Topics

Number Systems:
- Binary, Octal, Decimal, and Hexadecimal number systems.
- Complements of numbers (1’s complement, 2’s complement).
- Weighted and Non-weighted codes (BCD, Gray code, Excess-3 code).
Boolean Algebra:
- Basic theorems and properties.
- Switching functions: Canonical (SOP, POS) and Standard forms.
- Algebraic simplification of Boolean expressions.
Digital Logic Gates:
- AND, OR, NOT, NAND, NOR, XOR, XNOR gates.
- Universal gates (NAND, NOR).
- Multilevel NAND/NOR realizations.

Number Systems in Digital Electronics

In digital electronics, number systems are used to represent and manipulate numerical data. The most common number systems include:

Binary (Base-2)
Octal (Base-8)
Decimal (Base-10)
Hexadecimal (Base-16)

Each system has a different base (radix) and is used in different computing applications.

1. Binary Number System (Base-2)

Uses only two digits: 0 and 1.
Fundamental system in digital circuits (since computers work with ON/OFF states).
Each digit is called a bit (Binary Digit).
Example:
$101 1_{2} = 1 \times 2^{3} + 0 \times 2^{2} + 1 \times 2^{1} + 1 \times 2^{0} = 8 + 0 + 2 + 1 = 1 1_{10}$

Applications:

Used in processors, memory, and digital logic circuits.

2. Octal Number System (Base-8)

Uses 8 digits: 0 to 7.
Convenient for representing 3-bit binary numbers (since $8 = 2^{3}$ ).
Example:
$4 7_{8} = 4 \times 8^{1} + 7 \times 8^{0} = 32 + 7 = 3 9_{10}$

Applications:

Early computing systems (now less common).
Sometimes used in file permissions (Unix/Linux).

3. Decimal Number System (Base-10)

Uses 10 digits: 0 to 9.
The standard system for human counting.
Example:
$25 6_{10} = 2 \times 1 0^{2} + 5 \times 1 0^{1} + 6 \times 1 0^{0}$

Applications:

Everyday arithmetic, financial calculations.

4. Hexadecimal Number System (Base-16)

Uses 16 digits: 0 to 9 and A (10), B (11), C (12), D (13), E (14), F (15).
Represents 4-bit binary numbers compactly (since $16 = 2^{4}$ ).
Example:
$2 F_{16} = 2 \times 1 6^{1} + 15 \times 1 6^{0} = 32 + 15 = 4 7_{10}$

Applications:

Memory addressing, color codes (HTML/CSS), assembly language.

Conversion Between Number Systems

Conversion Type	Method
Binary ↔ Decimal	Use weighted positional expansion (powers of 2).
Octal ↔ Binary	Group 3 bits per octal digit.
Hexadecimal ↔ Binary	Group 4 bits per hex digit.
Decimal ↔ Binary	Repeated division by 2 (for integer part).
Decimal ↔ Hex/Octal	Divide by 16/8 and collect remainders.

Example Conversions

Binary to Decimal:
$110 1_{2} = 1 \times 2^{3} + 1 \times 2^{2} + 0 \times 2^{1} + 1 \times 2^{0} = 1 3_{10}$
Decimal to Binary:
$2 5_{10} \to 25 \div 2 = 12$ (Remainder 1)
$12 \div 2 = 6$ (Remainder 0)
$6 \div 2 = 3$ (Remainder 0)
$3 \div 2 = 1$ (Remainder 1)
$1 \div 2 = 0$ (Remainder 1)
→ Read backwards: $1100 1_{2}$
Binary to Hexadecimal:
$1101101 0_{2}$ → Group into 4 bits: $11011010$ → $D A_{16}$
Hexadecimal to Binary:
$3 E_{16}$ → $3 = 0011$ , $E = 1110$ → $0011111 0_{2}$

Summary Table

Number System	Base	Digits	Binary Grouping	Example
Binary	2	0,1	-	$101 0_{2}$
Octal	8	0-7	3 bits	$7 5_{8}$
Decimal	10	0-9	-	$9 9_{10}$
Hexadecimal	16	0-9, A-F	4 bits	$A 3 F_{16}$

Why Different Number Systems?

Binary: Used by computers (simplest for digital logic).
Octal/Hex: Compact representation of binary (easier for humans).
Decimal: Natural for human counting.

CC

Weighted and Non-Weighted Codes in Digital Electronics

In digital systems, binary codes are used to represent numbers, letters, or symbols. These codes can be classified into two main categories:

Weighted Codes
Non-Weighted Codes

Each type has different properties and applications in digital circuits and computing.

