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School: W. Alabama
E&CE471: Electromagnetic Waves Final Examination (Spring 2008) Aug 12, 2008, Instructor: A. H. Majedi Duration: 2.5 hours. Calculators are permitted. Full mark constitutes 100. -Name: Student ID#: 1- Using the generalized scattering parameter, find the sc
School: W. Alabama
. .J . . - 7 2 J - # - @. $ # $ D m/s EM Plane-Wave Propagation EM Wave Equations in Lossless Media 0 . H 0H Good Conductor cfw_ -cfw_ cfw_ -cfw_ H General Relation between E & H in TEM Wave . 0H 0 H cfw_ cfw_ Linear Polarization or Circular Polarizat
School: W. Alabama
Solution_Set7 1 Solution_Set7 2 Solution_Set7 3 Solution_Set7 4 Solution_Set7 5 Solution_Set7 6 Solution_Set7 7
School: W. Alabama
Solution_Set6 1 Solution_Set6 2 Solution_Set6 3 Solution_Set6 4 Solution_Set6 5 Solution_Set6 6 Solution_Set6 7 Solution_Set6 8 Solution_Set6 9
School: W. Alabama
ECE309 THERMODYNAMICS & HEAT TRANSFER Spring 2008 Instructor: T.A.s: Richard Culham Syeda Tasnim Omid Nemati Shuwen Wang E3-2133J E2-3369 E3-2136B E2-4404/3 ext. 33839 ext. 33583 ext. 33885 ext. 35266 Email: rix@mhtlab.uwaterloo.ca Email: shtasnim@en
School: W. Alabama
Midterm Review Questions (Ch. 3 & 4) January 24th, 2005 ECE493 Topic 2 1) What are the differences and tradeoffs between physical etching and chemical etching? Wet etching and dry etching? Why isnt there such a thing as wet physical etching? Physical
School: W. Alabama
Inverter Sizing Inverter Chain In Out CL If CL is given: - How many stages are needed to minimize the delay? - How to size the inverters? May need some additional constraints. Inverter Delay Minimum length devices, L=0.25m Assume that for W P =
School: W. Alabama
Pass-Transistor Logic 1 EE141 Pass-Transistor Logic B Switch Network B Out A Out B Inputs N transistors No static consumption Primary inputs drive the gate terminals + source-drain terminals. In contrast to static CMOS primary inputs drive gate
School: W. Alabama
1.0 Introduction 1.1 Nanotechnology Historical Developments Around 370BC, Democritus a Greek philosopher developed the atomic theory of matter. Nano in GREEK means DWARF. The prefix nano means a billionth (10-9)! Nanotechnology deals with structur
School: W. Alabama
E&CE471: Electromagnetic Waves Final Examination (Spring 2008) Aug 12, 2008, Instructor: A. H. Majedi Duration: 2.5 hours. Calculators are permitted. Full mark constitutes 100. -Name: Student ID#: 1- Using the generalized scattering parameter, find the sc
School: W. Alabama
. .J . . - 7 2 J - # - @. $ # $ D m/s EM Plane-Wave Propagation EM Wave Equations in Lossless Media 0 . H 0H Good Conductor cfw_ -cfw_ cfw_ -cfw_ H General Relation between E & H in TEM Wave . 0H 0 H cfw_ cfw_ Linear Polarization or Circular Polarizat
School: W. Alabama
E&CE471: Electromagnetic Waves Final Examination (Spring 2005) Aug 11, 2005, Instructor: A. H. Majedi Duration: 2.5 hours Calculators are permitted. Answer all questions & problems. Full mark constitutes 100. Extra 2 marks is bonus. -Name: Student ID#: 1-
School: W. Alabama
UNIVERSITY OF WATERLOO Department of Electrical and Computer Engineering Introduction to Thermodynamics and Heat Transfer E&CE 309 Mid-Term Examination R. Karimi Spring 2002 June 10, 02, 9:30-11:15 A.M. 1. (8) A closed, rigid tank contains 0.5 kg of
School: W. Alabama
ECE309 Thermodynamics & Heat Transfer Quiz #2: Name: ID #: Problem: A hair dryer is basically a duct in which a few layers of electric resistors are placed. A small fan pulls the air in and forces it through the resistors where it is heated. Air en
School: W. Alabama
University of Waterloo Department of Electrical & Computer Engineering E&CE 223 Digital Circuits and Systems Solution for Final Examination April 5, 2004 Total Time = 3 hours, Total Marks = 100 Student Name: Student ID: 1. 2. 3. 4. 5. 6. 7. 8
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Problem Set #7 1. 2. 3. 4. 5. 6. 7. Problem 9.4 from the text book. Problem 9.10 from the text book. Problem 9.13 from the text book. Problem 9.14 from the text book. Problem 9.17 from the text book. Problem 9.20 from the text book. Problem 9.22 from the
School: W. Alabama
MICROWAVE and RF ENGINEERING E&CE 493/671 Assignment #1 Due Date: Feb 5th, 2007 Problem #1 (25 marks) (a) (b) Obtain expressions for the voltage and current standing-wave patterns on a lossless opencircuited transmission line. Sketch these patterns.
