l gica combinacional revisi n 1 n.
Download
Skip this Video
Loading SlideShow in 5 Seconds..
Lógica combinacional (Revisión) 1 PowerPoint Presentation
Download Presentation
Lógica combinacional (Revisión) 1

Loading in 2 Seconds...

play fullscreen
1 / 22

Lógica combinacional (Revisión) 1 - PowerPoint PPT Presentation


  • 64 Views
  • Uploaded on

Lógica combinacional (Revisión) 1. Funciones lógicas, Tablas de verdad, y switches NOT, AND, OR, NAND, NOR, XOR, . . . Minimal set Teoremas y axiomas del algebra booleana Puertas lógicas Redes de funciones booleanas Comportamiento de timing Formas canónicas Dos niveles

loader
I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.
capcha
Download Presentation

PowerPoint Slideshow about 'Lógica combinacional (Revisión) 1' - rodney


An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.


- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript
l gica combinacional revisi n 1
Lógica combinacional (Revisión)1
  • Funciones lógicas, Tablas de verdad, y switches
    • NOT, AND, OR, NAND, NOR, XOR, . . .
    • Minimal set
  • Teoremas y axiomas del algebra booleana
  • Puertas lógicas
    • Redes de funciones booleanas
    • Comportamiento de timing
  • Formas canónicas
    • Dos niveles
    • Funciones incompletamente especificadas
  • Minimización de dos niveles

1Nota: Material tomado del curso: “Components and Design Techniques for Digital Systems”, U. de Berkeley. (libro: Contemporarylogicdesign)

representation of digital designs
Representation of Digital Designs
  • Physical devices (transistors, relays)
  • Switches
  • Truth tables
  • Boolean algebra
  • Gates
  • Waveforms
  • Finite state behavior
  • Register-transfer behavior
  • Concurrent abstract specifications
combinational vs sequential digital circuits
Combinational vs. Sequential Digital Circuits
  • Simple model of a digital system is a unit with inputs and outputs:
  • Combinational means "memory-less"
    • Digital circuit is combinational if its output valuesonly depend on its inputs

inputs

outputs

system

combinational logic symbols
Combinational Logic Symbols
  • Common combinational logic systems have standard symbols called logic gates
    • Buffer, NOT
    • AND, NAND
    • OR, NOR

A

Z

A

Easy to implementwith CMOS transistors(the switches we haveavailable and use most)

Z

B

A

Z

B

sequential logic
Sequential Logic
  • Sequential systems
    • Exhibit behaviors (output values) that depend on current as well as previous inputs
  • Time response of real circuits are sequential
    • Outputs do not change instantaneously after an input change
    • Why not, and why is it then sequential?
  • Fundamental abstraction of digital design is to reason (mostly) about steady-state behaviors
    • Examine outputs only after sufficient time has elapsed for the systemto make its required changes and settle down
synchronous sequential digital systems
Synchronous Sequential Digital Systems
  • Combinational outputs depend only on current inputs
    • After sufficient time has elapsed
  • Sequential circuits have memory
    • Even after waiting for transient activity to finish
  • Steady-state abstraction: most designers use it when constructing sequential circuits
    • Memory of system is its state
    • Changes in system state only allowed at specific times controlled by external periodic signal (the clock)
    • Clock period is time between state changes sufficiently long so that system reaches steady-state before next state change
possible logic functions of two variables

0

1

X

not X

Y

not Y

X xor Y

X = Y

X and Y

X nand Ynot (X and Y)

X or Y

X nor Ynot (X or Y)

Possible Logic Functions of Two Variables
  • 16 possible functions of 2 input variables:
    • 2**(2**n) functions of n inputs