1. Weighted Codes

Each bit position has a specific weight (like in traditional binary numbers).
The decimal equivalent is obtained by summing the weighted bits.
Examples:
- Binary Coded Decimal (BCD)
- 8421 Code
- 2421 Code
- Excess-3 Code (XS-3) (partially weighted but has a bias)

(a) Binary Coded Decimal (BCD)

A 4-bit code where each decimal digit (0-9) is represented by its 4-bit binary equivalent.
Weights: 8, 4, 2, 1 (like standard binary).
Invalid codes: 1010 (10) to 1111 (15) are unused.

Decimal	BCD (8421)
0	0000
1	0001
…	…
9	1001

Example:
$5 3_{10}$ in BCD → 0101 0011 (5 = 0101, 3 = 0011)

Applications:

Digital displays (7-segment LEDs).
Financial and calculator systems.

(b) 2421 Code

A self-complementing weighted code (9’s complement can be obtained by inverting bits).
Weights: 2, 4, 2, 1.
Example:
$5_{10} = 1011$ (since $2 \times 1 + 4 \times 0 + 2 \times 1 + 1 \times 1 = 5$ )

Applications:

Used in some arithmetic circuits.

2. Non-Weighted Codes

No positional weights are assigned to bits.
Used for error detection, sequencing, and encoding.
Examples:
- Gray Code (Reflected Binary Code)
- Excess-3 Code (XS-3) (has a fixed bias but is non-weighted in structure)

(a) Gray Code

Only one bit changes between consecutive numbers.
Prevents glitches in digital circuits (e.g., encoders, Karnaugh maps).

Decimal	Binary	Gray Code
0	0000	0000
1	0001	0001
2	0010	0011
3	0011	0010

Conversion (Binary → Gray):

MSB remains the same.
XOR each bit with the next higher bit. $G_{i} = B_{i} \oplus B_{i + 1}$

Example:
Binary: $101 0_{2}$ → Gray:

$G_{3} = 1$
$G_{2} = 1 \oplus 0 = 1$
$G_{1} = 0 \oplus 1 = 1$
$G_{0} = 1 \oplus 0 = 1$
→ Gray Code = 1111

Applications:

Rotary encoders (position sensing).
Error correction in digital communication.

(b) Excess-3 Code (XS-3)

A biased code where each digit is represented by BCD + 3.
Non-weighted but derived from BCD.
Self-complementing (9’s complement is obtained by inverting bits).

Decimal	BCD (8421)	Excess-3
0	0000	0011
1	0001	0100
…	…	…
9	1001	1100

Example:
$5_{10}$ in XS-3 → BCD (0101) + 3 (0011) = 1000

Applications:

Older computing systems.
Some arithmetic operations (simplifies subtraction).

Comparison: Weighted vs. Non-Weighted Codes

Feature	Weighted Codes (BCD, 8421)	Non-Weighted Codes (Gray, XS-3)
Bit Position Weight	Yes (e.g., 8-4-2-1)	No
Arithmetic Use	Common (e.g., BCD adders)	Rare (except XS-3)
Error Detection	No	Yes (Gray code reduces glitches)
Sequential Changes	Multiple bits may change	Only one bit changes (Gray)

Key Takeaways

Weighted Codes (BCD, 8421, 2421) → Used in arithmetic operations.
Non-Weighted Codes (Gray, XS-3) → Used in sequencing, error detection.
Gray Code → Prevents glitches in digital transitions.
Excess-3 → Simplifies subtraction in some cases.

Boolean Algebra in Digital Electronics

Boolean algebra is the mathematical foundation of digital logic circuits. It deals with binary variables (0 and 1) and logical operations (AND, OR, NOT). Below is a structured breakdown of key concepts:

Basic Theorems and Properties

Boolean algebra follows fundamental laws similar to conventional algebra but with binary operations.

(a) Fundamental Operations

Operation	Symbol	Expression
AND	$\cdot$ or none	$A \cdot B$
OR	$+$	$A + B$
NOT	$\overline{A}$ or $A^{'}$	$\overline{A}$

(b) Boolean Postulates

Identity Law
- $A + 0 = A$
- $A \cdot 1 = A$
Null Law
- $A + 1 = 1$
- $A \cdot 0 = 0$
Idempotent Law
- $A + A = A$
- $A \cdot A = A$
Inverse Law
- $A + \overline{A} = 1$
- $A \cdot \overline{A} = 0$
Commutative Law
- $A + B = B + A$
- $A \cdot B = B \cdot A$
Associative Law
- $A + (B + C) = (A + B) + C$
- $A \cdot (B \cdot C) = (A \cdot B) \cdot C$
Distributive Law
- $A \cdot (B + C) = A \cdot B + A \cdot C$
- $A + (B \cdot C) = (A + B) \cdot (A + C)$
De Morgan’s Theorem
- $\overline{A + B} = \overline{A} \cdot \overline{B}$
- $\overline{A \cdot B} = \overline{A} + \overline{B}$

(c) Duality Principle

Any Boolean expression remains valid if:
- $+$ and $\cdot$ are swapped.
- $0$ and $1$ are swapped.

