1. CSE 246: Computer Arithmetic Algorithms and Hardware Design Instructor: Prof. Chung-Kuan Cheng Lecture 6

2. CSE 2462 Topics: Homework Multiplication & Booth recoding The Project

3. CSE 2463 Multiplication & Booth recoding (6) (5) (30) x (6) (5) (30) x (A)(B) (A)6 x 5 (B)6 x 5 with 5 Booth recoded

4. CSE 2464 Reformulation of multiplication eliminating sign extension The above is the summation of a six digit multiplicand (the sixth digit is the sign digit) x multiplicator digits. The S1 in the first row (one without negation) is the sign extension, an arbitrary number of S1s can be added in the first row next to S1 without affecting the result. The S terms on the right turn the one’s complement representation of the above row into two’s complement.

5. CSE 2465 Reformulation of multiplication eliminating sign extension (cont.) The “formula” on the previous page with the – at the bottom replaced with the equivalent -1 0 0 0 0 0 -1 -1 -1 -1 -1

6. CSE 2466 Arbitrary sign extension -S S S S S Z 4 Z 3 Z 2 Z 1 Z 0 is equivalent to -S S S S Z 4 Z 3 Z 2 Z 1 Z 0 which is in turn equivalent to any of the following -S S S Z 4 Z 3 Z 2 Z 1 Z 0 -S S Z 4 Z 3 Z 2 Z 1 Z 0 -S Z 4 Z 3 Z 2 Z 1 Z 0 which is in turn equivalent to (S – 1) Z 4 Z 3 Z 2 Z 1 Z 0 the -1 in which is used to produce - on the slide before the previous -1 -1 -1 -1 -1

7. CSE 2467 Another multiplication example with Booth recoded multiplicator (22) (19) (418) x

8. CSE 2468 Same method, negative multiplicator (3) (-1) (-3)

9. CSE 2469 Essentially the same method, radix 4

10. CSE 24610 Shortcut to multiplication by digits in {-2,-1,0,1,2}

11. CSE 24611 x {-2,-1,0,1,2} examples 5 x 2 -> 0101 << 1 1010 5 x 0 -> 0101 0000 5 x -1 -> 000101 111011 5 x -2 -> 000101 110110

12. CSE 24612 The project 1)What: what is the problem? 2)Where: where is the problem? 3)How: how to solve the problem? We have reached the conclusion that the order above is the correct order to tackle a problem of the sort that we need to solve for the project.

13. CSE 24613 The project (cont.) * Could I solve the problem? * I could not solve the problem! * We could not solve the problem! We have reached the conclusion that since the problem we will be dealing with is in the engineering domain, we can either solve it as it’s originally stated or a redefinition of it. Unlike the scientists we have the “redefinition” degree of freedom.

14. CSE 24614 The project (cont.) For the project we may, for instance, … * Come up with a clever way to use transistors and wires to improve (an) aspect/s of performance (perhaps we can go lower than transistors by adopting a different technology) * Come up with a idea to reduce power (low power) * Use optical computing or nanotechnology (quantum, molecular) to a good end * Come up with a fault tolerant system * Come up with a system with a reconfigurable datapath * Write a survey about using nanotechnology or optical computing to improve area, power, execution speed, and etc. of a circuit. If choose to do this for the final project, must have a comprehensive understanding of the topic.