CENG 241 Digital Design 1 Lecture 1

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CENG 241Digital Design 1Lecture 1

• Amirali Baniasadi
• amirali_at_ece.uvic.ca

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CENG 241 Digital Design 1

• Instructor
• Amirali Baniasadi (Amir)
• Office hours EOW 441, Only by
  appt.
• Email amirali_at_ece.uvic.ca
  Office Tel [phone]
• Web Page for this class will be
  at
• http//www.ece.uvic.ca/amirali/c
  ourses/CENG241/ceng241.html
• Text Digital Design
• Fourth edition,
• by Morris Mano, Prentice
  Hall Publishers

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Course Structure

• Lectures Mostly follow textbook.
• Reading assignments posted on the web for each
  week.
• Homework Some from the book some will be posted
  on the web site.
• Quizzes 3 in class exams. Dates will be
  announced in advance.
• Note that the above is approximate.

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Course Problems

• Late homework 10 penalty per day up to maximum
  of 5 days (after that Homework will not be
  accepted)
• Guide to completing assignments
• Studying together in groups is encouraged
• Discussion (only)
• Work submitted must be your own

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Course Philosophy

• Book to be used as supplement for lectures (If a
  topic is not covered in the class, or a detail
  not presented in the class, that means I expect
  you to read on your own to learn those details)
• Regular Homework (10)
• Lab (30)- Attend orientation _at_ ELW A359.
• Two Midterms (30)- Dates will be announced in
  advance.
• Final Exam(30)
• To pass the course you should also pass the lab
  and the final exam.

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What are my expectations?

• Stay Positive and Enjoy.

• CommitmentRegular study and homework submission

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This Lecture

• Digital Design?
• Binary Systems

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Binary storage registers

• How do we store binary information?
• Binary cell place to store one bit of
  information. 0 or 1.
• Register a group of binary cells.
• Register transfer An operation in a digital
  system

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Binary storage registers
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Binary information processing
Example Add two 10-bit binary numbers
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Binary logic

• Binary logic deals with variables that take on
  two discrete values and operations that assume
  logical meaning.
• Logic gates electronic circuits that operate on
  one or more input signals to produce an output
  signal.
• Example
• x y x AND y
• 0 0 0
• 0 1 0
• 1 0 0
• 1 1 1

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Electrical signals
Two values 0 or 1
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Symbols for digital logic circuits
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Input-Output signals for gates
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Gates with multiple inputs
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Boolean Algebra

• Basic definitions
• x00xx
• x.11.xx
• x.(yz)(x.y)(x.z)
• x(y.z)(xy).(xz)
• xx1
• x.x0

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Boolean Algebra Theorems

• xxx
• x.xx
• x11
• x.00
• xx.yx
• x.(xy)x

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Boolean Algebra Functions

• examples
• F1xy.z
• F2x.y.zx.y.zx.y
• x.z(yy)x.y
• F2x.zx.y

A Boolean Function can be represented in many
algebraic forms
We look for the most simple form
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Boolean Function Example

• Truth table
• x y z F1
  F2
• 0 0 0 0
  0
• 0 0 1 1
  1
• 0 1 0 0
  0
• 0 1 1 0
  1
• 1 0 0 1
  1
• 1 0 1 1
  1
• 1 1 0 1
  0
• 1 1 1 1
  0

A Boolean Function can be represented in only one
truth table forms
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Boolean Function Implementation
y
Y.z
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Boolean Function Implementation
X.y.z
X.y.z
X.y
X.y
X.z
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Complement of a function

• DeMorgans theorem
• (xy)x.y (x.y)xy
• What about three variables?
• (xyz)?
• Let Axy (Az)A.z(xy).zx.y.z
• (x.y.z)xyz

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Canonical Standard Forms

• Consider two binary variables x, y and the AND
  operation
• four combinations are possible x.y, x.y, x.y,
  x.y
• each AND term is called a minterm or standard
  products
• for n variables we have 2n minterms
• Consider two binary variables x, y and the OR
  operation
• four combinations are possible xy, xy, xy,
  xy
• each OR term is called a maxterm or standard sums
• for n variables we have 2n maxterms
• Canonical Forms
• Boolean functions expressed as a sum of minterms
  or product of maxterms.

