Assignment

Directorate of Online Education

SESSION APR 2023

PROGRAM BACHELOR OF COMPUTER APPLICATION (BCA)
SEMESTER II
COURSE CODE & NAME DCA1205-DIGITAL LOGIC
NAME: THAHSHIN SHAFRIYA
ROLL NO: 2214509750
 SET-1
1.
A certain collection of symbols are used to represent numbers in a mathematical notation
technique called a number system. The most common method of representing numbers is the
base-10 system, sometimes known as the decimal system, which uses ten digits (0–9). There are
other number systems, such as binary (base 2), octal (base 8), and hexadecimal (base 16).
A decimal number can be converted into its corresponding binary, octal, and hexadecimal
representations by following these steps:
1. Converting a decimal value from binary to decimal: Begin by repeatedly dividing the decimal
value by 2, noting the outcome of each division. Continue doing this until the quotient
approaches zero. The binary number is produced by writing the remainders backwards. Consider
converting the decimal value 25 to binary:
- 25 ÷ 2 = 12 remainder 1
- 12 ÷ 2 = 6 remainder 0
- 6 ÷ 2 = 3 remainder 0
- 3 ÷ 2 = 1 remainder 1
- 1 ÷ 2 = 0 remainder 1
So, the binary equivalent of 25 is 11001.
Similar to the binary conversion, divide the decimal number by 8 several times and record the
remainders. The remainders are written in the octal number's reversed sequence. Let's translate
the number 49 from decimal to octal:
- 49 ÷ 8 = 6 remainder 1
- 6 ÷ 8 = 0 remainder 6
So, the octal equivalent of 49 is 61.
3. Decimal to Hexadecimal Conversion: Divide the decimal integer by 16 several times, noting
the residue after each division. The hexadecimal number is produced when the quotient is equal
to zero by writing the remainders in reverse order. For remainders greater than 9, the decimal
numerals 10, 11, 12, 13, 14, and 15 are denoted by the letters A, B, C, D, E, and F, respectively.
Let's convert 157 from decimal to hexadecimal form:
- 157 ÷ 16 = 9 remainder 13 (D in hexadecimal)
- 9 ÷ 16 = 0 remainder 9
So, the hexadecimal equivalent of 157 is 9D.
You can change a decimal number into its binary, octal, and hexadecimal representations by
following these steps. These conversions are crucial in many different disciplines, but they are
particularly crucial in computer science since binary, octal, and hexadecimal numbers are
frequently employed to represent data and carry out calculations.
2.
Logic gates, which are electronic components that carry out logical operations based on input
signals and generate equivalent output signals, are necessary components of digital circuits.
These operations make use of logical connections like AND, OR, NOT, XOR, NAND, NOR,
and XNOR. Logic gates are connected in a variety of ways, each of which has a specific function
and behaviour, to create sophisticated digital circuits.
Let's examine the various logic gates and their in-depth descriptions:
1. AND Gate: An AND gate is a device with one output terminal and two or more input
terminals. Only when all input signals are high (logic 1) does it produce an output signal. If any
of the input signals are low, the output signal is also low (logic 0). The behaviour of an AND
gate can be described by the Boolean equation Output = A AND B, where A and B are the input
signals.
2. OR Gate: The OR gate has one output terminal and two or more input terminals, just like the
AND gate. It generates an output signal when at least one of the input signals is high (logic 1).
The output signal is only low (logic 0) when all of the input signals are low.
3. NOT Gate: Also known as an inverter, the NOT gate has just one input terminal and one
output terminal. The output signal is the opposite of the input signal. When the input signal is
high (logic 1), the output signal is low (logic 0), and vice versa. The behaviour of the NOT gate
can be described by the Boolean expression Output = NOT A.
4. XOR Gate: There is a single output terminal and two input terminals on the exclusive OR
(XOR) gate. It generates a high (logic 1) output signal if the ratio of high inputs to high inputs is
unusual. Under the assumption that there are an equal number of high inputs, the output signal is
low (logic 0). It is possible to utilise the Boolean expression Output = A XOR B.
5. NAND Gate: This combination of an AND gate and a NOT gate is known as a NAND gate. It
has one output terminal and two or more input terminals. When compared to the AND gate's
output signal, the output signal is inverted. Only when all input signals are high (logic 1) are all
output signals low (logic 0). The Boolean equation Output = NOT(A AND B) can be used to
describe the behaviour of the NAND gate.
6. NOR Gate: The NOR gate is created by adding an OR gate and a NOT gate. It has one output
terminal and two or more input terminals. When compared to the OR gate's output signal, the
output signal is inverted.
