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Digital Computers and Digital Systems - Lecture Slides | ENGR 160, Study notes of Microelectronic Circuits

Eastern Washington University (EWU)Microelectronic Circuits

Material Type: Notes; Class: DIGITAL CIRCUITS; Subject: Engineering; University: Eastern Washington University; Term: Spring 2010;

Typology: Study notes

Pre 2010

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1-1 Digital Computers and Digital Systems
Digital Systems
§ A digital system manipulates discrete elements of information.
§ meaningful symbols: alphanumeric characters, arithmetic operators, etc.
§ signals
electrical signals – voltages and currents
signals are binary in nature – two states
more states would reduce reliability
§ examples: digital watch, electronic calculator, remote control, digital instruments,
etc.
Analog Systems
§ An analog system manipulates a continuous signal.
§ An analog signal is a continuous signal analogous to a process or quantity.
electrical signals – voltages and currents
signals are continuous in nature – an infinite number of states between min and
max
§ example: The electrical signal supplied to a speaker system is an analog of the
sound waves.
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Download Digital Computers and Digital Systems - Lecture Slides | ENGR 160 and more Study notes Microelectronic Circuits in PDF only on Docsity!

1-1 Digital Computers and Digital Systems

Digital Systems

ß A digital system manipulates discrete elements of information.

ß meaningful symbols: alphanumeric characters, arithmetic operators, etc.

ß signals

electrical signals – voltages and currents

signals are binary in nature – two states

more states would reduce reliability

ß examples: digital watch, electronic calculator, remote control, digital instruments,

etc.

Analog Systems

ß An analog system manipulates a continuous signal.

ß An analog signal is a continuous signal analogous to a process or quantity.

electrical signals – voltages and currents

signals are continuous in nature – an infinite number of states between min and

max

ß example: The electrical signal supplied to a speaker system is an analog of the

sound waves.

Digital Computers

ß A digital computer is a general purpose machine. It becomes a specific

machine, e.g., a word processor, internet browser, CD player, etc., when it is

programmed as such.

ß Before electronic digital computers, human beings who calculated for a living

were called computers.

ß block diagram

control unit – internal control

processor – arithmetic and logic operations

storage (memory) – short term (RAM), long term (ROM, magnetic media)

input – keyboard, mouse, etc.

output – monitor, printer, speakers, etc.

ß programs

ß data

ß binary digit – bit

ß byte – 8 bits

ß word – 8, 16, 32, or 64 bits

The octal number system (base 8)

ß eight symbols – 0, 1, 2, 3, 4, 5, 6, 7

ß 8 n^ combinations, where n is the number of digits

ß example:

10728 can be written as

The hexadecimal (hex) number system (base 16)

ß sixteen symbols – 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F

ß 16 n^ combinations, where n is the number of digits

ß example:

B65F 16 can be written as

Table 1-1 Numbers with Different Bases

Decimal

(base 10)

Binary

(base 2)

Octal

(base 8)

Hexadecimal

(base 16)

10 1010 12 A

11 1011 13 B

12 1100 14 C

13 1101 15 D

14 1110 16 E

15 1111 17 F

ß Arithmetic operations on numbers with different bases

examples:

Binary Octal Decimal Hexadecimal

111 777 999 FFF

Addition Subtraction Multiplication

+10001101 -10001101 x 101

+1010__

Convert decimal 173 to octal

Integer

Quotient

Remainder Coefficient

173 / 8 = 21 + 5 a 0 = 5 21 / 8 = 2 + 5 a 1 = 5 2 / 8 = 0 + 2 a 2 = 2

Convert decimal 1019 to hex

Integer

Quotient

Remainder Coefficient

1019 / 16 = 63 + 11 a 0 = 11 63 / 16 = 3 + 15 a 1 = 15 3 / 16 = 0 + 3 a 2 = 3

101910 = 3FB 16

Convert hex F0A5 to binary

F = 1111 0 = 0000 A = 1010 5 = 0101

F0A5 16 = 11110000101001012

Convert octal 7025 to binary

Convert decimal 0.375 to binary

Integer

Product

Remainder Coefficient

0.375 ⋅ 2 =^0 +^ 0.75^ a-1^ =^0

0.75 ⋅ 2 =^1 +^ 0.50^ a-2^ =^1

0.50 ⋅ 2 =^1 +^ 0.00^ a-3^ =^1

Convert decimal 6.25 to binary

step 1 – convert the integer part

Integer

Quotient

Remainder Coefficient

6 / 2 = 3 + 0 a 0 = 0 3 / 2 = 1 + 1 a 1 = 1 1 / 2 = 0 + 1 a 2 = 1

step 2 – convert the fractional part

Integer

Product

Remainder Coefficient

0.25 ⋅ 2 =^0 +^ 0.50^ a-1^ =^0

0.50 ⋅ 2 =^1 +^ 0.00^ a-2^ =^1

step 3 – add the results from the first two steps

To check the answer, convert the binary number back to decimal.

