PRESENTED BY

DAVIS OOMMEN ABRAHAMRICHU JOSE CYRIAC

BINARY MULTIPLICATION USING BOOTH’S RADIX-4

ALGORITHM

MICROELECTRONICS & VLSI DESIGN NIT CALICUT

WINTER 2012

WHY BOOTH’S ALGORITHM?

In ALU, only add/subtract/shift operations are possible.

Multiplication involves 2 basic operations - generation of partial products + their accumulation

2 ways to speed up - reducing number of partial products and/or accelerating accumulation

Fewer partial products generated for groups of consecutive 0’s and 1’s in Booth's algorithm

RADIX-2 : AN OVERVIEW

----------------------------------------------------------------------Xi Xi–1 Yi Explanation

---------------------------------------------------------------------- 0 0 0 No string of 1s in sight 0 1 1 End of string of 1s in x 1 0 -1 Beginning of string of 1s in x 1 1 0 Continuation of string of 1s in x

------------------------------------------------------------------------

EXAMPLE

1 1 0 1 0 1 1 1 0 Operand x 0 -1 1 -1 1 0 0 -1 0 Recoded version y

(0)

TIP: Yi=Xi-1 -Xi

RADIX-2 : AN EXAMPLE

M 0110 +6 X Y 0010(0) +2 Z 0 1 -1 0 RECODED MULTIPLIER

ACCUMULATOR Y Yn-1 Z OPERATIONS

0000 0010 0 0

0000 0001 0 -1 SHIFT

1010

1101

0001

0000

0

1

1

A<-A-M

SHIFT

0011

0001

0000

1000

1

0 0

A<-A+M

SHIFT

0000 1100 0

DRAWBACKS OF RADIX-2 ALGORITHM

Algorithm inefficient with isolated 1'se.g. 001010101(0) recoded as 0 1-1 1-1 1-1 1 -1, requiring 8 instead of 4 operations

RADIX-4 : CODING TECHNIQUE

––––––––––––––––––––––––––––––––––––––––––––––––––––xi+1 xi xi–1 zi/2 Explanation–––––––––––––––––––––––––––––––––––––––––––––––––––– 0 0 0 0 No string of 1s in sight 0 0 1 1 End of string of 1s 0 1 0 1 Isolated 1 0 1 1 2 End of string of 1s 1 0 0 -2 Beginning of string of 1s 1 0 1 -1 End a string, begin new one 1 1 0 -1 Beginning of string of 1s 1 1 1 0 Continuation of string of 1s––––––––––––––––––––––––––––––––––––––––––––––––––––

Example1 0 0 1 1 1 0 1 1 0 1 0 1 1 1 0 (0) Operand x

-2 2 -1 2 -1 -1 0

-2 Radix-4 version z

RADIX 4 : AN EXAMPLE

VHDL SIMULATION

VHDL code simulation for the multiplication of two binary numbers

A=00010001(17)B=11110111(-9)

CONCLUSION

In radix-4 algorithm , n/2=3 steps are used ie. 2 multiplier bits in each step

All shift operations are 2 bit position shifts Compared to radix-2 Booth's algorithm - less

patterns with more partial products; Smaller increase in number of operations

Algorithms can be extended for higher radices also

THANK YOU

APPENDIX