6.003: Signals and Systems. Modulation
|
|
|
- Ισίδωρα Μανιάκης
- 5 χρόνια πριν
- Προβολές:
Transcript
1 6.3: Signals and Sysems Modulaion December 6, 2
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 unil Friday, December 6, a noon. You will be able o view quaniaive resuls a hp://web.mi.edu/subjecevaluaion/resuls.hml and suden-wrien summaries a hp://hkn.mi.edu/ug sel.php
3 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, phonograph, CD, cell phone, MP3 elevision, cinema, HDTV, DVD coax, wised pair, cable TV, DSL, opical fiber, E/M Modulaion can improve mach based on frequency.
4 Ampliude Modulaion Ampliude modulaion can be used o mach audio frequencies o radio frequencies. I allows parallel ransmission of muliple channels. x () z () cos w x 2 () z 2 () z() LPF y() cos w 2 cos w c x 3 () z 3 () cos w 3
5 Superheerodyne Receiver Edwin Howard Armsrong invened he superheerodyne receiver, which made broadcas AM pracical. Edwin Howard Armsrong also invened and paened he regeneraive (posiive feedback) circui for amplifying radio signals (while he was a junior a Columbia Universiy). He also invened wide-band FM.
6 Ampliude, Phase, and Frequency Modulaion There are many ways o embed a message in a carrier. Ampliude Modulaion (AM) + carrier: y () = ( x() + C ) cos(ω c ) Phase Modulaion (PM): Frequency Modulaion (FM): y 2 () = cos(ω c + kx()) ( y 3 () = cos ω c + k ) x(τ)dτ PM: signal modulaes insananeous phase of he carrier. y 2 () = cos(ω c + kx()) FM: signal modulaes insananeous frequency of carrier. ( ) y 3 () = cos ω c + k x(τ)dτ } {{ } φ() ω i () = ω c + d d φ() = ω c + kx()
7 Frequency Modulaion Compare AM o FM for x() = cos( ). AM: y () = ( x() + C ) cos(ω c ) = (cos( ) +.) cos(ω c ) FM: y 3 () = cos ( ω c + k x(τ)dτ) = cos(ω c + k sin( )) Advanages of FM: consan power no need o ransmi carrier (unless DC imporan) bandwidh?
8 Frequency Modulaion Early invesigaors hough ha narrowband FM could have arbirarily narrow bandwidh, allowing more channels han AM. ( ) y 3 () = cos ω c + k x(τ)dτ } {{ } φ() ω i () = ω c + d d φ() = ω c + kx() Small k small bandwidh. Righ?
9 Frequency Modulaion Early invesigaors hough ha narrowband FM could have arbirarily narrow bandwidh, allowing more channels han AM. Wrong! ( y 3 () = cos ω c + k ) x(τ)dτ ( = cos(ω c ) cos k If k ( hen ) cos k x(τ)dτ ( ) sin k x(τ)dτ k x(τ)dτ ( y 3 () cos(ω c ) sin(ω c ) k ) ( ) x(τ)dτ sin(ω c ) sin k x(τ)dτ ) x(τ)dτ Bandwidh of narrowband FM is he same as ha of AM! (inegraion does no change he highes frequency in he signal)
10 Phase/Frequency Modulaion Find he Fourier ransform of a PM/FM signal. y() = cos(ω c + mx()) = cos(ω c + m sin( )) = cos(ω c ) cos(m sin( ))) sin(ω c ) sin(m sin( ))) x() is periodic in T = 2π, herefore cos(m sin( )) is periodic in T. sin( ) cos( sin( ))
11 Phase/Frequency Modulaion Find he Fourier ransform of a PM/FM signal. y() = cos(ω c + mx()) = cos(ω c + m sin( )) = cos(ω c ) cos(m sin( ))) sin(ω c ) sin(m sin( ))) x() is periodic in T = 2π, herefore cos(m sin( )) is periodic in T. 2 sin( ) 2 2 cos(2 sin( ))
12 Phase/Frequency Modulaion Find he Fourier ransform of a PM/FM signal. y() = cos(ω c + mx()) = cos(ω c + m sin( )) = cos(ω c ) cos(m sin( ))) sin(ω c ) sin(m sin( ))) x() is periodic in T = 2π, herefore cos(m sin( )) is periodic in T. 3 sin( ) 3 3 cos(3 sin( ))
13 Phase/Frequency Modulaion Find he Fourier ransform of a PM/FM signal. y() = cos(ω c + mx()) = cos(ω c + m sin( )) = cos(ω c ) cos(m sin( ))) sin(ω c ) sin(m sin( ))) x() is periodic in T = 2π, herefore cos(m sin( )) is periodic in T. 4 sin( ) 4 4 cos(4 sin( ))
14 Phase/Frequency Modulaion Find he Fourier ransform of a PM/FM signal. y() = cos(ω c + mx()) = cos(ω c + m sin( )) = cos(ω c ) cos(m sin( ))) sin(ω c ) sin(m sin( ))) x() is periodic in T = 2π, herefore cos(m sin( )) is periodic in T. 5 sin( ) 5 5 cos(5 sin( ))
15 Phase/Frequency Modulaion Find he Fourier ransform of a PM/FM signal. y() = cos(ω c + mx()) = cos(ω c + m sin( )) = cos(ω c ) cos(m sin( ))) sin(ω c ) sin(m sin( ))) x() is periodic in T = 2π, herefore cos(m sin( )) is periodic in T. 6 sin( ) 6 6 cos(6 sin( ))
