Systems Multipliers and Other Circuits Shankar Balachandran* - - PowerPoint PPT Presentation

Shankar Balachandran* Associate Professor, CSE Department Indian Institute of Technology Madras

*Currently a Visiting Professor at IIT Bombay

Digital Circuits and Systems

Spring 2015 Week 9 Module 52

Multipliers and Other Circuits

Arithmetic Circuits 2

Arithmetic and Logic Unit (ALU)

 An ALU is a combinational circuit that can perform any of

a number of different arithmetic and logical operation on a pair of n-bit operands.

 Operation to be performed is specified by a set of control

(function-select) inputs.

 ALUs are designed in a bit-slice fashion and slices can be

cascaded to form larger ALUs. Some ALUs are designed as an m-bit group instead of a single bit slice.

 m-bit group enables us to improve carry propagation

within a group using carry lookahead or other similar enhancements.

Arithmetic Circuits 3

Unsigned Binary Multiplication

 Consider unsigned m- and n-bit integers, X and Y

   

    1 1

2 2

n j j j m i i i

y Y x X

 

                        

            1 1 1 1 1

2 2 2 2

m i j i n j j i n j j j m i i i n m k k k

y x y x p Y X P

Arithmetic Circuits 4

Array Multiplier

x1 x0 x2 x3 y0 p0

FA HA HA FA

y1 p1

FA HA FA FA

y2 p2

FA HA FA FA

y3 p3 p4 p5 p6 p7

Arithmetic Circuits 5

x1 x0 x2 x3 y0 p0

FA HA HA FA

y1 p1

FA HA FA FA

y2 p2

FA HA FA FA

y3 p3 p4 p5 p6 p7

m

Performance of Array Multiplier

n

Critical path 1 Critical path 2

        sum

carry and mult

t n t n m t t 1 2 1       

Arithmetic Circuits 7

Signed Multiplication

0 0 0 0 0 0 0 0 0

Positive Multiplicand

0 1 1 1 0 x 0 1 0 1 1

(+14) (+11) (+154)

0 1 1 1 0 0 1 1 1 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 + 0 1 0 0 1 1 0 1 0 1 1 1 1 1 1 1 0 0 1

Negative Multiplicand

1 0 0 1 0 x 0 1 0 1 1

(-14) (+11) (-154)

1 0 0 1 0 1 0 0 1 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 + 1 0 1 1 0 0 1 1 0

Negative Multiplier: Negate multiplier and multiplicand and use the above scheme for positive multiplier.

Arithmetic Circuits 8

Modular Multiplication



Using small multipliers to build larger multipliers

x

A ≡ a7a6a5a4a3a2a1a0 ≡ A7-4 A3-0 A B B ≡ b7b6b5b4b3b2b1b0 ≡ B7-4 B3-0 A3-0*B3-0 A7-4*B3-0 A3-0*B7-4 A7-4*B7-4 A7-4 A3-0 B7-4 B3-0 B3-0 A3-0 *

x

B3-0 A7-4 * B7-4 A3-0 * B7-4 A7-4 *

4×4 multiplier 4×4 multiplier 4×4 multiplier 4×4 multiplier

P3-0 P7-4 P11-8 P15-12 B3-0 A3-0 * B3-0 A7-4 * B7-4 A3-0 * B7-4 A7-4 *

a more convenient arrangement

Arithmetic Circuits 9

Adding Three 4-Bit Numbers (X, Y, Z)

FA x0 y0 z0 FA x1 y1 z1 FA x2 y2 z2 FA x3 y3 z3 FA s1 FA s2 FA s3 s0

Arithmetic Circuits 10

Comparator – Equality Check

 Iterative algorithm:

     

if then else if then go to step else end;

1 1

1. 0; 1; 2. 1 1; 0; 3. 1; 4. 2;

i i i i i

i EQ x y EQ EQ EQ i i i n

 

         

yi xi

EQi EQi+1 y0 x0

EQ0 EQ1 x y

EQO EQI

EQUIV x y

EQO EQI

EQUIV x y

EQO EQI

EQUIV x y

EQO EQI

EQUIV

y1 x1

EQ2

y2 x2

EQ3

yn-1 xn-1

EQn EQn-1

   

?

1 2 1 1 2 1

... ...

n n

x x x x y y y y

 



Arithmetic Circuits 11

4-Bit Comparator

 Magnitude comparator for

n-bit unsigned binary numbers

 Inputs: a3a2a1a0

b3b2b1b0 altbin aeqbin agtbin

 Outputs:

altbout aeqbout agtbout

 Logic Equations:

         

> = = < =

3 0 3 0 3 0 3 0 3 0 3 0 3 0 3 0 3 0 3 0

• ut
• ut
• ut

in in in

a b a b a b ag agtb a tb aeqb altb a b a b eqb altb

         

       

b3-0 a3-0 agtbin altbin aeqbin agtbin altbin aeqbin

4 4

COMP4

 

        

 

    

     

3 0 3 0 3 3 3 3 2 2 3 3 2 2 1 1 3 3 2 2 1 1 3 0 3 0 3 3 2 2 1 1 3 0 3 0 3 0 3 0 3 0 3 0

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 a b a b a b

         

                    

Arithmetic Circuits 12

12-Bit Comparator (unsigned)

b3-0 a3-0 agtbin altbin aeqbin agtbin altbin aeqbin

4 4

COMP4 XLTY4 XEQY4 XGTY4 b3-0 a3-0 agtbin altbin aeqbin agtbin altbin aeqbin

4 4

COMP4 b3-0 a3-0 agtbin altbin aeqbin agtbin altbin aeqbin

4 4

COMP4 XLTY8 XEQY8 XGTY8 XLTY XEQY XGTY 1 x11…0 y11…0 x3…0 x7…4 x11…8 y3…0 y7…4 y11…8

Arithmetic Circuits 13

MIN/MAX Circuit

 Use a 4-bit comparator to design an 8-bit MIN/MAX circuit.

 Inputs:

unsigned inputs, X ≡ x7..0 , Y ≡ y7..0 and a control signal MINMAX

 Output:

   

if if min , 1 max , X Y MINMAX Z X Y MINMAX        

MUX 1 12

Z

X>Y MINMAX Z S Y

1 1

X

1

X

1 1

Y

1

MINMAX COMP

12 12

X Y

X>Y

S

End of Week 9: Module 52

Thank You

Multipliers+Others 14