MIT6_004s09_lec09

MIT6_004s09_lec09 - MIT OpenCourseWare http:/ocw.mit.edu...

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MIT OpenCourseWare http://ocw.mit.edu For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms . 6.004 Computation Structures Spring 2009
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L09 - Multipliers 1 6.004 – Spring 2009 3/5/09 Cost/Performance Tradeoffs: a case study Digital Systems Architecture 1.01 modified 2/23/09 10:44 Lab #3 due tonight! L09 - Multipliers 2 6.004 – Spring 2009 3/5/09 Binary Multiplication Engineering Principle: Exploit STRUCTURE in problem. a b a b x n bits n bits 2n bits since (2 n -1) 2 < 2 2n EASY PROBLEM: design combinational circuit to multiply tiny (1-, 2-, 3-bit) operands. .. HARD PROBLEM: design circuit to multiply BIG (32-bit, 64-bit) numbers We can make big multipliers out of litle ones! L09 - Multipliers 3 6.004 – Spring 2009 3/5/09 Making a 2n-bit multiplier using n-bit multipliers Given n-bit multipliers: Synthesize 2n-bit multipliers: x ab a H a L b H b L a L b L a L b H a H b L a H b H a x b = ab n bits n bits 2n bits x a b 2n bits 2n bits ab 4n bits L09 - Multipliers 4 6.004 – Spring 2009 3/5/09 Our Basis: n=1: minimalist starting point Multiplying two 1-bit numbers is preTy simple: a x b = ab 0 Of course, we could start with optimized combinational multipliers for larger operands; e.g. 2 a 1 a 0 2 b 1 b 0 4 c 3 c 2 c 1 c 0 2-bit Multiplier the logic gets more complex, but some optimizations are possible. ..
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Our induction step: 2n-bit by 2n-bit multiplication: 1. Divide multiplicands into n-bit pieces 2. Form 2n-bit partial products, using n-bit by n-bit multipliers. 3. Align appropriately 4. Add. Induction: we can use the same structuring principle to build a 4n-bit multiplier from our newly-constructed 2n-bit ones. .. 6.004 – Spring 2009 3/5/09 L09 - Multipliers 5 x a • b a H a L b H b L a L b L a L b H a H b L REGROUP partial products - 2 additions rather than 3! a H b H Brick Wall view of partial products Making 4n-bit multipliers from n-bit ones: 2 “induction steps” 6.004 – Spring 2009 3/5/09 L09 - Multipliers 6 b 3 2 1 0 x bbb 3 0 a a a a a 0 b 2 a 0 b 3 a 1 b 2 a 1 b 3 a 0 b 0 a 0 b 1 a 1 b 0 a 1 b 1 a 2 b 2 a 2 b 3 a 3 b 2 a 3 b 3 a 2 b 0 a 2 b 1 a 3 b 0 a 3 b 1 Multiplier Cookbook: Chapter 1 Step 1: Form (& arrange) Given problem: Partial Products: Subassemblies: • Partial Products • Adders Step 2: Sum 6.004 – Spring 2009 3/5/09 L09 - Multipliers 7 3 1 a 0 a a a b 0 x MULT ADD a 0 b 2 a 0 b 3 a 1 b 2 a 1 b 3 a 0 b 0 a 0 b 1 a 1 b 0 a 1 b 1 a 2 b 2 a 2 b 3 a 3 b 2 a 3 b 3 a 2 b 0 a 2 b 1 a 3 b 0 a 3 b 1 Performance/Cost Analysis 2 Partial Products: n = 2 ± (n ) Things to Add: 2n -2 = ± (n) Adder Width: 2n = ± 2 Hardware Cost: ? =
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MIT6_004s09_lec09 - MIT OpenCourseWare http:/ocw.mit.edu...

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