Problem 10.1 1.1 Normal-order
Διαβάστε περισσότερα2 Composition. Invertible Mappings
Arkansas Tech University MATH 4033: Elementary Modern Algebra Dr. Marcel B. Finan Composition. Invertible Mappings In this section we discuss two procedures for creating new mappings from old ones, namely,
Διαβάστε περισσότεραFractional Colorings and Zykov Products of graphs
Fractional Colorings and Zykov Products of graphs Who? Nichole Schimanski When? July 27, 2011 Graphs A graph, G, consists of a vertex set, V (G), and an edge set, E(G). V (G) is any finite set E(G) is
Διαβάστε περισσότεραInverse trigonometric functions & General Solution of Trigonometric Equations. ------------------ ----------------------------- -----------------
Inverse trigonometric functions & General Solution of Trigonometric Equations. 1. Sin ( ) = a) b) c) d) Ans b. Solution : Method 1. Ans a: 17 > 1 a) is rejected. w.k.t Sin ( sin ) = d is rejected. If sin
Διαβάστε περισσότεραMain source: "Discrete-time systems and computer control" by Α. ΣΚΟΔΡΑΣ ΨΗΦΙΑΚΟΣ ΕΛΕΓΧΟΣ ΔΙΑΛΕΞΗ 4 ΔΙΑΦΑΝΕΙΑ 1
Main source: "Discrete-time systems and computer control" by Α. ΣΚΟΔΡΑΣ ΨΗΦΙΑΚΟΣ ΕΛΕΓΧΟΣ ΔΙΑΛΕΞΗ 4 ΔΙΑΦΑΝΕΙΑ 1 A Brief History of Sampling Research 1915 - Edmund Taylor Whittaker (1873-1956) devised a
Διαβάστε περισσότεραHOMEWORK 4 = G. In order to plot the stress versus the stretch we define a normalized stretch:
HOMEWORK 4 Problem a For the fast loading case, we want to derive the relationship between P zz and λ z. We know that the nominal stress is expressed as: P zz = ψ λ z where λ z = λ λ z. Therefore, applying
Διαβάστε περισσότεραSection 7.6 Double and Half Angle Formulas
09 Section 7. Double and Half Angle Fmulas To derive the double-angles fmulas, we will use the sum of two angles fmulas that we developed in the last section. We will let α θ and β θ: cos(θ) cos(θ + θ)
Διαβάστε περισσότεραLecture 21: Scattering and FGR
ECE-656: Fall 009 Lecture : Scattering and FGR Professor Mark Lundstrom Electrical and Computer Engineering Purdue University, West Lafayette, IN USA Review: characteristic times τ ( p), (, ) == S p p
Διαβάστε περισσότεραGeorge S. A. Shaker ECE477 Understanding Reflections in Media. Reflection in Media
Geoge S. A. Shake C477 Udesadg Reflecos Meda Refleco Meda Ths hadou ages a smplfed appoach o udesad eflecos meda. As a sude C477, you ae o equed o kow hese seps by hea. I s jus o make you udesad how some
Διαβάστε περισσότεραThe choice of an optimal LCSCR contract involves the choice of an x L. such that the supplier chooses the LCS option when x xl
EHNIA APPENDIX AMPANY SIMPE S SHARIN NRAS Proof of emma. he choice of an opimal SR conrac involves he choice of an such ha he supplier chooses he S opion hen and he R opion hen >. When he selecs he S opion
Διαβάστε περισσότεραe t e r Cylindrical and Spherical Coordinate Representation of grad, div, curl and 2
Cylindical and Spheical Coodinate Repesentation of gad, div, cul and 2 Thus fa, we have descibed an abitay vecto in F as a linea combination of i, j and k, which ae unit vectos in the diection of inceasin,
Διαβάστε περισσότεραFourier Series. MATH 211, Calculus II. J. Robert Buchanan. Spring Department of Mathematics
Fourier Series MATH 211, Calculus II J. Robert Buchanan Department of Mathematics Spring 2018 Introduction Not all functions can be represented by Taylor series. f (k) (c) A Taylor series f (x) = (x c)
Διαβάστε περισσότεραMatrices and Determinants
