MIT6_004s09_lec09

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

This preview shows pages 1–4. Sign up to view the full content.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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. ..
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: ? =

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### Page1 / 8

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

This preview shows document pages 1 - 4. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online