Transcript Digital Computers Chapter 1:

Summary Page
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
-Addition
-Subtraction
BCD Binary Coded Decimal
4 bit code represents number 0-9
Base 10
BCD
0
0000
1
0001
9
1001
Parity Bit (checks for transmission errors
Checks if total number of bits is even or odd
Number
even parity
1000001
01000001
1010100
11010100
Digital Computers
Chapter 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.
Orange Boxes include information not in your text
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
+5V
R2
1K
Hi
R1
1K
+5V
And Gate
A
10K
Lo
Off
Hi
On
LED
Lo
Out
A
+5V
10K
S1 LED
B
B
+5V
Out
+5V
4.7K
+5V
+5V
Digital Computers
Chapter 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.
HIGH (H) or
True (T) or 1
LOW (L) or
False (F) or 0
Parity Bits: Used to detect errors
Output
INPUT
5.0
4.0
3.0
2.0
1.0
0.0
HIGH
LOW
(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)
Digital Computers
Chapter 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.
American Standard Code for Information Interchange (7-bit code)
0000
0001
0010
0011
0100
0101
0110
(pg 25 text)
Digital Computers
Chapter 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
Digital Computers
Chapter 1: Number Systems
Octal: (base 8)
83
Use symbols 0,1,2,3,4,5,6,7
82
81
80
Hexadecimal: (base 16):
Base 10
Base 2
Base 8
Base 16
Value
100 10 1
8 4 2 1
64 8 1
256 16 1
Power
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
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
Arithmetic Operations:
Example 2:
Example 1:
Base 10
Power 10100
Base 2
10 1
16 8 4 2 1
Carries
Base 10
Base 2
100 10 1
32 16 8 4 2 1
Carries
0 0000
12
0 1100
+ 17
+1 0001
-------
-------------
29
1 1101
10 1100
22
1 0110
+ 23
+1 0111
------45
------------1 0 1101
Digital Computers
Chapter 1: Number Systems
The Rules for subtraction are the same in decimal. A borrow here adds 2 (in the decimal system a borrow adds 10)
Example: (Base 10)
Borrow
1
1
1
13
15
245
-1 - 9
7
- 197
---- ----
----
4
8
------48
Explanation
Column 3
Example 1:
Base 10
Power 10100
Base 2
10 1
16 8 4 2 1
Borrows
Starting
Problem
0 0110
22
1 0110
- 19
-1 0011
-------
-------------
03
0 0011
Column 2
0
1
-1
-1
Column 1
Column 1
0
-1
But we needed to
borrow due to the
second column, so
This would
normally be 0 (1-1)
10 borrow 1 = 1
But we needed to
borrow due to the
first column, so
Cant take 1 from 0
so we borrow from
the next column
becomes
11 borrow 1 = 10
Result
1
10
10
-1
-1
- 1
----
----
------
0
1
1
Digital Computers
Chapter 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
Example:
Result
positive
25
00011001
-18
00010010
=
B Register
25=00011001
2s compliment
Complementer
11101110
2s compliment
+18
B Register
18=00010010
00011001
adder
00010010
Reverse the digits 11101101
Then add 1
+ 11101110
Overflow
ignored
+1
------------11101110
Example: 18
-25
00010010
= -18
00010010
00011001 2s complement 11100111 + 11100111
-------------Have to reverse process
011111001
--------------7=100000111
11111001
reverse digits 00000110
+1
Final answer 00000111 =-7