Chapter 1: Digital Computers and Information

Illustration at beginning of each Chapter

Base 10 Binary Base 2 Octal Base 8 Hex bas 16

08 1000 10 8

15 1111 17 F

BCD Binary Coded Decimal

4 bit code represents number 0-9

Base 10 BCD

0 0000

1 0001

9 1001

-Addition

-Subtraction

Parity Bit (checks for transmission errors

Checks if total number of bits is even or odd

Number even parity

1000001 01000001

1010100 11010100

Summary Page

Digital ComputersChapter 1:

Logic Design deals with the basic concepts and tools used to design digital hardware consisting of logic circuits.

Computer Design deals with the additional concepts and tools used to design computers and other complex digital hardware.

Computers and digital hardware in general are referred to as digital systems.

Characteristics of a digital system is the manipulation of discrete elements of information. Any set that is restricted to a finite number of elements contains discrete information.

Examples of discrete sets are the 10 decimal digits, the 26 letters of the alphabet etc.

Discrete elements of information are represented in a digital system by physical quantities called signals. Electrical signals such as voltages and currents are most common. Transistors dominate the circuitry that implements these signals. Signals in most present day electronic digitals systems ase just two discrete values and are therefore said to be binary.

A Bipolar Transistor is a 3 terminal semiconductor sevice in which a small current at one terminal can control a much larger current flowing between the 2nd and 3rd terminal. Transistors can function both as amplifiers ans switches.

+5V

R21K

Hi

Lo

+5V

R11K

LEDS1 LED

Lo Off

Hi On

And Gate +5V

A

10K

B

10K

4.7K

Out

A B Out

+5V

+5V

+5V +5V +5V

Orange Boxes include information not in your text

Digital ComputersChapter 1:

We typically represent two discrete values by ranges of voltages values called HIGH and LOW.

The HIGH output voltage value ranges between 4.0 and 5.5 Volts

The LOW output voltages ranges between -0.5 and 5.5 voltages

The HIGH input range allows 3.0-5.5 volts to be recognized

The LOW input ranges allow -0.5 to 2.0 volts.

The fact that the input ranges are longer than the output ranges allows the circuits to function correctly in spite of variation in their behavior and undesirable noise voltages that may be added or subtracted from their outputs.

Parity Bits: Used to detect errors (if there is excessive noise or errors, how would you detect it?)

An additional bit is sometimes added to a binary code to make the total number of 1’s in the resulting code word even or odd.

Original message(7 bits) Modified with Even Parity (8 Total bits)

1000001 (two 1’s) 01000001 (total # bits is even no change)

1010100 (three 1’s) 11010100 (total # bits is odd, so add a 1 so total is even four 1’s)

Output

0.0

1.0

2.0

3.0

4.0

5.0

LOW (L) or

False (F) or 0

HIGH (H) or

True (T) or 1

INPUT

LOW

HIGH

Digital ComputersChapter 1:

Why is Binary used?

Consider a system with 10 values. The voltages between 0 and 5.0 volts would be divided into 10 ranges. Each of length 0.5 volt. A circuit with have to provide an output with each of these ranges. An input circuit would have to determine which of these belonged to each of these 10 ranges.

If we wanted to compensate for noise that each range would be 0.25 volts. And the boundaries would be less than 0.25 volts

This would require costly and complex electronic circuits and still would be disturbed by small noise voltages.

Instead binary circuits are used with significant variation in output and input ranges. The resulting transistor circuit is simple, easy to design and extremely reliable.

Information Representation:

A Binary Digit is referred to as a bit.

Information is represented as groups of bits.

By using various coding schemes groups of bits can represent discrete symbols.

0000

0001

0010

0011

0100

0101

0110

American Standard Code for Information Interchange (7-bit code) (pg 25 text)

Digital ComputersChapter 1: Codes

Unicode:

A 16 bit code for representing the symbols and ideographs for the worlds languages.

Gray Code:

A code having the property that only one bit at a time changes between codes during counting is a Gray Code.

Binary Coded Decimal: (BCD)

Most commonly used code to represent decimal digits: (binary combinations 1010-1111 not used)

Decimal BCD

0 0000

1 0001

2 0010

3 0011

4 0100

5 0101

6 0110

7 0111

8 1000

9 1001

Octal: (base 8)

Use symbols 0,1,2,3,4,5,6,7 83 82 81 80

Hexadecimal: (base 16):

Use symbols 0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F 163 162 161 160

Decimal: (base 10)

Use symbols 0,1,2,3,4,5,6,7,8,9 104 103 102 100

Base 10 Base 2 Base 8 Base 16

100 10 1 8 4 2 1 64 8 1 256 16 1

102 101 100 23 22 21 20 82 81 80 163 161 160

1 0 0 0 1 0 0 1 0 0 1

10 1 0 1 0 0 1 2 0 0 A

Value

Power

Digital ComputersChapter 1: Number Systems

Power 10

Arithmetic Operations:

Example 1:

Base 10 Base 2

100 10 1 16 8 4 2 1

Carries 0 0 0 0 0

1 2 0 1 1 0 0

+ 1 7 + 1 0 0 0 1

------- -------------

2 9 1 1 1 0 1

Example 2:

Base 10 Base 2

100 10 1 32 16 8 4 2 1

Carries 1 0 1 1 0 0

2 2 1 0 1 1 0

+ 2 3 + 1 0 1 1 1

------- -------------

4 5 1 0 1 1 0 1

Digital ComputersChapter 1: Number Systems

Power 10

Example 1:

Base 10 Base 2

100 10 1 16 8 4 2 1

Borrows 0 0 1 1 0

2 2 1 0 1 1 0

- 1 9 - 1 0 0 1 1

------- -------------

0 3 0 0 0 1 1

The Rules for subtraction are the same in decimal. A borrow here adds 2 (in the decimal system a borrow adds 10)

Column 1

ExplanationColumn 1

0

- 1

Cant take 1 from 0 so we borrow from the next column becomes

10

- 1

------

1

Column 2

1

- 1

This would normally be 0 (1-1)

But we needed to borrow due to the first column, so

11 borrow 1 = 10

10

-1

----

1

Example: (Base 10)

1 13 15 245

-1 - 9 7 - 197

---- ---- ---- -------

4 8 48

Borrow 1 1

Column 3

0

- 1

But we needed to borrow due to the second column, so

10 borrow 1 = 1

1

-1

----

0

Starting

Problem

Result

Digital ComputersChapter 1: Number Systems

Two’s Complement: (used to subtract two numbers by adding) (Hardware simpler)

Subtract a number by converting the subtrahend to a two complement form then adding.

Take the boolean complement of each bit, including the sign bit.

-That is set each 1 to 0 and each 0 to 1. Then add 1

+18 = 00010010

Reverse the digits 11101101

Then add 1 +1

-------------

11101110 = -18

Example: 25 00011001

-18 00010010

B Register18=00010010

Complementer11101110

adder

B Register25=00011001

2s compliment

2s compliment

00011001

+ 11101110

--------------- 7=100000111

Overflow ignored

Example: 18 00010010 00010010 11111001

-25 00011001 2s complement 11100111 + 11100111 reverse digits 00000110

-------------- +1

011111001 Final answer 00000111 =-7Have to reverse process

Result

positive