L04-Comb3QM - Fall 2009 ECE 18-240 Fundamentals of Computer...

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Fall 2009 ECE 18-240 Fundamentals of Computer Engineering LEC 4: Systematic Simplification: Logic Minimization via QM Don Thomas & William Nace Electrical & Computer Engineering Carnegie Mellon University LEC 4 Fall-09 18-240 LEC2 — 18-240: Where are we...? 1 Handout: Lec4 HW1 due in your recit section. Short QUIZ in your recit, too Lab1 starts this week, part 1A 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-flops and FSMs Chapter 7.1~7.8
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Fall-09 18-240 L4 — Systematic Simplification via QM What you know Boolean algebra, Kmaps for simplification, some Verilog Good for small stuff, up to ~ 6 variables What you don’t know Techniques for larger simplification, up to tens of vars Techniques you could implement as a computer program Systematic techniques that don’t depend on you having a “good eye” Now what? Quine McCluskey algorithm: Historically important, exact, slow But gives insight on how it’s done automatically 3 Fall-09 18-240 L4 — Definition: Better==? A minimum (optimum) solution for a 2-level SOP …has the fewest number of product terms (i.e. AND gates) …and, among all solutions with the same number of product terms, optimum solution has the fewest total literals (i.e. AND gate inputs) Example: suppose all 3 networks below implement same function F()… 4 Product Terms, 10 literals 3 Prod. Terms, 8 literals 3 Prod. Terms, 6 literals A minimum solution 4
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Fall-09 18-240 L4 — Towards Systematic Methods... Question: are there “bad” ways to group in a Kmap? Of course — you never want to group something small when you can group something big Remember: BIG smaller product term and fewer inputs on AND gate cd ab 00 01 11 10 00 01 11 10 1 1 1 1 cd ab 00 01 11 10 00 01 11 10 1 1 1 1 1 1 1 BAD SIMPLIFICATION 1 1 1 GOOD SIMPLIFICATION 5 Fall-09 18-240 L4 — Terminology: Implicants Informally, any product term you could legitimately circle in the Kmap Formally : For a function F(), a product term that has the property that any pattern of input values that makes the product term == 1 also makes the function F() == 1 In other words, if we know that this product term, call it “p”, is == 1, then this implies that we also know the function F() == 1 here …hence, the name.
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