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Multiplication. Multiplier Notation. Partial Products Logical-AND. Shift and Add Paradigm. Shift and Add Examples. Programmed Multiplication. Programmed Multiplication (cont.). Hardware Shift and Add (right). Hardware Shift and Add. Hardware Shift and Add (left).

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multiplier notation
Multiplier Notation

Partial Products

Logical-AND

booth s recoding or encoding
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
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
Booth’s Recoding
  • Maps Words With Digit Set [0,1] to Those With [-1,1]
sequential multiplication
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
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
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
sign extension example
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 example1
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
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 4 multiplication
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
Partial Product Selection
  • 0, a and 2a are easy
  • 3a=a+2a Requies an Adder!
  • Need a Way to Compute 3a Efficiently
computing 3 a
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
using radices 4
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
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
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]