CSE 246: Computer Arithmetic Algorithms and Hardware Design

1
CSE 246 Computer Arithmetic Algorithms and
Hardware Design
Winter 2004 Lecture 6

• Instructor
• Prof. Chung-Kuan Cheng

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Topics

• Homework
• Multiplication Booth recoding
• The Project

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Homework

• Homework 2 recap
• Need to come up with a number system
  different from the conventional binary system.
  Inventing a new number system would be ideal.
  Show that the number system is better than the
  binary system in certain aspect/s. Also,
  design/describe the basic operations such as
  addition, subtraction, and comparison.
• Due 02/03/2004
• Homework 3
• Textbook problems 4.2, 4.9, 4.11, 4.12, and 4.26
• Due 02/10/2004

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Multiplication Booth recoding
(6)
(A)
(B)
(6)
(5)
x

x
(5)
(30)
(30)

• 6 x 5
• 6 x 5 with 5 Booth recoded

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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 ones complement
representation of the above row into twos
complement.
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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
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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
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Another multiplication example with Booth recoded
multiplicator
(22)
x
(19)
(418)
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Same method, negative multiplicator
(3)
(-1)
(-3)
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Essentially the same method, radix 4
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Shortcut to multiplication by digits in
-2,-1,0,1,2
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x -2,-1,0,1,2 examples
0101 ltlt 1 1010

• 5 x 2 -gt

0101 0000
5 x 0 -gt
000101 111011
5 x -1 -gt
000101 110110
5 x -2 -gt
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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.

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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
  its originally stated or a redefinition of it.
  Unlike the scientists we have the redefinition
  degree of freedom.

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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.