ee1a2 2011 parta 1 intro and numbers

8/6/2019 Ee1a2 2011 Parta 1 Intro and Numbers

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EE1A2Microprocessor Systems and Digital Logic

Introduction and Numbers

Originally written by Dr. Tim Collins

Edited and presented by Dr Sandra I. Woolley

"Whether you believe you can, or you can't, you are right

Henry Ford


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Content

Binary Arithmetic

Addition and subtraction.

Arithmetic circuits.

Arithmetic Logic Units (ALUs)

Microcontrollers

Microcontroller architecture.

Essential building blocks of a computer.

Programming


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Binary Arithmetic

Number Systems

Decimal

Binary

Hexadecimal

Addition

Long addition

Full adder circuits

Parallel adders

Carry-look-ahead circuitry

Subtraction

Twos complement

Subtraction using a parallel adder


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Arithmetic Logic Units (ALUs)

Adder-Subtraction Circuit

Combining addition and subtraction in a single controllablecircuit

Arithmetic Logic Units

General-purpose arithmetic and logic calculation units

Registers

How memory circuits can simplify the connections to an ALU


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The Anatomy of a Microcontroller

Tri-State Ports and Busses

How a bus can interconnect many different registers withouthuge wiring difficulties.

Connecting an ALU to a Bus

The Building Blocks of a Computer

ALU

Registers

I/O Ports

Program Memory

Programs


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Number Systems Decimal

Base 10

Ten digits, 0-9

Columns represent (from right to left) units, tens, hundreds etc.

123

1v102 + 2v101 + 3v100

or

1 hundred, 2 tens and 3units


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Bases

When counting upwards in base-10, we increase the units digit until

we get to 10 when we reset the units to zero and increase the tensdigit.

So, in base-n, we increase the units until we get to n when we reset

the units to zero and increase the ns digit.

Consider hours-minutes-seconds as an example of a base-60

number system:

Eg. 12:58:43+00:03:20 = 13:02:03

NB. The base of a number is often indicated by a subscript. E.g.(123)

10indicates the base-10 number123.


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Binary

Base-2

Two digits, 0 & 1

Columns represent (from right to left) units, twos, fours, eights

etc.

1111011

1v26 + 1v25 + 1v24 + 1v23 + 0v22 + 1v21 + 1v20

= 1v64 + 1v32 + 1v16 + 1v8 + 0v4 + 1v2 + 1v1

= 123


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Binary Numbers Terminology

Each digit in a binary

number is known as abit.

A group of eight bits

makes a binary number

known as a byte. A group of more than

eight bits is known as a

word.

Typical word lengths are12, 16, 32, 64.

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Decimal to Binary Conversion

123z 2 = 61 remainder1



15z 2 = 7 remainder17z 2 = 3 remainder1

3z 2 = 1 remainder1

1z 2 = 0 remainder1

Least significant bit (rightmost)

Most significant bit (leftmost)

Answer: (123)10

= (1111011)2

Example Converting (123)10 into binary


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Twos Complement

One byte (eight bits) can be used to represent the decimal number

range0 to 255 (unsigned) or

-128 to 127 (signed)

i.e.,

Unsigned: 0 1 2 3 ... 126 127 128 129 130 ... 253 254 255

or

Signed : 0 1 2 3 ... 126 127 -128 -127 -126 ... -3 -2 -1

Negative binary numbers are formed by subtracting from a number

one greater than the maximum possible (i.e. 2

n

or 256 for a byte) For example,

(123)10

= (01111011)2

(-123)10

= (10000101)2

= (133)10

= (256-123)10


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Frequently Asked Question

So how can you tell the difference between:

(-123)10

= (10000101)2

and

(133)10 = (10000101)2

The answer is you cant unless you know whether youre using

signed or unsigned arithmetic:

127128Signed

2550Unsigned

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x

x


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Hexadecimal

Base-16

Sixteen digits, 0-9 and A-F (ten to fifteen)

Columns represent (from right to left) units, 16s, 256s, 4096s

etc.

7B

7v161 + 11v160 = 123


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Decimal to HexConversion

123z

16 = 7 remainder 11 (orB

)7 z 16 = 0 remainder 7

Answer : (123)10 = (7B)16

Converting (123)10 into hex


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Binary to Hex/Hex to Binary

Each group of four binary bits maps on to a single hex digit.

Even very long numbers can be converted easily, treating each hexdigit independently.

0111 1011

7 B

1011 1001 0110 1111 1010

B 9 6 F A

E.g.


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Binary Arithmetic - Addition

Binary long addition works just like decimal long addition.

1 0 0 1 1 10 0 1 1 1 0

0

1

0

1

1

0

1

1

1

0

0

1

+

Carried digits

Result

When we work on

paper we usually

leave less space for

the carried digits.


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Full Adder

Each column of the sum has three inputs

Digits from the two numbers to add (A and B)

Carry bit from previous column

It also has two outputs

SUM Result bit

COUT Carry bit to next column

These are the logical operations performed by a full adder circuit.

ABBCACC

CBASUM

ININOUT

IN

!

!

Full Adder

B A CIN

COUT SUM


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Parallel Adder

To add two n-bit numbers together, n full-adders should be

cascaded. Each full-adder represents a column in the long addition.

ull dder

B A CI

COUT

SUM

Full r

B A CI

COUT

SUM

Full r

B A CI

COUT

SUM

B A B A B A

CI

Q Q Q


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Summary

In digital electronics, numbers are represented using base-2

(binary). Base-16 (hex) is often used by human programmers because

binary to hex conversion is very easy.

Binary numbers may be unsigned or signed (twos complement).

Binary addition works in a similar way to decimal long addition.


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Recommended Private Study

Practice binary and hexadecimal conversions.

Use the binary values of A and B from our binary long additionexample and check the answers using the Cout and SUM formulae.


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Thank You

ee1a2 2011 parta 1 intro and numbers

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