7. XNOR Gate: The XNOR gate, also known as exclusive NOR, is created by combining the
XOR and NOT gates. It has one output terminal and two input terminals. If the number of high
inputs is even, it produces a high (logic 1) output signal. The output signal is low (logic 0) if the
ratio of high inputs to low inputs is unusual. The Boolean equation Output = NOT(A XOR B)
can be used to describe the behaviour of the XNOR gate.
In digital circuits, these logic gates act as the basic building blocks that can be joined to carry out
intricate logical processes. One can create circuits for electronic devices like computers and
smartphones that can execute arithmetic operations, memory functions, and control actions by
integrating various logic gates.
3.
In digital logic design, a Karnaugh map, also known as a K-map, is used to simplify Boolean
statements and minimise the number of logic gates required in a circuit. It provides a way to
manipulate and comprehend Boolean functions visually. K-maps are quite useful when working
with functions with a small number of variables.
The Boolean function f(a, b, c, d) = m(0, 2, 4, 6, 7, 8, 9, 11, 12, 14) is used in the example. Here,
the input variables are represented by letters a, b, c, and d, while the symbol m stands for the
minterms' sum.
To simplify this function using a K-map, we first construct a 4-variable Karnaugh map.
```
cd\ab 00 01 11 10
+---+---+---+---+ 00 | 1 | | 1 | |
+---+---+---+---+ 01 | | | 1 | 1 |
+---+---+---+---+ 11 | 1 | 1 | 1 | 1 |
+---+---+---+---+ 10 | 1 | 1 | | 1 |
+---+---+---+---+ ```
Now that we know where we may simplify, we can do so by grouping cells that are nearby and
have a value of 1. The groups should be in powers of two (1, 2, 4, 8, etc.) and can be created
either horizontally or vertically.
Looking at the K-map reveals two groups: one with four cells (minterms 0, 2, 8, 10), and another
with only two cells (minterms 4, 12). In order to produce the condensed Boolean expression, we
use the following groups:
f(a, b, c, d) = ∑m(0, 2, 8, 10) + ∑m(4, 12)
To further simplify the expression, we can observe that both groups have the same values for
variables a, b, and d. We can combine them to eliminate the variable c:
f(a, b, c, d) = ∑m(0, 2, 8, 10, 4, 12)
This can be further simplified by realizing that the remaining minterms cover all possible
combinations of a, b, and d. Therefore, we can eliminate the variables a, b, and d, resulting in the
simplified Boolean expression:
f(a, b, c, d) = 1
Hence, the simplified form of the Boolean function f(a, b, c, d) = ∑m(0, 2, 4, 6, 7, 8, 9,
11, 12, 14) is simply f(a, b, c, d) = 1, indicating that the function is always true (logic 1)
regardless of the input values.
SET-2
4.
A sequential circuit is a specific kind of digital circuit that incorporates memory components that
can store and process data over time. Contrary to combinational circuits, which only take into
account current inputs when producing outputs, sequential circuits also take into account the
circuit's previous state. They are frequently utilised in memory units, counters, shift registers,
and control units. A crucial element of sequential circuits is the flip-flop, a digital circuit
component that can store a single bit of information expressed as a binary value (0 or 1). One of
the most common flip-flops is the JK flip-flop, which has a number of functions based on the
inputs.
The operation of a JK flip-flop can be explained as follows:
1. Set (S) Mode:
- When J = 1 and K = 0, the flip-flop enters the set mode.
- Irrespective of the previous state, the output Q becomes 1, and Q' becomes 0. - In this mode,
the flip-flop is "set" or "forced" to the 1 state.
2. Reset (R) Mode:
- When J = 0 and K = 1, the flip-flop enters the reset mode.
- Irrespective of the previous state, the output Q becomes 0, and Q' becomes 1. - In this mode,
the flip-flop is "reset" or "forced" to the 0 state.
3. Toggle (T) Mode:
- When J = 1 and K = 1, the flip-flop enters the toggle mode.
- The output Q changes to the complement of its previous state. - If the previous state was 0, Q
becomes 1, and vice versa.
- Q' also changes to the complement of Q.
- In this mode, the flip-flop "toggles" its state.

5.
A digital counter is a sequential circuit that counts in accordance with a specified pattern or
sequence. Frequency divisors, timers, and address generators are just a few applications where it
commonly shows up. Counters commonly use flip-flops or other memory elements to store and
transmit the count value. One type of counter with a unique configuration and behaviour is the
ring counter.
