Frequency Modulation. Dr. Hwee-Pink Tan

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "Frequency Modulation. Dr. Hwee-Pink Tan http://www.cs.tcd.ie/hweepink.tan"

Transcription

1 Frequency Modulaion Dr. Hwee-Pink Tan hp:// Lecure maerial was absraced from "Communicaion Sysems" by Simon Haykin.

2 Ouline Day 1 Day 2 Day 3 Angle Modulaion Frequency Modulaion (FM) Narrowband and Wideband FM Transmission bandwidh FM Sereo Mix Phase-locked Loop (PLL) Non-linear effecs in FM receivers Summary Tuorial

3 Recall Wha is modulaion? Message m() Highes freq W ransmier Sinusoidal carrier c()= A c cos [2f c ] modulaion Direc ransmission Unsuiable/ inefficien Transmission channel Modulaed signal s()=a()cos () receiver m() demodulaion s() obained by varying characerisic of c() according o m() Ampliude A()<-> Ampliude Modulaion Angle () <-> Angle Modulaion

4 Recall Ampliude Modulaion s() = A c [1+k a m()]cos 2f c Envelope of s() has same shape as m() provided: k a m() <1 f c >>W Easy and cheap o generae s() and reverse

5 Recall Ampliude Modulaion Drawbacks of AM waseful of power ransmission of carrier waseful of bandwidh ransmission bandwidh, B T = 2W Improved resource uilizaion (power or bandwidh) raded-off wih increased sysem complexiy Angle modulaion offers pracical means of rading-off beween power and bandwidh

6 Angle modulaion s() = A c cos[ i ()], f i () = 1 d i () 2 d Phase Modulaion (PM) i () = 2 f c + k p m() s() = A c cos[2 f c + k p m()] Frequency Modulaion (FM) f i () = f c + k f m() s() = A c cos[2 f c + 2k f m()d] 0

7 PM vs FM Frequency modulaor Inegraor Phase modulaor FM wave Modulaing wave, m() A c cos(2f c ) s() = A c cos[2 f c + 2k f m()d] 0 Differeniaor Phase modulaor Frequency modulaor PM wave s() = A c cos[2 f c + k p m()]

8 Frequency modulaion Non-linear funcion of m() FM signal: s() = A c cos[2 f c + 2k f m()d] 0 Consider single one signal: m()=a m cos2f m f i () = f c + k f m() = f c + k f A m cos[2 f m ] = f c + f cos[2 f m ] i () = 2 0 f i ()d = 2 f c + f sin[2 f m ] f m = 2 f c + sin[2 f m ] Frequency deviaion Modulaion index s() = A c cos[2 f c + sin(2 f m )]

9 Narrowband FM (<<1) )] sin(2 )sin[ sin(2 )] sin(2 )cos[ cos(2 ) ( we have :, Expanding f f A f f A s s() m c c m c c = 1 sin(2f m ) ]} ) ( cos[2 ] ) ( {cos[2 2 1 ) cos(2 ) )sin(2 sin(2 ) cos(2 ) s( 1, If f f f f A f A f f A f A m c m c c c c m c c c c + + << carrier Upper side-frequency Lower side-frequency Resulan

10 Comparison wih AM AM Signal carrier Upper side-frequency Lower side-frequency Resulan Narrow band FM Signal Resulan carrier Lower side-frequency Upper side-frequency

11 Wideband FM s() = A c cos[2 f c + sin(2 f m )] f c >> f m = Re[A c exp( j2 f c + j sin(2 f m ))] = Re[ s()exp( j2 f c )] (*) s() = A c exp[ j sin(2 f m )] = A c J n ()exp( j2n f m ) n= nh order Bessel funcion of firs kind Subs. ino (*), and applying FT: S(f) = A c 2 n= J n ()[( f f c nf m ) + ( f + f c + nf m )] = A c J 0 ()( f f c ) + A c 2 J ±1()( f f c f m ) + A c 2 J ±2()( f f c 2 f m ) + Carrier componen Side freq f c ±f m Side freq f c ±2f m

12 Observaions J 0 () = 1 Ampliude of carrier varies wih J 0 () Wih AM, ampliude of carrier = A c J 1 () = 2 J n2 () 0 n= J n 2 () = 1 Special case: << 1 Only J 0 (), J 1 () f c ± f m significan (narrowband FM) nh order Bessel funcion of firs kind S(f) = A c 2 n= J n ()[( f f c nf m ) + ( f + f c + nf m )] = A c J 0 ()( f f c ) + A c 2 J ±1 ()( f f c f m ) + A c 2 J ±2 ()( f f c 2 f m ) + Carrier componen Side freq f c ±f m Side freq f c ±2f m

13 Example - fixed f m, variable A m m() = A m cos[2 f m ], f = k f A m, = f f m S(f) = A c J 0 ()( f f c ) + A c 2 J ±1()( f f c f m ) + A c 2 J ±2()( f f c 2 f m ) + Carrier componen Side freq f c ±f m Side freq f c ±2f m f m f m A m

14 Example - fixed A m, variable f m m() = A m cos[2 f m ], f = k f A m, = f f m S(f) = A c J 0 ()( f f c ) + A c 2 J ±1()( f f c f m ) + A c 2 J ±2()( f f c 2 f m ) + f m f m f m

15 Example - fixed A m, variable f m m() = A m cos[2 f m ], f = k f A m, = f f m S(f) = A c J 0 ()( f f c ) + A c 2 J ±1()( f f c f m ) + A c 2 J ±2()( f f c 2 f m ) + f m f m As, number of specral lines wihin f c f < f < f c + f As, he bandwidh of s() approaches he limiing value of 2f!! [Noe: For 1,bandwidh of s() 2 f m (As in AM)]

16 Transmission bandwidh S(f) = A c 2 n= J n ()[( f f c nf m ) + ( f + f c + nf m )] ransmission bandwidh =!!! Bu, effecively, finie number of side frequencies are significan Large : Falls rapidly owards 0 for f-f c >2f Small : Significan sidebands wihin f c ± f m Carson's rule: B T,Carson 2f (1 + 1 )

17 Transmission bandwidh S(f) = A c 2 n= J n ()[( f f c nf m ) + ( f + f c + nf m )] ransmission bandwidh =!!! Bu, effecively, finie number of side frequencies are significan reain up o n max side frequencies s.. J nmax () J 0 () B T = 2n max f m B T,1% = 1 % bandwidh wih = 0.01

