Digital Logic Design: Number Systems, Codes, Circuits, and Applications

Posted on Jun 5, 2024 in Design and Engineering

Number Systems and Codes

Binary, Octal, and Hexadecimal Conversions

  • Binary: Base-2 numeral system using 0s and 1s. Conversion between binary and decimal involves place values.
  • Octal: Base-8 numeral system using digits 0 to 7. Conversion to and from binary can be done by grouping 3 bits at a time.
  • Hexadecimal: Base-16 numeral system using digits 0 to 9 and A to F. Conversion to binary can be done by converting each hex digit to a 4-bit binary representation.

ASCII Code

  • ASCII (American Standard Code for Information Interchange): A character encoding standard for representing text using 7 or 8 bits. Each character is represented by a unique code.

Excess-3 Code

  • Excess-3 code: A self-complementing BCD (Binary Coded Decimal) code. To obtain the Excess-3 code for a decimal number, add 3 to the decimal value and convert the result to BCD.

Gray Code

  • Gray code: A binary code where adjacent numbers differ by only one bit. It finds applications in digital encoders and error detection systems.

BCD (Binary Coded Decimal)

  • BCD: A binary representation of decimal numbers. Each decimal digit is represented using 4 bits. BCD is commonly used in digital systems for handling decimal arithmetic.

Error Detection Codes – Parity Method

  • Parity: A simple error detection method. It involves adding an extra bit to a binary number to ensure the total number of 1s (or 0s) is even (even parity) or odd (odd parity).

Signed Numbers

Signed numbers represent positive and negative values. In binary, the most significant bit (MSB) is used to indicate the sign, where 0 represents positive and 1 represents negative.

Representation, Addition, and Subtraction of Signed Numbers

  • Signed Magnitude Representation: The MSB represents the sign, and the remaining bits represent the magnitude.
  • One’s Complement Representation: To obtain the one’s complement, invert all the bits of the positive number.
  • Two’s Complement Representation: To obtain the two’s complement, invert all the bits of the positive number and then add 1.

Fixed Point and Floating-Point Representation

  • Fixed-Point: Used for representing numbers with a fixed number of digits after the decimal point.
  • Floating-Point: Used for representing large and small numbers with varying precision.

Logic Gates and Families

  • Logic Gates: Basic building blocks of digital circuits. Examples include AND, OR, NOT, NAND, NOR, XOR, and XNOR gates.
  • Universal Gates: NAND and NOR gates are universal gates because any digital circuit can be constructed using only these gates.
  • TTL (Transistor-Transistor Logic) and CMOS (Complementary Metal-Oxide-Semiconductor): Two common logic families with different characteristics.
  • Internal Diagram of TTL NAND Gate: A TTL NAND gate is typically composed of transistors arranged in a way that implements the NAND logic function.
  • Internal Diagram of CMOS NOR Gate: A CMOS NOR gate uses complementary pairs of NMOS and PMOS transistors to implement the NOR logic function.
  • Comparison of CMOS and TTL Performance: CMOS is known for low power consumption, while TTL has faster switching speeds but higher power consumption. The choice depends on the application requirements.

Boolean Laws and Theorems

  • Commutative Law: A ∨ B = B ∨ A, A ∧ B = B ∧ A
  • Associative Law: (A ∨ B) ∨ C = A ∨ (B ∨ C), (A ∧ B) ∧ C = A ∧ (B ∧ C)
  • Distributive Law: A ∨ (B ∧ C) = (A ∨ B) ∧ (A ∨ C), A ∧ (B ∨ C) = (A ∧ B) ∨ (A ∧ C)
  • Identity Law: A ∨ 0 = A, A ∧ 1 = A
  • Null Law: A ∨ 1 = 1, A ∧ 0 = 0
  • Inverse Law: A ∨ ¬A = 1, A ∧ ¬A = 0
  • De Morgan’s Theorem: ¬(A ∧ B) = ¬A ∨ ¬B, ¬(A ∨ B) = ¬A ∧ ¬B

Sum of Products (SOP) Method

  • SOP: A method to represent Boolean expressions using the OR operation (sum) of ANDed terms (products). It helps in simplifying complex Boolean expressions.

Product of Sums (POS) Method

  • POS: A method to represent Boolean expressions using the AND operation (product) of ORed terms (sums). It can also simplify Boolean expressions in a different form.

Karnaugh Map (K-map) Representation and Simplification

  • K-map: A graphical tool used for simplifying Boolean expressions with up to four variables. It is based on grouping adjacent cells with 1s to identify simplified terms.

K-map Groups

  • Pairs: Grouping of adjacent cells in pairs (2×1 group) to form a product term with two variables.
  • Quads: Grouping of four adjacent cells (2×2 group) to form a product term with three variables.
  • Octets: Grouping of eight adjacent cells (2×4 group or 4×2 group) to form a product term with four variables.