Matrices and Determinants SUBJECTIVE PROBLEMS: Q 1. For what value of k do the following system of equations possess a non-trivial (i.e., not all zero) solution over the set of rationals Q? x + ky + 3z
Διαβάστε περισσότεραExercise, May 23, 2016: Inflation stabilization with noisy data 1
Monetay Policy Henik Jensen Depatment of Economics Univesity of Copenhagen Execise May 23 2016: Inflation stabilization with noisy data 1 Suggested answes We have the basic model x t E t x t+1 σ 1 ît E
Διαβάστε περισσότεραLifting Entry (continued)
ifting Entry (continued) Basic planar dynamics of motion, again Yet another equilibrium glide Hypersonic phugoid motion Planar state equations MARYAN 1 01 avid. Akin - All rights reserved http://spacecraft.ssl.umd.edu
Διαβάστε περισσότεραMean-Variance Analysis
Mean-Variance Analysis Jan Schneider McCombs School of Business University of Texas at Austin Jan Schneider Mean-Variance Analysis Beta Representation of the Risk Premium risk premium E t [Rt t+τ ] R1
Διαβάστε περισσότεραApproximation of distance between locations on earth given by latitude and longitude
Approximation of distance between locations on earth given by latitude and longitude Jan Behrens 2012-12-31 In this paper we shall provide a method to approximate distances between two points on earth
Διαβάστε περισσότεραOther Test Constructions: Likelihood Ratio & Bayes Tests
Other Test Constructions: Likelihood Ratio & Bayes Tests Side-Note: So far we have seen a few approaches for creating tests such as Neyman-Pearson Lemma ( most powerful tests of H 0 : θ = θ 0 vs H 1 :
Διαβάστε περισσότεραCalculating the propagation delay of coaxial cable
Your source for quality GNSS Networking Solutions and Design Services! Page 1 of 5 Calculating the propagation delay of coaxial cable The delay of a cable or velocity factor is determined by the dielectric
Διαβάστε περισσότεραΚΥΠΡΙΑΚΗ ΕΤΑΙΡΕΙΑ ΠΛΗΡΟΦΟΡΙΚΗΣ CYPRUS COMPUTER SOCIETY ΠΑΓΚΥΠΡΙΟΣ ΜΑΘΗΤΙΚΟΣ ΔΙΑΓΩΝΙΣΜΟΣ ΠΛΗΡΟΦΟΡΙΚΗΣ 6/5/2006
Οδηγίες: Να απαντηθούν όλες οι ερωτήσεις. Ολοι οι αριθμοί που αναφέρονται σε όλα τα ερωτήματα είναι μικρότεροι το 1000 εκτός αν ορίζεται διαφορετικά στη διατύπωση του προβλήματος. Διάρκεια: 3,5 ώρες Καλή
Διαβάστε περισσότεραOrdinal Arithmetic: Addition, Multiplication, Exponentiation and Limit
Ordinal Arithmetic: Addition, Multiplication, Exponentiation and Limit Ting Zhang Stanford May 11, 2001 Stanford, 5/11/2001 1 Outline Ordinal Classification Ordinal Addition Ordinal Multiplication Ordinal
Διαβάστε περισσότεραUniform Convergence of Fourier Series Michael Taylor
Uniform Convergence of Fourier Series Michael Taylor Given f L 1 T 1 ), we consider the partial sums of the Fourier series of f: N 1) S N fθ) = ˆfk)e ikθ. k= N A calculation gives the Dirichlet formula
Διαβάστε περισσότεραEcon 2110: Fall 2008 Suggested Solutions to Problem Set 8 questions or comments to Dan Fetter 1
Eon : Fall 8 Suggested Solutions to Problem Set 8 Email questions or omments to Dan Fetter Problem. Let X be a salar with density f(x, θ) (θx + θ) [ x ] with θ. (a) Find the most powerful level α test
Διαβάστε περισσότεραElectronic Companion to Supply Chain Dynamics and Channel Efficiency in Durable Product Pricing and Distribution
i Eleconic Copanion o Supply Chain Dynaics and Channel Efficiency in Duable Poduc Picing and Disibuion Wei-yu Kevin Chiang College of Business Ciy Univesiy of Hong Kong wchiang@ciyueduh I Poof of Poposiion
