L02-Comb2Kmap.key

# L02-Comb2Kmap.key - Fall 2009 ECE 18-240 Fundamentals of...

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Fall 2009 ECE 18-240 Fundamentals of Computer Engineering LEC 2: Combinational Logic 2 Don Thomas & William Nace Electrical & Computer Engineering Carnegie Mellon University LEC 2 Fall 09 18-240 LEC2 — 18-240: Where are we. ..? 1 Handout: Lec2; HW1 also out on Blackboard Labs and recitations start this 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-fops and FSMs Chapter 7.1~7.8 9/25 Recitation: HW2 and Quiz

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Fall 09 18-240 LEC2 — Combinational Logic 2 What you know so far Gates Boolean algebra — useful for simplification, unsystematic SOP, POS, a little Verilog Where we are headed Systematic techniques for simplifications — amenable to automation Today: Simplification techniques Big idea 1: Uniting theorem in Boolean algebra Big idea 2: Karnaugh maps (Kmaps) as useful tool for simplification of small-ish functions Big idea 3: Incompletely specified functions 3 A B F 0 0 0 0 1 0 1 0 1 1 1 1 Fall 09 18-240 LEC2 — Big Idea #1: Uniting Theorem Key Tool: The Uniting Theorem A (B' + B) = A F = A's values don't change within the ON-set rows B's values change within the rows where F==1 (“ON set”) G = B's values stay the same within the ON-set rows A's values change within the ON-set rows B is eliminated, A remains A is eliminated, B remains Essence of Simplification: Find two element subsets of the ON-set where only one variable changes its value. This single varying variable can be eliminated! 4 If an input (B) can change without changing the output, that input value is not needed A B F 0 0 1 0 1 0 1 0 1 1 1 0
Fall 09 18-240 LEC2 — Reflections. .. Example from last time ƒ = A' B C + A B' C' + A B' C + A B C' + A B C Problem Easy to see how to apply Uniting Theorem… Hard to know if you applied it in all the right places… …especially in a function of many variables Question: Is there is an easier way to SEE potential simplifications? i.e., potential applications of Uniting Theorem…? Answer Need an intrinsically geometric representation for Boolean f() Something we can draw, see… New Question Can you graph a Boolean function. ..? 5 A B C F 0 0 0 0 0 0 1 1 0 1 0 1 0 1 1 1 1 0 0 0 1 0 1 1 1 1 0 1 1 1 1 1 A way to see all the possible adjacencies of variables Fall 09 18-240 LEC2 — Boolean N-cubes If you tried to “graph” a Boolean eqn in 3 vars, you’d only have values at the corners where a var =0 or =1 Conventionally drawn like this for 3 vars. Value in the circle is values at this corner. This is a 3-cube .

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## This note was uploaded on 05/10/2011 for the course ECE 18240 taught by Professor Hoe during the Fall '08 term at Carnegie Mellon.

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L02-Comb2Kmap.key - Fall 2009 ECE 18-240 Fundamentals of...

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