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Minterms

• x y z
  Terms Designation
• 0 0 0
  x.y.z m0
• 0 0 1
  x.y.z m1
• 0 1 0
  x.y.z m2
• 0 1 1
  x.y.z m3
• 1 0 0
  x.y.z m4
• 1 0 1
  x.y.z m5
• 1 1 0
  x.y.z m6
• 1 1 1
  x.y.z m7

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Maxterms

• x y z
  Designation Terms
• 0 0 0 M0
  xyz
• 0 0 1 M1
  xyz
• 0 1 0 M2
  xyz
• 0 1 1 M3
  xyz
• 1 0 0 M4
  xyz
• 1 0 1 M5
  xyz
• 1 1 0 M6
  xyz
• 1 1 1 M7
  xyz

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Boolean Function ExamplHow to express
algebraically

• Question How do we find the function using the
  truth table?
• Truth table example
• x y z F1
  F2
• 0 0 0 0
  0
• 0 0 1 1
  1
• 0 1 0 0
  0
• 0 1 1 0
  1
• 1 0 0 1
  1
• 1 0 1 1
  1
• 1 1 0 1
  0
• 1 1 1 1
  0

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Boolean Function ExamplHow to express
algebraically

• 1.Form a minterm for each combination forming a 1
• 2.OR all of those terms
• Truth table example
• x y z F1
  minterm
• 0 0 0 0
  
• 0 0 1 1
  x.y.z m1
• 0 1 0 0
  
• 0 1 1 0
  
• 1 0 0 1
  x.y.z m4
• 1 0 1 0
  
• 1 1 0 0
  
• 1 1 1 1
  x.y.z m7
• F1m1m4m7x.y.zx.y.zx.y.zS(1,4,7)

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Boolean Function ExamplHow to express
algebraically

• Truth table example
• x y z F2
  minterm
• 0 0 0 0
  m0
• 0 0 1 0
  m1
• 0 1 0 0
  m2
• 0 1 1 1
  m3
• 1 0 0 0
  m4
• 1 0 1 1
  m5
• 1 1 0 1
  m6
• 1 1 1 1
  m7
• F2m3m5m6m7x.y.zx.y.zx.y.zx.y.zS(3,5,6,
  7)

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Boolean Function ExamplHow to express
algebraically

• 1.Form a maxterm for each combination forming a 0
• 2.AND all of those terms
• Truth table example
• x y z F1
  maxterm
• 0 0 0 0
  xyz M0
• 0 0 1 1
  
• 0 1 0 0
  xyz M2
• 0 1 1 0
  xyz M3
• 1 0 0 1
  
• 1 0 1 0
  xyz M5
• 1 1 0 0
  xyz M6
• 1 1 1 1
  
• F1M0.M2.M3.M5.M6 ?(0,2,3,5,6)

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Boolean Function ExamplHow to express
algebraically

• Truth table example
• x y z F2
  maxterm
• 0 0 0 0
  xyz M0
• 0 0 1 0
  xyz M1
• 0 1 0 0
  xyz M2
• 0 1 1 1
  
• 1 0 0 0
  xyz M4
• 1 0 1 1
• 1 1 0 1
• 1 1 1 1
• FM0.M1.M2.M4?(0,1,2,4)(xyz).(xyz).(xyz)
  .(xyz)

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Maxterms Minterms Intuitions

• Minterms
• If a function is expressed as SUM of PRODUCTS,
  then if a single product is 1 the function would
  be 1.
• Maxterms
• If a function is expressed as PRODUCT of SUMS,
  then if a single product is 0 the function would
  be 0.
• Canonical Forms
• Boolean functions expressed as a sum of minterms
  or product of maxterms.

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Standard Forms
Standard From Sum of Product or Product of Sum
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Nonstandard Forms
Nonstandard From Neither a Sum of Product nor
Product of Sum
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Implementations
Three-level implementation vs. two-level
implementation
Two-level implementation normally preferred due
to delay importance.
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Digital Logic Gates
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Summary?

• Read textbook readings
• Be up-to-date
• Solve exercises
• Come back with your input questions for
  discussion
• Binary systems, Binary logic.