18 1 percen bandwidh of FM wave As, n max B T,1%

19 1 percen bandwidh of FM wave Small Large Small values of more exravagan in B T han larger!! Pracically, B T,Carson B T B T,1%

20 FM Sereo Transmi wo separae signals via same carrier 2 differen secions of orchesra, e.g., vocalis and accompanis, o give spaial dimension o is percepion Requiremens Mus operae wihin allocaed FM broadcas channels Mus be compaible wih monophonic radio receivers

21 FM Sereo Mux m l +m r : Monophonic recepion m l -m r 2f c =38kHz f c =19kHz m() = [m l () + m r ()] +[m l () m r ()]cos(2[2 f c ]) +K cos(2 f c )

22 FM Sereo Demux m l () + m r () [m l () m r ()]cos(2[2 f c ]) K cos(2 f c ) 2f c =38kHz m() = [m l () + m r ()] +[m l () m r ()]cos(2[2 f c ]) +K cos(2 f c )

23 Phase Locked Loop (PLL) s() = A c sin[2 f c + 1 ()] X Loop filer v() r() = A v cos[2 f c + 2 ()] 2 () = 2k v 0 v()d Volage Conrolled Oscillaor PLL for freq. demod If s() is FM wave, obain m() from v() Require o <=> Phase lock!

24 Phase Locked Loop (PLL) HF :4f c erm e() = { s() = A c sin[2 f c + 1 ()] LF : K sin[ 1 () 2 ()] e ( ) X H(f) Loop filer v() = e( )h( )d r() = A v cos[2 f c + 2 ()] 2 () = 2k v 0 v()d Volage Conrolled Oscillaor Dynamic behavior of PLL d e () = d () 1 2K o d d K o = k m k v A c A v sin[ e ( )]h( )d,

25 Non-linear PLL model Sinusoidal non-lineariy makes i difficul o analyze PLL Dynamic behavior of PLL d e () = d () 1 2K o d d K o = k m k v A c A v sin[ e ( )]h( )d,

26 Non-linear PLL model Assume e () << 1 sin e () e () Linearized behavior of PLL d e () d = d 1 () d 2K o e ( )h( )d, FT can be applied! K o = k m k v A c A v

27 Linear PLL model 1 e ( f ) = 1 + L( f ) ( f ), 1 H ( f ) L( f ) = K o [Open-loop ransfer funcion] jf V( f ) = K o k v H ( f ) e ( f ) = jf k v L( f ) e ( f ) V( f ) = ( jf / k v )L( f ) 1 + L( f ) 1 ( f )

28 Phase-locked Linear PLL V( f ) = ( jf / k )L( f ) v 1 ( f ) 1 + L( f ) e ( f ) = 1 1+ L( f ) 1 ( f ) L( f ) 1 V( f ) jf k v 1 ( f ) v() = 1 2k v d 1 () d e ( f ) 0 Phase lock!!!

29 Phase locked linear PLL as frequency demodulaor If s() is FM signal, 1 () = 2k f 0 m()d v() = k f k v m() If L(f) >>1 Linearized PLL model Phase lock saisfied [ e 0] v() = 1 d 1 () 2k v d Deermines complexiy of PLL Bandwidh of s() >> bandwidh of H(f) [m()]

30 Design of H(f) Firs order H(f)=1 L(f) =K o /f Drawback K o conrols boh loop bandwidh and L(f) >>1 hold-in frequency range, f H f H (K o =1) f H (K o =3)

31 Second-order PLL H ( f ) = 1 + a jf ( jf / f e ( f ) = n ) ( jf / f n ) + ( jf / f n ) 2 ( f ), 1 f n = ak o Naural frequency = K o 4a Damping facor For m() = A m cos2 f m, 1 () = sin(2 f m ) e () = e0 cos(2 f m + )

32 Second-order PLL Wih appropriae choice of (,f n ), we can mainain e small Linear PLL model Rule of humb: Loop should remain locked if e0 (f m =f n )<90 o

33 Non-linear effecs in FM sysems v i () v i () = A c cos[2 f c + ()], () = 2k f 0 m()d memoryless communicaions channel v o () = a 1 v i () + a 2 v i 2 () + a 3 v i 3 () v o () = A 0 + A 1 cos[2 f c + ()] + A 2 cos[4 f c + 2()] + A 3 cos[6 f c + 3()] Assume v i () is FM signal f = frequency deviaion W = highes freq. comp. of m()

34 Non-linear effecs in FM sysems v i () = A c cos[2 f c + ()], () = 2k f 0 m()d Carson s Rule 2f memoryless communicaions channel v o () = A 0 + A 1 cos[2 f c + ()] + A 2 cos[4 f c + 2()] + A 3 cos[6 f c + 3()] f c -W f c f c +W 2f c -W 2f c 2f c +W 2f 4f

35 Non-linear effecs in FM sysems To separae ou desired FM signal f c +f+w < 2f c -W-2f =>f c > 3f + 2W Apply bandpass filer [f c -f-w, f c +f+w] v o '() = (a a 3 A c2 )v i () Unlike AM, FM no affeced by disorion due o channel wih amp. non-lineariies f c -W f c f c +W 2f c -W 2f c 2f c +W 2f 4f

36 Summary Unlike AM, FM is non-linear modulaion process Specral analysis is more difficul Developed insigh by sudying single-one FM Carson s rule for ransmission bandwidh B T = 2f(1+1/) Phase-Locked Loop for frequency demodulaion

Lecture 3: FM Transmitters & Receivers

Lecture 3: FM Transmitters & Receivers Leure 3: FM Transmiers & Reeivers Review of Previous Leures Frequeny Modulaion FM Transmier FM Demodulaion Tehniques FM Reeiver Conlusions & Disussions Leure Noes are available from: hp://homepages.unl.a.uk/~aghizas

More information

Modulation for Analog Communication. Yao Wang Polytechnic University, Brooklyn, NY11201 http://eeweb.poly.edu/~yao

Modulation for Analog Communication. Yao Wang Polytechnic University, Brooklyn, NY11201 http://eeweb.poly.edu/~yao Modulaion or Analog Communiaion Yao Wang Polyehni Universiy, Brooklyn, NY11201 hp://eeweb.poly.edu/~yao Ouline Baseband ommuniaion: bandwidh requiremen Modulaion o oninuous signals Ampliude modulaion Quadraure

