Digital Logic, Electronics, and Communication Systems

Posted on Jun 11, 2026 in Systems Computer Engineering

Number Systems and Binary Arithmetic

Number systems are based on different radices: Decimal (base 10, digits 0–9), Binary (base 2, digits 0 and 1), Octal (base 8, digits 0–7), and Hexadecimal (base 16, digits 0–9 and A–F). Conversion is very important:

• To convert decimal to binary, repeatedly divide by 2 and write remainders in reverse order.
• To convert binary to decimal, multiply each bit by powers of 2 and sum them.
• Binary to octal conversion is done by grouping bits in 3s.
• Binary to hexadecimal uses grouping in 4s.

Complements are used in subtraction: 1’s complement is obtained by flipping all bits, and 2’s complement is obtained by adding 1 to the 1’s complement. Basic logic gates include:

• AND: Output 1 only if both inputs are 1.
• OR: Output 1 if at least one input is 1.
• NOT: Inverts the input.
• XOR: Output 1 if inputs are different.

Coding Systems and Internal Codes

In coding systems, ASCII is a 7-bit character encoding scheme representing 128 characters (e.g., ‘A’ = 65 = 1000001), while EBCDIC is an 8-bit coding system used mainly in IBM systems with 256 characters. Internal codes refer to coding schemes used within computer systems, including ASCII, EBCDIC, and modern Unicode.

• BCD (Binary Coded Decimal): Represents each decimal digit using 4 bits (e.g., 59 = 0101 1001), which makes decimal conversion easy but uses more bits.
• Excess-3 code: Obtained by adding 3 to the decimal number and then converting to binary (e.g., 4 becomes 7 → 0111); it is a self-complementing code.
• Gray code: Designed so that only one bit changes between consecutive values, reducing errors. To convert binary to Gray code, keep the MSB the same and XOR adjacent bits (e.g., 1011 becomes 1110).

Boolean Algebra and Logic Simplification

Boolean Algebra deals with binary variables (0 and 1) and basic operations: AND (·), OR (+), and NOT (‾). Important laws include:

• Identity: A+0=A, A·1=A
• Null: A+1=1, A·0=0
• Idempotent: A+A=A, A·A=A
• Complement: A+Ā=1, A·Ā=0
• Commutative, Associative, and Distributive laws.

The Principle of Duality states that in any Boolean expression, swapping AND with OR and 0 with 1 gives the dual expression (both are valid). De Morgan’s Theorems are crucial: (A·B)’ = A’ + B’ and (A + B)’ = A’·B’, used for simplification and gate conversion.

Universal Gates and Canonical Forms

NAND and NOR gates are universal gates, meaning any Boolean function can be implemented using only NAND or only NOR gates. NAND implementation is often preferred in digital circuits; for example, NOT can be made by connecting both inputs of a NAND together, and similarly for NOR. AND, OR, and other gates can be derived using combinations of NAND or NOR gates.

Canonical forms represent Boolean functions using all variables in each term. The two types are:

• Sum of Products (SOP): The function is expressed as a sum (OR) of minterms.
• Product of Sums (POS): The function is a product (AND) of maxterms.

Standard forms are simplified versions where terms may not include all variables but still follow SOP or POS structure.

Minimization Using Karnaugh Maps (K-maps)

Minimization of Boolean functions is done to reduce the number of terms and literals, leading to simpler circuits. Algebraic simplification uses Boolean laws, but a more systematic method is the Karnaugh Map (K-map). A K-map is a graphical tool used to simplify expressions up to 4–5 variables by grouping adjacent 1s (for SOP) or 0s (for POS) in powers of 2 (1, 2, 4, 8…). Larger groups lead to greater simplification. Gray code ordering is used in K-maps to ensure only one-bit change between adjacent cells.

Applications of K-map Minimization

Applications include designing efficient digital circuits, reducing hardware cost, minimizing power consumption, and improving speed. SOP is generally used for implementing logic circuits with AND-OR structures, while POS is used for OR-AND structures. For exams, focus on applying De Morgan’s theorems, converting expressions using duality, implementing circuits using NAND/NOR gates, writing canonical forms from truth tables, and solving K-map minimization problems accurately.

Semiconductor Devices and Diodes

A diode is a semiconductor device that allows current to flow in only one direction. Its key characteristics include forward bias (conducts after threshold voltage ~0.7V for silicon, 0.3V for germanium) and reverse bias (very small leakage current until breakdown). The V-I characteristics show a nonlinear curve with a sharp rise in the forward region. A Zener diode is specially designed to operate in the reverse breakdown region and maintain a constant voltage (Zener voltage), making it useful as a voltage regulator.