110.01 2 = 1 ⋅ 22 + 1 ⋅ 21 + 0 ⋅ 20 + 0 ⋅ 2 -1^ + 1 ⋅ 2 -

Some decimal fractional numbers cannot be represented in binary exactly. One example

is 0.7.

Integer

Product

Remainder Coefficient

0.7 ⋅ 2 =^1 +^ 0.4^ a-1^ =^1

0.4 ⋅ 2 =^0 +^ 0.8^ a-2^ =^0

0.8 ⋅ 2 =^1 +^ 0.6^ a-3^ =^1

0.6 ⋅ 2 =^1 +^ 0.2^ a-4^ =^1

0.2 ⋅ 2 =^0 +^ 0.4^ a-5^ =^0

0.4 ⋅ 2 =^0 +^ 0.8^ a-6^ =^0

0.8 ⋅ 2 =^1 +^ 0.6^ a-7^ =^1

1-5 Complements

Diminished Radix Complement

Given a number N in base r , having n digits, the ( r – 1)’s complement of N is

defined as ( rn^ – 1) – N.

For decimal numbers:

ß r = 10 and ( r – 1) = 9

ß 10 n^ represents a number that consists of a single 1 followed by n 0 ’s.

example: 104 = 10000

ß 10 n^ – 1 is a number represented by n 9 ’s

ß 9 ’s complement of N is (10 n^ – 1) – N

example: 9 ’s complement of 43210 is 99999 – 43210 = 56789

ß The 9 ’s complement of a decimal number is obtained by subtracting

each digit from 9.

For binary numbers:

ß r = 2 and ( r – 1) = 1

ß 2 n^ represents a number that consists of a single 1 followed by n 0 ’s.

example: 24 = 10000

ß 2 n^ – 1 is a number represented by n 1 ’s

ß 1 ’s complement of N is (2 n^ – 1) – N

example: 1 ’s complement of 1010 is 1111 – 1010 = 01012

The 1 ’s complement of a binary number is obtained by subtracting each

digit from 1.

Radix Complement

The r ’s complement of an n -digit number in base r is defined as rn^ – N for N ≠ 0

and 0 for N = 0.

The 10 ’s complement of 43210 = 100000 – 43210 = 56790.

Observe that the 10 ’s complement of N is 1 more than the 9 ’s complement.

The 2 ’s complement of 1010 = 10000 – 1010 = 0110.

Observe that the 2 ’s complement of N is 1 more than the 1 ’s complement.

To take the 2 ’s complement the easy way – invert all digits and add 1.

example: the 2 ’s complement of 1001 is:

Addition of signed binary numbers using 2’s complement:

The extra bit is called an overflow and is stored separately from the byte

representing the sum.

Subtraction of binary numbers using 2’s complement:

(±A) – (+B) = (±A) + (-B)

(±A) – (-B) = (±A) + (+B)

The minuend is 0000 0101 (5 10 ) and the subtrahend is 1111 0010 (-14 10 )

Step 1: Take the 2’s complement of the subtrahend 1111 0010

Step 2: Add the two numbers

How does the computer know if 1100 1000 means 200 or –56?

It doesn’t – it is up to the programmer to interpret the results.

unsigned arithmetic signed arithmetic

Error Detection Code

Parity – an additional bit added to the data to make the total number of ones odd

(odd parity) or even (even parity).

Parity Bit

Odd Parity Even Parity

Message P Message P

Gray Code

Four-bit Gray code

Gray

code

Decimal

equivalent

1-9 Binary Logic

Definition of Binary Logic

Binary logic consists of binary variables and logic operations.

ß Boolean algebra

ß variables take on two discrete values:

true / false yes / no on /off 1 / 0

5v / 0v hot / cold

ß Three basic logic operations:

AND

X ⋅ Y = Z or XY = Z “ X AND Y is equal to Z ”

Z is true if and only if X is true and Y is true; otherwise Z is false.

The alarm will sound if the system is armed and someone touches

the car.

OR

X + X = Z “ X OR Y is equal to Z ”

Z is true if X is true or Y is true; otherwise Z is false.

NOT

X’ = Z “not X is equal to Z ”

Z is true if X is false. Z is false if X is true.

unary operator

Logic Gates

ß A logic gate performs one and only one logic operation (not strictly true –

some also include NOT operations).

ß A logic circuit is composed of one or more logic gates and all

interconnections.

Timing Diagrams