More information

Signal Rectification

Signal Rectification 9/3/25 Signal Recificaion.doc / Signal Recificaion n imporan applicaion of juncion diodes is signal recificaion. here are wo ypes of signal recifiers, half-wae and fullwae. Le s firs consider he ideal

More information

RC, RL and RLC circuits

RC, 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 information

Analogue and Digital Signal Processing. First Term Third Year CS Engineering By Dr Mukhtiar Ali Unar

Analogue and Digital Signal Processing. First Term Third Year CS Engineering By Dr Mukhtiar Ali Unar Analogue and Digial Signal Processing Firs Term Third Year CS Engineering By Dr Mukhiar Ali Unar Recommended Books Haykin S. and Van Veen B.; Signals and Sysems, John Wiley& Sons Inc. ISBN: 0-7-380-7 Ifeachor

More information

Pulse-Width Modulation Inverters

Pulse-Width Modulation Inverters SECTION 3.6 INVERTERS 189 Pulse-Widh Modulaion Inverers Pulse-widh modulaion is he process of modifying he widh of he pulses in a pulse rain in direc proporion o a small conrol signal; he greaer he conrol

More information

µ r of the ferrite amounts to 1000...4000. It should be noted that the magnetic length of the + δ

µ r of the ferrite amounts to 1000...4000. It should be noted that the magnetic length of the + δ Page 9 Design of Inducors and High Frequency Transformers Inducors sore energy, ransformers ransfer energy. This is he prime difference. The magneic cores are significanly differen for inducors and high

More information

CHARGE AND DISCHARGE OF A CAPACITOR

CHARGE AND DISCHARGE OF A CAPACITOR REFERENCES RC Circuis: Elecrical Insrumens: Mos Inroducory Physics exs (e.g. A. Halliday and Resnick, Physics ; M. Sernheim and J. Kane, General Physics.) This Laboraory Manual: Commonly Used Insrumens:

More information

Capacitors and inductors

Capacitors and inductors Capaciors and inducors We coninue wih our analysis of linear circuis by inroducing wo new passive and linear elemens: he capacior and he inducor. All he mehods developed so far for he analysis of linear

More information

Lecture 080 All Digital PPLs (5/15/03) Page 080-1

Lecture 080 All Digital PPLs (5/15/03) Page 080-1 Lecure 8 All Digial PPLs (5/15/3) Page 8-1 LECTURE 8 ALL DIGITAL PHASE LOCK LOOPS (ADPLL) (Reference [2]) Ouline Building Blocks of he ADPLL Examples of ADPLL Implemenaion ADPLL Design ADPLL Sysem Simulaion

More information

Form measurement systems from Hommel-Etamic Geometrical tolerancing in practice DKD-K-02401. Precision is our business.

Form measurement systems from Hommel-Etamic Geometrical tolerancing in practice DKD-K-02401. Precision is our business. Form measuremen sysems from Hommel-Eamic Geomerical olerancing in pracice DKD-K-02401 Precision is our business. Drawing enries Tolerance frame 0.01 0.01 Daum leer Tolerance value in mm Symbol for he oleranced

More information

PHYS245 Lab: RC circuits

PHYS245 Lab: RC circuits PHYS245 Lab: C circuis Purpose: Undersand he charging and discharging ransien processes of a capacior Display he charging and discharging process using an oscilloscope Undersand he physical meaning of

More information

Voltage level shifting

Voltage level shifting rek Applicaion Noe Number 1 r. Maciej A. Noras Absrac A brief descripion of volage shifing circuis. 1 Inroducion In applicaions requiring a unipolar A volage signal, he signal may be delivered from a bi-polar

More information

LLC Resonant Converter Reference Design using the dspic DSC

LLC Resonant Converter Reference Design using the dspic DSC LLC Resonan Converer Reference Design using he dspic DSC 2010 Microchip Technology Incorporaed. All Righs Reserved. LLC Resonan Converer Webinar Slide 1 Hello, and welcome o his web seminar on Microchip

More information

RFIC Design and Testing for Wireless Communications

RFIC Design and Testing for Wireless Communications RFIC Design and Tesing for Wireless Communicaions PragaTI (TI India Technical Universiy Course July 8,,, 008 Lecure 6: Basic Conceps Lineariy, noise figure, dynamic range By Vishwani D. grawal Fa Foser

More information

Chapter 7. Response of First-Order RL and RC Circuits

Chapter 7. Response of First-Order RL and RC Circuits Chaper 7. esponse of Firs-Order L and C Circuis 7.1. The Naural esponse of an L Circui 7.2. The Naural esponse of an C Circui 7.3. The ep esponse of L and C Circuis 7.4. A General oluion for ep and Naural

More information

The Transport Equation

The Transport Equation The Transpor Equaion Consider a fluid, flowing wih velociy, V, in a hin sraigh ube whose cross secion will be denoed by A. Suppose he fluid conains a conaminan whose concenraion a posiion a ime will be

More information

TSG-RAN Working Group 1 (Radio Layer 1) meeting #3 Nynashamn, Sweden 22 nd 26 th March 1999

TSG-RAN Working Group 1 (Radio Layer 1) meeting #3 Nynashamn, Sweden 22 nd 26 th March 1999 TSG-RAN Working Group 1 (Radio Layer 1) meeing #3 Nynashamn, Sweden 22 nd 26 h March 1999 RAN TSGW1#3(99)196 Agenda Iem: 9.1 Source: Tile: Documen for: Moorola Macro-diversiy for he PRACH Discussion/Decision

More information

Acceleration Lab Teacher s Guide

Acceleration Lab Teacher s Guide Acceleraion Lab Teacher s Guide Objecives:. Use graphs of disance vs. ime and velociy vs. ime o find acceleraion of a oy car.. Observe he relaionship beween he angle of an inclined plane and he acceleraion

More information

5.8 Resonance 231. The study of vibrating mechanical systems ends here with the theory of pure and practical resonance.

5.8 Resonance 231. The study of vibrating mechanical systems ends here with the theory of pure and practical resonance. 5.8 Resonance 231 5.8 Resonance The sudy of vibraing mechanical sysems ends here wih he heory of pure and pracical resonance. Pure Resonance The noion of pure resonance in he differenial equaion (1) ()

