ECE 3110: Introduction to Digital Systems
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Transcript of ECE 3110: Introduction to Digital Systems
ECE 3110: Introduction to Digital Systems
Combinational Logic Design Principles
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
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
Dr. Xubin He ECE 3110: Introduction to Digital systems 4
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.
Dr. Xubin He ECE 3110: Introduction to Digital systems 7
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 8
Chapter 4 Combinational Logic Design
Principles Analyze Synthesis Fundamental Theory: Switching
Algebra
Dr. Xubin He ECE 3110: Introduction to Digital systems 9
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 10
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 11
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 12
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 13
Boolean operators
Complement: X (opposite of X) AND: X Y OR: X + Y
binary operators, describedfunctionally by truth table.
Dr. Xubin He ECE 3110: Introduction to Digital systems 14
Logic symbols
Dr. Xubin He ECE 3110: Introduction to Digital systems 15
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
Dr. Xubin He ECE 3110: Introduction to Digital systems 17
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.
Dr. Xubin He ECE 3110: Introduction to Digital systems 19
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 20
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 21
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 22
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 23
Next…
N-variables theorems Representations of logic fucntions Read Chapter 4.2 and take notes Combinational circuit analysis