COE 202 notes - Airbus5717

resources
Data Representation
Binary Logic and Gates
TODO Standard & Canonical Forms
TODO Other Gate types
TODO Verilog (1/3)

IMPORTANT NOTICE:

Mobile screens may not display the web page properly due to alignment issues

These notes are not enough for high grade (u need to practice and read the slides)

Notes does not cover the whole material

Every Professor teaches with a different ordering for courses

These notes are according to Dr. Al-Suwaiyan’s order of sections

source of the notes are on https://github.com/airbus5717/coe202

This document is generated by orgmode with the emacs text editor

resources

GOOGLE DRIVE (Exams and Materials)

click here

Videos

Dr. Aiman El-Maleh

Microsoft stream videos: click here

Dr. M. Mudawar

YouTube playlist: click here

Dr. Ali Al-Suwaiyan

YouTube playlist: click here

UNIT (For Suwaiyan videos)	YT video lecture
1. Numbering systems	1-5
2. Boolean algebra	6-8
3. Std and canonical forms	9-10
4. Verilog(1/3)	11, 11.5
5. K-Map simplification	12-15(till min 24)
6. Other gate types	(contine)15-17
7. Combination Logic Design Procedure	18 (till min 24)
8. Arithmetic Circuits	(continue) 18-23
9. Functional blocks	24-28
10. Verilog(2/3)	29.1, 29.2, homework 1-2
11. Analysis & Design of Sequential Circuits	30-34
12. Registers and counters	35-38(till min 20)
13. Verilog(3/3)	(contine)38-39
14. review	40

Data Representation

Introduction

computers represent data in binary numbers (1 and 0)
all data must be represented in binary format
data could be numbers, alphanumeric characters, images, sounds and many more.
in general they are (numbers and characters)

Numbering Systems

Numbering systems are characterized by their base number (also called radix or r for short)
a number with base n will have digits from 0 to (n-1)
for example base 2 includes : 0 and 1
the widely used numbering systems are:

Numbering system Base digits set

Binary 2 0, 1

Octal 8 0, …, 7

Decimal 10 0, 1, …, 9

Hexadecimal 16 0, … 9, A, … F

Numbering system	Base	digits set
Binary	2	0, 1
Octal	8	0, …, 7
Decimal	10	0, 1, …, 9
Hexadecimal	16	0, … 9, A, … F

Weighted Number Systems

a number D consists of n digits with each digit having a particular position.
Every Digit has a fixed weight

(from el-maleh slides)

for example in base 10: 10 = 1 * 10 + 0 * 10

The Radix (`Base`)

the allowed set of digits are from 0 to r-1 for example in base 8: 0...7

revise the (El-Maleh’s) slides (7-12)

Digit weight

example a number in base 8: 34556

the most significant digit (MSD) is : 3
the least significant digit (LSD) is : 6

Binary System

the r = 2
each digit is either 1 or 0
each bit represents a power of 2

2ⁿ Decimal value

2⁰ 1

2¹ 2

2² 4

2³ 8

2⁴ 16

2⁵ 32

2⁶ 64

2⁷ 128

2⁸ 256

2⁹ 512

2¹⁰ 1024
example of conversion from binary to decimal binary number (101) = 1 * 2² + 0 * 2¹ + 1 * 2⁰ = 5 in decimal
see a YouTube video on conversion from decimal to binary click here
another one on from binary to decimal click here

2ⁿ	Decimal value
2⁰	1
2¹	2
2²	4
2³	8
2⁴	16
2⁵	32
2⁶	64
2⁷	128
2⁸	256
2⁹	512
2¹⁰	1024

Octal System

r = 8
Octal digits = {0, 1, 2, 3, 4, 5, 6, 7}
conversion videos click here for octal 2 decimal and click here for decimal 2 octal

Hexadecimal System

r = 16
Digits are 0..9 then A, B, C, D, E, F
F is equavilant to 15 in decimal
conversion videos click here for deci 2 hex and click here for hex 2 deci also click here for hex 2 binary and click here for binary 2 hex.