Διαβάστε περισσότεραStrain gauge and rosettes
Strain gauge and rosettes Introduction A strain gauge is a device which is used to measure strain (deformation) on an object subjected to forces. Strain can be measured using various types of devices classified
Διαβάστε περισσότεραω = radians per sec, t = 3 sec
Secion. Linear and Angular Speed 7. From exercise, =. A= r A = ( 00 ) (. ) = 7,00 in 7. Since 7 is in quadran IV, he reference 7 8 7 angle is = =. In quadran IV, he cosine is posiive. Thus, 7 cos = cos
Διαβάστε περισσότεραD Alembert s Solution to the Wave Equation
D Alembert s Solution to the Wave Equation MATH 467 Partial Differential Equations J. Robert Buchanan Department of Mathematics Fall 2018 Objectives In this lesson we will learn: a change of variable technique
Διαβάστε περισσότεραPractice Exam 2. Conceptual Questions. 1. State a Basic identity and then verify it. (a) Identity: Solution: One identity is csc(θ) = 1
Conceptual Questions. State a Basic identity and then verify it. a) Identity: Solution: One identity is cscθ) = sinθ) Practice Exam b) Verification: Solution: Given the point of intersection x, y) of the
Διαβάστε περισσότερα3.4 SUM AND DIFFERENCE FORMULAS. NOTE: cos(α+β) cos α + cos β cos(α-β) cos α -cos β
3.4 SUM AND DIFFERENCE FORMULAS Page Theorem cos(αβ cos α cos β -sin α cos(α-β cos α cos β sin α NOTE: cos(αβ cos α cos β cos(α-β cos α -cos β Proof of cos(α-β cos α cos β sin α Let s use a unit circle
Διαβάστε περισσότεραΑναγνώριση Προτύπων. Baysian Θεωρία Αποφάσεων (DETECTION)
Αναγνώριση Προτύπων Baysian Θεωρία Αποφάσεων (DETECTION) Χριστόδουλος Χαμζάς Τα περιεχόμενα των παρουσιάσεων προέρχονται από τις παρουσιάσεις του αντίστοιχου διδακτέου μαθήματος του καθ. Παναγιώτη Τσακαλίδη,
Διαβάστε περισσότερα26 28 Find an equation of the tangent line to the curve at the given point Discuss the curve under the guidelines of Section
SECTION 5. THE NATURAL LOGARITHMIC FUNCTION 5. THE NATURAL LOGARITHMIC FUNCTION A Click here for answers. S Click here for solutions. 4 Use the Laws of Logarithms to epand the quantit.. ln ab. ln c. ln
Διαβάστε περισσότεραAreas and Lengths in Polar Coordinates
Kiryl Tsishchanka Areas and Lengths in Polar Coordinates In this section we develop the formula for the area of a region whose boundary is given by a polar equation. We need to use the formula for the
Διαβάστε περισσότεραderivation of the Laplacian from rectangular to spherical coordinates
derivation of the Laplacian from rectangular to spherical coordinates swapnizzle 03-03- :5:43 We begin by recognizing the familiar conversion from rectangular to spherical coordinates (note that φ is used
Διαβάστε περισσότεραBayes Rule and its Applications
Bayes Rule and its Applications Bayes Rule: P (B k A) = P (A B k )P (B k )/ n i= P (A B i )P (B i ) Example : In a certain factory, machines A, B, and C are all producing springs of the same length. Of
Διαβάστε περισσότεραFinite Field Problems: Solutions
Finite Field Problems: Solutions 1. Let f = x 2 +1 Z 11 [x] and let F = Z 11 [x]/(f), a field. Let Solution: F =11 2 = 121, so F = 121 1 = 120. The possible orders are the divisors of 120. Solution: The
Διαβάστε περισσότερα( )( ) ( ) ( )( ) ( )( ) β = Chapter 5 Exercise Problems EX α So 49 β 199 EX EX EX5.4 EX5.5. (a)
hapter 5 xercise Problems X5. α β α 0.980 For α 0.980, β 49 0.980 0.995 For α 0.995, β 99 0.995 So 49 β 99 X5. O 00 O or n 3 O 40.5 β 0 X5.3 6.5 μ A 00 β ( 0)( 6.5 μa) 8 ma 5 ( 8)( 4 ) or.88 P on + 0.0065
