ECE 3110: Introduction to Digital Systems
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Transcript of ECE 3110: Introduction to Digital Systems
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ECE 3110: Introduction to Digital Systems
Combinational Logic Design Principles
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Dr. Xubin He ECE 3110: Introduction to Digital systems 2
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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Dr. Xubin He ECE 3110: Introduction to Digital systems 3
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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Dr. Xubin He ECE 3110: Introduction to Digital systems 4
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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