Integers in (Binary, Octal, Decimal and Hexadecimal)

Binary(2)	Octal(8)	Decimal(10)	Hexadecimal(16)
0000	0	0	0
0001	1	1	1
0010	2	2	2
0011	3	3	3
0100	4	4	4
0101	5	5	5
0110	6	6	6
0111	7	7	7
1000	10	8	8
1001	11	9	9
1010	12	10	A
1011	13	11	B
1100	14	12	C
1101	15	13	D
1110	16	14	E
1111	17	15	F

Binary coded decimal

every number is represented as 4 bits
there are different ways to represent it
- BCD8421 way is like for 1: 0001 and so forth
- XS-3 is like BCD8421 but add 3 to BCD8421
example 103
- BCD8421: 0001 0000 0011
- XS-3: 0100 0011 0110

ASCII Chars

each number/char/symbol is represented with a number from 0 to 127
extended ascii has to 256 numbers
ascii table link

Error detection by parity bit

Sender tries to send data to reciever which is encodeded in binary
Data could get corrupted during transmission
so basically we construct a basic error checker that would help reduce errors (but not always the case)

Parity Bit

We choose an even or odd parity bit
Even parity: number of 1s is even
- add zero to keep the 1s even
Odd parity: number of 1s is odd
- add zero to keep the 1s odd

check slides for example

Binary Logic and Gates

Boolean Algebra

Digital circuts use binary Systems, signals can be either logic ’0’ or ’1’ corresponding to High(1) or Low(0) voltages.

Digital Circuts take binary signals and produce binary signals
Binary logic algebra is commonly referred to as binary algebra
Algebric structure with
- Booleans = {0 , 1}
- Two Binary Operators = {., +}
  - the dot . represents AND
  - the plus sign + represents OR
- One Unary operator = (written as a Dash above the symbol or as ' after the symbol)
Example F = A.B + 1

Operator precedence

Parethesis ( Expression ) {Highest priority}
NOT operator Expr'
AND operator Expr1 . Expr2
OR operator Expr1 + Expr2 {Lowest priority}

Logic Gates & Operations

Gate drawings are from wikimedia (C)

`AND` Gate

Truth table for (AB) or (A*B)

A	B	AB
0	0	0
0	1	0
1	0	0
1	1	1

`OR` Gate

Truth Table for (A+B)

A	B	A+B
0	0	0
0	1	1
1	0	1
1	1	1

`NOT` Gate

Truth Table for (A’)

A	A’
0	1
1	0

`NAND` Gate

Truth Table for (AB)’

A	B	(AB)’
0	0	1
0	1	1
1	0	1
1	1	0

`NOR` Gate

Truth Table for (A+B)’

A	B	(A+B)’
0	0	1
0	1	0
1	0	0
1	1	0

`XOR` Gate

Truth Table for (A XOR B)

A	B	A XOR B
0	0	0
0	1	1
1	0	1
1	1	0

`XNOR` Gate

Truth Table for (A XNOR B)

A	B	A XNOR B
0	0	1
0	1	0
1	0	0
1	1	1

NOTE: solve a few problems on slides and etc.

Basic Identities and Duality principle

Identities Table

Prop.Name	Expression	Dual
Identity	X + 0 = X	X . 1 = X
NULL	X + 1 = 1	X . 0 = 0
Idempotence	X + X = X	XX = X
Complement	X + X’ = 1	XX’= 0
Commutative	X+Y = Y+X	XY = XY
Associative	(X+Y)+Z = X+(Y+Z)	(XY)Z = X(YZ)
Distributive	X(Y+Z) = XY + XZ	X+YZ = (X+Y)(X+Z)
Absorption	X + XY = X	X(X + Y) = X
Minimization	XY + X’Y = Y	(X+Y)(X’ + Y) = Y
Simplification	X + X’Y = X+Y	X(X’+ Y) = XY
Consensus	XY+X’Z+YZ = XY+X’Z	(X+Y)(X’+Z)(Y+Z)=(X+Y)(X’+Z)
DeMorgan’s	(X+Y)’ = X’Y’	(XY)’ = X’ + Y’

to get a dual

Change every AND to OR (. to +)
Change every OR to AND (+ to .)
Complement variables
- change 1 to 0
- change 0 to 1

Dual can be used to simplify a function but must be dualed back and the simplification

Algerbraic Manipulations

Solve problems in Suwaiyan slides from [BooleanAlgebra slides]
from page 11 to 16

also practice more

TODO Standard & Canonical Forms

Canonical form

Minterms

Minterm is a product term which all literals show up the product in true or compliment form

ABC is a minterm
AB’C’ is a minterm
AB is not a minterm (missing C)

example: M2 = A’BC’ = because 2 is 010 M6 = ABC’ = 110

minterms are boolean functions

a minterm will be 1 only once in the Truth Table (TT)