School: W. Alabama
Solutions 2 1. a class 1000 clean room is defined as one in which the number of dust particles smaller than 0.5 micrometer is less than 1000 per cubic (1) inch, (2) foot, (3) meter. 2. the dust particle size used in defining clean room air quality is
School: W. Alabama
ECE-223, Solutions for Assignment #3 Chapter 3, Digital Design, M. Mano, 3rd Edition 3.3) Simplify the following Boolean functions, using three-variable maps: a) xy + xyz + xyz b) xy + yz + xyz c) AB + BC + BC a) b) xy+xz x+yz c) C + AB 3.5) Sim
School: W. Alabama
Introduction to Thermodynamics and Heat Transfer (ECE 309) Suggested Problems for Chapter 2 1. A 0.5-m3 rigid vessel initially contains saturated liquid-vapor mixture of water at 100 C. The water is now heated until it reaches the critical state. Det
School: W. Alabama
Introduction to Thermodynamics and Heat Transfer (ECE 309) Suggested Problems for Chapter 1 1. Both a gage and a manometer are attached to a gas tank to measure its pressure. If the reading on the pressure gage is 80 kPa, determine the distance betwe
School: W. Alabama
E&CE471: Electromagnetic Waves Final Examination (Spring 2008) Aug 12, 2008, Instructor: A. H. Majedi Duration: 2.5 hours. Calculators are permitted. Full mark constitutes 100. -Name: Student ID#: 1- Using the generalized scattering parameter, find the sc
School: W. Alabama
. .J . . - 7 2 J - # - @. $ # $ D m/s EM Plane-Wave Propagation EM Wave Equations in Lossless Media 0 . H 0H Good Conductor cfw_ -cfw_ cfw_ -cfw_ H General Relation between E & H in TEM Wave . 0H 0 H cfw_ cfw_ Linear Polarization or Circular Polarizat
School: W. Alabama
Solution_Set7 1 Solution_Set7 2 Solution_Set7 3 Solution_Set7 4 Solution_Set7 5 Solution_Set7 6 Solution_Set7 7
School: W. Alabama
Solution_Set6 1 Solution_Set6 2 Solution_Set6 3 Solution_Set6 4 Solution_Set6 5 Solution_Set6 6 Solution_Set6 7 Solution_Set6 8 Solution_Set6 9
School: W. Alabama
Problem Set #7 1. 2. 3. 4. 5. 6. 7. Problem 9.4 from the text book. Problem 9.10 from the text book. Problem 9.13 from the text book. Problem 9.14 from the text book. Problem 9.17 from the text book. Problem 9.20 from the text book. Problem 9.22 from the
School: W. Alabama
E&CE471: Electromagnetic Waves Final Examination (Spring 2005) Aug 11, 2005, Instructor: A. H. Majedi Duration: 2.5 hours Calculators are permitted. Answer all questions & problems. Full mark constitutes 100. Extra 2 marks is bonus. -Name: Student ID#: 1-
School: W. Alabama
ECE 492B Symposium Presentation Grading Sheet Group Number: Marker Name: Marker Signature: Comments (Please provide point form feedback!) 1. Organization - Introductory remarks - Presentation structure - Concluding remarks - Use of available time
School: W. Alabama
School: W. Alabama
E&CE 427: Digital System Engineering Mark Aagaard University of Waterloo Dept of Electrical and Computer Engineering 2003t1Winter March 24, 2003 E&CE 427: 2003t1Winter 0 ii Contents I Lecture Notes 1 VHDL LEC-02: Introduction to VHDL . . . . . .
