CENG 241 Digital Design 1 Lecture 3 - PowerPoint PPT Presentation

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CENG 241 Digital Design 1 Lecture 3

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... Exampl How to express algebraically 1.Form a minterm for each combination forming a 1 ... * Digital Logic Gates ... PowerPoint Presentation ... – PowerPoint PPT presentation

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Title: CENG 241 Digital Design 1 Lecture 3


1
CENG 241Digital Design 1Lecture 3
  • Amirali Baniasadi
  • amirali_at_ece.uvic.ca

2
This Lecture
  • Review of last lecture
  • Boolean Algebra
  • Lab Location ELW A359
  • B01 Monday, 9/21 130 - 300 pm
  • B02 Tuesday, 9/22 130-300 pm
  • B03 Wednesday, 9/23 130-300 pm
  • B04  Thursday, 9/24 130 300 am
  • B05  Friday, 9/25 130 - 300 pm
  • B06 Thursday 9/24 330-500 pm
  • B07 Monday 9/21 430-600 pm
  • HW 1 announced. Due Friday September 25th.

3
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

4
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

5
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

6
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)

7
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)

8
Implementations
Three-level implementation vs. two-level
implementation
Two-level implementation normally preferred due
to delay importance.
9
Digital Logic Gates
10
Integrated Circuits (ICs)
  • Levels of Integration
  • SSI fewer than 10 gates on chip
  • MSI10 to 1000 gates on chip
  • LSI thousands of gates on chip
  • VLSIMillions of gates on chip
  • Digital Logic Families
  • TTL transistor-transistor logic
  • ECL emitter-coupled logic
  • MOS metal-oxide semiconductor
  • CMOS complementary metal-oxide semiconductor

11
Digital Logic Parameters
  • Fan-out maximum number of output signals
  • Fan-in number of inputs
  • Power dissipation
  • Propagation delay
  • Noise margin maximum noise

12
Gate-Level Minimization
  • The Map Method
  • A simple method for minimizing Boolean functions
  • Map diagram made up of squares
  • Each square represents a minterm

13
Two-Variable Map
14
Two-Variable Map
Maps representing x.y and xy
15
Three-Variable Map
16
Three-Variable Map
Minterms are not arranged in a binary sequence
Minterms arranged in gray code Only one bit
changes from one column to the next
17
Three-Variable Map
Each variable is 1 in 4 squares, 0 in 4 squares
Each variable is 1 in 4 squares, 0 in 4 squares
Variable appears unprimed in squares equal to
1 Variable appears primed in squares equal to 0
18
Three-Variable Map-example 1
Sum of two adjacent minterms can be simplified
to a single AND term consisting of two literals
19
Three-Variable Map-example 2
20
Three-Variable Map-example 3
21
Three-Variable Map-example 4
22
Four-Variable Map
23
Four-Variable Map-example 1
1
24
Four-Variable Map-example 2
25
HW 1
  • HW 1- Due Friday, September 25th (400 PM)
  • Solve the following problems from the textbook
    2-20, 2-21. 3-2, 3-3, 3-4, 3-5 and 3-12.

26
Summary
  • Extension to multiple inputs
  • Positive Negative Logic
  • Integrated Circuits
  • Gate Level Minimization
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