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L01-Comb1Reps handout

# L01-Comb1Reps handout - Spring 2009 ECE 18-240 Fundamentals...

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LEC 1 Spring 2009 ECE 18-240 Fundamentals of Computer Engineering LEC 1: Representing Combinational Logic Don Thomas & William Nace Electrical & Computer Engineering Carnegie Mellon University Fall-09 18-240 LEC1 18-240: Where are we...? 1 Handouts: Lec1 No recitations, no labs this week. Start next week. 2 Week Date Lecture Reading Lab HW 1 8/25 L0 Introduction, Comb. Logic Chapter1 (optional) & Chapter2 (optional) No Lab 8/27 L1 Boolean Algebra Ch4.1~4.2 8/28 No Friday recitations 2 9/1 L2 Karnaugh Maps Ch4.3 Lab 0 HW1 9/3 L3 Verilog HDL and simulation Ch5.1 and Ch5.4 9/4 Recitation 3 9/8 L4 Logic Minimization (Q-M Algorithm) Ch4.3~4.4 Lab 1A 9/10 L5 Structured Logic Implementation 9/11 Recitation: HW1 and Quiz 4 9/15 L6 Synthesizable Verilog Chapter 6 Lab 1B HW2 9/17 L7 Comb. Logic Wrap-up Chapter 6 9/18 Recitation: Lab Quiz 5 9/22 L8 Numbers and Arithmetic Chapter 2; HW3 Out Lab 2A 9/24 L9 Flip-flops and FSMs Chapter 7.1~7.8 9/25 Recitation: HW2 and Quiz

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Fall-09 18-240 L1 — Representing Combinational Logic What you (probably) know Overview: Combinational Logic Boolean Algebra, Defining laws (Axioms), some useful manipulations Basic Gates: AND, OR, NOT, NAND, NOR, XOR, … Why simplification? A little bit of motivation What you (probably) don’t know Industrial-strength representation and manipulation techniques Boolean algebra beyond simple stuff Canonical forms for Boolean equations This lecture Some basic methods for representing combinational logic: Boolean algebra, canonical forms, hardware description language form 3 Fall-09 18-240 L1 — Boolean Algebra: Now What? Can do simplification Given an expression, apply Boolean laws to the equation(s) … until the result looks “simple enough” Try this one — 1-bit adder’s carry-out function: Cout := A' B Cin + A B' Cin + A B Cin' + A B Cin Hmm, same problem as regular algebra — how do you known you’re done? A B Cin Cout Sum Hint X = X + X Cout := A Cin + B Cin + A B From last time Cout := (A' B Cin + A B Cin ) + (A B' Cin + A B Cin ) + (A B Cin' + A B Cin) Cout := Cin (A' B + A B' ) + A B Cout := A' B Cin + A B' Cin + A B Cout := A' B Cin + A B' Cin + A B Cin' + A B Cin + A B Cin + A B Cin 4
Fall-09 18-240 L1 — Boolean Algebra — Duality Interesting observation — All the axioms come in “dual” form Anything true for an expression also true for its dual So any derivation you could make that is true, can be flipped into dual form, and it stays true. Duality — More formally A dual of a Boolean expression is derived by replacing Every AND operation by an OR operation Every OR operation by an AND Every constant 1 with a constant 0 Every constant 0 with a constant 1 But don’t change any of the literals, or play with the complements! Example a + ( b c ) = ( a + b ) • ( a + c ) a • (b + c) = (a • b) + (a • c) From last time 5 Fall-09 18-240 L1 — Duality Reloaded Suppose by my Boolean algebra has elements vs.

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L01-Comb1Reps handout - Spring 2009 ECE 18-240 Fundamentals...

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