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School: W. Alabama
MICROWAVE and RF ENGINEERING E&CE 493/671 Assignment #1 Due Date: Feb 5th, 2007 Problem #1 (25 marks) (a) (b) Obtain expressions for the voltage and current standing-wave patterns on a lossless opencircuited transmission line. Sketch these patterns.
School: W. Alabama
Solutions 2 1. a class 1000 clean room is defined as one in which the number of dust particles smaller than 0.5 micrometer is less than 1000 per cubic (1) inch, (2) foot, (3) meter. 2. the dust particle size used in defining clean room air quality is
School: W. Alabama
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School: W. Alabama
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Electrical & Computer Engineering Department University of Waterloo E&CE 477 Photonic Communication Systems & Devices Winter 2006 Instructor: Office: Email: Phone: Office Hours: Lecture: Tutorial: TAs: Hamed Majedi CEIT 4006 ahmajedi@maxwell.uwaterl
School: W. Alabama
ECE 492B Symposium Presentation Grading Sheet Group Number: Marker Name: Marker Signature: Comments (Please provide point form feedback!) 1. Organization - Introductory remarks - Presentation structure - Concluding remarks - Use of available time
School: W. Alabama
Chapter 2 Optical Fibers: Structures, Waveguiding & Fabrication Theories of Optics Light is an electromagentic phenomenon described by the same theoretical principles that govern all forms of electromagnetic radiation. Maxwell's equations are in t
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School: W. Alabama
ECE-223, Solutions for Assignment #3 Chapter 3, Digital Design, M. Mano, 3rd Edition 3.3) Simplify the following Boolean functions, using three-variable maps: a) xy + xyz + xyz b) xy + yz + xyz c) AB + BC + BC a) b) xy+xz x+yz c) C + AB 3.5) Sim
School: W. Alabama
ECE 309 Project #2: Heat Transfer Assigned: 16 July 2004 Due: July 23, 2004 All questions must be completed individually. All projects must be handed in during the lecture period on the Due date. No extensions are possible. Failure to hand the pr
School: W. Alabama
ECE309 INTRODUCTION TO THERMODYNAMICS & HEAT TRANSFER 3 August 2004 Final Examination R. Culham This is a 3 hour, closed-book examination. You are permitted to use one 8.5 in. 11 in. crib sheet (both sides), Conversion Factors (inside cover of tex
School: W. Alabama
Introduction to Thermodynamics and Heat Transfer (ECE 309) Suggested Problems for Chapter 8,9 1. A 0.2-cm-thick, 10-cm-high, and 15-cm-long circuit board houses electronic components on one side that dissipate a total of 15 W of heal uniformly. The b
School: W. Alabama
Introduction to Thermodynamics and Heat Transfer (ECE 309) Suggested Problems for Chapter 3 1. A 4 m 5 m 7 m room is heated by the radiator of a steam-heating system as shown in Fig. 1. The seam radiator transfers heat at a rate of 10,000 kJ /h, an
School: W. Alabama
Introduction to Thermodynamics and Heat Transfer (ECE 309) Suggested Problems for Chapter 2 1. A 0.5-m3 rigid vessel initially contains saturated liquid-vapor mixture of water at 100 C. The water is now heated until it reaches the critical state. Det
School: W. Alabama
Introduction to Thermodynamics and Heat Transfer (ECE 309) Suggested Problems for Chapter 1 1. Both a gage and a manometer are attached to a gas tank to measure its pressure. If the reading on the pressure gage is 80 kPa, determine the distance betwe
School: W. Alabama
UNIVERSITY OF WATERLOO Department of Electrical and Computer Engineering Introduction to Thermodynamics and Heat Transfer E&CE 309 Mid-Term Examination R. Karimi Spring 2002 June 10, 02, 9:30-11:15 A.M. 1. (8) A closed, rigid tank contains 0.5 kg of
School: W. Alabama
ECE 316 Probability Theory and Random Processes Chapter 7 Solutions (Part 1) Xinxin Fan Problems 3. If X and Y are independent uniform (0, 1) random variables, show that E[|X - Y | ] = 2 , f or > 0. ( + 1)( + 2) Solution. From the definition of t
School: W. Alabama
ECE 316 Probability Theory and Random Processes Chapter 6 Solutions (Part 2) Xinxin Fan Problems 18. Two points are selected randomly on a line of length L so as to be on opposite sides of the midpoint of the line. [In other words, the two points X
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ECE 316 Probability Theory and Random Processes Chapter 5 Solutions (Part 1) Xinxin Fan Problems 2. A system consisting of one original unit plus a spare can function for a random amount of time X. If the density of X is given (in units of months)
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ECE 316 Probability Theory and Random Processes Chapter 4 Solutions (Part 3) Xinxin Fan Problems 53. Approximately 80, 000 marriages took place in the state of New York last year. Estimate the probability that for at least one of these couples (a)