16 Phase/Frequency Modulaion Find he Fourier ransform of a PM/FM signal. y() = cos(ω c + mx()) = cos(ω c + m sin( )) = cos(ω c ) cos(m sin( ))) sin(ω c ) sin(m sin( ))) x() is periodic in T = 2π, herefore cos(m sin( )) is periodic in T. 7 sin( ) 7 7 cos(7 sin( ))
17 Phase/Frequency Modulaion Find he Fourier ransform of a PM/FM signal. y() = cos(ω c + mx()) = cos(ω c + m sin( )) = cos(ω c ) cos(m sin( ))) sin(ω c ) sin(m sin( ))) x() is periodic in T = 2π, herefore cos(m sin( )) is periodic in T. 8 sin( ) 8 8 cos(8 sin( ))
18 Phase/Frequency Modulaion Find he Fourier ransform of a PM/FM signal. y() = cos(ω c + mx()) = cos(ω c + m sin( )) = cos(ω c ) cos(m sin( ))) sin(ω c ) sin(m sin( ))) x() is periodic in T = 2π, herefore cos(m sin( )) is periodic in T. 9 sin( ) 9 9 cos(9 sin( ))
19 Phase/Frequency Modulaion Find he Fourier ransform of a PM/FM signal. y() = cos(ω c + mx()) = cos(ω c + m sin( )) = cos(ω c ) cos(m sin( ))) sin(ω c ) sin(m sin( ))) x() is periodic in T = 2π, herefore cos(m sin( )) is periodic in T. sin( ) cos( sin( ))
20 Phase/Frequency Modulaion Find he Fourier ransform of a PM/FM signal. y() = cos(ω c + mx()) = cos(ω c + m sin( )) = cos(ω c ) cos(m sin( ))) sin(ω c ) sin(m sin( ))) x() is periodic in T = 2π, herefore cos(m sin( )) is periodic in T. 2 sin( ) 2 2 cos(2 sin( ))
21 Phase/Frequency Modulaion Find he Fourier ransform of a PM/FM signal. y() = cos(ω c + mx()) = cos(ω c + m sin( )) = cos(ω c ) cos(m sin( ))) sin(ω c ) sin(m sin( ))) x() is periodic in T = 2π, herefore cos(m sin( )) is periodic in T. 5 sin( ) 5 5 cos(5 sin( ))
22 Phase/Frequency Modulaion Find he Fourier ransform of a PM/FM signal. y() = cos(ω c + mx()) = cos(ω c + m sin( )) = cos(ω c ) cos(m sin( ))) sin(ω c ) sin(m sin( ))) x() is periodic in T = 2π, herefore cos(m sin( )) is periodic in T. m sin( ) m m cos(m sin( )) increasing m
23 Phase/Frequency Modulaion Find he Fourier ransform of a PM/FM signal. y() = cos(ω c + mx()) = cos(ω c + m sin( )) = cos(ω c ) cos(m sin( ))) sin(ω c ) sin(m sin( ))) x() is periodic in T = 2π, herefore cos(m sin( )) is periodic in T. cos(m sin( )) m = a k k
24 Phase/Frequency Modulaion Find he Fourier ransform of a PM/FM signal. y() = cos(ω c + mx()) = cos(ω c + m sin( )) = cos(ω c ) cos(m sin( ))) sin(ω c ) sin(m sin( ))) x() is periodic in T = 2π, herefore cos(m sin( )) is periodic in T. cos(m sin( )) m = a k k
25 Phase/Frequency Modulaion Find he Fourier ransform of a PM/FM signal. y() = cos(ω c + mx()) = cos(ω c + m sin( )) = cos(ω c ) cos(m sin( ))) sin(ω c ) sin(m sin( ))) x() is periodic in T = 2π, herefore cos(m sin( )) is periodic in T. cos(m sin( )) m = 2 a k k
26 Phase/Frequency Modulaion Find he Fourier ransform of a PM/FM signal. y() = cos(ω c + mx()) = cos(ω c + m sin( )) = cos(ω c ) cos(m sin( ))) sin(ω c ) sin(m sin( ))) x() is periodic in T = 2π, herefore cos(m sin( )) is periodic in T. cos(m sin( )) m = 5 a k k
27 Phase/Frequency Modulaion Find he Fourier ransform of a PM/FM signal. y() = cos(ω c + mx()) = cos(ω c + m sin( )) = cos(ω c ) cos(m sin( ))) sin(ω c ) sin(m sin( ))) x() is periodic in T = 2π, herefore cos(m sin( )) is periodic in T. cos(m sin( )) m = a k k
28 Phase/Frequency Modulaion Find he Fourier ransform of a PM/FM signal. y() = cos(ω c + mx()) = cos(ω c + m sin( )) = cos(ω c ) cos(m sin( ))) sin(ω c ) sin(m sin( ))) x() is periodic in T = 2π, herefore cos(m sin( )) is periodic in T. cos(m sin( )) m = 2 a k k
29 Phase/Frequency Modulaion Find he Fourier ransform of a PM/FM signal. y() = cos(ω c + mx()) = cos(ω c + m sin( )) = cos(ω c ) cos(m sin( ))) sin(ω c ) sin(m sin( ))) x() is periodic in T = 2π, herefore cos(m sin( )) is periodic in T. cos(m sin( )) m = 3 a k k
30 Phase/Frequency Modulaion Find he Fourier ransform of a PM/FM signal. y() = cos(ω c + mx()) = cos(ω c + m sin( )) = cos(ω c ) cos(m sin( ))) sin(ω c ) sin(m sin( ))) x() is periodic in T = 2π, herefore cos(m sin( )) is periodic in T. cos(m sin( )) m = 4 a k k