Rectifiers: Converting AC to DC

Rectifiers are circuits that convert AC to DC:

• A half-wave rectifier uses a single diode and conducts only during one half-cycle of AC, resulting in low efficiency and high ripple.
• A full-wave rectifier uses two diodes (center-tapped transformer) and converts both halves of AC into DC, improving efficiency.
• A bridge rectifier uses four diodes and does not require a center-tapped transformer; it provides better output and is widely used.

Ripple factor decreases and efficiency increases from half-wave to full-wave to bridge rectifiers.

Transistors: BJT, JFET, and MOSFET

A BJT (Bipolar Junction Transistor) is a three-terminal device (Emitter, Base, Collector) that operates using both electrons and holes. It works in regions: cutoff (OFF), active (amplification), and saturation (ON). In the active region, a small base current controls a large collector current (Ic = βIb). BJTs are used as amplifiers and switches.

• JFET (Junction Field Effect Transistor): A voltage-controlled device where current flows through a channel controlled by gate voltage; it has high input impedance and operates in depletion mode.
• MOSFET (Metal Oxide Semiconductor FET): An advanced FET with even higher input impedance; it operates in enhancement or depletion modes and is widely used in digital circuits.

CMOS Logic and Integrated Circuits

CMOS (Complementary MOS) logic uses both NMOS and PMOS transistors to achieve low power consumption and high noise immunity. In CMOS logic gates, one type of transistor conducts while the other is off, minimizing power loss. Basic logic gates using CMOS include the inverter (NOT gate using one PMOS and one NMOS), NAND (parallel PMOS + series NMOS), and NOR (series PMOS + parallel NMOS). CMOS is widely used in modern ICs due to efficiency and scalability.

Fundamentals of Communication Systems

A communication system is used to transfer information from a source to a destination through a medium. The basic elements include:

1. Information source: Generates the message.
2. Input transducer: Converts physical signal to electrical.
3. Transmitter: Processes and modulates the signal.
4. Channel: Medium like air or cable.
5. Noise: Unwanted disturbances.
6. Receiver: Demodulates and processes the signal.
7. Output transducer: Converts electrical signal back to physical form.
8. Destination: The final point of the message.

Modulation and Demodulation Techniques

Modulation is the process of varying a high-frequency carrier signal according to the low-frequency message signal to enable efficient transmission over long distances, reduce antenna size, and avoid mixing of signals.

• Amplitude Modulation (AM): The amplitude of the carrier wave is varied while frequency and phase remain constant. The AM wave contains a carrier and two sidebands (USB and LSB). The modulation index (m) = Vm/Vc (should be ≤ 1 for no distortion).
• AM Demodulation (Detection): Used to recover the original message signal. The most common method is the envelope detector, which uses a diode, capacitor, and resistor.
• Frequency Modulation (FM): The frequency of the carrier is varied according to the message signal while amplitude remains constant. It provides better noise immunity and higher fidelity.
• Phase Modulation (PM): The phase of the carrier signal is varied in accordance with the message signal. FM and PM are types of angle modulation.

Comparison of AM and FM

AM is simpler, cheaper, and requires less bandwidth but is more susceptible to noise and has lower sound quality. FM provides better noise immunity, higher sound quality, and constant amplitude transmission but requires more bandwidth and complex circuitry.

Reference and Examples

Number System Conversions

• Decimal → Binary: Divide by 2, remainders read upwards. Example: 13 → 1101
• Binary → Decimal: Σ (bit × 2^position). Example: 1011 → 11
• Binary ↔ Octal: Group 3 bits. Example: 111001 → 71₈
• Binary ↔ Hex: Group 4 bits. Example: 101011 → 2B₁₆

Complements and Logic Gates

• 1’s complement = flip bits
• 2’s complement = 1’s comp + 1 (Used for subtraction)

Logic Gates Truth Table (Outputs):

NOT Gate: 0 → 1, 1 → 0

Coding Examples

• ASCII: 7-bit (128 chars). ‘A’ = 65 → 1000001
• EBCDIC: 8-bit (256 chars). Used in IBM.
• BCD: Each digit → 4 bits. Example: 59 → 0101 1001
• Excess-3: Add 3, then binary. Example: 4 → 7 → 0111
• Gray Code: Only 1 bit changes. Binary → Gray: MSB same, Next = XOR(prev bits). Example: 1011 → 1110

Number System Summary