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ECE 316 Probability Theory and Random Processes Chapter 4 Solutions (Part 2) Xinxin Fan Problems 20. A gambling book recommends the following winning strategy for the game of roulette. It recommends that a gambler bet $1 on red. If red appears (whi
School: W. Alabama
ECE 316 Probability Theory and Random Processes Chapter 4 Solutions (Part 1) Xinxin Fan Problems 5. Let X represent the dierence between the number of heads and the number of tails obtained when a coin is tossed n times. What are the possible value
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ECE 316 Probability Theory and Random Processes Chapter 3 Solutions (Part 2) Xinxin Fan Problems 57. A simplified model for the movement of the price of a stock supposes that on each day the stock's price either moves up 1 unit with probability p o
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ECE 316 Probability Theory and Random Processes Chapter 2 Solutions (Part 2) Xinxin Fan Problems 17. If 8 castles (that is, rooks) are randomly placed on a chessboard, compute the probability that none of the rooks can capture any of the others. Th
School: W. Alabama
ECE309 THERMODYNAMICS & HEAT TRANSFER Spring 2008 Instructor: T.A.s: Richard Culham Syeda Tasnim Omid Nemati Shuwen Wang E3-2133J E2-3369 E3-2136B E2-4404/3 ext. 33839 ext. 33583 ext. 33885 ext. 35266 Email: rix@mhtlab.uwaterloo.ca Email: shtasnim@en
School: W. Alabama
Midterm Review Questions (Ch. 3 & 4) January 24th, 2005 ECE493 Topic 2 1) What are the differences and tradeoffs between physical etching and chemical etching? Wet etching and dry etching? Why isnt there such a thing as wet physical etching? Physical
School: W. Alabama
Inverter Sizing Inverter Chain In Out CL If CL is given: - How many stages are needed to minimize the delay? - How to size the inverters? May need some additional constraints. Inverter Delay Minimum length devices, L=0.25m Assume that for W P =
School: W. Alabama
Pass-Transistor Logic 1 EE141 Pass-Transistor Logic B Switch Network B Out A Out B Inputs N transistors No static consumption Primary inputs drive the gate terminals + source-drain terminals. In contrast to static CMOS primary inputs drive gate
School: W. Alabama
1.0 Introduction 1.1 Nanotechnology Historical Developments Around 370BC, Democritus a Greek philosopher developed the atomic theory of matter. Nano in GREEK means DWARF. The prefix nano means a billionth (10-9)! Nanotechnology deals with structur
School: W. Alabama
E&CE471: Electromagnetic Waves Final Examination (Spring 2008) Aug 12, 2008, Instructor: A. H. Majedi Duration: 2.5 hours. Calculators are permitted. Full mark constitutes 100. -Name: Student ID#: 1- Using the generalized scattering parameter, find the sc
School: W. Alabama
. .J . . - 7 2 J - # - @. $ # $ D m/s EM Plane-Wave Propagation EM Wave Equations in Lossless Media 0 . H 0H Good Conductor cfw_ -cfw_ cfw_ -cfw_ H General Relation between E & H in TEM Wave . 0H 0 H cfw_ cfw_ Linear Polarization or Circular Polarizat
School: W. Alabama
E&CE471: Electromagnetic Waves Final Examination (Spring 2005) Aug 11, 2005, Instructor: A. H. Majedi Duration: 2.5 hours Calculators are permitted. Answer all questions & problems. Full mark constitutes 100. Extra 2 marks is bonus. -Name: Student ID#: 1-
School: W. Alabama
UNIVERSITY OF WATERLOO Department of Electrical and Computer Engineering Introduction to Thermodynamics and Heat Transfer E&CE 309 Mid-Term Examination R. Karimi Spring 2002 June 10, 02, 9:30-11:15 A.M. 1. (8) A closed, rigid tank contains 0.5 kg of
School: W. Alabama
ECE309 Thermodynamics & Heat Transfer Quiz #2: Name: ID #: Problem: A hair dryer is basically a duct in which a few layers of electric resistors are placed. A small fan pulls the air in and forces it through the resistors where it is heated. Air en
School: W. Alabama
University of Waterloo Department of Electrical & Computer Engineering E&CE 223 Digital Circuits and Systems Solution for Final Examination April 5, 2004 Total Time = 3 hours, Total Marks = 100 Student Name: Student ID: 1. 2. 3. 4. 5. 6. 7. 8
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Problem Set #7 1. 2. 3. 4. 5. 6. 7. Problem 9.4 from the text book. Problem 9.10 from the text book. Problem 9.13 from the text book. Problem 9.14 from the text book. Problem 9.17 from the text book. Problem 9.20 from the text book. Problem 9.22 from the
School: W. Alabama
MICROWAVE and RF ENGINEERING E&CE 493/671 Assignment #1 Due Date: Feb 5th, 2007 Problem #1 (25 marks) (a) (b) Obtain expressions for the voltage and current standing-wave patterns on a lossless opencircuited transmission line. Sketch these patterns.
