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Arithmetic Circuits (Part II) Randy H. Katz University of California, Berkeley Spring 2007. BCD Circuits Combinational Multiplier Circuit Design Case Study: 8 Bit Multiplier Sequential Multiplier Circuit. Overview. BCD Addition. BCD Number Representation.

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overview
BCD Circuits

Combinational Multiplier Circuit

Design Case Study: 8 Bit Multiplier

Sequential Multiplier Circuit

Overview
bcd addition
BCD Addition

BCD Number Representation

Decimal digits 0 thru 9 represented as 0000 thru 1001 in binary

Addition:

5 = 0101

3 = 0011

1000 = 8

5 = 0101

8 = 1000

1101 = 13!

Problem

when digit

sum exceeds 9

Solution: add 6 (0110) if sum exceeds 9!

5 = 0101

8 = 1000

1101

6 = 0110

1 0011 = 1 3 in BCD

9 = 1001

7 = 0111

1 0000 = 16 in binary

6 = 0110

1 0110 = 1 6 in BCD

bcd addition4
BCD Addition

Adder Design

A

B

A

B

A

B

A

B

3

3

2

2

1

1

0

0

CO

F

A

CI

CO

F

A

CI

CO

F

A

CI

CO

F

A

CI

Cin

S

S

S

S

A1

1

1XX

A2

1X1X

CO

F

A

CI

CO

F

A

CI

0

S

S

Cout

S

S

S

S

3

2

1

0

Add 0110 to sum whenever it exceeds 1001 (11XX or 1X1X)

combinational multiplier
Combinational Multiplier

Basic Concept

multiplicand

multiplier

1101 (13)

1011 (11)

1101

product of 2 4-bit numbers

is an 8-bit number

*

1101

Partial products

0000

1101

10001111

(143)

combinational multiplier6
Combinational Multiplier

Partial Product Accumulation

A1

B1

A1 B0

A0 B1

A0

B0

A0 B0

A3

B3

A3 B0

A2 B1

A1 B2

A0 B3

A2

B2

A2 B0

A1 B1

A0 B2

A3 B1

A2 B2

A1 B3

A3 B3

A3 B2

A2 B3

S7

S6

S4

S5

S3

S2

S1

S0

combinational multiplier7
Combinational Multiplier

Partial Product Accumulation

A

B

A

B

A

B

A

B

A

B

A

B

A

B

A

B

A

B

A

B

A

B

A

B

A

B

A

B

A

B

A

B

3

3

3

2

2

3

3

1

2

2

1

3

3

0

2

1

1

2

0

3

2

0

1

1

0

2

0

1

1

0

0

0

F

A

HA

HA

HA

F

A

F

A

F

A

F

A

F

A

HA

F

A

F

A

S

S

S

S

S

S

S

S

7

6

5

4

3

2

1

0

Note use of parallel carry-outs to form higher order sums

12 Adders, if full adders, this is 6 gates each = 72 gates

16 gates form the partial products

total = 88 gates!

combinational multiplier8
Combinational Multiplier

Another Representation of the Circuit

Sum In

Building block: full adder + and

X

Y

F

A

A

B

Cin

Cout

CO

CI

S

Sum Out

A3

A2

A0

A1

B0

A3 B0

A2 B0

A1 B0

A0 B0

0

C

C

C

C

S

S

S

S

B1

A3 B1

A2 B1

A1 B1

A0 B1

0

C

C

C

C

S

S

S

S

B2

A3 B2

A2 B2

A1 B2

A0 B2

0

C

C

C

C

S

S

S

S

B3

A3 B3

A2 B3

A1 B3

A0 B3

0

C

S

S

S

S

P7

P6

P5

P4

P3

P2

P1

P0

4 x 4 array of building blocks

case study 8 x 8 multiplier
Case Study: 8 x 8 Multiplier

Problem Decomposition

How to implement 8 x 8 multiply in terms of 4 x 4 multiplies?

A7-4

B7-4

A3-0

B3-0

*

A3-0 * B3-0

= PP0

8 bit products

A7-4 * B3-0

= PP1

A3-0 * B7-4

= PP2

A7-4 * B7-4

= PP3

P15-12 P11-8 P7-4 P3-0

P3-0 = PP0

P7-4 = PP0 + PP1 + PP2

P11-8 = PP1 + PP2 + PP3

P15-12 = PP3

3-0

+ Carry-in

3-0

3-0

3-0

+ Carry-in

7-4

7-4

3-0

+ Carry-in

7-4

case study 8 x 8 multiplier10
Case Study: 8 x 8 Multiplier

Calculation of Partial Products

Use 4 4x4 multipliers to create the 4 partial products

case study 8 x 8 multiplier11
Case Study: 8 x 8 Multiplier

Three-At-A-Time Adder

Clever use of the Carry Inputs

Sum A[3-0], B[3-0], C[3-0]:

Two Level Full Adder Circuit

Note: Carry lookahead schemes also possible!

case study 8 x 8 multiplier12
Case Study: 8 x 8 Multiplier

Three-At-A-Time Adder with TTL Components

Full Adders

(2 per package)

