1. Multiplication

2. Multiplier Notation
Partial Products
Logical-AND

3. Shift and Add Paradigm

4. Shift and Add Examples

5. Programmed Multiplication

6. Programmed Multiplication (cont.)

7. Hardware Shift and Add (right)

8. Hardware Shift and Add

9. Hardware Shift and Add (left)

10. Signed Number Multiplication (positive case)

11. Signed Number Multiplication (negative case)

12. Booth’s Recoding (or encoding)
Developed for Speeding Up Multiplication in Early Computers
When a Partial Product of 0 Occurs, Can Skip Addition and Just Shift
Doesn’t Help Multipliers Where Datapaths Go Through Adder Such as Previous Examples
Does Help Designs for Asynchronous Implementation or Microprogramming Since Shifting is Faster Than Addition
Variable Delay – Depends on Number of One’s in
Booth Observed that a String of 1’s May be Replaced as:

13. Booth’s Recoding Example
xn xn xi xi x0 (0)
yi=xi-1 - xi
yn yi y0
EXAMPLE
(0)

14. Booth’s Recoding
Maps Words With Digit Set [0,1] to Those With [-1,1]

15. Sequential Multiplication
X (-310)
Y (recoded)
(-1) Add –A
Shift
(+1) Add +A
11011
Shift
001111
Shift (+1510)

16. Booth Multiplier Example

17. Booth’s Recoding Drawbacks
Number of add/sub Operations are Variable
Some Inefficiencies
EXAMPLE
(0)
Can Use Modified Booth’s Recoding to Prevent
Will Look at This in Later Class

18. Sign Extension Consider 6-bit 2’s Complement Number
s=0 Positive Value; s=1 Negative Value
Show Sign Extension Works:
Definition of 2’s Complement

19. Sign Extension Example
Y (recoding)
(neg. A)
(0 A)
(neg. A)
(0 A)
(neg. A)
(0 A)
(24210)

20. Sign Extension Example
Same Trick as Before, Complement Original Sign Bit
Add 1 to Column 5 1
(neg. A)
(0 A)
(neg. A)
(0 A)
(neg. A)
(0 A)
(24210)

21. Methods for Fast Multiplication
Reduce Number of Partial Products to be Added
Group Multiplier Bits Together
Higher Radix Multiplier
Add the Partial Products Faster

22. Radix-r Shift and Add

23. Radix-4 Multiplication
Shifter is Multi-bit
No Longer a Simple AND of xi with a
Need 4:1 MUX with 0, a, 2a, 3a as Inputs

24. Partial Product Selection
0, a and 2a are easy
3a=a+2a  Requies an Adder!
Need a Way to Compute 3a Efficiently

25. Example With 3a Availability

26. Computing 3a
One Way is to Precompute 3a and Store in Register Initially
Another Way is When 3a Occurs Add -a
Send Carry of 1 to Next into Next Radix-4 Digit of Multiplier
Causes Incoming Multiple to be [0,4] Versus [0,3]
– 4 Because incoming carry to 112 Causes Digit 1002
Multiples 0, 1, 2 Handled Easily
Multiple 3 Converted to –1 With Outgoing Carry of 1
Multiple 4 Converted to 0 With Outgoing Carry of 1
Requires Extra Cycle of Computation Since MSD May Have Carry

27. Example With 3a Availability

28. Using Radices >4 Could Also Use Radices of 8, 16, ...
Bit Groupings of Size 3, 4, ...
Multiple Generation Hardware Becomes More Complex
Must Precompute 3a, 5a, 7a, ....
Or Use 3a With a Carry Scheme
Carry Scheme Converts Multipliers 5a, 6a, 7a to –3a, -2a, -a, etc.
Carry Digit in This Form Becomes a 1

29. Booth Recoding
Modern Arithmetic Circuits DO NOT Apply Booth Recoding Directly
Useful in Understanding Higher-radix Versions of Booth Recoding
No Consecutive 1’s or –1’s Occur Using Previously Seen Booth Recoding
Booth Recoding in Radix-4 Results in the Following:
Only Multiples of a or 2a are Required
These are Easily Obtained Using Shifting and Complementation

30. Modified Booth Recoding
Booth Recoding Results From xi and xi-1
Radix-4 Multiplier Digits Implies Booth Recoding Based on xi+1, xi and xi-1
Similar to Classical Booth Recoding, Modified Booth Recoding Encodes Multipliers into [-2,2]

31. Modified Booth Recoding

32. Example Modified Booth Recoding

33. Example Multiplication with MBR

34. Hardware MBR Example