31 Phase/Frequency Modulaion Find he Fourier ransform of a PM/FM signal. y() = cos(ω c + mx()) = cos(ω c + m sin( )) = cos(ω c ) cos(m sin( ))) sin(ω c ) sin(m sin( ))) x() is periodic in T = 2π, herefore cos(m sin( )) is periodic in T. cos(m sin( )) m = 5 a k k
32 Phase/Frequency Modulaion Fourier ransform of firs par. x() = sin( ) y() = cos(ω c + mx()) = cos(ω c + m sin( )) = cos(ω c ) cos(m sin( ))) sin(ω }{{} c ) sin(m sin( ))) ya() Y a (jω) m = 5 ω c ω c ωm ω
33 Phase/Frequency Modulaion Find he Fourier ransform of a PM/FM signal. y() = cos(ω c + mx()) = cos(ω c + m sin( )) = cos(ω c ) cos(m sin( ))) sin(ω c ) sin(m sin( ))) x() is periodic in T = 2π, herefore sin(m sin( )) is periodic in T. m sin( ) m m sin(m sin( )) increasing m increasing m
34 Phase/Frequency Modulaion Find he Fourier ransform of a PM/FM signal. y() = cos(ω c + mx()) = cos(ω c + m sin( )) = cos(ω c ) cos(m sin( ))) sin(ω c ) sin(m sin( ))) x() is periodic in T = 2π, herefore sin(m sin( )) is periodic in T. sin(m sin( )) m = b k k
35 Phase/Frequency Modulaion Find he Fourier ransform of a PM/FM signal. y() = cos(ω c + mx()) = cos(ω c + m sin( )) = cos(ω c ) cos(m sin( ))) sin(ω c ) sin(m sin( ))) x() is periodic in T = 2π, herefore sin(m sin( )) is periodic in T. sin(m sin( )) m = b k k
36 Phase/Frequency Modulaion Find he Fourier ransform of a PM/FM signal. y() = cos(ω c + mx()) = cos(ω c + m sin( )) = cos(ω c ) cos(m sin( ))) sin(ω c ) sin(m sin( ))) x() is periodic in T = 2π, herefore sin(m sin( )) is periodic in T. sin(m sin( )) m = 2 b k k
37 Phase/Frequency Modulaion Find he Fourier ransform of a PM/FM signal. y() = cos(ω c + mx()) = cos(ω c + m sin( )) = cos(ω c ) cos(m sin( ))) sin(ω c ) sin(m sin( ))) x() is periodic in T = 2π, herefore sin(m sin( )) is periodic in T. sin(m sin( )) m = 5 b k k
38 Phase/Frequency Modulaion Find he Fourier ransform of a PM/FM signal. y() = cos(ω c + mx()) = cos(ω c + m sin( )) = cos(ω c ) cos(m sin( ))) sin(ω c ) sin(m sin( ))) x() is periodic in T = 2π, herefore sin(m sin( )) is periodic in T. sin(m sin( )) m = b k k
39 Phase/Frequency Modulaion Find he Fourier ransform of a PM/FM signal. y() = cos(ω c + mx()) = cos(ω c + m sin( )) = cos(ω c ) cos(m sin( ))) sin(ω c ) sin(m sin( ))) x() is periodic in T = 2π, herefore sin(m sin( )) is periodic in T. sin(m sin( )) m = 2 b k k
40 Phase/Frequency Modulaion Find he Fourier ransform of a PM/FM signal. y() = cos(ω c + mx()) = cos(ω c + m sin( )) = cos(ω c ) cos(m sin( ))) sin(ω c ) sin(m sin( ))) x() is periodic in T = 2π, herefore sin(m sin( )) is periodic in T. sin(m sin( )) m = 3 b k k
41 Phase/Frequency Modulaion Find he Fourier ransform of a PM/FM signal. y() = cos(ω c + mx()) = cos(ω c + m sin( )) = cos(ω c ) cos(m sin( ))) sin(ω c ) sin(m sin( ))) x() is periodic in T = 2π, herefore sin(m sin( )) is periodic in T. sin(m sin( )) m = 4 b k k
42 Phase/Frequency Modulaion Find he Fourier ransform of a PM/FM signal. y() = cos(ω c + mx()) = cos(ω c + m sin( )) = cos(ω c ) cos(m sin( ))) sin(ω c ) sin(m sin( ))) x() is periodic in T = 2π, herefore sin(m sin( )) is periodic in T. sin(m sin( )) m = 5 b k k
43 Phase/Frequency Modulaion Fourier ransform of second par. x() = sin( ) y() = cos(ω c + mx()) = cos(ω c + m sin( )) = cos(ω c ) cos(m sin( ))) sin(ω }{{} c ) sin(m sin( ))) }{{} ya() y b () Y b (jω) m = 5 ω c ω c ωm ω
44 Phase/Frequency Modulaion Fourier ransform. x() = sin( ) y() = cos(ω c + mx()) = cos(ω c + m sin( )) = cos(ω c ) cos(m sin( ))) sin(ω }{{} c ) sin(m sin( ))) }{{} ya() y b () Y (jω) m = 5 ω c ω c ωm ω
45 Frequency Modulaion Wideband FM is useful because i is robus o noise. AM: y () = (cos( ) +.) cos(ω c ) FM: y 3 () = cos(ω c + m sin( )) FM generaes a redundan signal ha is resilien o addiive noise.
46 Summary Modulaion is useful for maching signals o media. Examples: commercial radio (AM and FM) Close wih unconvenional applicaion of modulaion in microscopy.
