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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

Dr. Xubin He ECE 3110: Introduction to Digital systems

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

- 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

- 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

- Combinational Logic Design Principles
- Analyze
- Synthesis
- Fundamental Theory: Switching Algebra

Dr. Xubin He ECE 3110: Introduction to Digital systems

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:
- 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

- 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

- 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

Complement: X¢ (opposite of X)

AND: X × Y

OR: X + Y

Boolean operatorsbinary operators, describedfunctionally by truth table.

Dr. Xubin He ECE 3110: Introduction to Digital systems

Logic symbols

Dr. Xubin He ECE 3110: Introduction to Digital systems

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)

- 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)

- Proofs by perfect induction

Dr. Xubin He ECE 3110: Introduction to Digital systems

Two- and three- variable Theorems

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

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

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

- 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

- 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

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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