School: W. Alabama
Solutions 2 1. a class 1000 clean room is defined as one in which the number of dust particles smaller than 0.5 micrometer is less than 1000 per cubic (1) inch, (2) foot, (3) meter. 2. the dust particle size used in defining clean room air quality is
School: W. Alabama
ECE-223, Solutions for Assignment #3 Chapter 3, Digital Design, M. Mano, 3rd Edition 3.3) Simplify the following Boolean functions, using three-variable maps: a) xy + xyz + xyz b) xy + yz + xyz c) AB + BC + BC a) b) xy+xz x+yz c) C + AB 3.5) Sim
School: W. Alabama
Introduction to Thermodynamics and Heat Transfer (ECE 309) Suggested Problems for Chapter 2 1. A 0.5-m3 rigid vessel initially contains saturated liquid-vapor mixture of water at 100 C. The water is now heated until it reaches the critical state. Det
School: W. Alabama
Introduction to Thermodynamics and Heat Transfer (ECE 309) Suggested Problems for Chapter 1 1. Both a gage and a manometer are attached to a gas tank to measure its pressure. If the reading on the pressure gage is 80 kPa, determine the distance betwe
School: W. Alabama
ECE 316 Probability Theory and Random Processes Chapter 7 Solutions (Part 1) Xinxin Fan Problems 3. If X and Y are independent uniform (0, 1) random variables, show that E[|X - Y | ] = 2 , f or > 0. ( + 1)( + 2) Solution. From the definition of t
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ECE 316 Probability Theory and Random Processes Chapter 6 Solutions (Part 2) Xinxin Fan Problems 18. Two points are selected randomly on a line of length L so as to be on opposite sides of the midpoint of the line. [In other words, the two points X
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ECE 316 Probability Theory and Random Processes Chapter 5 Solutions (Part 1) Xinxin Fan Problems 2. A system consisting of one original unit plus a spare can function for a random amount of time X. If the density of X is given (in units of months)
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ECE 316 Probability Theory and Random Processes Chapter 4 Solutions (Part 3) Xinxin Fan Problems 53. Approximately 80, 000 marriages took place in the state of New York last year. Estimate the probability that for at least one of these couples (a)
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ECE 316 Probability Theory and Random Processes Chapter 4 Solutions (Part 2) Xinxin Fan Problems 20. A gambling book recommends the following winning strategy for the game of roulette. It recommends that a gambler bet $1 on red. If red appears (whi
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ECE 316 Probability Theory and Random Processes Chapter 4 Solutions (Part 1) Xinxin Fan Problems 5. Let X represent the dierence between the number of heads and the number of tails obtained when a coin is tossed n times. What are the possible value
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ECE 316 Probability Theory and Random Processes Chapter 3 Solutions (Part 2) Xinxin Fan Problems 57. A simplified model for the movement of the price of a stock supposes that on each day the stock's price either moves up 1 unit with probability p o
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ECE 316 Probability Theory and Random Processes Chapter 2 Solutions (Part 2) Xinxin Fan Problems 17. If 8 castles (that is, rooks) are randomly placed on a chessboard, compute the probability that none of the rooks can capture any of the others. Th
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ECE-223, Assignment #7 Digital Design, M. Mano, 3rd Edition, Chapter 6 6.6) Design a 4-bit shift register with parallel load using D flip-flops. These are two control inputs: shift and load. When shift = 1, the content of the register is shifted by
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ECE-223, Assignment #3 Chapter 3, Digital Design, M. Mano, 3rd Edition 3.3) Simplify the following Boolean functions, using three-variable maps: a) xy + xyz + xyz b) xy + yz + xyz c) AB + BC + BC 3.5) Simplify the following Boolean functions, using
