Hardware Implementations

Gates and Circuits

Three Main Gates

AND OR NOT

Gate Diagrams

Example 1: [(today is Monday) AND (it is raining)] OR (it is snowing)

Gate Diagrams

Example: What does it represent?

Gate Diagrams

Example: {[(today is Monday) AND (it is raining)] OR (it is snowing)} AND {NOT [(it is raining) AND (it is snowing)]}

Truth Table to Gates

First, build the Boolean algebra expression that gives Z Z = AB + A’B’ Z = (A AND B) OR

(NOT A AND NOT B)

A B Z

T T T

T F F

F T F

F F T

Truth Table to Gates

Z = AB + A’B’ Next, build the

circuit that goes with the Boolean algebra expression Z

A B Z

T T T

T F F

F T F

F F T

Z = AB + A’B’

Binary Arithmetic

We can add binary numbers just like decimal numbers only using base two arithmetic.

For example:

5 101

1110

101

+ 7 + 111 + 111

12 1100 1100

Binary Addition

Notice in addition:

0 0 1 1

+ 0 + 1 + 0 + 1

0 1 1 10

False True True False

False False False True

Sum

Carry

A B Sum

(1) T (1) T (0) F

(1) T (0) F (1) T

(0) F (1) T (1) T

(0) F (0) F (0) F

Sum and Carry

A B Carry

1 1 1

1 0 0

0 1 0

0 0 0

A B Sum

1 1 0

1 0 1

0 1 1

0 0 0

Sum Circuit

A B Sum

1 1 0

1 0 1

0 1 1

0 0 0

Sum = AB’ + A’B

Carry Circuit

A B Carry

1 1 1

1 0 0

0 1 0

0 0 0

Carry = AB

Half Adder - Sum and Carry

Half Adder

The sum digit is 0 if the sum is even. The sum digit is 1 if the sum is odd. The carry is 1 if the sum is greater than 1. Handles the case where we add two binary

digits with no inward carry.

Full Adder

Takes a carry in and produces the result and carry out.

So, we have 3 inputs and two outputs. Combine two half-adders together with an OR gate

to get a full adder for each binary digit. How many half adders would we need to add two

8-digit binary numbers? How many gates?

Full Adder

Subtraction

A B Sub

1 1 0

1 0 1

0 1 1

0 0 0

A B Borrow

1 1 0

1 0 0

0 1 1

0 0 0

Binary Subtraction

We do binary subtraction like decimal subtraction only the borrowing is done in 2’s instead of 10’s.

122 01111010

- 7 - 00000111

115 01110011

Subtraction as Addition

If A = 01111010, B = 00000111, then using the twos-complement representation for –B, we have –B = 11111000 + 1 = 11111001 so

122 01111010

- 7 + 11111001

115 01110011

Binary Multiplication

Again, just like decimal except we add and multiply in binary.

* 0 1

0 0 0

1 0 1

5 101

x 7 x 111

35 100011

NAND Gates and NOT

This gate represents (A NAND NOT B).

NAND Truth Table

A B A NAND B

T T F

T F T

F T T

F F T

NAND

Fact: All other gates (AND, OR, NOT) can be constructed using only NAND gates

Verification:

Exercises

Fill in a truth table and give a Boolean expression for the following circuits.

Exercises

- How would you create a one binary digit multiplier? A two-digit by one-digit multiplier? A two-digit by two-digit multiplier? * 0 1

0 0 0

1 0 1