More information

Pi Principles i of Communications

Pi Principles i of Communications Pi Priniples i o Communiaions i Chaper 3: Analog Modulaion (oninued Texbook: Ch 3, Ch 4.-4.4, Ch 6.-6. 3E 3. Ee o Noise on AMS Sysems Baseband sysem (a basis or omparison o various modulaion sysems: No

More information

Module 3 Design for Strength. Version 2 ME, IIT Kharagpur

Module 3 Design for Strength. Version 2 ME, IIT Kharagpur Module 3 Design for Srengh Lesson 2 Sress Concenraion Insrucional Objecives A he end of his lesson, he sudens should be able o undersand Sress concenraion and he facors responsible. Deerminaion of sress

More information

Lecture 9 Analog and Digital I/Q Modulation

Lecture 9 Analog and Digital I/Q Modulation Lecure 9 Analog and Digial I/Q Modulaion Analog I/Q Modulaion Time Domain View Polar View Frequency Domain View Digial I/Q Modulaion Phase Shi Keying Consellaions /4/26 Coheren Deecion Transmier Oupu x()

More information

Inductance and Transient Circuits

Inductance and Transient Circuits Chaper H Inducance and Transien Circuis Blinn College - Physics 2426 - Terry Honan As a consequence of Faraday's law a changing curren hrough one coil induces an EMF in anoher coil; his is known as muual

More information

9. Capacitor and Resistor Circuits

9. Capacitor and Resistor Circuits ElecronicsLab9.nb 1 9. Capacior and Resisor Circuis Inroducion hus far we have consider resisors in various combinaions wih a power supply or baery which provide a consan volage source or direc curren

More information

Photo Modules for PCM Remote Control Systems

Photo Modules for PCM Remote Control Systems Phoo Modules for PCM Remoe Conrol Sysems Available ypes for differen carrier frequencies Type fo Type fo TSOP173 3 khz TSOP1733 33 khz TSOP1736 36 khz TSOP1737 36.7 khz TSOP1738 38 khz TSOP174 4 khz TSOP1756

More information

Entropy: From the Boltzmann equation to the Maxwell Boltzmann distribution

Entropy: From the Boltzmann equation to the Maxwell Boltzmann distribution Enropy: From he Bolzmann equaion o he Maxwell Bolzmann disribuion A formula o relae enropy o probabiliy Ofen i is a lo more useful o hink abou enropy in erms of he probabiliy wih which differen saes are

More information

Part II Converter Dynamics and Control

Part II Converter Dynamics and Control Par II onverer Dynamics and onrol 7. A equivalen circui modeling 8. onverer ransfer funcions 9. onroller design 1. Inpu filer design 11. A and D equivalen circui modeling of he disconinuous conducion mode

More information

17 Laplace transform. Solving linear ODE with piecewise continuous right hand sides

17 Laplace transform. Solving linear ODE with piecewise continuous right hand sides 7 Laplace ransform. Solving linear ODE wih piecewise coninuous righ hand sides In his lecure I will show how o apply he Laplace ransform o he ODE Ly = f wih piecewise coninuous f. Definiion. A funcion

More information

USE OF EDUCATION TECHNOLOGY IN ENGLISH CLASSES

USE OF EDUCATION TECHNOLOGY IN ENGLISH CLASSES USE OF EDUCATION TECHNOLOGY IN ENGLISH CLASSES Mehme Nuri GÖMLEKSİZ Absrac Using educaion echnology in classes helps eachers realize a beer and more effecive learning. In his sudy 150 English eachers were

More information

1. y 5y + 6y = 2e t Solution: Characteristic equation is r 2 5r +6 = 0, therefore r 1 = 2, r 2 = 3, and y 1 (t) = e 2t,

1. y 5y + 6y = 2e t Solution: Characteristic equation is r 2 5r +6 = 0, therefore r 1 = 2, r 2 = 3, and y 1 (t) = e 2t, Homework6 Soluions.7 In Problem hrough 4 use he mehod of variaion of parameers o find a paricular soluion of he given differenial equaion. Then check your answer by using he mehod of undeermined coeffiens..

More information

Idealistic characteristics of Islamic Azad University masters - Islamshahr Branch from Students Perspective

Idealistic characteristics of Islamic Azad University masters - Islamshahr Branch from Students Perspective Available online a www.pelagiaresearchlibrary.com European Journal Experimenal Biology, 202, 2 (5):88789 ISSN: 2248 925 CODEN (USA): EJEBAU Idealisic characerisics Islamic Azad Universiy masers Islamshahr

More information

Automatic measurement and detection of GSM interferences

Automatic measurement and detection of GSM interferences Auomaic measuremen and deecion of GSM inerferences Poor speech qualiy and dropped calls in GSM neworks may be caused by inerferences as a resul of high raffic load. The radio nework analyzers from Rohde

More information

RC Circuit and Time Constant

RC Circuit and Time Constant ircui and Time onsan 8M Objec: Apparaus: To invesigae he volages across he resisor and capacior in a resisor-capacior circui ( circui) as he capacior charges and discharges. We also wish o deermine he

More information

Mobile and Ubiquitous Compu3ng. Mul3plexing for wireless. George Roussos. g.roussos@dcs.bbk.ac.uk

Mobile and Ubiquitous Compu3ng. Mul3plexing for wireless. George Roussos. g.roussos@dcs.bbk.ac.uk Mobile and Ubiquious Compu3ng Mul3plexing for wireless George Roussos g.roussos@dcs.bbk.ac.uk Overview Sharing he wireless (mul3plexing) in space by frequency in 3me by code PuEng i all ogeher: cellular

More information

Example What is the minimum bandwidth for transmitting data at a rate of 33.6 kbps without ISI?