Don’t Care Conditions

  • Don’t care conditions: Used when certain input combinations are not expected to occur in the circuit. They are marked as “X” in the K-map, and their values can be either 0 or 1 depending on the desired output.

Combinational Circuits

Adders

  • Half Adder: A combinational circuit that adds two binary digits and produces the sum (S) and carry (C) outputs.
  • Full Adder: A combinational circuit that adds three binary digits (two inputs and a carry) and produces the sum (S) and carry (C) outputs.

Subtractors

  • Half Subtractor: A combinational circuit that subtracts two binary digits and produces the difference (D) and borrow (B) outputs.
  • Full Subtractor: A combinational circuit that subtracts three binary digits (two inputs and a borrow) and produces the difference (D) and borrow (B) outputs.
  • 4-Bit Parallel Binary Adder/Subtractor: A circuit that can perform addition or subtraction of two 4-bit binary numbers based on a control input.
  • Carry Look-Ahead Adders: An advanced adder design that reduces the carry propagation delay by generating carry signals for multiple bits in parallel.

Flip-Flops and Counters

Types of Flip-Flops and Their Applications

  • SR Flip-Flop (Set-Reset Flip-Flop): It has two inputs, Set (S) and Reset (R). SR flip-flop is used for basic memory storage, latch circuits, and asynchronous applications.
  • JK Flip-Flop: It has three inputs, J, K, and a clock input. JK flip-flops are widely used for sequential circuits, frequency division, and in shift registers.
  • D Flip-Flop (Data Flip-Flop): It has a single data (D) input and a clock input. D flip-flops are used in shift registers, frequency division, and to store data in sequential circuits.
  • T Flip-Flop (Toggle Flip-Flop): It has a single input T and a clock input. T flip-flops are used in frequency division and as an oscillator.

Application Areas

Flip-flops are essential components in digital circuits and are used in various applications, including:

  • Memory elements in registers and counters.
  • State storage in sequential circuits like shift registers, counters, and memory units.
  • Frequency division in clock dividers and frequency dividers.
  • Data synchronization and signal processing.
  • Control and timing circuits in microprocessors and microcontrollers.

Design of Various Counter Circuits

  • Asynchronous Counters: Built with flip-flops, where the output of one flip-flop serves as the clock input to the next. Also known as ripple counters.
  • Synchronous Counters: Use a common clock signal for all flip-flops, ensuring that all flip-flops change states simultaneously.
  • Decade Counter: A type of synchronous counter that counts from 0 to 9 and then resets.
  • Up/Down Counters: Counters that can count up or down depending on the control signal.
  • Modulus Counters: Counters designed to reset at a specific count value, useful for generating specific sequences.

Level-Triggered Flip-Flop vs. Edge-Triggered Flip-Flop

  • Level-Triggered Flip-Flop: Responds to the input continuously while the clock input is at a specific level (e.g., high or low). Its output changes whenever the input is at the specified level.
  • Edge-Triggered Flip-Flop: Changes its output state only when the clock input transitions from one edge to another (e.g., rising or falling edge). It captures the input data at the moment of the clock edge transition.

Master-Slave Flip-Flop

A master-slave flip-flop is a combination of two flip-flops. The first flip-flop is the master, which only works when the clock signal is high. The second flip-flop is the slave, which only works when the clock signal is low. This configuration helps avoid timing issues and glitches in sensitive applications.

Design of Mod-7 Asynchronous Counter using J-K Flip-Flop

To design a mod-7 asynchronous counter using J-K flip-flops, we need three J-K flip-flops. The outputs of the flip-flops are connected to form the desired counting sequence from 0 to 6 (mod-7). The clock signal is applied directly to the first flip-flop (J-K0) and connected through appropriate logic gates to the clock inputs of the subsequent flip-flops (J-K1 and J-K2).

Ring Counter vs. Johnson Counter

  • Ring Counter: A type of shift register where only one flip-flop output is high at a time, creating a circular pattern. It requires fewer flip-flops compared to other counters and finds applications in digital clocks and frequency dividers.
  • Johnson Counter: Also known as a twisted ring counter, it is a shift register that goes through a sequence of states, producing a single 1 bit that “moves” through the register. It finds applications in sequence generation and shift register applications.

Various Types of Shift Registers

  • Serial-In Serial-Out (SISO) Shift Register: Shifts data in and out serially, bit by bit.
  • Serial-In Parallel-Out (SIPO) Shift Register: Shifts data in serially and outputs it in parallel form.
  • Parallel-In Serial-Out (PISO) Shift Register: Loads parallel data and shifts it out serially.
  • Parallel-In Parallel-Out (PIPO) Shift Register: Performs parallel loading and parallel output of data.
  • Bidirectional Shift Register: Can shift data in both left and right directions.

Difference Between a Counter and a Shift Register

  • Counter: A digital circuit that counts the number of clock pulses and produces a binary count as its output. It is used for counting and generating specific sequences.
  • Shift Register: A digital circuit that shifts data in a serial manner, either left or right, based on clock pulses. It is used for data storage, conversion, and time delay applications.

