Unit1 - Subjective Questions

Difference between Analog and Digital Systems:

Advantages of Digital Systems:

1. Reliability: They are more reliable than analog systems.
2. Programmability: Operations can be changed easily by changing the program.
3. Noise Immunity: Digital circuits are less affected by noise.
4. Storage: Information storage is easy and compact.

A Positional Number System is a system where the value of a digit depends not only on the digit itself but also on its position within the number.

It is characterized by a Radix (or base) .

Determination of Value:
A number with base having digits is evaluated as:

Example:
For a decimal number :

1. to Decimal:

2. to Octal:

• with remainder 5
• with remainder 6
• with remainder 3
  Reading remainders bottom-up:

3. to Binary:
Convert each hex digit to 4-bit binary:

• 
  Result:

Gray Code:
Gray code is a non-weighted binary code where two successive values differ by only one bit.

Why it is called Unit Distance Code:
Because the Hamming distance between any two consecutive code words is exactly one (only one bit changes).

Conversion to Gray:
Let Binary bits be and Gray bits be .

1. MSB is same:

Result:

Signed numbers allow computers to represent both positive and negative integers.

1. Sign-Magnitude Representation:

• The MSB is the sign bit (0 for +, 1 for -). The remaining bits are magnitude.

2. 1's Complement Representation:

• Negative numbers are obtained by inverting all bits of the positive number.
• Invert bits:

3. 2's Complement Representation:

• Obtained by adding 1 to the 1's complement.
• 1's Comp of 5 = $1010$
• Add 1:

Note: 2's complement is preferred in arithmetic circuits as it eliminates the double zero problem (+0 and -0).

Operation: , where and .
This is equivalent to .

Step 1: Find 2's Complement of B (1100)

• 1's complement of
• Add 1:
• 2's Comp of

Step 2: Add A to 2's Comp of B

Step 3: Analyze Result

• There is no carry generated out of the MSB.
• This indicates the result is negative and is in its 2's complement form.

Step 4: Get final magnitude

• Take 2's complement of the result .
• 1's comp = $0001$
• Add 1 = $0010$ (which is 2 in decimal).
• Since result was negative: Result = or .

BCD (Binary Coded Decimal):
It is a way to represent decimal digits (0-9) using 4-bit binary sequences. It uses the weights 8-4-2-1. Codes $1010$ through $1111$ are invalid in BCD.

BCD Addition Rules:

1. Perform binary addition.
2. If the sum of a 4-bit group is , it is valid.
3. If the sum OR if a carry is generated from the group, add (decimal 6) to correct it.

Example:

Step 1: Binary Add

Step 2: Correction
Since (9), add $0110$:

Step 3: Format
Split into 4-bit groups:
Result =

Hamming Code:
A linear error-correcting code that can detect up to two-bit errors or correct one-bit errors. It places parity bits at positions that are powers of 2 ($1, 2, 4, 8...$).

Generating Code for Data $1010$ (Odd Parity):
Data bits: .
Format (7 bits): .
Bits: .

Calculate Parity Bits (Odd Parity implies total 1s must be odd):

1. checks positions 1, 3, 5, 7:
   • Bits: . Existing 1s = 2 (Even). To make odd, .
2. checks positions 2, 3, 6, 7:
   • Bits: . Existing 1s = 1 (Odd). To keep odd, .
3. checks positions 4, 5, 6, 7:
   • Bits: . Existing 1s = 2 (Even). To make odd, .

Final Code:

Result: $1001101$

Comparison:

• Fixed Point: The decimal point is fixed. Limited range of numbers. Higher precision for small integers. Hardware implementation is simpler.
• Floating Point: The decimal point can 'float'. Can represent extremely large or small numbers (large dynamic range). More complex hardware.

IEEE 754 Floating Point (Single Precision - 32 bit):
It consists of three fields:

1. Sign (S) - 1 bit: 0 for positive, 1 for negative.
2. Exponent (E) - 8 bits: Represents the scale using Excess-127 notation (Biased exponent).
3. Mantissa/Significand (M) - 23 bits: Represents the fractional part of the number (normalized).

Value formula:

1. Octal Addition:

• . In Octal: Sum 2, Carry 1.
• . In Octal: Sum 1, Carry 1.
• . In Octal: Sum 1, Carry 1.
• Final Carry 1 comes down.
• Result:

2. Hexadecimal Subtraction:

• (10 - 11): Cannot do. Borrow from 3. $3$ becomes $2$. becomes .
• . Hex for 15 is F.
• .
• Result:

Parity:
A parity bit is an extra bit added to a binary code to ensure that the total number of 1-bits in the string is even or odd. It is used for error detection.