Example What is the minimum bandwidth for transmitting data at a rate of 33.6 kbps without ISI? Emple Wh is he minimum ndwidh for rnsmiing d re of 33.6 kps wihou ISI? Answer: he minimum ndwidh is equl o he yquis ndwidh. herefore, BW min W R / 33.6/ 6.8 khz oe: If % roll-off chrcerisic is used, ndwidh

More information

Challenges in the Design of High-Speed Clock and Data Recovery Circuits

Challenges in the Design of High-Speed Clock and Data Recovery Circuits TOPICS IN CIRCUITS FOR COMMUNICTIONS Challenges in he esign of High-Speed Clock and aa Recovery Circuis ehzad Razavi, Universiy of California, Los ngeles STRCT This aricle describes he challenges in he

More information

Module 4. Single-phase AC circuits. Version 2 EE IIT, Kharagpur

Module 4. Single-phase AC circuits. Version 2 EE IIT, Kharagpur Module 4 Single-phase A circuis ersion EE T, Kharagpur esson 5 Soluion of urren in A Series and Parallel ircuis ersion EE T, Kharagpur n he las lesson, wo poins were described:. How o solve for he impedance,

More information

RC (Resistor-Capacitor) Circuits. AP Physics C

RC (Resistor-Capacitor) Circuits. AP Physics C (Resisor-Capacior Circuis AP Physics C Circui Iniial Condiions An circui is one where you have a capacior and resisor in he same circui. Suppose we have he following circui: Iniially, he capacior is UNCHARGED

More information

RECOMMENDATION ITU-R S Coordination identification between geostationary-satellite orbit fixed-satellite service networks

RECOMMENDATION ITU-R S Coordination identification between geostationary-satellite orbit fixed-satellite service networks Rec. ITU-R S.1524 1 RECOMMENDATION ITU-R S.1524 Coordinaion idenificaion beween geosaionary-saellie orbi fixed-saellie service neworks (2001) The ITU Radiocommunicaion Assembly, considering a) ha he Plenipoeniary

More information

A Mathematical Description of MOSFET Behavior

A Mathematical Description of MOSFET Behavior 10/19/004 A Mahemaical Descripion of MOSFET Behavior.doc 1/8 A Mahemaical Descripion of MOSFET Behavior Q: We ve learned an awful lo abou enhancemen MOSFETs, bu we sill have ye o esablished a mahemaical

More information

CLOCK SKEW CAUSES CLOCK SKEW DUE TO THE DRIVER EROSION OF THE CLOCK PERIOD

CLOCK SKEW CAUSES CLOCK SKEW DUE TO THE DRIVER EROSION OF THE CLOCK PERIOD DESIGNING WITH HIGH SPEED CLOCK DRIERS CONFERENCE PAPER CP-19 Inegraed Device Technology, Inc. By Sanley Hronik ABSTRACT Today s high speed sysems are encounering problems wih clocking ha were no consideraions

More information

Constant Data Length Retrieval for Video Servers with Variable Bit Rate Streams

Constant Data Length Retrieval for Video Servers with Variable Bit Rate Streams IEEE Inernaional Conference on Mulimedia Compuing & Sysems, June 17-3, 1996, in Hiroshima, Japan, p. 151-155 Consan Lengh Rerieval for Video Servers wih Variable Bi Rae Sreams Erns Biersack, Frédéric Thiesse,

More information

Fourier Series & The Fourier Transform

Fourier Series & The Fourier Transform Fourier Series & The Fourier Transform Wha is he Fourier Transform? Fourier Cosine Series for even funcions and Sine Series for odd funcions The coninuous limi: he Fourier ransform (and is inverse) The

More information

Molding. Injection. Design. GE Plastics. GE Engineering Thermoplastics DESIGN GUIDE

Molding. Injection. Design. GE Plastics. GE Engineering Thermoplastics DESIGN GUIDE apple GE Plasics GE Engineering Thermoplasics DESIGN GUIDE Wall Thickness Paring Lines Ejecion Appearance Pars Ribs/Gusses Bosses Holes Depressions Radii, Filles and Corners Molding Design Injecion s for

More information

PROFIT TEST MODELLING IN LIFE ASSURANCE USING SPREADSHEETS PART ONE

PROFIT TEST MODELLING IN LIFE ASSURANCE USING SPREADSHEETS PART ONE Profi Tes Modelling in Life Assurance Using Spreadshees PROFIT TEST MODELLING IN LIFE ASSURANCE USING SPREADSHEETS PART ONE Erik Alm Peer Millingon 2004 Profi Tes Modelling in Life Assurance Using Spreadshees

More information

The Application of Multi Shifts and Break Windows in Employees Scheduling

The Application of Multi Shifts and Break Windows in Employees Scheduling The Applicaion of Muli Shifs and Brea Windows in Employees Scheduling Evy Herowai Indusrial Engineering Deparmen, Universiy of Surabaya, Indonesia Absrac. One mehod for increasing company s performance

More information

Trends in TCP/IP Retransmissions and Resets

Trends in TCP/IP Retransmissions and Resets Trends in TCP/IP Reransmissions and Reses Absrac Concordia Chen, Mrunal Mangrulkar, Naomi Ramos, and Mahaswea Sarkar {cychen, mkulkarn, msarkar,naramos}@cs.ucsd.edu As he Inerne grows larger, measuring

More information

Information Systems for Business Integration: ERP Systems

Information Systems for Business Integration: ERP Systems Informaion Sysems for Business Inegraion: ERP Sysems (December 3, 2012) BUS3500 - Abdou Illia, Fall 2012 1 LEARNING GOALS Explain he difference beween horizonal and verical business inegraion. Describe

More information

Having an average input power of 1 Watt, compute the S/N ratio at the output of the link, after 4 Km.

Having an average input power of 1 Watt, compute the S/N ratio at the output of the link, after 4 Km. Exerise 1 A miro-oaxial able wih aenuaion A(f) = 9 onnes wo elephone saions. The able arries a PCM binary signal a 4 Mbi/s and roll-off faor of he raised osine filer equal o 0.5. The able is a a noise

More information

On Jitter by Dan Lavry, Lavry Engineering, Inc. email: dan@lavryengineering.com

On Jitter by Dan Lavry, Lavry Engineering, Inc. email: dan@lavryengineering.com On Jier by Dan Lavry, Lavry Engineering, Inc. email: dan@lavryengineering.com Copyrigh 1997 Lavry Engineering The firs par of his aricle for people who wan o gain some inuiive undersanding abou clock jier

More information

The Greek financial crisis: growing imbalances and sovereign spreads. Heather D. Gibson, Stephan G. Hall and George S. Tavlas

The Greek financial crisis: growing imbalances and sovereign spreads. Heather D. Gibson, Stephan G. Hall and George S. Tavlas The Greek financial crisis: growing imbalances and sovereign spreads Heaher D. Gibson, Sephan G. Hall and George S. Tavlas The enry The enry of Greece ino he Eurozone in 2001 produced a dividend in he

More information

Mathematics in Pharmacokinetics What and Why (A second attempt to make it clearer)

Mathematics in Pharmacokinetics What and Why (A second attempt to make it clearer) Mahemaics in Pharmacokineics Wha and Why (A second aemp o make i clearer) We have used equaions for concenraion () as a funcion of ime (). We will coninue o use hese equaions since he plasma concenraions

More information

As widely accepted performance measures in supply chain management practice, frequency-based service

As widely accepted performance measures in supply chain management practice, frequency-based service MANUFACTURING & SERVICE OPERATIONS MANAGEMENT Vol. 6, No., Winer 2004, pp. 53 72 issn 523-464 eissn 526-5498 04 060 0053 informs doi 0.287/msom.030.0029 2004 INFORMS On Measuring Supplier Performance Under

More information

Duration and Convexity ( ) 20 = Bond B has a maturity of 5 years and also has a required rate of return of 10%. Its price is $613.

