CSE 246: Computer Arithmetic Algorithms and Hardware Design

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

2. Topics: • Homework • Multiplication & Booth recoding • The Project

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

4. 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. 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 -1

6. Arbitrary sign extension -S S S S S Z4 Z3 Z2 Z1 Z0 is equivalent to -S S S S Z4 Z3 Z2 Z1 Z0 which is in turn equivalent to any of the following -S S S Z4 Z3 Z2 Z1 Z0 -S S Z4 Z3 Z2 Z1 Z0 -S Z4 Z3 Z2 Z1 Z0 which is in turn equivalent to (S – 1) Z4 Z3 Z2 Z1 Z0 the -1 in which is used to produce - on the slide before the previous -1 -1 -1 -1 -1 -1

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

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

9. Essentially the same method, radix 4

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

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

12. The project • What: what is the problem? • Where: where is the problem? • 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. 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. 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.