Example:
$A + 0 = A$ → Dual is $A \cdot 1 = A$ .

Switching functions: Canonical (SOP, POS) and Standard forms

1. Switching Functions

A switching function is a Boolean expression that defines the output of a digital circuit based on its input variables. It can be represented as: $F (A, B, C, \dots) = Boolean expression$

Example:
$F (A, B) = A + \overline{A} B$

2. Canonical Forms

Canonical forms are unique and standardized ways of representing Boolean functions using minterms (for SOP) or maxterms (for POS).

(a) Sum of Products (SOP) Form (Minterm Canonical Form)

Represents a function as a sum (OR) of minterms (AND terms).
A minterm is a product term where all variables appear exactly once (either in true or complemented form).
Each minterm corresponds to a row in the truth table where the output is 1.

Example:

Consider a function $F (A, B)$ with the truth table:

A	B	F
0	0	1
0	1	0
1	0	1
1	1	1

The SOP form is: $F (A, B) = \overline{A} \overline{B} + A \overline{B} + A B$ (OR of all minterms where $F = 1$ )

Minterm Notation:

$F (A, B) = \sum m (0, 2, 3)$ (where $m_{0} = \overline{A} \overline{B}, m_{2} = A \overline{B}, m_{3} = A B$ )

(b) Product of Sums (POS) Form (Maxterm Canonical Form)

Represents a function as a product (AND) of maxterms (OR terms).
A maxterm is a sum term where all variables appear exactly once (either true or complemented).
Each maxterm corresponds to a row in the truth table where the output is 0.

Example:

Using the same truth table as above:

A	B	F
0	0	1
0	1	0
1	0	1
1	1	1

The POS form is: $F (A, B) = (A + \overline{B})$ (Only one maxterm where $F = 0$ , i.e., $M_{1} = A + \overline{B}$ )

Maxterm Notation:

$F (A, B) = \prod M (1)$

3. Standard Forms

Standard forms are simplified versions of canonical forms where not all variables need to appear in every term.

(a) Standard SOP Form

A simplified version of the canonical SOP.
Each product term may not contain all variables.

Example: $F (A, B, C) = A B + \overline{B} C$
(Not all terms have all variables.)

(b) Standard POS Form

A simplified version of the canonical POS.
Each sum term may not contain all variables.

Example: $F (A, B, C) = (A + B) (\overline{B} + C)$
(Not all terms have all variables.)

Key Differences: Canonical vs. Standard Forms

Feature	Canonical Form	Standard Form
Uniqueness	Unique for a given truth table	Not unique (multiple possible forms)
Completeness	All variables appear in every term	Some variables may be missing
Complexity	More complex (more terms)	Simplified (fewer terms)

3. Algebraic Simplification of Boolean Expressions

Simplification reduces the number of logic gates required in a circuit.

(a) Techniques

Using Boolean Theorems
- Apply Distributive, De Morgan’s, and Identity Laws.
- Example:
  $A \overline{B} + A B = A (\overline{B} + B) = A \cdot 1 = A$
Consensus Theorem
- $A B + \overline{A} C + BC = A B + \overline{A} C$
Karnaugh Maps (K-Maps) (Graphical method for 2-6 variables).

(b) Example Simplifications

Simplify $F = \overline{A} B + A \overline{B} + A B$
- $F = \overline{A} B + A (\overline{B} + B)$
- $F = \overline{A} B + A$ (since $\overline{B} + B = 1$ )
- $F = A + B$ (Absorption Law)
Simplify $F = (A + B) (A + \overline{B})$
- $F = A + B \overline{B}$ (Distributive Law)
- $F = A + 0 = A$

Summary Table

Concept	Key Points
Boolean Theorems	Identity, Null, Distributive, De Morgan’s Laws.
Canonical Forms	SOP (minterms), POS (maxterms).
Standard Forms	Simplified SOP/POS.
Simplification	Boolean algebra rules, K-Maps.

Applications

Logic gate minimization (fewer transistors, lower power).
Digital circuit design (adders, multiplexers).
Error detection and correction.

Digital Logic Gates: Fundamentals and Applications

Logic gates are the basic building blocks of digital circuits. They perform Boolean operations on binary inputs (0 and 1) to produce a single output. Below is a detailed breakdown of logic gates, universal gates, and their implementations.

1. Basic Logic Gates

(a) AND Gate

Truth Table:

A	B	Y = A·B
0	0	0
0	1	0
1	0	0
1	1	1

Function: Output is 1 only if all inputs are 1.

(b) OR Gate

Truth Table:

A	B	Y = A+B
0	0	0
0	1	1
1	0	1
1	1	1

Function: Output is 1 if at least one input is 1.