(mi) will be 1 at row i of TT and 0 otherwise

example:

A	B	C	m3	m5	m6
0	0	0	0	0	0
0	0	1	0	0	0
0	1	0	0	0	0
0	1	1	1	0	0
1	0	0	0	0	0
1	0	1	0	1	0
1	1	0	0	0	1
1	1	1	0	0	0

Sum of minterms

All boolean functions can be written in sum-of-minterms(SOM) form

x’y = 01
xy’ = 10

F(x, y) = m1 + m2 = x'y + xy'

Example: express G(a, b, c) in TT below in SOM canonical form using SIGMA(Σ) notation
a b c G

0 0 0 1

0 0 1 0

0 1 0 1

0 1 1 0

1 0 0 0

1 0 1 1

1 1 0 0

1 1 1 0
- G(a, b, c) = Σm(0, 2, 5)
- G(a, b, c) = a’b’c’ + a’bc’ + ab’c

a	b	c	G
0	0	0	1
0	0	1	0
0	1	0	1
0	1	1	0
1	0	0	0
1	0	1	1
1	1	0	0
1	1	1	0

Maxterms

Maxterm: Sum term where all literals show up the sum in true or complement form

maxterm examples

A + B + C’
A’+ B’ + C’
A + C (WARN: NOT A MAXTERM (B is missing))

A	B	C	F	M1	M4	M7
0	0	0	1	1	1	1
0	0	1	0	0	1	1
0	1	0	1	1	1	1
0	1	1	1	1	1	1
1	0	0	0	1	0	1
1	0	1	1	1	1	1
1	1	0	1	1	1	1
1	1	1	0	1	1	0

F(A, B, C) = M1*M4*M7 = πM(1, 4, 7)

Use DeMorgan’s law to convert between the two canonical forms

Example: G(A, B, C) = Σm(1, 3, 4)

G(A, B, C) = m1 + m3 + m4 G(A, B, C) = πM(0, 2, 5, 6, 7)
Another example: F(A,B,C,D) = πM(0, 1, 2, 3, 4, 5, 6, 8, 10, 11, 12, 13, 14)

convert to SOM F(A,B,C,D) = Σm(7, 19, 15)
Example3: express F’ in POM and SOM

F’(A,B,C,D) = πM(7,9,15) F’(A,B,C,D) = Σm(0,1,2,3,4,5,6,8,10,11,12,13,14)

Operations on Canonical forms

ANDing two SOMs

take the intersection of indecies
ORing two SOMs

take union indecies
ANDing two POMs

union
ORing two POMs

intersection
Example
F(A,B,C) = Σm(0, 1, 3, 5) G(A,B,C) = πM(1, 2, 3, 4)
- F * G = (Σm(0, 1, 3, 5)) * (πM(1, 2, 3, 4))
  
  = πM(2,4,6,7) * πM(1,2,3,4)
  
  = πM(1,2,3,4,6,7)
- F + G = (Σm(0, 1, 3, 5)) + (Σm(0, 5, 6, 7))
  
  = Σm(0,1,3,5,6,7)
- F + G in POM? = πM(2, 4)

Standard form

a product term is an ANDing of one or more literals

TODO Other Gate types

TODO Verilog (1/3)

// an example file with test benchmark
// check EL-Rabaa slides for more examples
// the slides includes the info on how to use eda playground page
module fun1(output f1, input a, b, c);
 and(m0, ~a, ~b, ~c);
 and(m3, ~a, b ,c);
 and(m6, a, b , ~c);
 and(m7, a, b, c);
 or(f1, m0, m3, m6, m7);
endmodule

module fun2(output f2, input a, b, c);
 assign M1 = a | b | ~c;
 assign M2 = a | ~b | c;
 assign M4 = ~a | b | c;
 assign M5 = ~a | b | ~c;
 assign f2 = M1 & M2 & M4 & M5;
endmodule

module tb;
 reg a, b, c;
 wire f1, f2, EQ;
 fun1 fun10(f1, a, b, c);
 fun2 fun20(f2, a, b, c);
 xnor(EQ, f1, f2);
 integer t;
 initial begin
$dumpfile("dump.vcd"); $dumpvars;
  for (t=0; t<8; t=t+1) begin
   #1 {a, b, c} = t[2:0];
   $display("%b, %b, %b\n", a, b, c);
  end
  $finish();
 end
endmodule