Duration and Convexity ( ) 20 = Bond B has a maturity of 5 years and also has a required rate of return of 10%. Its price is $613. Graduae School of Business Adminisraion Universiy of Virginia UVA-F-38 Duraion and Convexiy he price of a bond is a funcion of he promised paymens and he marke required rae of reurn. Since he promised

More information

A Note on Using the Svensson procedure to estimate the risk free rate in corporate valuation

A Note on Using the Svensson procedure to estimate the risk free rate in corporate valuation A Noe on Using he Svensson procedure o esimae he risk free rae in corporae valuaion By Sven Arnold, Alexander Lahmann and Bernhard Schwezler Ocober 2011 1. The risk free ineres rae in corporae valuaion

More information

11/6/2013. Chapter 14: Dynamic AD-AS. Introduction. Introduction. Keeping track of time. The model s elements

11/6/2013. Chapter 14: Dynamic AD-AS. Introduction. Introduction. Keeping track of time. The model s elements Inroducion Chaper 14: Dynamic D-S dynamic model of aggregae and aggregae supply gives us more insigh ino how he economy works in he shor run. I is a simplified version of a DSGE model, used in cuing-edge

More information

Lecture 2: Telegrapher Equations For Transmission Lines. Power Flow.

Lecture 2: Telegrapher Equations For Transmission Lines. Power Flow. Whies, EE 481 Lecure 2 Page 1 of 13 Lecure 2: Telegraher Equaions For Transmission Lines. Power Flow. Microsri is one mehod for making elecrical connecions in a microwae circui. I is consruced wih a ground

More information

Single-machine Scheduling with Periodic Maintenance and both Preemptive and. Non-preemptive jobs in Remanufacturing System 1

Single-machine Scheduling with Periodic Maintenance and both Preemptive and. Non-preemptive jobs in Remanufacturing System 1 Absrac number: 05-0407 Single-machine Scheduling wih Periodic Mainenance and boh Preempive and Non-preempive jobs in Remanufacuring Sysem Liu Biyu hen Weida (School of Economics and Managemen Souheas Universiy

More information

Principal components of stock market dynamics. Methodology and applications in brief (to be updated ) Andrei Bouzaev, bouzaev@ya.

Principal components of stock market dynamics. Methodology and applications in brief (to be updated ) Andrei Bouzaev, bouzaev@ya. Principal componens of sock marke dynamics Mehodology and applicaions in brief o be updaed Andrei Bouzaev, bouzaev@ya.ru Why principal componens are needed Objecives undersand he evidence of more han one

More information

Stochastic Recruitment: A Limited-Feedback Control Policy for Large Ensemble Systems

Stochastic Recruitment: A Limited-Feedback Control Policy for Large Ensemble Systems Sochasic Recruimen: A Limied-Feedback Conrol Policy for Large Ensemble Sysems Lael Odhner and Harry Asada Absrac This paper is abou sochasic recruimen, a conrol archiecure for cenrally conrolling he ensemble

More information

Using RCtime to Measure Resistance

Using RCtime to Measure Resistance Basic Express Applicaion Noe Using RCime o Measure Resisance Inroducion One common use for I/O pins is o measure he analog value of a variable resisance. Alhough a buil-in ADC (Analog o Digial Converer)

More information

Multiprocessor Systems-on-Chips

Multiprocessor Systems-on-Chips Par of: Muliprocessor Sysems-on-Chips Edied by: Ahmed Amine Jerraya and Wayne Wolf Morgan Kaufmann Publishers, 2005 2 Modeling Shared Resources Conex swiching implies overhead. On a processing elemen,

More information

3 Runge-Kutta Methods

3 Runge-Kutta Methods 3 Runge-Kua Mehods In conras o he mulisep mehods of he previous secion, Runge-Kua mehods are single-sep mehods however, muliple sages per sep. They are moivaed by he dependence of he Taylor mehods on he

More information

DYNAMIC MODELS FOR VALUATION OF WRONGFUL DEATH PAYMENTS

DYNAMIC MODELS FOR VALUATION OF WRONGFUL DEATH PAYMENTS DYNAMIC MODELS FOR VALUATION OF WRONGFUL DEATH PAYMENTS Hong Mao, Shanghai Second Polyechnic Universiy Krzyszof M. Osaszewski, Illinois Sae Universiy Youyu Zhang, Fudan Universiy ABSTRACT Liigaion, exper

More information

ARCH 2013.1 Proceedings

ARCH 2013.1 Proceedings Aricle from: ARCH 213.1 Proceedings Augus 1-4, 212 Ghislain Leveille, Emmanuel Hamel A renewal model for medical malpracice Ghislain Léveillé École d acuaria Universié Laval, Québec, Canada 47h ARC Conference

More information

Present Value Methodology

Present Value Methodology Presen Value Mehodology Econ 422 Invesmen, Capial & Finance Universiy of Washingon Eric Zivo Las updaed: April 11, 2010 Presen Value Concep Wealh in Fisher Model: W = Y 0 + Y 1 /(1+r) The consumer/producer

More information

The Phase Modulator In NBFM Voice Communication Systems

The Phase Modulator In NBFM Voice Communication Systems The Phase Modulator In NBFM Voice Communication Systems Virgil Leenerts 8 March 5 The phase modulator has been a point of discussion as to why it is used and not a frequency modulator in what are called

More information

Angle Modulation, II. Lecture topics FM bandwidth and Carson s rule. Spectral analysis of FM. Narrowband FM Modulation. Wideband FM Modulation