47 6.3 Microscopy Dennis M. Freeman Sanley S. Hong Jekwan Ryu Michael S. Mermelsein Berhold K. P. Horn
48 6.3 Model of a Microscope microscope Microscope = low-pass filer
49 Phase-Modulaed Microscopy microscope
50
51
52
53
54
55
56
57
58
59
60
61
6.003: 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,
6.003: 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
Lecture 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
ΣΤΟΧΑΣΤΙΚΑ ΣΥΣΤΗΜΑΤΑ & ΕΠΙΚΟΙΝΩΝΙΕΣ 1o Τμήμα (Α - Κ): Αμφιθέατρο 4, Νέα Κτίρια ΣΗΜΜΥ Διαμόρφωση Γωνίας (Angle Modulation) - 2
ΣΤΟΧΑΣΤΙΚΑ ΣΥΣΤΗΜΑΤΑ & ΕΠΙΚΟΙΝΩΝΙΕΣ 1o Τμήμα (Α - Κ): Αμφιθέατρο 4, Νέα Κτίρια ΣΗΜΜΥ Διαμόρφωση Γωνίας (Angle Modulaion) - 4.3: Διαμόρφωση Συχνότητας (Frequency Modulaion FM) καθ. Βασίλης Μάγκλαρης maglaris@nemode.nua.gr
3 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
( ) ( ) ( ) 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,
ΣΤΟΧΑΣΤΙΚΑ ΣΥΣΤΗΜΑΤΑ & ΕΠΙΚΟΙΝΩΝΙΕΣ 1o Τμήμα (Α - Κ): Αμφιθέατρο 4, Νέα Κτίρια ΣΗΜΜΥ Διαμόρφωση Γωνίας (Angle Modulation) - 3
ΣΤΟΧΑΣΤΙΚΑ ΣΥΣΤΗΜΑΤΑ & ΕΠΙΚΟΙΝΩΝΙΕΣ 1o Τμήμα (Α - Κ): Αμφιθέατρο 4, Νέα Κτίρια ΣΗΜΜΥ Διαμόρφωση Γωνίας (Angle Modulaion) - 3 4.4: Βρόχος Κλειδωμένης Φάσης (Phase-Locked Loop - PLL) 4.5: Μη Γραμμικά Φαινόμενα
ΣΤΟΧΑΣΤΙΚΑ ΣΥΣΤΗΜΑΤΑ & ΕΠΙΚΟΙΝΩΝΙΕΣ 1o Τμήμα (Α - Κ): Αμφιθέατρο 4, Νέα Κτίρια ΣΗΜΜΥ Διαμόρφωση Γωνίας (Angle Modulation) - 1
ΣΤΟΧΑΣΤΙΚΑ ΣΥΣΤΗΜΑΤΑ & ΕΠΙΚΟΙΝΩΝΙΕΣ 1o Τμήμα (Α - Κ): Αμφιθέατρο 4, Νέα Κτίρια ΣΗΜΜΥ Διαμόρφωση Γωνίας (Angle Modulaion) - 1 0.0: Μετάδοση Αναλογικής & Ψηφιακής Πληροφορίας (Baseband, Bandpass) Σύντομη
Exercises 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
Example 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
Outline 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),
Durbin-Levinson recursive method
Durbin-Levinson recursive method A recursive method for computing ϕ n is useful because it avoids inverting large matrices; when new data are acquired, one can update predictions, instead of starting again
Sampling Basics (1B) Young Won Lim 9/21/13
Sampling Basics (1B) Copyright (c) 2009-2013 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any
Section 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.
Homework 8 Model Solution Section
MATH 004 Homework Solution Homework 8 Model Solution Section 14.5 14.6. 14.5. Use the Chain Rule to find dz where z cosx + 4y), x 5t 4, y 1 t. dz dx + dy y sinx + 4y)0t + 4) sinx + 4y) 1t ) 0t + 4t ) sinx
CHAPTER 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) =
Anti-aliasing Prefilter (6B) Young Won Lim 6/8/12
ni-aliasing Prefiler (6B) Copyrigh (c) Young W. Lim. Permission is graned o copy, disribue and/or modify his documen under he erms of he GNU Free Documenaion License, Version. or any laer version published
Math221: 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
Main 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
ω = 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
Στεγαστική δήλωση: Σχετικά με τις στεγαστικές υπηρεσίες που λαμβάνετε (Residential statement: About the residential services you get)
Νόμος περί Αναπηριών 2006 (Disability Act 2006) Στεγαστική δήλωση: Σχετικά με τις στεγαστικές υπηρεσίες που λαμβάνετε (Residential statement: About the residential services you get) Greek Νόμος περί Αναπηριών
( ) ( 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
4.6 Autoregressive Moving Average Model ARMA(1,1)
84 CHAPTER 4. STATIONARY TS MODELS 4.6 Autoregressive Moving Average Model ARMA(,) This section is an introduction to a wide class of models ARMA(p,q) which we will consider in more detail later in this
ΕΡΓΑΣΙΑ ΜΑΘΗΜΑΤΟΣ: ΘΕΩΡΙΑ ΒΕΛΤΙΣΤΟΥ ΕΛΕΓΧΟΥ ΦΙΛΤΡΟ KALMAN ΜΩΥΣΗΣ ΛΑΖΑΡΟΣ
ΑΡΙΣΤΟΤΕΛΕΙΟ ΠΑΝΕΠΙΣΤΗΜΙΟ ΘΕΣΣΑΛΟΝΙΚΗΣ ΤΜΗΜΑ ΜΑΘΗΜΑΤΙΚΩΝ ΜΕΤΑΠΤΥΧΙΑΚΟ ΠΡΟΓΡΑΜΜΑ ΣΠΟΥΔΩΝ ΘΕΩΡΗΤΙΚΗ ΠΛΗΡΟΦΟΡΙΚΗ ΚΑΙ ΘΕΩΡΙΑ ΣΥΣΤΗΜΑΤΩΝ & ΕΛΕΓΧΟΥ ΕΡΓΑΣΙΑ ΜΑΘΗΜΑΤΟΣ: ΘΕΩΡΙΑ ΒΕΛΤΙΣΤΟΥ ΕΛΕΓΧΟΥ ΦΙΛΤΡΟ KALMAN ΜΩΥΣΗΣ
HFC SERIES High Freq. Wound Ceramic Chip Inductors
FEATURES High frequency applications. Low DC resistance and high allowable DC current. Close tolerance application.2% tolerence is available for particular inductance values. Small footprint as well as
Phys460.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
Second 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
Fourier 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)
Ψηφιακή Επεξεργασία Φωνής
ΕΛΛΗΝΙΚΗ ΔΗΜΟΚΡΑΤΙΑ ΠΑΝΕΠΙΣΤΗΜΙΟ ΚΡΗΤΗΣ Ψηφιακή Επεξεργασία Φωνής Διάλεξη: Προσαρμόσιμο Αρμονικό Μοντέλο Παρουσίαση: Gilles Degottex Στυλιανού Ιωάννης Τμήμα Επιστήμης Υπολογιστών A Full-Band Adaptive Harmonic
What happens when two or more waves overlap in a certain region of space at the same time?
