Reliable and Fault-Resilient Schemes for Efficient Radix-4 ...
Radix 4 booth
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PRESENTED BY
DAVIS OOMMEN ABRAHAMRICHU JOSE CYRIAC
BINARY MULTIPLICATION USING BOOTH’S RADIX-4
ALGORITHM
MICROELECTRONICS & VLSI DESIGN NIT CALICUT
WINTER 2012
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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
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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
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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
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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
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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
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RADIX 4 : AN EXAMPLE
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VHDL SIMULATION
VHDL code simulation for the multiplication of two binary numbers
A=00010001(17)B=11110111(-9)
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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
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THANK YOU
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APPENDIX