Angle Modulation, II. Lecture topics FM bandwidth and Carson s rule. Spectral analysis of FM. Narrowband FM Modulation. Wideband FM Modulation Angle Modulation, II EE 179, Lecture 12, Handout #19 Lecture topics FM bandwidth and Carson s rule Spectral analysis of FM Narrowband FM Modulation Wideband FM Modulation EE 179, April 25, 2014 Lecture

More information

IR Receiver Module for Light Barrier Systems

IR Receiver Module for Light Barrier Systems IR Receiver Module for Ligh Barrier Sysems MECHANICAL DATA Pinning: 1 = OUT, 2 = GND, 3 = V S 19026 APPLICATIONS Reflecive sensors for hand dryers, owel or soap dispensers, waer fauces, oile flush Vending

More information

LECTURE 9. C. Appendix

LECTURE 9. C. Appendix LECTURE 9 A. Buck-Boos Converer Design 1. Vol-Sec Balance: f(d), seadysae ransfer funcion 2. DC Operaing Poin via Charge Balance: I(D) in seady-sae 3. Ripple Volage / C Spec 4. Ripple Curren / L Spec 5.

More information

Why Did the Demand for Cash Decrease Recently in Korea?

Why Did the Demand for Cash Decrease Recently in Korea? Why Did he Demand for Cash Decrease Recenly in Korea? Byoung Hark Yoo Bank of Korea 26. 5 Absrac We explores why cash demand have decreased recenly in Korea. The raio of cash o consumpion fell o 4.7% in

More information

Communication Networks II Contents

Communication Networks II Contents 3 / 1 -- Communicaion Neworks II (Görg) -- www.comnes.uni-bremen.de Communicaion Neworks II Conens 1 Fundamenals of probabiliy heory 2 Traffic in communicaion neworks 3 Sochasic & Markovian Processes (SP

More information

IR Receiver Modules for Remote Control Systems

IR Receiver Modules for Remote Control Systems TSOP382.., TSOP384.. IR Receiver Modules for MECHANICAL DATA Pinning: = OUT, 2 =, 3 = V S 926 FEATURES Very low supply curren Phoo deecor and preamplifier in one package Inernal filer for PCM frequency

More information

4 Convolution. Recommended Problems. x2[n] 1 2[n]

4 Convolution. Recommended Problems. x2[n] 1 2[n] 4 Convoluion Recommended Problems P4.1 This problem is a simple example of he use of superposiion. Suppose ha a discree-ime linear sysem has oupus y[n] for he given inpus x[n] as shown in Figure P4.1-1.

More information

Analysis of Pricing and Efficiency Control Strategy between Internet Retailer and Conventional Retailer

Analysis of Pricing and Efficiency Control Strategy between Internet Retailer and Conventional Retailer Recen Advances in Business Managemen and Markeing Analysis of Pricing and Efficiency Conrol Sraegy beween Inerne Reailer and Convenional Reailer HYUG RAE CHO 1, SUG MOO BAE and JOG HU PARK 3 Deparmen of

More information

Communication Systems

Communication Systems AM/FM Receiver Communication Systems We have studied the basic blocks o any communication system Modulator Demodulator Modulation Schemes: Linear Modulation (DSB, AM, SSB, VSB) Angle Modulation (FM, PM)

More information

Hedging with Forwards and Futures

Hedging with Forwards and Futures Hedging wih orwards and uures Hedging in mos cases is sraighforward. You plan o buy 10,000 barrels of oil in six monhs and you wish o eliminae he price risk. If you ake he buy-side of a forward/fuures

More information

The front end of the receiver performs the frequency translation, channel selection and amplification of the signal.

The front end of the receiver performs the frequency translation, channel selection and amplification of the signal. Many receivers must be capable of handling a very wide range of signal powers at the input while still producing the correct output. This must be done in the presence of noise and interference which occasionally

More information

Mobile Broadband Rollout Business Case: Risk Analyses of the Forecast Uncertainties

Mobile Broadband Rollout Business Case: Risk Analyses of the Forecast Uncertainties ISF 2009, Hong Kong, 2-24 June 2009 Mobile Broadband Rollou Business Case: Risk Analyses of he Forecas Uncerainies Nils Krisian Elnegaard, Telenor R&I Agenda Moivaion Modelling long erm forecass for MBB

More information

THE PRESSURE DERIVATIVE

THE PRESSURE DERIVATIVE Tom Aage Jelmer NTNU Dearmen of Peroleum Engineering and Alied Geohysics THE PRESSURE DERIVATIVE The ressure derivaive has imoran diagnosic roeries. I is also imoran for making ye curve analysis more reliable.

More information

Smooth Priorities for Multi-Product Inventory Control

Smooth Priorities for Multi-Product Inventory Control Smooh rioriies for Muli-roduc Invenory Conrol Francisco José.A.V. Mendonça*. Carlos F. Bispo** *Insiuo Superior Técnico - Universidade Técnica de Lisboa (email:favm@mega.is.ul.p) ** Insiuo de Sisemas e

More information

Appendix A: Area. 1 Find the radius of a circle that has circumference 12 inches.

Appendix A: Area. 1 Find the radius of a circle that has circumference 12 inches. Appendi A: Area worked-ou s o Odd-Numbered Eercises Do no read hese worked-ou s before aemping o do he eercises ourself. Oherwise ou ma mimic he echniques shown here wihou undersanding he ideas. Bes wa

More information

Dopamine, dobutamine, digitalis, and diuretics during intraaortic balloon support

Dopamine, dobutamine, digitalis, and diuretics during intraaortic balloon support Dopamine, dobuamine, digialis, and diureics during inraaoric balloon suppor Sephen Slogoff, M.D. n his presenaion, should like o discuss some conceps of drug herapy for inraaoric balloon paiens. Figure

More information

GoRA. For more information on genetics and on Rheumatoid Arthritis: Genetics of Rheumatoid Arthritis. Published work referred to in the results:

GoRA. For more information on genetics and on Rheumatoid Arthritis: Genetics of Rheumatoid Arthritis. Published work referred to in the results: For more informaion on geneics and on Rheumaoid Arhriis: Published work referred o in he resuls: The geneics revoluion and he assaul on rheumaoid arhriis. A review by Michael Seldin, Crisopher Amos, Ryk

More information

YEN FUTURES: EXAMINING HEDGING EFFECTIVENESS BIAS AND CROSS-CURRENCY HEDGING RESULTS ROBERT T. DAIGLER FLORIDA INTERNATIONAL UNIVERSITY SUBMITTED FOR

