Computer Architecture: From Number Systems to Parallelism

Codes: Gray, BCD, Excess‑3, ASCII, Parity

Digital systems require specialized encoding schemes to represent decimal digits, characters, and detect transmission errors. This guide covers five fundamental binary codes that form the backbone of digital communication and computation systems.

Overview of Binary Codes

Binary codes serve different purposes in digital systems. Each code is optimized for specific applications. Understanding their properties, conversion methods, and applications is crucial for designing efficient digital systems.

Code Classification Summary

• BCD (8421): Base-10, weighted, not self-complementary, no error detection; used in decimal display systems.

• Excess-3: Base-10, non-weighted, self-complementary, limited error detection; used in arithmetic operations.

• Gray Code: Base-2, non-weighted, not self-complementary, no error detection; used in position encoding.

• ASCII: Not weighted, not self-complementary, no error detection; used in character representation.

• Parity: Not weighted, not self-complementary, used for error detection.

1. Binary Coded Decimal (BCD – 8421 Code)

Definition and Properties
BCD represents each decimal digit (0 to 9) using exactly 4 bits with weights of 8, 4, 2, and 1. It is a weighted code.

Conversion Methods

• Decimal to BCD: Convert each decimal digit separately to its 4-bit binary form and concatenate.
  Example: 527 → 5 → 0101, 2 → 0010, 7 → 0111 → Result: 0101 0010 0111.

• Binary to BCD: First convert the binary number to decimal, then apply decimal-to-BCD conversion.
  Example: 11101₂ → 29₁₀ → 2 → 0010, 9 → 1001 → Result: 0010 1001.

BCD Arithmetic

• Addition: If the result exceeds 9 or generates a carry, add 0110 (6) to correct.
  Example: 8 (1000) + 7 (0111) = 1111 (invalid) + 0110 = 1 0101 (15₁₀).

• Subtraction: If a borrow occurs, subtract 0110 from the digit receiving the borrow.

2. Gray Code (Reflected Binary Code)

Definition and Properties
Gray code ensures that only one bit changes between consecutive code words. This minimizes transition errors, especially in mechanical systems.

Conversion Methods

• Binary to Gray:

  1. The most significant bit (MSB) of Gray code is the same as the MSB of the binary.

  2. Each following Gray bit is the XOR of the current and previous binary bits.
     Example: 1011 → Gray = 1110.

• Gray to Binary:

  1. MSB of Binary = MSB of Gray.

  2. Each following Binary bit is the XOR of the previous binary bit and the current Gray bit.

Applications
Used in analog-to-digital converters, rotary encoders, clock domain crossing, and genetic algorithms.

3. Excess‑3 Code

Definition and Properties
Excess-3 is a non-weighted, self-complementary code that represents decimal digits by adding 3 to their binary values. This simplifies digital arithmetic and complements.

Conversion Methods

• Method 1 – Direct Addition:
  Add 3 to each decimal digit and convert to 4-bit binary.
  Example: 27 → 2+3 = 5 → 0101; 7+3 = 10 → 1010 → Result: 0101 1010.

• Method 2 – BCD Addition:
  Convert the decimal to BCD, then add 0011 (3) to each group.
  Example: 45 → 0100 0101 + 0011 → 0111 1000.

• Excess-3 to Decimal:
  Subtract 3 from each 4-bit group.
  Example: 0110 1001 → 6 → 3; 9 → 6 → Result: 36.

Self-Complementary Property
Taking the 1’s complement of an Excess-3 code gives the Excess-3 representation of the 9’s complement of the original digit.
Example: 5 (1000) → 1’s comp = 0111 → Represents 4 (which is 9−5).

4. ASCII Code

Definition and Structure
ASCII is a 7-bit character encoding standard representing 128 characters, including control characters, digits, uppercase/lowercase letters, and symbols.

Examples of ASCII Codes

• Control Characters: 0 = NUL, 1 = SOH, 8 = BS (Backspace), 9 = HT (Tab), 10 = LF (Line Feed), 13 = CR (Carriage Return), 27 = ESC.

• Digits: 0 to 9 → ASCII 48 to 57 → Binary starts from 0110000 to 0111001.

• Letters:
  Uppercase A-Z = 65 to 90
  Lowercase a-z = 97 to 122

Conversion Methods

• Text to ASCII:
  Example: "HELLO" → H = 1001000, E = 1000101, L = 1001100, O = 1001111.

• ASCII to Text:
  Example: 1000011 1000001 1010100 → 67, 65, 84 → CAT.

ASCII Patterns

• Digits: start with 0011

• Uppercase: start with 100

• Lowercase: start with 110

• Toggle case: flip the 5th bit (±32).

5. Parity Code

Definition and Types
Parity adds an extra bit to a binary message to make the number of 1s even (even parity) or odd (odd parity). It is a basic form of error detection.

Even Parity
If the number of 1s in the data is even, parity bit = 0; if odd, parity = 1.
Example:

• 0000000 → Even → Add 0 → 00000000

• 0000001 → Odd → Add 1 → 10000001

Odd Parity
If the number of 1s is even, parity bit = 1; if odd, parity = 0.
Example:

• 0000000 → Even → Add 1 → 10000000

• 0000001 → Odd → Add 0 → 00000001

Implementation

• Count the number of 1s in the data.

• Add a parity bit to maintain even or odd parity as required.

• Used in error checking during transmission.

Error Detection
At the transmitter, the parity bit is calculated and added. At the receiver, the bits are re-evaluated. If the parity doesn’t match, an error is detected.
Limitation: Parity cannot detect errors if an even number of bits are flipped.

Code Comparison and Applications

• BCD: Excellent for digital displays; acceptable for arithmetic with correction logic.

• Excess-3: Great for arithmetic operations due to self-complementary nature.

• Gray Code: Best for applications needing glitch-free transitions, like position sensors.

• ASCII: Industry standard for representing characters in computing and communication.

• Parity: Basic error detection in communication systems.

Practice Problems

1. BCD Addition
   Add 0111 (7) + 1001 (9) = 10000 (invalid) → Add 0110 → Result: 1 0110 (16).

2. Gray Code Sequence (3-bit)
   Sequence: 000, 001, 011, 010, 110, 111, 101, 100.

3. ASCII Encoding Example
   "Code‑123":

• C = 67 → 1000011

• o = 111 → 1101111

• d = 100 → 1100100

• e = 101 → 1100101

• • = 45 → 0101101

• 1 = 49 → 0110001

• 2 = 50 → 0110010

• 3 = 51 → 0110011

Key Takeaways

1. BCD is suitable for decimal display systems but needs correction for sums above 9.

2. Excess‑3 code's self‑complementing nature helps in subtraction.

3. Gray code minimizes errors in hardware transitions by changing only one bit.

4. ASCII enables consistent text representation in digital systems.

5. Parity bits provide simple error detection but fail against even-bit errors.

6. The appropriate binary code should be chosen based on the application’s requirements—whether speed, accuracy, simplicity, or fault tolerance.