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.

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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 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