Διαβάστε περισσότεραCurvilinear Systems of Coordinates
A Cuvilinea Systems of Coodinates A.1 Geneal Fomulas Given a nonlinea tansfomation between Catesian coodinates x i, i 1,..., 3 and geneal cuvilinea coodinates u j, j 1,..., 3, x i x i (u j ), we intoduce
Διαβάστε περισσότεραJesse Maassen and Mark Lundstrom Purdue University November 25, 2013
Notes on Average Scattering imes and Hall Factors Jesse Maassen and Mar Lundstrom Purdue University November 5, 13 I. Introduction 1 II. Solution of the BE 1 III. Exercises: Woring out average scattering
Διαβάστε περισσότεραLaplace s Equation in Spherical Polar Coördinates
Laplace s Equation in Spheical Pola Coödinates C. W. David Dated: Januay 3, 001 We stat with the pimitive definitions I. x = sin θ cos φ y = sin θ sin φ z = cos θ thei inveses = x y z θ = cos 1 z = z cos1
Διαβάστε περισσότεραEE512: Error Control Coding
EE512: Error Control Coding Solution for Assignment on Finite Fields February 16, 2007 1. (a) Addition and Multiplication tables for GF (5) and GF (7) are shown in Tables 1 and 2. + 0 1 2 3 4 0 0 1 2 3
Διαβάστε περισσότεραMATH 38061/MATH48061/MATH68061: MULTIVARIATE STATISTICS Solutions to Problems on Matrix Algebra
MATH 38061/MATH48061/MATH68061: MULTIVARIATE STATISTICS Solutios to Poblems o Matix Algeba 1 Let A be a squae diagoal matix takig the fom a 11 0 0 0 a 22 0 A 0 0 a pp The ad So, log det A t log A t log
Διαβάστε περισσότεραOnline Appendix to the Paper No Claim? Your Gain: Design of Residual Value Extended Warranties under Risk Aversion and Strategic Claim Behavior
Online Appendix to the Pape No Claim? You Gain: Design of Residual Value Extended Waanties unde Risk Avesion and Stategic Claim Behavio Lemma Given any x ě y ě 0, pe x e y q{ is inceasing in Moeove, pe
Διαβάστε περισσότεραENGR 691/692 Section 66 (Fall 06): Machine Learning Assigned: August 30 Homework 1: Bayesian Decision Theory (solutions) Due: September 13
ENGR 69/69 Section 66 (Fall 06): Machine Learning Assigned: August 30 Homework : Bayesian Decision Theory (solutions) Due: Septemer 3 Prolem : ( pts) Let the conditional densities for a two-category one-dimensional
Διαβάστε περισσότεραVEKTORANALYS. CURVILINEAR COORDINATES (kroklinjiga koordinatsytem) Kursvecka 4. Kapitel 10 Sidor
VEKTORANALYS Kusvecka 4 CURVILINEAR COORDINATES (koklinjiga koodinatstem) Kapitel 10 Sido 99-11 TARGET PROBLEM An athlete is otating a hamme Calculate the foce on the ams. F ams F F ma dv a v dt d v dt
Διαβάστε περισσότεραΜηχανική Μάθηση Hypothesis Testing
ΕΛΛΗΝΙΚΗ ΔΗΜΟΚΡΑΤΙΑ ΠΑΝΕΠΙΣΤΗΜΙΟ ΚΡΗΤΗΣ Μηχανική Μάθηση Hypothesis Testing Γιώργος Μπορμπουδάκης Τμήμα Επιστήμης Υπολογιστών Procedure 1. Form the null (H 0 ) and alternative (H 1 ) hypothesis 2. Consider
Διαβάστε περισσότεραME 374, System Dynamics Analysis and Design Homework 9: Solution (June 9, 2008) by Jason Frye
ME 374, System Dynamics Analysis and Design Homewk 9: Solution June 9, 8 by Jason Frye Problem a he frequency response function G and the impulse response function ht are Fourier transfm pairs herefe,
Διαβάστε περισσότεραLecture 15 - Root System Axiomatics
Lecture 15 - Root System Axiomatics Nov 1, 01 In this lecture we examine root systems from an axiomatic point of view. 1 Reflections If v R n, then it determines a hyperplane, denoted P v, through the
Διαβάστε περισσότεραHW 3 Solutions 1. a) I use the auto.arima R function to search over models using AIC and decide on an ARMA(3,1)
