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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

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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

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 operators

binary 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

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