Full

Adder

Full

Adder

Full

Adder

Full

Adder

Standard ALU configured as 4-bit

cascaded adder

(with internal carry lookahead)

4-bit ALU/Adder

Note the off-set in the outputs

case study 8 x 8 multiplier13
Case Study: 8 x 8 Multiplier

Accumulation of Partial Products

FA

FA

FA

FA

FA

FA

FA

FA

4-bit ALU/Adder

4-bit ALU/Adder

4-bit ALU/Adder

Just a case of cascaded three-at-a-time adders!

case study 8 x 8 multiplier14
Case Study: 8 x 8 Multiplier

The Complete System (Actual TTL packages)

case study 8 x 8 multiplier15
Case Study: 8 x 8 Multiplier

Package Count and Performance

4 74284/74285 pairs = 8 packages

4 74183, 3 74181, 1 74182 = 8 packages

16 packages total

Partial product calculation (74284/285) = 40 ns typ, 60 ns max

Intermediate sums (74183) = 9 ns/20ns = 15 ns average, 33 ns max

Second stage sums w/carry lookahead

74LS181: carry G and P = 20 ns typ, 30 ns max

74182: second level carries = 13 ns typ, 22 ns max

74LS181: formations of sums = 15 ns typ, 26 ns max

103 ns typ, 171 ns max

sequential multiplier
4-Bit Multiplier Example: 3 x 4 = 12

Four cycles to completion

Cycle

Multiplier

Multiplicand

Product

Initialize

0011

0000 0100

0000 0000

Cycle 0, Multiplier[0]=1

0001

0000 1000

0000 0100

Cycle 1, Multiplier[0]=1

0000

0001 0000

0000 1100

Cycle 2, Multiplier[0]=0

0000

0010 0000

0000 1100

Cycle 3, Multiplier[0]=0

0000

0100 0000

0000 1100

Sequential Multiplier

Product = 0

For i = 0 to 3 do

If Multiplier[0] = 1 then Product = Product + Multiplicand

Shift right the Multiplier

Shift left the Multiplicand

sequential multiplier17
Datapath and Control Block Diagram

External load signals for Multiplier and Multiplicand

Low order bit of multiplier: if 0, shift; if 1, shift and add

Sequential Multiplier

8

Load1

‘0’

Multiplier

Digit

Clock

Start

Digit

8

8

Load2

Multiplicand

‘0’

Shift

Control

16

16

Accumulator

Shift

Clear

Load3

16

Clear

Product

Load3

16

sequential multiplier18
State Diagram

One state for each bit

Sequential Multiplier

Reset

Start’

Idle

Start/Clear

S0

Digit/Load3,ShiftDigit’/Shift

S1

Digit/Load3,ShiftDigit’/Shift

S2

Digit/Load3,ShiftDigit’/Shift

S3

Digit/Load3,ShiftDigit’/Shift

S4

Digit/Load3,ShiftDigit’/Shift

S5

Digit/Load3,ShiftDigit’/Shift

S6

Digit/Load3,ShiftDigit’/Shift

S7

Digit/Load3,ShiftDigit’/Shift

sequential multiplier19
Control Timing BehaviorSequential Multiplier

Clk

State

Start

Clear

Load

Shift

Idle

S0

S1

S2

S3

S4

S5

S6

S7

Idle

Digit0

Digit1

Digit2

Digit3

Digit4

Digit5

Digit6

Digit7

sequential multiplier20
Slightly Revised Block Diagram

If multiplier low order bit is 0 than assert 0 into accumulator

Else pass multiplicand through to accumulator

Simplifies the control!

Sequential Multiplier

8

Load1

‘0’

Multiplier

Digit

8

8

Load2

Clock

Start

Multiplicand

‘0’

Load3-Shift

16

Control

Pass or Zero

Pass/Zero’

16

16

Accumulator

Clear

Load3-Shift

16

Clear

Product

16

sequential multiplier21
Revised State Diagram

Fewer control outputs!

Sequential Multiplier

Reset

Start’

Idle

Start/Clear

S0

Load3-Shift

S1

Load3-Shift

S2

Load3-Shift

S3

Load3-Shift

S4

Load3-Shift

S5

Load3-Shift

S6

Load3-Shift

S7

Load3-Shift

sequential multiplier22
Symbolic/Encoded State Transition Table

State assignment chosen as follows:

Idle state set to 0000, i.e., START resets the state FFs

States S0 to S7 set to 1000 to 1111, easy to implement as a counter

Sequential Multiplier

Current State Start Next State Clear Load3-Shift

Idle 0000 0 0000 Idle 0 0

Idle 0000 1 1000 S0 1 0

S0 1000 - 1001 S1 0 1

S1 1001 - 1010 S2 0 1

S2 1010 - 1011 S3 0 1

S3 1011 - 1100 S4 0 1

S4 1100 - 1101 S5 0 1

S5 1101 - 1110 S6 0 1

S6 1110 - 1111 S7 0 1

S7 1111 - 0000 Idle 0 1

booth multiplier
Tricky encoding scheme to reduce the number of stages in a binary multiplier