X

F

Y

  • X Y 16 possible functions (F0–F15)0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 10 1 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1
  • 1 0 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1
  • 1 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1
cost of different logic functions
Cost of Different Logic Functions
  • Some are easier, others harder, to implement
    • Each has a cost associated with the number of switches needed
    • 0 (F0) and 1 (F15): require 0 switches, directly connect output to low/high
    • X (F3) and Y (F5): require 0 switches, output is one of inputs
    • X' (F12) and Y' (F10): require 2 switches for "inverter" or NOT-gate
    • X nor Y (F4) and X nand Y (F14): require 4 switches
    • X or Y (F7) and X and Y (F1): require 6 switches
    • X = Y (F9) and X  Y (F6): require 16 switches
    • Because NOT, NOR, and NAND are the cheapest they are the functions we implement the most in practice
minimal set of functions

X Y X nor Y0 0 11 1 0

X Y X nand Y0 0 11 1 0

Minimal Set of Functions
  • Implement any logic functions from NOT, NOR, and NAND?
    • For example, implementing X and Yis the same as implementing not (X nand Y)
  • Do it with only NOR or only NAND
    • NOT is just a NAND or a NOR with both inputs tied together
    • and NAND and NOR are "duals", i.e., easy to implement one using the other
  • Based on the mathematical foundations of logic: Boolean Algebra

X

X

X

X

X

X

X • Y = X + Y

X nand Y not ( (not X) nor (not Y) ) X nor Y not ( (not X) nand (not Y) )

?

logic functions and boolean algebra
Logic Functions and Boolean Algebra
  • Any logic function that can be expressed as a truth table can be written as an expression in Boolean algebra using the operators: ', +, and •

X Y X • Y0 0 00 1 01 0 0

1 1 1

X Y X' X' • Y0 0 1 00 1 1 11 0 0 0

1 1 0 0

X Y X' Y' X • Y X' • Y' ( X • Y ) + ( X' • Y' )0 0 1 1 0 1 10 1 1 0 0 0 01 0 0 1 0 0 0

1 1 0 0 1 0 1

( X • Y ) + ( X' • Y' ) º X = Y

Boolean expression that is true when the variables X and Y have the same value

and false, otherwise

X, Y are Boolean algebra variables

simple example

A

S

B

Cout

Cin

A B Cin S Cout

0 0 0 0 0 1

0 1 0

0 1 1

1 0 0 1 0 1

1 1 0

1 1 1

0

1

1

0

1

0

0

1

0

0

0

1

0

1

1

1

Simple Example
  • 1-bit binary adder
    • Inputs: A, B, Carry-in
    • Outputs: Sum, Carry-out

S = A' B' Cin + A' B Cin' + A B' Cin' + A B Cin

Cout = A' B Cin + A B' Cin + A B Cin' + A B Cin

apply the theorems to simplify expressions
Apply the Theorems to Simplify Expressions
  • Theorems of Boolean algebra can simplify Boolean expressions
    • e.g., full adder's carry-out function (same rules apply to any function)

Cout = A' B Cin + A B' Cin + A B Cin' + A B Cin

= A' B Cin + A B' Cin + A B Cin' + A B Cin + A B Cin

= A' B Cin + A B Cin + A B' Cin + A B Cin' + A B Cin

= (A' + A) B Cin + A B' Cin + A B Cin' + A B Cin

= (1) B Cin + A B' Cin + A B Cin' + A B Cin

= B Cin + A B' Cin + A B Cin' + A B Cin + A B Cin

= B Cin + A B' Cin + A B Cin + A B Cin' + A B Cin

= B Cin + A (B' + B) Cin + A B Cin' + A B Cin

= B Cin + A (1) Cin + A B Cin' + A B Cin

= B Cin + A Cin + A B (Cin' + Cin)

= B Cin + A Cin + A B (1)

= B Cin + A Cin + A B

from boolean expressions to logic gates cont d
From Boolean Expressions to Logic Gates (cont’d)
  • More than one way to map expressions to gates
    • e.g., Z = A' • B' • (C + D) = (A' • (B' • (C + D)))

