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Booth encoding

reduce amount of partial products for

fast multiplication(Chap. 10)

Lecture 15

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Core Problem reduce amount of partial products in multiplication

*)

+)

partial products

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Question generate only twopartial products for this

multiplication

MD

1 1 0 00 0 1 1*) MR

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In Class-Exercise (The Answer) generate only twopartial products for this

multiplication

MD

1 1 0 00 0 1 1*) MR

+MD+)

-MD

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In Class-Exercise (The Answer) Q: the mathematical foundation?

MD

1 1 0 00 0 1 1*) MR

+MD+)

-MD

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Key idea of Booth encoding generate two partial products for each consecutive

sequence of 1s in MR

MD

1 1 0 00 0 1 1*) MR

+)

+MD

-MD

1 1 0 00 0 0 1

+MD

-MD

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Key idea of Booth encoding generate two partial products for each consecutive

sequence of 1s in MR

MD

1 1 0 00 0 1 1*) MR

+)

+MD

-MD

1 1 0 00 0 0 1

+MD

-MD

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How to implement the idea? its difficult to design a hardware to detect

variable length of consecutive 1s

=> scan fixed number of bits (say 2 bits) in

MR each time

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Deriving the Booth-2 encoding:

the first trial

an unsuccessful trial to identify the

problem

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Goal of Booth-2 encoding

generate only n/2 partial products for n-bitmultiplication

*)

+)

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A straight scheme

generate a partial product per 2-bit of MR

*)

+)

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A straight scheme

generate a partial product per 2-bit of MR

*)

+)

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A straight scheme

generate a partial product per 2-bit of MR

*)

+)

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A straight scheme

generate a partial product per 2-bit of MR

*)

+)

2 bits

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Lets look for the encoding rule

Its the MSB of a series of consecutive 1s

*) 0 1

+)

+2MD

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Lets look for the encoding rule

Its the LSB of a series of consecutive 1s

*) 1 0

+)

-2MD

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Lets look for the encoding rule

Q: What should I do?

*) 1 1

+)

???

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Lets look for the encoding rule

You have to look for another bit in MR to determine

*) 01 1

+)

-MD

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Lets look for the encoding rule

You have to look for another bit in MR to determine

*) 11 1

+)

0

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Lets look for the encoding rule

Q: What should I do?

*) 0 0

+)

???

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Lets look for the encoding rule

You have to look for another bit in MR to determine

*) 10 0

+)

+MD

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Booth-2 encoding table

the correct scheme

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Examining MR to generate

partial product

generate a partial product per 2-bit of MR

but examine 3consecutive bits in MR

*)

+)

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Examining MR to generate

partial product

generate a partial product per 2-bit of MR

but examine 3consecutive bits in MR

*)

+)

0

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Examining MR to generate

partial product

generate a partial product per 2-bit of MR

but examine 3consecutive bits in MR

*)

+)

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Examining MR to generate

partial product

generate a partial product per 2-bit of MR

but examine 3consecutive bits in MR

*)

+)

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The Booth-2 encoding table

0 0 0

0 0 1

0 1 0

0 1 1

1 0 01 0 1

1 1 0

1 1 1

0

MD

MD

2*MD

-2*MD-MD

-MD

0

MR[i+1:i-1] Partial Product

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Deriving the encoding table

0 0 0

0 0 1

0 1 0

0 1 1

1 0 0

1 0 1

1 1 0

1 1 1

0

MD

MD

2*MD

-2*MD

-MD

-MD

0

MR[i+1:i-1] Partial Product

*) 10 1

+)

+2MD

ending a series of consecutive ones

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Deriving the encoding table

0 0 0

0 0 1

0 1 0

0 1 1

1 0 0

1 0 1

1 1 0

1 1 1

0

MD

MD

2*MD

-2*MD

-MD

-MD

0

MR[i+1:i-1] Partial Product

*) 01 1

+)

-MD

beginning a series of consecutive ones

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In-Class Exercise (1)

derive the remaining part of the table

0 0 0

0 0 1

0 1 0

0 1 1

1 0 0

1 0 1

1 1 0

1 1 1

0

MD

MD

2*MD

-2*MD

-MD

-MD

0

MR[i+1:i-1] Partial Product

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In-Class Exercise (2)

apply Booth-2 encoding to generate partial products

0 0 0

0 0 1

0 1 0

0 1 1

1 0 0

1 0 1

1 1 0

1 1 1

0

MD

MD

2*MD

-2*MD

-MD

-MD

0

MR[i+1:i-1] Partial Product

MD

1 1 0 00 0 1 1*) MR1 1 0 00 0 0 1

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In-Class Exercise (3)

apply Booth-2 encoding with actual number MD Q: How to deal with negative number?

0 0 0

0 0 1

0 1 0

0 1 1

1 0 0

1 0 1

1 1 0

1 1 1

0

MD

MD

2*MD

-2*MD

-MD

-MD

0

MR[i+1:i-1] Partial Product

MD

1 1 0 00 0 1 1*) MR1 1 0 00 0 0 1

1 1 0 00 1 0 0 0 1 0 01 0 1 1

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Coming Up

fast multiplication in 1 cycle

Chap. 8 & Chap. 11

short encoding for negative partial products

Chap. 11

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Lab 04

Design a Booth-2 multiplier

accepts two 16-bit unsigned numbers A and B

come-out the 32-bit answer C=A*B after 9 cycles

Feel difficult to design a multi-cycle hardware?

Recall: RTL design in digital system course

multiplierA16

B16

START

32 C=A*B

FINISH