YEN FUTURES: EXAMINING HEDGING EFFECTIVENESS BIAS AND CROSS-CURRENCY HEDGING RESULTS ROBERT T. DAIGLER FLORIDA INTERNATIONAL UNIVERSITY SUBMITTED FOR YEN FUTURES: EXAMINING HEDGING EFFECTIVENESS BIAS AND CROSS-CURRENCY HEDGING RESULTS ROBERT T. DAIGLER FLORIDA INTERNATIONAL UNIVERSITY SUBMITTED FOR THE FIRST ANNUAL PACIFIC-BASIN FINANCE CONFERENCE The

More information

SINAMICS S120 drive system

SINAMICS S120 drive system SINAMICS S120 drive sysem Design PM340, frame sizes FSA o FSF The PM340 feaure he following connecions as sandard: DCP/R1 and DCN DC link Terminals DCP/R1 and R2 for connecion of an exernal braking PM-IF

More information

Journal Of Business & Economics Research September 2005 Volume 3, Number 9

Journal Of Business & Economics Research September 2005 Volume 3, Number 9 Opion Pricing And Mone Carlo Simulaions George M. Jabbour, (Email: jabbour@gwu.edu), George Washingon Universiy Yi-Kang Liu, (yikang@gwu.edu), George Washingon Universiy ABSTRACT The advanage of Mone Carlo

More information

AN Smart LED Lamp Driver IC with PFC Function. Introduction. Soft-Start Function. Internal Digital Reference by ZCD.

AN Smart LED Lamp Driver IC with PFC Function. Introduction. Soft-Start Function. Internal Digital Reference by ZCD. www.fairchildsemi.com AN-9744 Smar LED Lamp Driver IC wih PFC Funcion Inroducion The FL7701 is a PWM peak curren conroller for a buck converer opology operaing in Coninuous Conducion Mode (CCM) wih an

More information

PI4ULS5V202 2-Bit Bi-directional Level Shifter with Automatic Sensing & Ultra Tiny Package

PI4ULS5V202 2-Bit Bi-directional Level Shifter with Automatic Sensing & Ultra Tiny Package Feaures can be Less han, Greaer han or Equal o V CCB 1.2V o 5.5V on A Por and 1.2V o 5.5V on B Por High Speed wih 20 Mb/s Daa Rae for push-pull applicaion High Speed wih 2 Mb/s Daa Rae for open-drain applicaion

More information

ANALYSIS AND COMPARISONS OF SOME SOLUTION CONCEPTS FOR STOCHASTIC PROGRAMMING PROBLEMS

ANALYSIS AND COMPARISONS OF SOME SOLUTION CONCEPTS FOR STOCHASTIC PROGRAMMING PROBLEMS ANALYSIS AND COMPARISONS OF SOME SOLUTION CONCEPTS FOR STOCHASTIC PROGRAMMING PROBLEMS R. Caballero, E. Cerdá, M. M. Muñoz and L. Rey () Deparmen of Applied Economics (Mahemaics), Universiy of Málaga,

More information

Astable multivibrator using the 555 IC.(10)

Astable multivibrator using the 555 IC.(10) Visi hp://elecronicsclub.cjb.ne for more resources THE 555 IC TIMER The 555 IC TIMER.(2) Monosable mulivibraor using he 555 IC imer...() Design Example 1 wih Mulisim 2001 ools and graphs..(8) Lile descripion

More information

Packet-Oriented Communication Protocols for Smart Grid Services over Low-Speed PLC

Packet-Oriented Communication Protocols for Smart Grid Services over Low-Speed PLC Packe-Oriened Communicaion Proocols for Smar Grid Services over Low-Speed PLC Michael Bauer, Wolfgang Plapper, Chong Wang, Klaus Doser Insiue of Indusrial Informaion Technology Universiä Karlsruhe (TH)

More information

What is a swap? A swap is a contract between two counter-parties who agree to exchange a stream of payments over an agreed period of several years.

What is a swap? A swap is a contract between two counter-parties who agree to exchange a stream of payments over an agreed period of several years. Currency swaps Wha is a swap? A swap is a conrac beween wo couner-paries who agree o exchange a sream of paymens over an agreed period of several years. Types of swap equiy swaps (or equiy-index-linked

More information

Task is a schedulable entity, i.e., a thread

Task is a schedulable entity, i.e., a thread Real-Time Scheduling Sysem Model Task is a schedulable eniy, i.e., a hread Time consrains of periodic ask T: - s: saring poin - e: processing ime of T - d: deadline of T - p: period of T Periodic ask T

More information

3F4 Digital Modulation Course

3F4 Digital Modulation Course 3F4 Digial Modulaion Course (supervisor copy) 1 3F4 Digial Modulaion Course Nick Kingsbury February 10, 2016 Conens 1 Modulaion, Phasors and Bandlimied Noise 3 1.1 Types of Modulaion - AM, FM, PM........................

More information

Planning Demand and Supply in a Supply Chain. Forecasting and Aggregate Planning

Planning Demand and Supply in a Supply Chain. Forecasting and Aggregate Planning Planning Demand and Supply in a Supply Chain Forecasing and Aggregae Planning 1 Learning Objecives Overview of forecasing Forecas errors Aggregae planning in he supply chain Managing demand Managing capaciy

More information

The Grantor Retained Annuity Trust (GRAT)

The Grantor Retained Annuity Trust (GRAT) WEALTH ADVISORY Esae Planning Sraegies for closely-held, family businesses The Granor Reained Annuiy Trus (GRAT) An efficien wealh ransfer sraegy, paricularly in a low ineres rae environmen Family business

More information

CLASSIFICATION OF REINSURANCE IN LIFE INSURANCE

CLASSIFICATION OF REINSURANCE IN LIFE INSURANCE CLASSIFICATION OF REINSURANCE IN LIFE INSURANCE Kaarína Sakálová 1. Classificaions of reinsurance There are many differen ways in which reinsurance may be classified or disinguished. We will discuss briefly

More information

Signal Processing and Linear Systems I

Signal Processing and Linear Systems I Sanford Universiy Summer 214-215 Signal Processing and Linear Sysems I Lecure 5: Time Domain Analysis of Coninuous Time Sysems June 3, 215 EE12A:Signal Processing and Linear Sysems I; Summer 14-15, Gibbons

More information