(c) NOT Gate (Inverter)

Truth Table:

A	Y = A’
0	1
1	0

Function: Output is the complement of the input.

(d) NAND Gate (NOT-AND)

Truth Table:

A	B	Y = (A·B)‘
0	0	1
0	1	1
1	0	1
1	1	0

Function: Output is 0 only if all inputs are 1.

(e) NOR Gate (NOT-OR)

Truth Table:

A	B	Y = (A+B)‘
0	0	1
0	1	0
1	0	0
1	1	0

Function: Output is 1 only if all inputs are 0.

(f) XOR Gate (Exclusive-OR)

Truth Table:

A	B	Y = A⊕B
0	0	0
0	1	1
1	0	1
1	1	0

Function: Output is 1 if inputs are different.

(g) XNOR Gate (Exclusive-NOR)

Truth Table:

A	B	Y = A⊙B
0	0	1
0	1	0
1	0	0
1	1	1

Function: Output is 1 if inputs are the same.

2. Universal Gates

Universal gates can implement any Boolean function without needing other gates. The two universal gates are:

(a) NAND Gate as Universal

Can implement NOT, AND, OR:
- NOT: A NAND A = A'
- AND: (A NAND B) NAND (A NAND B) = A·B
- OR: (A NAND A) NAND (B NAND B) = A+B

(b) NOR Gate as Universal

Can implement NOT, AND, OR:
- NOT: A NOR A = A'
- OR: (A NOR B) NOR (A NOR B) = A+B
- AND: (A NOR A) NOR (B NOR B) = A·B

3. Multilevel NAND/NOR Realizations

(a) NAND-NAND Realization (SOP Form)

Convert Sum of Products (SOP) to NAND gates only.

Example:

F = A B + C D

→ Implement as:

AB → NAND + NAND (invert twice)  
CD → NAND + NAND  
Final OR → NAND (De Morgan’s Law)

(b) NOR-NOR Realization (POS Form)

Convert Product of Sums (POS) to NOR gates only.

Example:

F = (A + B) (C + D)

→ Implement as:

A+B → NOR + NOR (invert twice)  
C+D → NOR + NOR  
Final AND → NOR (De Morgan’s Law)

Summary Table

Gate	Function	Universal?
AND	A·B	No
OR	A+B	No
NOT	A’	No
NAND	(A·B)‘	Yes
NOR	(A+B)‘	Yes
XOR	A⊕B	No
XNOR	A⊙B	No

Applications

NAND/NOR: Used in IC design (simplifies fabrication).
XOR: Used in adders, parity checkers, cryptography.
Universal Gates: Help in minimizing circuit cost.

Harsh RB

Explorer

UNIT 1 Number Systems and Boolean Algebra

Topics

Number Systems in Digital Electronics

1. Binary Number System (Base-2)

Applications:

2. Octal Number System (Base-8)

Applications:

3. Decimal Number System (Base-10)

Applications:

4. Hexadecimal Number System (Base-16)

Applications:

Conversion Between Number Systems

Example Conversions

Summary Table

Why Different Number Systems?

CC

Weighted and Non-Weighted Codes in Digital Electronics

1. Weighted Codes

(a) Binary Coded Decimal (BCD)

(b) 2421 Code

2. Non-Weighted Codes

(a) Gray Code

(b) Excess-3 Code (XS-3)

Comparison: Weighted vs. Non-Weighted Codes

Key Takeaways

Boolean Algebra in Digital Electronics

Basic Theorems and Properties

(a) Fundamental Operations

(b) Boolean Postulates

(c) Duality Principle

Switching functions: Canonical (SOP, POS) and Standard forms

1. Switching Functions

2. Canonical Forms

(a) Sum of Products (SOP) Form (Minterm Canonical Form)

Example:

Minterm Notation:

(b) Product of Sums (POS) Form (Maxterm Canonical Form)

Example:

Maxterm Notation:

3. Standard Forms

(a) Standard SOP Form

(b) Standard POS Form

Key Differences: Canonical vs. Standard Forms

3. Algebraic Simplification of Boolean Expressions

(a) Techniques

(b) Example Simplifications

Summary Table

Applications

Digital Logic Gates: Fundamentals and Applications

1. Basic Logic Gates

(a) AND Gate

(b) OR Gate

(c) NOT Gate (Inverter)

(d) NAND Gate (NOT-AND)

(e) NOR Gate (NOT-OR)

(f) XOR Gate (Exclusive-OR)

(g) XNOR Gate (Exclusive-NOR)

2. Universal Gates

(a) NAND Gate as Universal

(b) NOR Gate as Universal

3. Multilevel NAND/NOR Realizations

(a) NAND-NAND Realization (SOP Form)

(b) NOR-NOR Realization (POS Form)

Summary Table

Applications

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