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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.

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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

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

slide23
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

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