Analog-to-Digital and Digital-to-Analog Conversion

  1. The number of output voltages that can be produced by an 8-bit ADC is 2^8, which is 256. The ADC converts an analog input signal into a digital output, and with 8 bits, it can represent 256 discrete voltage levels (2^8 = 256).

Advantage of R-2R Ladder DAC over Weighted Resistor DAC

  • Simplicity: The R-2R ladder DAC uses only two types of resistor values (R and 2R), making it easier to implement and manufacture compared to the weighted resistor type DAC that requires multiple resistor values.
  • Better matching: R-2R ladder DAC provides better resistor matching, resulting in improved linearity and accuracy of the analog output.
  1. The Flash ADC (also known as parallel ADC) is the fastest ADC. It operates by comparing the analog input against multiple voltage reference levels in parallel and directly provides the digital output without the need for successive approximation. Flash ADCs are suitable for high-speed applications due to their parallel architecture and minimal conversion time.

PLA (Programmable Logic Array) vs. PAL (Programmable Array Logic)

  • PLA: It is a type of digital circuit that contains both programmable AND and OR arrays. It offers full programmability, allowing the designer to implement any combinational logic function. PLA is more flexible but requires more programming resources.
  • PAL: It is a type of programmable logic device that uses a fixed OR array and a programmable AND array. PALs are easier to program and can efficiently implement relatively smaller logic functions. They are less flexible compared to PLA but offer a more economical solution for smaller logic designs.

Programmable Logic Array (PLA)

  • A PLA is a programmable logic device with a fixed OR array and a programmable AND array. It provides full programmability for creating various combinational logic functions.
  • It consists of multiple input AND gates connected to a programmable array of product terms, followed by an OR gate array that sums these terms to produce the final output.
  • PLA allows users to implement custom logic functions by programming the connections between inputs, AND gates, and OR gates.

Differentiation from ROM (Read-Only Memory)

  • Both PLA and ROM are programmable logic devices, but their key differences lie in their functionality and usage.
  • PLA: Provides full programmability for implementing various combinational logic functions by programming AND and OR connections.
  • ROM: Acts as a memory device that stores pre-programmed data and cannot be reprogrammed in the field. It is used to store data or instructions, and its outputs depend on the address inputs. ROM is not flexible for logic implementation like PLA but is useful for data retrieval and storage purposes.

Finite State Machines

Difference between Moore and Mealy Machine

Both Moore and Mealy machines are types of finite state machines used in digital systems. The key difference lies in how they produce their outputs:

  • Moore Machine:
    • The output of a Moore machine depends only on the current state of the machine.
    • The outputs are associated with states, meaning the output remains constant during the state until a transition occurs.
    • It is more suitable for control applications where the output is only determined by the current state.
  • Mealy Machine:
    • The output of a Mealy machine depends on the current state and the inputs to the machine.
    • The outputs are associated with transitions, meaning the output can change during a state based on the input.
    • It is more suitable for applications where the output needs to react to input changes immediately.

Function of Mealy Machine

The Mealy machine is a type of finite state machine that produces outputs based on both its current state and the inputs. When a transition occurs in the machine, it not only changes its state but also generates an output based on the combination of the current state and the input signals.

The Mealy machine’s output function allows for more responsive and efficient designs, as the output can change more frequently based on the input signals, leading to potentially smaller and faster circuits.

Code Implementation of Simple Circuits using Verilog

Here’s an example of a simple Verilog code to implement a 2-input AND gate:

   “`verilog

   module AND_gate(input a, b, output reg y);

       always @(a or b) begin

           y = a & b;

       end

   endmodule

   “`

   In this code, we define an AND gate module that takes two input signals `a` and `b` and produces the output `y`. The `always` block is used to describe the behavior of the AND gate, where `y` is assigned the result of the AND operation between `a` and `b`.

4. **FPGA (Field-Programmable Gate Array):**

   FPGA is a type of programmable logic device that contains an array of configurable logic blocks, interconnects, and I/O pads. Unlike ASICs (Application-Specific Integrated Circuits), FPGAs can be programmed or reprogrammed to implement various digital logic functions and circuits.

   **Applications of FPGA:**

   – **Digital Signal Processing (DSP):** FPGAs are widely used in DSP applications for their parallel processing capabilities, making them suitable for filtering, audio and video processing, and communications.

   – **Prototyping and Emulation:** FPGAs are used for rapid prototyping of ASIC designs and testing complex systems before their final implementation in hardware.

   – **High-Performance Computing:** FPGAs can be used as accelerators in high-performance computing systems, especially for specific tasks that require hardware acceleration.

   – **Networking and Communication:** FPGAs are used in networking equipment like routers and switches to handle high-speed data packet processing and network protocol offloading.

   – **Embedded Systems:** FPGAs are integrated into various embedded systems to provide custom logic and interface options, enabling flexible and efficient system designs.