MULTIPLIERS

B. Baas 188

Multipliers

• Multiplies are widely used in digital signal processing, generally more so than in general-purpose workloads

• Major categories of multiplier types– Signed 2’s complement × Signed 2’s complement

Very useful for fixed-point 2’s complement data

– Unsigned × UnsignedVery useful for sign-magnitude data

• Hardware is typically built in a manner broadly similar to how you would do it with paper and pencil

• The naming convention is somewhat unfortunate:

multiplicand

multiplierx

a3 a2 a1 a0

× b3 b2 b1 b0

s s s p30 p20 p10 p00 b0

s s p31 p21 p11 p01 0 b1

s p32 p22 p12 p02 0 0 b2

p33 p23 p13 p03 0 0 0 b3

B. Baas 189

Multipliers

• Example: 4-bit signed 2’s complement multiplicand “a” times 4-bit multiplier “b”

• b could be signed or unsigned

• s = partial product sign extension bits

• pxy = ax × by

= ax AND by

B. Baas 190

3 Main Steps in Every Multiplier

1) Generation of partial products

2) Reduction or “compression” of the partial product array (normally using carry-save addition) so that the product is composed of two words

– Linear array addition

– Tree addition (Wallace tree)

3) Final adder: Carry-propagate adder (CPA)

– Converts the product in carry-save form into a single word form

– Any style of CPA is fine though we probably favor faster ones

multiplicand

multiplierx

product

Carry-Propagate Adder

Carry-Save

Adders

Partial-

Product

Array

multiplier

multiplicand

B. Baas 191

Straight-forward Partial Product Generation

Yi Partial product

0 0

1 +x (= multiplicand)

• This is the simplest method to generate partial products

• Hardware looks at one bit of the multiplier (Yi) at a time

• Partial products are copies of the multiplicand AND’d by bits of the multiplier

• Number of bits in the multiplier= Number of partial products= Number of terms/words/rows that

must be added

multiplicand

multiplierx

Carry-Propagate Adder

Carry-Save

Adders

Partial-

Product

Array

multiplier

multiplicand

B. Baas 192

Straight-forward Partial Product Generation

Yi Partial product

0 0

1 +x (= multiplicand)

• There are only two possible partial products whose hardware could be built using:

– a row of 2:1 muxes

– a row of AND gates (this should be more efficient)

multiplicand

00

+x

BOOTH ENCODING OF THE “MULTIPLIER” INPUT

B. Baas 194

Booth Encoding

• Method to reduce the number of partial products

• Named after Andrew Booth (1918-2009) who published the algorithm in 1951 while at Birkbeck College, London

• Booth-n

– Examines n+1 bits of the multiplier

– Encodes n bits

– n × reduction in the number of partial products

• But partial products must then be more complex than simply 0 or +multiplicand

Partial-

Product

Array

multiplier

Partial Product

Array

multiplier

B. Baas 195

Booth Encoding:Booth-2 or “Modified Booth”

Yi+1 Yi Yi-1 Partial product Comment

0 0 0 0 no string of 1’s

0 0 1 +x end of string of 1’s

0 1 0 +x a string of 1’s

0 1 1 +2x end of string of 1’s

1 0 0 –2x beginning of string of 1’s

1 0 1 – x –2x + x

1 1 0 – x beginning of string of 1’s

1 1 1 0 center of string of 1’s

[Waser and Flynn]

• Examine multiplier bits Yi+1, Yi, and Yi-1

• Can view multiplier as being built of strings of 1’s– Examine multiplier bits Yi+1, Yi, and Yi-1

– Perspective of moving left towards MSB

• There are 𝑁+2

2= 𝑁

2+ 1 partial products in the worst case

B. Baas 196

Booth Encoding:Booth-2 or “Modified Booth”

Yi+1 Yi Yi-1 Partial product Comment

0 0 0 0 no string of 1’s

0 0 1 +x end of string of 1’s

0 1 0 +x a string of 1’s

0 1 1 +2x end of string of 1’s

1 0 0 –2x beginning of string of 1’s

1 0 1 – x –2x + x

1 1 0 – x beginning of string of 1’s

1 1 1 0 center of string of 1’s

[Waser and Flynn]

• There are five possible partial products compared to two with non-Booth encoding

+2x

+x

0

–x

–2x

B. Baas 197

Booth Encoding:Booth-2 or “Modified Booth”

• Fortunately, these five possible partial products are very easy to generate

• Correctly generating the –x and –2x PPs requires a little care

– The key issue is to not separate 1) negation and 2) adding “1” LSB during the inversion process

multiplicand 0

s

0

multiplicand

0

~multiplicand 1

~s

1

~multiplicand

1

0 0

0

–2x

–x

0

+2x

+x

B. Baas 198

Booth Encoding:Booth-2 or “Modified Booth”

• Example: multiplier = 0010 = 2– Add 0 to right of LSB since first group has no group with

which to overlap

– Examine 3 bits at a time

– Encode 2 bits at a time

Overlap one bit between partial products

– 2x

+x 0 0 1 0 0

–2x

+x

s

0

s

0

4 × (+x) –2x

= +2x

B. Baas 199

Booth Encoding:Booth-2 or “Modified Booth”

• Example: multiplier = 1001 = –7– Add 0 to right of LSB since first group has no group with

which to overlap

– Examine 3 bits at a time

– Encode 2 bits at a time

Overlap one bit between partial products

+x

–2x 1 0 0 1 0

+x

–2x

s

0

s

0

4 × (–2x) +x

= –7x

B. Baas 200

Booth Encoding:Booth-2 or “Modified Booth”

• Example: multiplier = 01111111 = +127– Nice example of encoding a long string of 1’s

– Examine 3 bits at a time

– Encode 2 bits at a time

0

+2x

0 1 1 1 1 1 1 1 0

+2x

–x

s

0

s

0

64 × (+2x) + 16 × (0) + 4 × (0) – x

= +127x

0

0

0s s

–xs ss s

s s

s s

00

00

00

00

00

B. Baas 201

Booth Encoding:Booth-2 or “Modified Booth”

• End of a string of 1’s

• Beginning of a string of 1’s

0 1 1 1 1 1 1 1

+2x

0

0

0 0 1 1 1 1 1 1

+x

0

0

......

1 1 1 1 1 0 0

–2x

0

0

...1 1 1 1 1 1 0

–x

0

0

...

B. Baas 202

Booth Encoding: Booth-3

Yi+2 Yi+1 Yi Yi-1 Partial product

0 0 0 0 0

0 0 0 1 +x

0 0 1 0 +x

0 0 1 1 +2x

0 1 0 0 +2x

0 1 0 1 +3x

0 1 1 0 +3x

0 1 1 1 +4x

1 0 0 0 –4x

1 0 0 1 –3x

1 0 1 0 –3x

1 0 1 1 –2x

1 1 0 0 –2x

1 1 0 1 –x

1 1 1 0 –x

1 1 1 1 0 [Waser and Flynn]