Wave Superposition What happens when two or more waves overlap in a certain region of space at the same time? To find the resulting wave according to the principle of superposition we should sum the fields
ΜΕΛΕΤΗ ΚΑΙ ΠΡΟΣΟΜΟΙΩΣΗ ΙΑΜΟΡΦΩΣΕΩΝ ΣΕ ΨΗΦΙΑΚΑ ΣΥΣΤΗΜΑΤΑ ΕΠΙΚΟΙΝΩΝΙΩΝ.
Τµήµα Ηλεκτρονικής ΜΕΛΕΤΗ ΚΑΙ ΠΡΟΣΟΜΟΙΩΣΗ ΙΑΜΟΡΦΩΣΕΩΝ ΣΕ ΨΗΦΙΑΚΑ ΣΥΣΤΗΜΑΤΑ ΕΠΙΚΟΙΝΩΝΙΩΝ. Σπουδαστής: Γαρεφαλάκης Ιωσήφ Α.Μ. 3501 Επιβλέπων καθηγητής : Ασκορδαλάκης Παντελής. -Χανιά 2010- ΠΕΡΙΛΗΨΗ : Η παρούσα
Fourier transform of continuous-time signals
Fourier ransform of coninuous-ime signals Specral represenaion of non-periodic signals Fourier ransform: aperiodic signals repeiion of a finie-duraion signal x()> periodic signals. x x T x kt x kt k k
CT Correlation (2B) Young Won Lim 8/15/14
CT Correlation (2B) 8/5/4 Copyright (c) 2-24 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version.2 or any
Fourier Series. Fourier Series
ECE 37 Z. Aliyazicioglu Elecrical & Compuer Egieerig Dep. Cal Poly Pomoa Periodic sigal is a fucio ha repeas iself every secods. x() x( ± ) : period of a fucio, : ieger,,3, x() 3 x() x() Periodic sigal
3.4 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
ECE 308 SIGNALS AND SYSTEMS FALL 2017 Answers to selected problems on prior years examinations
ECE 308 SIGNALS AND SYSTEMS FALL 07 Answers to selected problems on prior years examinations Answers to problems on Midterm Examination #, Spring 009. x(t) = r(t + ) r(t ) u(t ) r(t ) + r(t 3) + u(t +
Uniform 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
[1] P Q. Fig. 3.1
![[1] P Q. Fig. 3.1](/thumbs/79/80362156.jpg "[1] P Q. Fig. 3.1")
1 (a) Define resistance....... [1] (b) The smallest conductor within a computer processing chip can be represented as a rectangular block that is one atom high, four atoms wide and twenty atoms long. One
Διπλωματική Εργασία του φοιτητή του Τμήματος Ηλεκτρολόγων Μηχανικών και Τεχνολογίας Υπολογιστών της Πολυτεχνικής Σχολής του Πανεπιστημίου Πατρών
ΠΑΝΕΠΙΣΤΗΜΙΟ ΠΑΤΡΩΝ ΤΜΗΜΑ ΗΛΕΚΤΡΟΛΟΓΩΝ ΜΗΧΑΝΙΚΩΝ ΚΑΙ ΤΕΧΝΟΛΟΓΙΑΣ ΥΠΟΛΟΓΙΣΤΩΝ ΤΟΜΕΑΣ:ΗΛΕΚΤΡΟΝΙΚΗΣ ΚΑΙ ΥΠΟΛΟΓΙΣΤΩΝ ΕΡΓΑΣΤΗΡΙΟ ΗΛΕΚΤΡΟΝΙΚΩΝ ΕΦΑΡΜΟΓΩΝ Διπλωματική Εργασία του φοιτητή του Τμήματος Ηλεκτρολόγων
ECE Spring Prof. David R. Jackson ECE Dept. Notes 2
ECE 634 Spring 6 Prof. David R. Jackson ECE Dept. Notes Fields in a Source-Free Region Example: Radiation from an aperture y PEC E t x Aperture Assume the following choice of vector potentials: A F = =
University of Illinois at Urbana-Champaign ECE 310: Digital Signal Processing
University of Illinois at Urbana-Champaign ECE : Digital Signal Processing Chandra Radhakrishnan PROBLEM SET : SOLUTIONS Peter Kairouz Problem Solution:. ( 5 ) + (5 6 ) + ( ) cos(5 ) + 5cos( 6 ) + cos(
Concrete 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)
ME 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,
Q1a. HeavisideTheta x. Plot f, x, Pi, Pi. Simplify, n Integers
2 M2 Fourier Series answers in Mathematica Note the function HeavisideTheta is for x>0 and 0 for x
D 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
The Euler Equations! λ 1. λ 2. λ 3. ρ ρu. E = e + u 2 /2. E + p ρ. = de /dt. = dh / dt; h = h( T ); c p. / c v. ; γ = c p. p = ( γ 1)ρe. c v.