T2

T1

use of 3-input gate

A

Z

A

B

T1

B

Z

C

C

T2

D

D

waveform view of logic functions
Waveform View of Logic Functions
  • Just a sideways truth table
    • But note how edges don't line up exactly
    • It takes time for a gate to switch its output!

time

change in Y takes time to "propagate" through gates

choosing different realizations of a function

A B C Z0 0 0 0

0 0 1 1

0 1 0 0

0 1 1 1

1 0 0 0

1 0 1 1

1 1 0 1

1 1 1 0

Choosing Different Realizations of a Function

two-level realization(we don't count NOT gates)

multi-level realization(gates with fewer inputs)

XOR gate (easier to draw but costlier to build)

which realization is best
Which Realization is Best?
  • Reduce number of inputs
    • literal: input variable (complemented or not)
      • can approximate cost of logic gate as 2 transistors per literal
      • why not count inverters?
    • Fewer literals means less transistors
      • smaller circuits
    • Fewer inputs implies faster gates
      • gates are smaller and thus also faster
    • Fan-ins (# of gate inputs) are limited in some technologies
  • Reduce number of gates
    • Fewer gates (and the packages they come in) means smaller circuits
      • directly influences manufacturing costs
which is the best realization cont d
Which is the Best Realization? (cont’d)
  • Reduce number of levels of gates
    • Fewer level of gates implies reduced signal propagation delays
    • Minimum delay configuration typically requires more gates
      • wider, less deep circuits
  • How do we explore tradeoffs between increased circuit delay and size?
    • Automated tools to generate different solutions
    • Logic minimization: reduce number of gates and complexity
    • Logic optimization: reduction while trading off against delay
canonical forms
Canonical Forms
  • Truth table is the unique signature of a Boolean function
  • Many alternative gate realizations may have the same truth table
  • Canonical forms
    • Standard forms for a Boolean expression
    • Provides a unique algebraic signature
sum of products canonical forms

A B C F F'0 0 0 0 1

0 0 1 1 0

0 1 0 0 1

0 1 1 1 0

1 0 0 0 1

1 0 1 1 0

1 1 0 1 0

1 1 1 1 0

+ A'BC

+ AB'C

+ ABC'

+ ABC

A'B'C

Sum-of-Products Canonical Forms
  • Also known as disjunctive normal form
  • Also known as minterm expansion

F = 001 011 101 110 111

F =

F' = A'B'C' + A'BC' + AB'C'

sum of products canonical form cont d

A B C minterms0 0 0 A'B'C' m0

0 0 1 A'B'C m1

0 1 0 A'BC' m2

0 1 1 A'BC m3

1 0 0 AB'C' m4

1 0 1 AB'C m5

1 1 0 ABC' m6

1 1 1 ABC m7

Sum-of-Products Canonical Form (cont’d)
  • Product term (or minterm)
    • ANDed product of literals – input combination for which output is true
    • Each variable appears exactly once, in true or inverted form (but not both)
  • F in canonical form:F(A, B, C) =m(1,3,5,6,7)= m1 + m3 + m5 + m6 + m7
  • = A'B'C + A'BC + AB'C + ABC' + ABC
  • canonical formminimal formF(A, B, C) = A'B'C + A'BC + AB'C + ABC + ABC'
        • = (A'B' + A'B + AB' + AB)C + ABC'
        • = ((A' + A)(B' + B))C + ABC'= C + ABC'= ABC' + C
  • = AB + C

short-hand notation forminterms of 3 variables

four alternative two level implementations of f ab c
Four Alternative Two-level Implementations of F = AB + C

A

canonical sum-of-productsminimized sum-of-productscanonical product-of-sumsminimized product-of-sums

B

F1

C

F2

F3

F4

waveforms for the four alternatives
Waveforms for the Four Alternatives
  • Waveforms are essentially identical
    • Except for timing hazards (glitches)
    • Delays almost identical (modeled as a delay per level, not type of gate or number of inputs to gate)