HW 3 Solutions a) I use the autoarima R function to search over models using AIC and decide on an ARMA3,) b) I compare the ARMA3,) to ARMA,0) ARMA3,) does better in all three criteria c) The plot of the
Διαβάστε περισσότεραLecture 26: Circular domains
Introductory lecture notes on Partial Differential Equations - c Anthony Peirce. Not to be copied, used, or revised without eplicit written permission from the copyright owner. 1 Lecture 6: Circular domains
Διαβάστε περισσότερα6.003: Signals and Systems. Modulation
6.003: Signals and Systems Modulation May 6, 200 Communications Systems Signals are not always well matched to the media through which we wish to transmit them. signal audio video internet applications
Διαβάστε περισσότεραIf we restrict the domain of y = sin x to [ π, π ], the restrict function. y = sin x, π 2 x π 2
Chapter 3. Analytic Trigonometry 3.1 The inverse sine, cosine, and tangent functions 1. Review: Inverse function (1) f 1 (f(x)) = x for every x in the domain of f and f(f 1 (x)) = x for every x in the
Διαβάστε περισσότεραOutline Analog Communications. Lecture 05 Angle Modulation. Instantaneous Frequency and Frequency Deviation. Angle Modulation. Pierluigi SALVO ROSSI
Outline Analog Communications Lecture 05 Angle Modulation 1 PM and FM Pierluigi SALVO ROSSI Department of Industrial and Information Engineering Second University of Naples Via Roma 9, 81031 Aversa (CE),
Διαβάστε περισσότεραFinite difference method for 2-D heat equation
Finite difference method for 2-D heat equation Praveen. C praveen@math.tifrbng.res.in Tata Institute of Fundamental Research Center for Applicable Mathematics Bangalore 560065 http://math.tifrbng.res.in/~praveen
Διαβάστε περισσότεραEE101: Resonance in RLC circuits
EE11: Resonance in RLC circuits M. B. Patil mbatil@ee.iitb.ac.in www.ee.iitb.ac.in/~sequel Deartment of Electrical Engineering Indian Institute of Technology Bombay I V R V L V C I = I m = R + jωl + 1/jωC
Διαβάστε περισσότεραMathCity.org Merging man and maths
MathCity.org Merging man and maths Exercise 10. (s) Page Textbook of Algebra and Trigonometry for Class XI Available online @, Version:.0 Question # 1 Find the values of sin, and tan when: 1 π (i) (ii)
Διαβάστε περισσότερα6.3 Forecasting ARMA processes
122 CHAPTER 6. ARMA MODELS 6.3 Forecasting ARMA processes The purpose of forecasting is to predict future values of a TS based on the data collected to the present. In this section we will discuss a linear
Διαβάστε περισσότερα1 3D Helmholtz Equation
Deivation of the Geen s Funtions fo the Helmholtz and Wave Equations Alexande Miles Witten: Deembe 19th, 211 Last Edited: Deembe 19, 211 1 3D Helmholtz Equation A Geen s Funtion fo the 3D Helmholtz equation
Διαβάστε περισσότεραA Two-Sided Laplace Inversion Algorithm with Computable Error Bounds and Its Applications in Financial Engineering
Electronic Companion A Two-Sie Laplace Inversion Algorithm with Computable Error Bouns an Its Applications in Financial Engineering Ning Cai, S. G. Kou, Zongjian Liu HKUST an Columbia University Appenix
Διαβάστε περισσότεραΚΥΠΡΙΑΚΗ ΕΤΑΙΡΕΙΑ ΠΛΗΡΟΦΟΡΙΚΗΣ CYPRUS COMPUTER SOCIETY ΠΑΓΚΥΠΡΙΟΣ ΜΑΘΗΤΙΚΟΣ ΔΙΑΓΩΝΙΣΜΟΣ ΠΛΗΡΟΦΟΡΙΚΗΣ 19/5/2007
Οδηγίες: Να απαντηθούν όλες οι ερωτήσεις. Αν κάπου κάνετε κάποιες υποθέσεις να αναφερθούν στη σχετική ερώτηση. Όλα τα αρχεία που αναφέρονται στα προβλήματα βρίσκονται στον ίδιο φάκελο με το εκτελέσιμο
Διαβάστε περισσότερα