hp://www.nd.ed/~gryggva/cfd-corse/ The Eler Eqaions The Eler Eqaions The Eler eqaions for D flow: + + p = x E E + p where Define E = e + / H = h + /; h = e + p/ Gréar Tryggvason Spring 3 Ideal Gas: p =
Jesse 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
Can I open a bank account online? Ερώτηση αν μπορείτε να ανοίξετε τραπεζικό λογαριασμό μέσω του ίντερνετ
- Γενικά Can I withdraw money in [country] without paying fees? Πληροφόρηση σχετικά με το αν πρέπει να πληρώσετε ποσοστά προμήθειας όταν κάνετε ανάληψη σε μια συγκεκριμένη χώρα What are the fees if I use
Problem 7.19 Ignoring reflection at the air soil boundary, if the amplitude of a 3-GHz incident wave is 10 V/m at the surface of a wet soil medium, at what depth will it be down to 1 mv/m? Wet soil is
Econ 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
( )( ) ( ) ( )( ) ( )( ) 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
HMY 220: Σήματα και Συστήματα Ι
HMY : Σήματα και Συστήματα Ι ΔΙΑΛΕΞΗ # Μετασχηματισμοί Σημάτων Ενέργεια και Ισχύς Σήματος Βασικές κατηγορίες σημάτων Περιοδικά σήματα Άρτια και περιττά σήματα Εκθετικά σήματα Μετασχηματισμοί σημάτων (signal
Terabyte Technology Ltd
Terabyte Technology Ltd is a Web and Graphic design company in Limassol with dedicated staff who will endeavour to deliver the highest quality of work in our field. We offer a range of services such as
Συστήματα Επικοινωνιών Ι
+ Διδάσκων: Δρ. Κ. Δεμέστιχας e-mail: cdemestichas@uowm.gr Συστήματα Επικοινωνιών Ι Διαμορφώσεις γωνίας Διαμόρφωση Συχνότητας Στενής Ζώνης + Περιεχόμενα n Διαμορφώσεις γωνίας n Διαμόρφωση φάσης PM n Διαμόρφωση
Second Order Partial Differential Equations
Chapter 7 Second Order Partial Differential Equations 7.1 Introduction A second order linear PDE in two independent variables (x, y Ω can be written as A(x, y u x + B(x, y u xy + C(x, y u u u + D(x, y
Estimation for ARMA Processes with Stable Noise. Matt Calder & Richard A. Davis Colorado State University
Estimation for ARMA Processes with Stable Noise Matt Calder & Richard A. Davis Colorado State University rdavis@stat.colostate.edu 1 ARMA processes with stable noise Review of M-estimation Examples of
Forced Pendulum Numerical approach
Numerical approach UiO April 8, 2014 Physical problem and equation We have a pendulum of length l, with mass m. The pendulum is subject to gravitation as well as both a forcing and linear resistance force.
Mock Exam 7. 1 Hong Kong Educational Publishing Company. Section A 1. Reference: HKDSE Math M Q2 (a) (1 + kx) n 1M + 1A = (1) =
Mock Eam 7 Mock Eam 7 Section A. Reference: HKDSE Math M 0 Q (a) ( + k) n nn ( )( k) + nk ( ) + + nn ( ) k + nk + + + A nk... () nn ( ) k... () From (), k...() n Substituting () into (), nn ( ) n 76n 76n
ΣΤΟΧΑΣΤΙΚΑ ΣΥΣΤΗΜΑΤΑ & ΕΠΙΚΟΙΝΩΝΙΕΣ 1o Τμήμα (Α - Κ): Αμφιθέατρο 4, Νέα Κτίρια ΣΗΜΜΥ Διαμόρφωση Πλάτους - 1
ΣΤΟΧΑΣΤΙΚΑ ΣΥΣΤΗΜΑΤΑ & ΕΠΙΚΟΙΝΩΝΙΕΣ 1o Τμήμα (Α - Κ): Αμφιθέατρο 4, Νέα Κτίρια ΣΗΜΜΥ Διαμόρφωση Πλάτους - 1 3.2: Διαμόρφωση Πλάτους (Amplitude Modulation, AM) 3.3: Διαμόρφωση Πλευρικής Ζώνης με Καταπιεσμένο
Potential Dividers. 46 minutes. 46 marks. Page 1 of 11
Potential Dividers 46 minutes 46 marks Page 1 of 11 Q1. In the circuit shown in the figure below, the battery, of negligible internal resistance, has an emf of 30 V. The pd across the lamp is 6.0 V and
J. of Math. (PRC) u(t k ) = I k (u(t k )), k = 1, 2,, (1.6) , [3, 4] (1.1), (1.2), (1.3), [6 8]
![J. of Math. (PRC) u(t k ) = I k (u(t k )), k = 1, 2,, (1.6) , [3, 4] (1.1), (1.2), (1.3), [6 8]](/thumbs/91/107007576.jpg "J. of Math. (PRC) u(t k ) = I k (u(t k )), k = 1, 2,, (1.6) , [3, 4] (1.1), (1.2), (1.3), [6 8]")
Vol 36 ( 216 ) No 3 J of Mah (PR) 1, 2, 3 (1, 4335) (2, 4365) (3, 431) :,,,, : ; ; ; MR(21) : 35A1; 35A2 : O17529 : A : 255-7797(216)3-591-7 1 d d [x() g(, x )] = f(, x ),, (11) x = ϕ(), [ r, ], (12) x(
