ece 3110: introduction to digital systems

ECE 3110: Introduction to Digital Systems

Combinational Logic Design Principles

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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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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.

nb 2log

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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.

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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

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Chapter 4 Combinational Logic Design

Principles Analyze Synthesis Fundamental Theory: Switching

Algebra

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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.

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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.

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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

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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.)

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Boolean operators

Complement: X (opposite of X) AND: X Y OR: X + Y

binary operators, describedfunctionally by truth table.

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Logic symbols

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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

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Axioms (postulates)

A1) X=0 if X‡1 A1’ ) X=1 if X‡0 A2) if X=0, then X’=1 A2’ ) 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

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Theorems (Single variable)

Proofs by perfect induction

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Two- and three- variable Theorems

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

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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

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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!

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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,,

+, ‘)

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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

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Next…

N-variables theorems Representations of logic fucntions Read Chapter 4.2 and take notes Combinational circuit analysis