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Multiplication

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Multiplier Notation

Partial Products

Logical-AND

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Shift and Add Paradigm

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Shift and Add Examples

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Programmed Multiplication

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Programmed Multiplication (cont.)

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Hardware Shift and Add (right)

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Hardware Shift and Add

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Hardware Shift and Add (left)

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Signed Number Multiplication(positive case)

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Signed Number Multiplication(negative case)

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Booths Recoding (or encoding)

Developed for Speeding Up Multiplication in Early Computers

When a Partial Product of 0 Occurs, Can Skip Addition and

Just Shift

Doesnt Help Multipliers Where Datapaths Go ThroughAdder

Such as Previous Examples Does Help Designs for Asynchronous Implementation or

Microprogramming Since Shifting is Faster Than Addition

Variable Delay Depends on Number of Ones in

Booth Observed that a String of 1s May be Replaced as:

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

! L

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Booths Recoding Example

xn

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

yn

... yi ... y0

yi=xi-1 - xi

xi xi-1 peration omments yi

0 0 shi t only string o zeros 0

1 1 shi t only string o ones 0

1 0 subtract shi t beg. string o ones -1

0 1 addition shi t end string o ones 1

EXAMPLE

0011110011(0)

0100010101

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Booths Recoding

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

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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

001111Shift 0001111 (+1510)

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Booth Multiplier Example

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Booths Recoding Drawbacks

Number of add/sub Operations are Variable

Some Inefficiencies

EXAMPLE

001010101(0)011111111

Can Use Modified Booths Recoding to Prevent

W

ill Look at Th

is in Later Class

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Sign Extension Consider 6-bit 2s Complement Number

s=0 Positive Value; s=1 NegativeValue

Show Sign Extension Works:

4 3 2 1 0

9 8 7 6 5 4 3 2 1 0

4 3 2 1 04

9 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

i

i

i

i

i

i

i

i

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 2s Complement

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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)

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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 BitAdd 1 to Column 5

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Methods for Fast Multiplication

Reduce Number of Partial Products to be Added

Group MultiplierBits Together

Higher Radix Multiplier

Add the Partial Products Faster

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Radix-rShift and Add

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Radix-4 Multiplication

Shifter is Multi-bit

No Longer a Simple AND ofxi witha

Need 4:1 MUX wit

h0,a

, 2a

, 3a

as Inputs

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Partial Product Selection

0, a and 2a are easy

3a=a+2ap Requies an Adder!

Need a Way to Compute 3a Efficiently

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Example With 3aAvailability

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Computing 3a

One Way is to Precompute 3a and Store in Register Initially

AnotherWay 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

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Example With 3aAvailability

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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 3aWith a Carry Scheme

Carry Scheme Converts Multipliers 5a, 6a, 7a

to 3a, -2a, -a, etc.

Carry Digit in This Form Becomes a 1

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Booth Recoding

Modern Arit

hmetic Circuits

DO NOTApply

Booth Recoding Directly

Useful in Understanding Higher-radixVersions of

Booth RecodingNo Consecutive 1s or 1s Occur Using Previously Seen

Booth Recoding

Booth Recoding in Radix-4 Results in the Following: Only Multiples ofsa ors2a are Required

These are Easily Obtained Using Shifting and Complementation

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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

Boot

hRecoding, Modified

Boot

h

Recoding Encodes Multipliers into [-2,2]

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Modified Booth Recoding

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Example Modified Booth Recoding

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Example Multiplication with MBR

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Hardware MBR Example