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Switching Theory / Schakeltechniek 5A050

Lab session Arithmetic

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A0 00 01 11 1

B0 01 01 01 1

C0 01 01 01 1

Z0 01 01 11 1

C AB

Z A

B Z:

1

1 1 1

1

C

smallest possible 2-level SOP expression / implementation truth

table

K-map

Z = AB + C

expression 2-level implementation

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A

B X:

x

1 x x

x C 1 1 D 1

1 1 1

1 1

X = A + B

• Don’t cares: extra freedom to reduce the circuit

• Multi-level optimization: smaller circuits but not exact

Block 3

• Number systems

• Arithmetic circuits

Number Systems

• Decimal system:

– Representation with base 10:

154 = 1 x 10²+ 5 x 10¹+ 4 x 10⁰

– 10 different symbols: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9

• Binary system:

– Representation with base 2:

11010 = 1 x 2⁴+ 1 x 2³+ 0 x 2²+ 1 x 2¹+ 0 x 2⁰ – 2 different symbols: 0, 1

Conversions

• From binary to decimal:

11010 = 1 x 2⁴+ 1 x 2³+ 0 x 2²+ 1 x 2¹+ 0 x 2⁰= 26

• From decimal to binary:

successive division 47 in binary ?

47 / 2 = 23 23 / 2 = 11 11 / 2 = 5 5 / 2 = 2 2 / 2 = 1 1/2 = 0

remainder 1

“ 1

“ 1

“ 1

“ 0“ 1

1 0 1 1 1 1 check: 32 + 8 + 4 + 2 + 1 = 47 !

Arithmetic operations: addition Decimal:

Binary:

1 1 1

+ 0 1 1

1 0 1 0 carries

Arithmetic operations: subtraction Decimal:

Binary:

borrows

Arithmetic circuits: halfadder

Adds two binary numbers:

1 1 + 1 1 0

+ 00 0 0

+ 10 0 1 (2 outputs needed)

+ 01 0 1

B 0 0 1 1 A 0 0 1 1

S 0 1 1 0

C 0 0 0

halfadder

1

HA A

B

S C

AB S

C

Arithmetic circuits: fulladder

B 0 0 1 0 1 0 1 1

Cin 0 0 1 0 1 0 1 1 A 0

0 0 0 1 1 1 1

S 0 1 1 0 1 0 0 1

Cout 0 0 0 0 1 1 1 1

fulladder FA

A

B S

Cin Cout

Arithmetic circuits: ripple carry adder

FA A3 B3

C S3

FA A2 B2

S2

FA A1 B1

S1

HA A0 B0

S0

A3 A2 A1 A0 + B3 B2 B1 B0 C S3 S2 S1 S0 Addition:

Ripple carry adder: example

FA A3 B3

C S3

FA A2 B2

S2

FA A1 B1

S1

HA A0 B0

S0

1 0 1 1 + 0 0 1 1 Addition:

(A)(B) 1 0 0

0 1 1 1

1

0

0

1

1 1

1 1

1 1

1 1 0

1 0

0

Negative numbers: two’s complement

000

001

010

011 100

101 110

111

Most significant bit:

• 0 for positive numbers

• 1 for negative numbers

0

-2

-3

-4

3 2 -1 1

Negative numbers: two’s complement

2’s complement representation of (decimal) number -p:

• take bit-wise complement of the binary representation of p

• add 1

example: 2’s complement representation of -3

• decimal number p: 3

• binary (2’s complement) representation: 011

• bit-wise complement: 100

• add 1: 101

(consistent with number wheel)

What do you obtain if you apply the conversion to 101 ?