Κατανοώντας και στηρίζοντας τα παιδιά που πενθούν στο σχολικό πλαίσιο
Κατανοώντας και στηρίζοντας τα παιδιά που πενθούν στο σχολικό πλαίσιο Δρ. Παναγιώτης Πεντάρης - University of Greenwich - Association for the Study of Death and Society (ASDS) Περιεχόµενα Εννοιολογικές
Spectrum Representation (5A) Young Won Lim 11/3/16
Spectrum (5A) Copyright (c) 2009-2016 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later
ΕΠΙΧΕΙΡΗΣΙΑΚΗ ΑΛΛΗΛΟΓΡΑΦΙΑ ΚΑΙ ΕΠΙΚΟΙΝΩΝΙΑ ΣΤΗΝ ΑΓΓΛΙΚΗ ΓΛΩΣΣΑ
Ανοικτά Ακαδημαϊκά Μαθήματα στο ΤΕΙ Ιονίων Νήσων ΕΠΙΧΕΙΡΗΣΙΑΚΗ ΑΛΛΗΛΟΓΡΑΦΙΑ ΚΑΙ ΕΠΙΚΟΙΝΩΝΙΑ ΣΤΗΝ ΑΓΓΛΙΚΗ ΓΛΩΣΣΑ Ενότητα 2: A Business Letter Το περιεχόμενο του μαθήματος διατίθεται με άδεια Creative Commons
HOMEWORK 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
Newborn Upfront Payment & Newborn Supplement
GREEK Newborn Upfront Payment & Newborn Supplement Female 1: Το μωρό μου θα ρθει σύντομα, θα πρέπει να κανονίσω τα οικονομικά μου. Άκουσα ότι η κυβέρνηση δεν δίνει πλέον το Baby Bonus. Ξέρεις τίποτα γι
Calculating 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
Matrices 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
7 Present PERFECT Simple. 8 Present PERFECT Continuous. 9 Past PERFECT Simple. 10 Past PERFECT Continuous. 11 Future PERFECT Simple
A/ Ονόματα και ένα παράδειγμα 1 Present Simple 7 Present PERFECT Simple 2 Present Continuous 8 Present PERFECT Continuous 3 Past Simple (+ used to) 9 Past PERFECT Simple she eats she is eating she ate
Lecture 6. Goals: Determine the optimal threshold, filter, signals for a binary communications problem VI-1
Lecue 6 Goals: Deemine e opimal esold, file, signals fo a binay communicaions poblem VI- Minimum Aveage Eo Pobabiliy Poblem: Find e opimum file, esold and signals o minimize e aveage eo pobabiliy. s s
ΣΤΟΧΑΣΤΙΚΑ ΣΥΣΤΗΜΑΤΑ & ΕΠΙΚΟΙΝΩΝΙΕΣ 1o Τμήμα (Α - Κ): Αμφιθέατρο 4, Νέα Κτίρια ΣΗΜΜΥ Διαμόρφωση Πλάτους - 1
ΣΤΟΧΑΣΤΙΚΑ ΣΥΣΤΗΜΑΤΑ & ΕΠΙΚΟΙΝΩΝΙΕΣ 1o Τμήμα (Α - Κ): Αμφιθέατρο 4, Νέα Κτίρια ΣΗΜΜΥ Διαμόρφωση Πλάτους - 1 3.2: Διαμόρφωση Πλάτους (Amplitude Modulation, AM) 3.3: Διαμόρφωση Πλευρικής Ζώνης με Καταπιεσμένο
= 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
SUPERPOSITION, MEASUREMENT, NORMALIZATION, EXPECTATION VALUES. Reading: QM course packet Ch 5 up to 5.6
SUPERPOSITION, MEASUREMENT, NORMALIZATION, EXPECTATION VALUES Readig: QM course packet Ch 5 up to 5. 1 ϕ (x) = E = π m( a) =1,,3,4,5 for xa (x) = πx si L L * = πx L si L.5 ϕ' -.5 z 1 (x) = L si
Balanced Slope Demodulator EEC 112. v o2
Balanced Slope Demodulator EEC 11 The circuit below isabalanced FM slope demodulator. ω 01 i i (t) C 1 L 1 1 Ideal +v o (t) 0 C 0 v o1 v o + + C 0 Ideal 0 ω 0 L C i i (t) It is the same as the circuit
LESSON 12 (ΜΑΘΗΜΑ ΔΩΔΕΚΑ) REF : 202/055/32-ADV. 4 February 2014
LESSON 12 (ΜΑΘΗΜΑ ΔΩΔΕΚΑ) REF : 202/055/32-ADV 4 February 2014 Somewhere κάπου (kapoo) Nowhere πουθενά (poothena) Elsewhere αλλού (aloo) Drawer το συρτάρι (sirtari) Page η σελίδα (selida) News τα νέα (nea)
LESSON 14 (ΜΑΘΗΜΑ ΔΕΚΑΤΕΣΣΕΡΑ) REF : 202/057/34-ADV. 18 February 2014
LESSON 14 (ΜΑΘΗΜΑ ΔΕΚΑΤΕΣΣΕΡΑ) REF : 202/057/34-ADV 18 February 2014 Slowly/quietly Clear/clearly Clean Quickly/quick/fast Hurry (in a hurry) Driver Attention/caution/notice/care Dance Σιγά Καθαρά Καθαρός/η/ο
University of Washington Department of Chemistry Chemistry 553 Spring Quarter 2010 Homework Assignment 3 Due 04/26/10
Universiy of Washingon Deparmen of Chemisry Chemisry 553 Spring Quarer 1 Homework Assignmen 3 Due 4/6/1 v e v e A s ds: a) Show ha for large 1 and, (i.e. 1 >> and >>) he velociy auocorrelaion funcion 1)
Appendix. 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
the total number of electrons passing through the lamp.