Types:

1. Even Parity: The total number of 1s (including the parity bit) must be even.
   • Data: $1011$ (three 1s). Parity bit added: $1$ $11011$ (four 1s).
2. Odd Parity: The total number of 1s must be odd.
   • Data: $1011$ (three 1s). Parity bit added: $0$ $01011$ (three 1s).

Limitations:

• It can only detect single-bit errors (or odd number of errors).
• If two bits flip (e.g., and ), the parity remains valid, and the error goes undetected.
• It cannot correct errors, only detect them.

Complements:
Complements are used in digital computers to simplify the subtraction operation and for logical manipulation. For a base- system, there are two types:

1. Radix-minus-one Complement ('s complement).
2. Radix Complement ('s complement).

Calculation for Decimal (Base 10):
Number , Digits .

1. 9's Complement ('s):

• Formula:
• Calculated by subtracting each digit from 9.
• 9's Comp = 543

2. 10's Complement ($10$'s):

• Formula: 9's Complement + 1
• 10's Comp = 544

Excess-3 Code:
It is a non-weighted BCD code obtained by adding 3 to the corresponding 8421 BCD code of a decimal digit.

Self-Complementing Property:
A code is self-complementing if the 9's complement of a decimal number can be obtained just by swapping 0s and 1s (1's complement) of the binary code.

Example:

• Decimal $2$ in Excess-3:
• 9's Complement of 2 is $7$.
• Take 1's complement of Excess-3(2) :
• Is $1010$ the Excess-3 code for 7?
• . Yes.
• Therefore, Excess-3 is self-complementing.

Classification of Binary Codes:

1. Weighted Codes: (e.g., 8421 BCD, 2421)
2. Non-Weighted Codes: (e.g., Excess-3, Gray Code)
3. Alphanumeric Codes: (e.g., ASCII, EBCDIC)
4. Error Detection/Correction Codes: (e.g., Parity, Hamming)

Weighted vs Non-Weighted:

Perform multiplication similar to decimal numbers.

Verification:

• Result .
  Matches.

1. Radix to Decimal:
Use the Positional Weight Expansion method.
Multiply each digit by its weight , where is the position index (0 for LSB integer, -1 for first fractional digit), and sum them up.

2. Decimal to Radix :
Split into Integer and Fractional parts.

• Integer Part (Successive Division):

  1. Divide the decimal integer by .
  2. Record the remainder.
  3. Divide the quotient by .
  4. Repeat until quotient is 0.
  5. Read remainders from bottom to top.

• Fractional Part (Successive Multiplication):

  1. Multiply the fraction by .
  2. Record the integer part of the result.
  3. Take the fractional part for the next step.
  4. Repeat until fraction is 0 or desired precision is reached.
  5. Read recorded integers from top to bottom.

Alphanumeric Codes:
Codes that represent numbers, alphabetic characters (A-Z), and special symbols ( etc.). Used for data communication and IO.

1. ASCII (American Standard Code for Information Interchange):

• Most widely used standard.
• Typically 7-bit code (128 characters).
• Includes control characters (NULL, ACK) and printable characters.
• Example: 'A' is $65$ ($1000001$).

2. EBCDIC (Extended Binary Coded Decimal Interchange Code):

• 8-bit code (256 characters).
• Developed by IBM for mainframes.
• Different ordering than ASCII (e.g., letters are not contiguous blocks).

Operation: where , .

Step 1: Find 16's Complement of N (971)

• 15's Comp:
• 16's Comp:

Step 2: Add M to 16's Comp of N

• Calculation check: . . .

Step 3: Check Carry

• A carry of 1 is generated.
• This indicates the result is positive.
• Discard the carry.

Result:

1. Number: $00110101$

• 1's Complement: Invert all bits $11001010$
• 2's Complement: Add 1 to 1's Comp
  

2. Number: $11001000$

• 1's Complement: Invert all bits $00110111$
• 2's Complement: Add 1 to 1's Comp
  

Data Representation:
It refers to the methods used to represent information (numbers, text, images, audio) inside a computer using binary signals (0s and 1s).

Impact of Radix:

1. Binary (Base 2):
   • Storage: Matches the ON/OFF nature of transistors. Most efficient for hardware implementation.
   • Readability: Very poor for humans (long strings: $1101010...$).
2. Decimal (Base 10):
   • Readability: Natural for humans.
   • Storage: Inefficient for digital hardware (requires conversion or complex BCD circuits).
3. Hexadecimal (Base 16) & Octal (Base 8):
   • Compromise: They act as shorthand for binary. One Hex digit represents 4 binary bits.
   • Usage: Used by programmers to view memory addresses and data codes concisely.