Radix 2 and 4 booth algorithm with example

1. P R E S E N T E D B Y
DAVIS OOMMEN ABRAHAM
RICHU JOSE CYRIAC
BINARY MULTIPLICATION
USING BOOTH’S RADIX-4
ALGORITHM
MICROELECTRONICS & VLSI DESIGN
NIT CALICUT
WINTER 2012

2. 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

3. 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

4. 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

5. DRAWBACKS OF RADIX-2 ALGORITHM
 Algorithm inefficient with isolated 1's
e.g. 001010101(0) recoded as 0 1-1 1-1 1-1 1 -
1, requiring 8 instead of 4 operations

6. 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
––––––––––––––––––––––––––––––––––––––––––––––––––––
Example
1 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

7. RADIX 4 : AN EXAMPLE

8. VHDL SIMULATION
VHDL code simulation for the multiplication of two
binary numbers
A=00010001(17)
B=11110111(-9)

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

10. THANK YOU

11. APPENDIX