1. A 12 V 36 W lamp is lit to normal brightness using a 12 V car battery of negligible internal resistance. The lamp is switched on for one hour (3600 s). For the time of 1 hour, calculate (i) the energy
6.1. Dirac Equation. Hamiltonian. Dirac Eq.
6.1. Dirac Equation Ref: M.Kaku, Quantum Field Theory, Oxford Univ Press (1993) η μν = η μν = diag(1, -1, -1, -1) p 0 = p 0 p = p i = -p i p μ p μ = p 0 p 0 + p i p i = E c 2 - p 2 = (m c) 2 H = c p 2
ΑΝΙΧΝΕΥΣΗ ΓΕΓΟΝΟΤΩΝ ΒΗΜΑΤΙΣΜΟΥ ΜΕ ΧΡΗΣΗ ΕΠΙΤΑΧΥΝΣΙΟΜΕΤΡΩΝ ΔΙΠΛΩΜΑΤΙΚΗ ΕΡΓΑΣΙΑ
ΕΘΝΙΚΟ ΜΕΤΣΟΒΙΟ ΠΟΛΥΤΕΧΝΕΙΟ ΣΧΟΛΗ ΗΛΕΚΤΡΟΛΟΓΩΝ ΜΗΧΑΝΙΚΩΝ ΚΑΙ ΜΗΧΑΝΙΚΩΝ ΥΠΟΛΟΓΙΣΤΩΝ ΤΟΜΕΑΣ ΕΠΙΚΟΙΝΩΝΙΩΝ ΗΛΕΚΤΡΟΝΙΚΗΣ ΚΑΙ ΣΥΣΤΗΜΑΤΩΝ ΠΛΗΡΟΦΟΡΙΚΗΣ ΑΝΙΧΝΕΥΣΗ ΓΕΓΟΝΟΤΩΝ ΒΗΜΑΤΙΣΜΟΥ ΜΕ ΧΡΗΣΗ ΕΠΙΤΑΧΥΝΣΙΟΜΕΤΡΩΝ
1 String with massive end-points
1 String with massive end-points Πρόβλημα 5.11:Θεωρείστε μια χορδή μήκους, τάσης T, με δύο σημειακά σωματίδια στα άκρα της, το ένα μάζας m, και το άλλο μάζας m. α) Μελετώντας την κίνηση των άκρων βρείτε
ANSWERSHEET (TOPIC = DIFFERENTIAL CALCULUS) COLLECTION #2. h 0 h h 0 h h 0 ( ) g k = g 0 + g 1 + g g 2009 =?
Teko Classes IITJEE/AIEEE Maths by SUHAAG SIR, Bhopal, Ph (0755) 3 00 000 www.tekoclasses.com ANSWERSHEET (TOPIC DIFFERENTIAL CALCULUS) COLLECTION # Question Type A.Single Correct Type Q. (A) Sol least
Τηλεπικοινωνιακά Συστήματα Ι
Τηλεπικοινωνιακά Συστήματα Ι Διάλεξη 7: Διαμόρφωση Γωνίας (1/2) Δρ. Μιχάλης Παρασκευάς Επίκουρος Καθηγητής 1 Ατζέντα Διαμόρφωση γωνίας Ορισμοί Η έννοια της Στιγμιαίας Συχνότητας Διαμόρφωση Φάσης (Phase
derivation 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
D-Wave D-Wave Systems Inc.
D-Wave D-Wave sems Inc. Anaol Yu. mirnov D-Wave sems Inc. Vancouver Briish Columbia HE QUANUM COMPUING COMPANY M Decoherence and Noise Conrol in rongl Driven uperconducing Quanum Bis Collaboraion: M. Grajcar
Exercises to Statistics of Material Fatigue No. 5
Prof. Dr. Christine Müller Dipl.-Math. Christoph Kustosz Eercises to Statistics of Material Fatigue No. 5 E. 9 (5 a Show, that a Fisher information matri for a two dimensional parameter θ (θ,θ 2 R 2, can
Πώς μπορεί κανείς να έχει έναν διερμηνέα κατά την επίσκεψή του στον Οικογενειακό του Γιατρό στο Ίσλινγκτον Getting an interpreter when you visit your
Πώς μπορεί κανείς να έχει έναν διερμηνέα κατά την επίσκεψή του στον Οικογενειακό του Γιατρό στο Ίσλινγκτον Getting an interpreter when you visit your GP practice in Islington Σε όλα τα Ιατρεία Οικογενειακού
DESIGN OF MACHINERY SOLUTION MANUAL h in h 4 0.
DESIGN OF MACHINERY SOLUTION MANUAL -7-1! PROBLEM -7 Statement: Design a double-dwell cam to move a follower from to 25 6, dwell for 12, fall 25 and dwell for the remader The total cycle must take 4 sec
16. 17. r t te 2t i t 1. 18 19 Find the derivative of the vector function. 19. r t e t cos t i e t sin t j ln t k. 31 33 Evaluate the integral.
SECTION.7 VECTOR FUNCTIONS AND SPACE CURVES.7 VECTOR FUNCTIONS AND SPACE CURVES A Click here for answers. S Click here for soluions. Copyrigh Cengage Learning. All righs reserved.. Find he domain of he
How to register an account with the Hellenic Community of Sheffield.
How to register an account with the Hellenic Community of Sheffield. (1) EN: Go to address GR: Πηγαίνετε στη διεύθυνση: http://www.helleniccommunityofsheffield.com (2) EN: At the bottom of the page, click
Numerical 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) =
Galatia SIL Keyboard Information
Galatia SIL Keyboard Information Keyboard ssignments The main purpose of the keyboards is to provide a wide range of keying options, so many characters can be entered in multiple ways. If you are typing
) 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
Lifting 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
BandPass (4A) Young Won Lim 1/11/14
BandPass (4A) Copyright (c) 22 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version.2 or any later version
Block Ciphers Modes. Ramki Thurimella
Block Ciphers Modes Ramki Thurimella Only Encryption I.e. messages could be modified Should not assume that nonsensical messages do no harm Always must be combined with authentication 2 Padding Must be
MathCity.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)
Final Test Grammar. Term C'
Final Test Grammar Term C' Book: Starting Steps 1 & Extra and Friends Vocabulary and Grammar Practice Class: Junior AB Name: /43 Date: E xercise 1 L ook at the example and do the same. ( Κξίηα ηξ παοάδειγμα