Ece 3110 introduction to digital systems
This presentation is the property of its rightful owner.
Sponsored Links
1 / 23

ECE 3110: Introduction to Digital Systems PowerPoint PPT Presentation


  • 72 Views
  • Uploaded on
  • Presentation posted in: General

ECE 3110: Introduction to Digital Systems. Combinational Logic Design Principles. Other codes. Character codes (nonnumeric) ASCII (7-bit string) Codes for action/condition/states Codes for Detecting and Correcting Errors Codes for Serial Data Transmission.

Download Presentation

ECE 3110: Introduction to Digital Systems

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


Ece 3110 introduction to digital systems

ECE 3110: Introduction to Digital Systems

Combinational Logic Design Principles


Other codes

Other codes

  • Character codes (nonnumeric)

    • ASCII (7-bit string)

  • Codes for action/condition/states

  • Codes for Detecting and Correcting Errors

  • Codes for Serial Data Transmission

Dr. Xubin He ECE 3110: Introduction to Digital systems


Codes for actions conditions states

Codes for Actions/Conditions/States

  • If there are n different actions, conditions, or states, we can represent them with a b-bit binary code with

  • Ceiling function: the smallest integer greater than or equal to the bracketed quantity.

Dr. Xubin He ECE 3110: Introduction to Digital systems


Ece 3110 introduction to digital systems

Dr. Xubin He ECE 3110: Introduction to Digital systems


Codes for serial data transmission and storage

Codes for serial data transmission and storage

  • Parallel data: disk storage

  • Serial data: telephone network

  • Bit rates: bps, numerically equals to the clock frequency(Hz)

  • Bit time: reciprocal of bit rate

  • Bit cell: time occupied by each bit.

  • Line code: format of actual signal on the line, NRZ (Non-Return-to-Zero)

  • Synchronization signal: identify the significane of each bit in the stream.


Chapter summary

Chapter Summary

  • Positional Number Systems, 2, 8, 10, 16

  • Conversions

  • Representation of Negative Numbers

  • Addition/Subtraction for unsigned and signed numbers

  • Binary multiplication/division

  • BCD, Gray…codes

Dr. Xubin He ECE 3110: Introduction to Digital systems


Chapter 4

Chapter 4

  • Combinational Logic Design Principles

    • Analyze

    • Synthesis

    • Fundamental Theory: Switching Algebra

Dr. Xubin He ECE 3110: Introduction to Digital systems


Combinational logic circuit

Combinational logic circuit

  • Outputs depend only on the current inputs (Not on history)

  • Contain an arbitrary number of logic gates and inverters, but NO feedback loops.

Dr. Xubin He ECE 3110: Introduction to Digital systems


Analysis vs synthesis

Analysis vs. Synthesis

  • Analysis:

    • Start with a logic diagram and proceed to a formal description of the function performed by that circuit.

  • Synthesis:

    • Do the reverse, starting with a formal description and proceeding to a logic diagram.

Dr. Xubin He ECE 3110: Introduction to Digital systems


Combinational circuit analysis

Combinational-Circuit Analysis

  • Kinds of combinational analysis:

    • exhaustive (truth table)

    • algebraic (expressions)

    • simulation / test bench

      • Write functional description in HDL

      • Define test conditions / test vectors, including corner cases

      • Compare circuit output with functional description (or known-good realization)

      • Repeat for “random” test vectors

Dr. Xubin He ECE 3110: Introduction to Digital systems


Switching algebra

Switching algebra

  • a.k.a. “Boolean algebra”

    • deals with boolean values -- 0, 1

  • Positive-logic convention

    • analog voltages LOW, HIGH --> 0, 1

  • Negative logic -- seldom used

  • Signal values denoted by variables(X, Y, FRED, etc.)

Dr. Xubin He ECE 3110: Introduction to Digital systems


Boolean operators

Complement:X¢ (opposite of X)

AND:X × Y

OR:X + Y

Boolean operators

binary operators, describedfunctionally by truth table.

Dr. Xubin He ECE 3110: Introduction to Digital systems


Logic symbols

Logic symbols

Dr. Xubin He ECE 3110: Introduction to Digital systems


Some definitions

Some definitions

  • Literal: a variable or its complement

    • X, X¢, FRED¢, CS_L

  • Expression: literals combined by AND, OR, parentheses, complementation

    • X+Y

    • P × Q × R

    • A + B × C

    • ((FRED × Z¢) + CS_L × A × B¢× C + Q5) × RESET¢

  • Equation: Variable = expression

    • P = ((FRED × Z¢) + CS_L × A × B¢× C + Q5) × RESET¢

Dr. Xubin He ECE 3110: Introduction to Digital systems


Axioms postulates

Axioms (postulates)

  • A1) X=0 if X‡1 A1’ ) X=1 if X‡0

  • A2) if X=0, then X’=1A2’ ) if X=1, then X’=0

  • A3) 0 • 0=0 A3’ ) 1+1=1

  • A4) 1 • 1=1 A4’ ) 0+0=0

  • A5) 0 • 1= 1 • 0 =0 A5’ ) 1+0=0+1=1

Logic multiplication and addition

precedence


Theorems single variable

Theorems (Single variable)

  • Proofs by perfect induction

Dr. Xubin He ECE 3110: Introduction to Digital systems


Two and three variable theorems

Two- and three- variable Theorems

In all of the theorems, it is possible to replace each variable with an arbitrary logic expression.


Duality

Duality

  • Swap 0 & 1, AND & OR

    • Result: Theorems still true

  • Principle of Duality (Metatheorem)

    • Any theorem or identity in switching algebra remains true if 0 and 1 are swapped and • and + are swapped throughout.

  • Why?

    • Each axiom (A1-A5) has a dual (A1¢-A5¢)

Dr. Xubin He ECE 3110: Introduction to Digital systems


Duality1

Duality

  • Counterexample:X + X × Y = X (T9)X × X + Y = X (dual)X + Y = X (T3¢)????????????

X + (X×Y) = X (T9)X× (X + Y) = X (dual)(X× X) + (X× Y) = X (T8)X+ (X× Y) = X (T3¢)

parentheses,operator precedence!

Dr. Xubin He ECE 3110: Introduction to Digital systems


Dual of a logic expression

Dual of a logic expression

  • If F(X1, X2, X3,… Xn,, +, ‘) is a fully parenthesized logic expression involving variables X1, X2, X3,… Xn and the operators +,, and ‘, then the dual of F, written FD, is the same expression with + and  swapped.

  • FD(X1, X2, X3,… Xn, +,, ‘)=F(X1, X2, X3,… Xn,, +, ‘)

Dr. Xubin He ECE 3110: Introduction to Digital systems


Sumamry

Sumamry

  • Variables, expressions, equations

  • Axioms (A1-A5 pairs)

  • Theorems (T1-T15 pairs)

    • Single variable

    • 2- or 3- variable

  • Prime, complement, logic multiplication/addition, precedence

  • Duality

Dr. Xubin He ECE 3110: Introduction to Digital systems


Ece 3110 introduction to digital systems

Next…

  • N-variables theorems

  • Representations of logic fucntions

  • Read Chapter 4.2 and take notes

  • Combinational circuit analysis

Dr. Xubin He ECE 3110: Introduction to Digital systems


  • Login