Multiplication

Multiplier Notation

Partial ProductsLogical-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)

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:

1 1 12 2 2 2 2 2j j i i j i

Booth’s Recoding Example

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

yn ... yi ... y0

yi=xi-1 - xi

xi xi-1 Operation Comments yi 0 0 shift only string of zeros 0 1 1 shift only string of ones 0 1 0 subtract shift beg. string of ones -1 0 1 addition shift end string of ones 1

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 0101Shift 00101(+1) Add +A 1011 11011Shift 111011(-1) Add –A 0101 001111Shift 0001111 (+1510)

Booth Multiplier Example

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• Consider 6-bit 2’s Complement Number

s=0 Positive Value; s=1 Negative Value• Show Sign Extension Works:

4 3 2 1 0

9 8 7 6 5 4 3 2 1 04 3 2 1 0

49 8 7 6 5

0

49 9 5

0

45

0

2 2 2 2 2 2 2 2 2 2

2 (2 2 2 2 ) 2

2 (2 2 ) 2

2 2

ii

i

ii

i

ii

i

s s s s s p p p p p

s s s s s p p p p p

s s p

s s p

s p

• 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

1 001010 (neg. A) 100000 (0 A) 001010 (neg. A) 100000 (0 A) 110110 (neg. A) 100000 (0 A) 00011110010 (24210)

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

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]

Modified Booth Recoding

Example Modified Booth Recoding

Example Multiplication with MBR

Hardware MBR Example