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Signed Number Multiplication(positive case)

Signed Number Multiplication(negative case)

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:

Booth’s Recoding Example

xn xn-1 ... xi xi-1 ... x0 (0)

yi=xi-1 - xi

yn... yi ... y0

EXAMPLE

0011110011(0)

0100010101

Booth’s Recoding

- Maps Words With Digit Set [0,1] to Those With [-1,1]

Sequential Multiplication

A 1011 (-510)

X 1101 (-310)

Y 0111 (recoded)

(-1) Add –A 0101

Shift 00101

(+1) Add +A 1011

11011

Shift 111011

(-1) Add –A 0101

001111

Shift 0001111 (+1510)

Booth’s Recoding Drawbacks

- Number of add/sub Operations are Variable
- Some Inefficiencies

EXAMPLE

001010101(0)

011111111

- Can Use Modified Booth’s Recoding to Prevent
- Will Look at This in Later Class

Sign Extension Show Sign Extension Works:

- Consider 6-bit 2’s Complement Number
- s=0 Positive Value; s=1 Negative Value

- Definition of 2’s Complement

Sign Extension Example

A 010110 (+2210)

X 001011 (+1110)

Y 010101 (recoding)

11111101010 (neg. A)

0000000000 (0 A)

111101010 (neg. A)

00000000 (0 A)

0010110 (neg. A)

000000 (0 A)

00011110010 (24210)

Sign Extension Example

- Same Trick as Before, Complement Original Sign Bit
- Add 1 to Column 5

1

001010 (neg. A)

100000 (0 A)

001010 (neg. A)

100000 (0 A)

110110 (neg. A)

100000 (0 A)

00011110010 (24210)

Methods for Fast Multiplication

- Reduce Number of Partial Products to be Added
- Group Multiplier Bits Together
- Higher Radix Multiplier

- Add the Partial Products Faster

Radix-r Shift and Add

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

Partial Product Selection

- 0, a and 2a are easy
- 3a=a+2a Requies an Adder!
- Need a Way to Compute 3a Efficiently

Example With 3a Availability

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

Example With 3a Availability

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

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

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]

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