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ECE-223, Solution for Assignment #8 Digital Design, M. Mano, 3rd Edition, Chapter 7 7.9) A DRAM chip uses two dimensional address multiplexing. It has 13 common address pins with the row address having 1 bit longer than column address. What is the c
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ECE 316 Probability Theory and Random Processes Chapter 8 Solutions Xinxin Fan Problems 2. From past experience a professor knows that the test score of a student taking her nal examination is a random variable with mean 75. (a) Given an upper boun
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ECE 316 Probability Theory and Random Processes Assignment 1 Solutions Xinxin Fan Problems 1. (a) How many dierent 7-place license plates are possible if the rst 2 places are for letters and other 5 for numbers? Solution. By the generalized version
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ECE 438: Digital Integrated Circuits Assignment #4 The Inverter Text: Chapter 5, Digital Integrated Circuits 2nd Ed, Rabaey 1) Consider the CMOS inverter circuit in Figure P1 with the following parameters. Assume long channel transistors and no vel
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ECE 438 Spring 2004 Assignment #5 Combinational Logic Gates in CMOS Chapter 6, Digital Integrated Circuits 2nd 1) Implement the equation X = (A + B) (C + D + E) + F) G using complementary CMOS. Size the devices so that the output resistance is the
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ECE 316 Probability Theory and Random Processes Chapter 7 Solutions (Part 2) Xinxin Fan Problems 50. The joint density of X and Y is given by f (x, y) = Compute E[X 2 | Y = y]. Solution. We rst nd the conditional PDF fX|Y (x | y) as follows: fX|Y (
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ECE-223, Assignment #2 Chapter 2, Digital Design, M. Mano, 3rd Edition 2.2) Simplify the following Boolean expression to a minimum number literals: a) xy + xy b) (x + y)(x + y) c) xyz + xy + xyz d) (A+B)(A+B) 2.3) Simplify the following Boolean expr
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ECE-223, Solutions for Assignment #1 Chapter 1, Digital Design, M. Mano, 3rd Edition 1.4) Convert the following numbers with the indicated bases to decimal: (4310)5, and (198)12. (4310)5 = 4*53 + 3*52 +51+0 = 500 + 75 + 5 = 580 (198)12 = 1*122 + 9*
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ECE-223, Assignment #6 Digital Design, M. Mano, 3rd Edition, Chapter 5 5.2) Construct a JK flip-flop using a D Flip-flop, a 2-to-1 line multiplexer and an inverter. 5.4) A PN flip-flop has four operations: clear to 0, no change, complement, and set
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ECE-223, Solution for Assignment #7 Digital Design, M. Mano, 3rd Edition, Chapter 6 6.6) Design a 4-bit shift register with parallel load using D flip-flops. These are two control inputs: shift and load. When shift = 1, the content of the register i
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ECE-223, Solutions for Assignment #4 Chapter 4, Digital Design, M. Mano, 3rd Edition 4.4) Design a combinational circuit with three inputs and one output. The output is 1 when the binary value of the inputs is less than 3. The output is 0 otherwise.
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ECE-223, Solutions for Assignment #6 Digital Design, M. Mano, 3rd Edition, Chapter 5 5.2) Construct a JK flip-flop using a D Flip-flop, a 2-to-1 line multiplexer and an inverter. 5.4) A PN flip-flop has four operations: clear to 0, no change, compl
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ECE-223, Solutions for Assignment #2 Chapter 2, Digital Design, M. Mano, 3rd Edition 2.2) Simplify the following Boolean expression to a minimum number literals: a) xy + xy b) (x + y)(x + y) c) xyz + xy + xyz d) (A+B)(A+B) a) xy + xy = x ( y+y)= x.