Considers two bits at a time rather than one—this cuts the number of multiplier steps in half

Each step is slightly more complex compared to the simple multiplier, but is almost as fast as the basic multiplier stage that it replaces

Booth Multiplier

Searching for ways to speed up the basic multiply step

alternative multiply hardware
32-bit Multiplicand reg, 32 -bit ALU, 64-bit Product reg, (0-bit Multiplier reg)

Rather than shift multiplier right and multiplicand left, we can shift BOTH the product and the multiplier to the right …

Alternative Multiply Hardware

Multiplicand

32 bits

32-bit ALU

Shift Right

Product

(Multiplier)

Control

Write

64 bits

multiply control
Multiplicand Product0010 0000 0011

1a. Add multiplicand to the left half of product &

place the result in the left half of Product register

Multiply Control

Start

Product0 = 1

1.

Test

Product0 = 0

Product0

2. Shift the Product register right 1 bit.

32nd

repetition?

No: < 32 repetitions

Yes: 32 repetitions

Done

motivation for booth s algorithm
Example 2 x 6 = 0010 x 0110: 0010 x 0110 + 0000 shift (0 in multiplier) + 0010 add (1 in multiplier) + 0010 add (1 in multiplier) + 0000 shift (0 in multiplier) 00001100

If ALU can subtract as well as add, get same result as follows: 6 = – 2 + 8 0110 = – 00010 + 01000 = 11110 + 01000

For example

0010 x 0110 0000 shift (0 in multiplier)– 0010 sub(first 1 in multiplier)0000 shift (mid string of 1s) + 0010 add (prior step had last 1) 00001100

Motivation for Booth’s Algorithm

Just add 1111110

(twos complement of 2)

booth multiplier an introduction
Recode each 1 in multiplier as “+2-1”

Converts sequences of 1 to 10…0(-1)

Might reduce the number of 1’s

Booth Multiplier: an Introduction

0 0 1 1 11 1 1 0 0

+1 -1

+1 -1

+1 -1

+1 -1

+1 -1

+1 -1

0 1 0 0 0 0 0 -1 0 0

recoding encoding example
If you use the last row in multiplication, you should get exactly the same result as using the first row (after all, they represent the same number!)Recoding (Encoding) Example

0 1 1 0 111 0 0 0 1 0

(+1 -1) (+1 -1) (+1 -1)

(+1 -1)(+1 -1)

(+1 -1)

+1 0 -1 +1 0 0 -1 0 0 +1 -1 0

booth multiplication example

Sign extension

Booth Multiplication Example

0 0 1 1 0 6x

0 1 1 1 0 14

+1 0 0 -1 0

0 0 0 0 0

1 1 0 1 0 (-6)

0 0 0 0 0

0 0 0 0 0

0 0 1 1 0

0 0 1 0 1 0 1 0 0 84

1 1 1

booth s algorithm implementation approach
Current Bit Bit to the Right Explanation Example Op

1 0 Begins run of 1s 0001111000 sub

1 1 Middle of run of 1s 0001111000 none

0 1 End of run of 1s 0001111000 add

0 0 Middle of run of 0s 0001111000 none

Originally for Speed (when shift is faster than add, it is advantageous to replace adds and subs with shifts)

Basic idea: replace a string of 1s in multiplier with an initial subtract for rightmost 1 in a run of 1’s, then later add back a 1 for the bit to the left of the last 1 in the run

Booth’s Algorithm: Implementation Approach

end of run

beginning of run

middle of run

0 1 1 1 1 0

–1

+ 10000

01111

booth s example 2 x 7
1a. P = P - m 1110 + 1110 1110 01110 shift P (sign ext)

1b. 0010 1111 00111 11 -> nop, shift

2. 0010 1111 10011 11 -> nop, shift

3. 0010 1111 11001 01 -> add

4a. 0010 + 0010 0001 11001 shift

4b. 0010 0000 1110 0 done

Booth’s Example (2 x 7)

Operation Multiplicand Product next?

0. initial value 0010 0000 01110 10 -> sub

booth s example 2 x 3
1a. P = P - m 1110 +1110 1110 11010shift P (sign ext)

1b. 0010 1111 01101 01 -> add + 0010

2a. 0001 01101shift P

2b. 0010 0000 10110 10 -> sub + 1110

3a. 0010 1110 10110 shift

3b. 0010 1111 0101 111 -> nop

4a 1111 0101 1 shift

4b. 0010 1111 10101 done

Booth’s Example (2 x -3)

Operation Multiplicand Product next?

0. initial value 0010 0000 11010 10 -> sub

lecture review
Lecture Review

We have covered:

  • BCD Adders
  • Simple extension of binary adders
  • • Multipliers
  • 4 x 4 multiplier: partial product accumulation
  • extension to 8 x 8 case
  • Sequential Multipliers
  • Booth Multiply Step
  • Recoding to speed up the calculation