In a digital counter known as a ring counter, a ring-like structure is made by connecting the
output of one flip-flop in a circular pattern to the input of the following flip-flop. When the input
of the first flip-flop is linked with the output of the last flip-flop, a closed loop is created. The
Ring Counter can be constructed using D flip-flops or JK flip-flops.
The following explanation will help you understand how a ring counter functions:
1. Initialization: To provide a baseline, all flip-flops are initially reset to a known state (usually
all zeros or all ones).
2. Clocking: All flip-flops get the same clock signal at the same time, which causes them to all
change states at once.
3. Shift Operation: The Ring Counter executes a shift operation during each clock cycle, which
involves moving each flip-flop's output to the flip-flop after it in the ring. The flip-flops receive
the count value via this shift operation.
4. Count Sequence: The number of flip-flops employed determines the output pattern of the Ring
Counter. There are 2n potential states for an n-bit Ring Counter. The counter moves on to the
following state in the predefined order with each clock cycle. The feedback connections between
the flip-flops determine the precise count sequence.
5. Output: The flip-flops' outputs show the Ring Counter's current count value. The outputs can
be used immediately or put through additional processing depending on the application.
The Ring Counter stands out for just having one flip-flop change states during each clock cycle.
This is due to the feedback connection's effect on how the count value moves around the ring.
The straightforward design and regular construction of a Ring Counter promote it. Due to the
fact that it requires less components than other counter systems, it is more inexpensive and easier
to build. The Ring Counter, however, has the potential to quickly enter an illegal state or exhibit
non-sequential behaviour if the inputs are not carefully controlled.
For frequency division, ring counters can be employed, with the final flip-flop's output serving as
the divided clock signal. In sequential logic circuits, they can also be used to generate specific
patterns or sequences based on the count sequence.
In conclusion, a ring counter is a sort of digital counter that makes use of flip-flops arranged in a
ring shape, with the output of one flip-flop being connected to the input of the next.
6.
In order to manage and regulate both vehicular and foot traffic at crossings, modern
transportation infrastructure must be equipped with traffic signal systems. They provide a
deliberate and well-thought-out strategy to ensure the swift and secure circulation of vehicles and
pedestrians, lowering traffic congestion and enhancing general road safety. Numerous significant
elements and concepts are included in the operational principles of traffic signal systems.
1. Intersection Layout: Traffic signal systems are built in intersections, which are locations
where two or more roads converge. The architecture and design of the intersection have a
significant impact on the placement and arrangement of traffic signals. The number of lanes,
turning movements, pedestrian crossings, and traffic volume are all taken into consideration
while designing the layout of a junction.
2. Signal Phases: At an intersection, each phase of the operation of a traffic signal system
corresponds to a different movement or group of movements. The most common signal phases
are green, yellow, and red. The green phase allows traffic to move forward while the yellow
phase announces the shift from green to red and warns vehicles to prepare to stop. During the red
phase, one-way traffic is paused to make room for other movements.
3. Timing and Sequencing: To enhance traffic flow and minimise delays, the timing and order of
the signal phases are carefully arranged. Traffic engineers consider a number of factors, such as
traffic demand, pedestrian activity, and synchronisation with adjacent junctions, to establish an
effective timing plan. The length of each phase and their order are chosen based on traffic
patterns and peak-hour requirements.
4. Detectors and Sensors: Traffic signal systems use detectors and sensors to monitor traffic
conditions and adjust the signal timings as necessary. Inductive loops implanted in the road
surface, video cameras, infrared sensors, and radar are routinely used to detect the presence of
vehicles or humans. The signal timings are dynamically changed to enhance traffic flow and
minimise delays.
5. Synchronisation and Coordination: Traffic signal systems at a network of intersections
typically operate together to synchronise and coordinate their signal timings. Slowdowns and
delays are minimised as traffic travels through a corridor smoothly thanks to coordinated traffic
flow. By synchronising the signal timings to produce a "green wave" that lets automobiles to go
at a steady speed, traffic engineers can reduce the number of red lights.
6. Pedestrian Safety: Traffic signal systems feature pedestrian signals and crosswalks to ensure
that pedestrians can cross intersections safely. Pedestrian lights specify precise crossing times
and, in some circumstances, pedestrian-only periods when it is safe to cross the roadway.
7. Emergency Vehicle Preemption: Traffic signal systems may allow emergency vehicles to
move ahead of other vehicles. Special sensors or transmitters on emergency vehicles may
produce a shift in the signal phase, giving them priority and allowing quick and secure passage
through the intersection.