Related Concepts: 05 Excess-3 Code & Self-Complementing Logic | 06 Gray Code & Code Conversions

04 BCD and Weighted Digital Codes

Concept Overview

Binary-Coded Decimal (BCD) is a family of binary encoding schemes where each individual decimal digit ($0$ through $9$) is represented separately by a 4-bit binary group.

The 8421 code is the standard form of BCD. It is a weighted code, meaning each bit position within a 4-bit group carries a specific positional weight ($8$, $4$, $2$, and $1$).

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1. Binary Codes Classification

graph TD
    Codes[Binary Codes] --> Weighted[Weighted Codes]
    Codes --> NonWeighted[Non-Weighted Codes]
    
    Weighted -->|8421 Positional Weights| StandardBCD[8421 Standard BCD]
    Weighted -->|Self-Complementing| Code2421[2421 Code]
    Weighted -->|Self-Complementing| Code84n2n1[84-2-1 Code]
    Weighted --> Code5421[5421 Code]
    Weighted --> Code5211[5211 Code]
    
    NonWeighted -->|Self-Complementing| XS3[Excess-3 Code]
    NonWeighted -->|Single Bit Change| Gray[Gray / Reflected Code]

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2. Characteristics of 8421 BCD Code

Advantages (Merits)

• Simple & Intuitive: Provides a direct mapping between decimal numbers and binary representations.
• No Multi-Digit Conversions: Converting decimal to BCD simply replaces each decimal digit with its 4-bit binary code word.
• Display System Friendly: Ideal for interfacing with 7-segment displays and decimal output devices {calculators, digital clocks, multimeters}.
• Industry Standard: Standardized across human-machine interfaces (HMIs).

Disadvantages (Demerits)

• Memory Inefficient: Requires 4 bits per decimal digit. For large numbers, BCD uses significantly more storage than pure binary {decimal $15_{10}$ needs 8 bits in BCD: 0001 0101, but only 4 bits in pure binary: 1111}.
• Complex Arithmetic Operations: Binary adders cannot directly add BCD numbers without special correction steps.
• Extra Hardware Requirements: BCD addition requires correction logic {adding $+6$ or binary 0110} whenever a 4-bit sum exceeds $9$ ($1001_2$) or generates a carry.

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3. Weighted BCD Codes & Comparison Table

A code is classified as weighted if every bit position $i$ is assigned a fixed numerical weight $w_i$. The decimal value $N$ of a 4-bit word $b_3{b}_2{b}_1{b}_0$ is evaluated using:

$N={\sum }_{i=0}^3(b_i\cdot w_i)=b_3{w}_3+b_2{w}_2+b_1{w}_1+b_0{w}_0$

Comparison Table of 4-Bit Weighted Codes:

Decimal Digit	8421 (Standard BCD)	2421 (Self-Complementing)	5421 Code	84-2-1 Code (Self-Complementing)	5211 Code
0	0000	0000	0000	0000	0000
1	0001	0001	0001	0111	0001
2	0010	0010	0010	0110	0011
3	0011	0011	0011	0101	0101
4	0100	0100	0100	0100	0111
5	0101	1011	1000	1011	1000
6	0110	1100	1001	1010	1001
7	0111	1101	1010	1001	1011
8	1000	1110	1011	1000	1101
9	1001	1111	1100	1111	1111

Verification of Bit Weights (Decimal 5)

• 8421 Code (0101): $8(0)+4(1)+2(0)+1(1)=4+1=5$
• 2421 Code (1011): $2(1)+4(0)+2(1)+1(1)=2+0+2+1=5$
• 5421 Code (1000): $5(1)+4(0)+2(0)+1(0)=5$
• 84-2-1 Code (1011): $8(1)+4(0)+(-2)(1)+(-1)(1)=8-2-1=5$
• 5211 Code (1000): $5(1)+2(0)+1(0)+1(0)=5$

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4. Invalid BCD Combinations & Don’t Care States

A 4-bit binary group yields $2^4=16$ possible combinations (0000 through 1111). Because BCD only uses the first 10 states (0000 to 1001 representing decimals 0–9), there are 6 invalid states:

$\text{Invalid 8421 States: }1010_2(10),1011_2(11),1100_2(12),1101_2(13),1110_2(14),1111_2(15)$

Use of Invalid States in Circuit Design

When designing combinational BCD circuits using K-maps, these 6 invalid states are assigned as “Don’t Care” (X) conditions. This allows logic minimization to form larger K-map groupings, yielding simpler Boolean equations.

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Past Year Questions (PYQs) & Solutions

1. Number System vs. Code & Multi-Conversion (2018 - 12 Marks)

Question: Explain the distinction between number system and code with example. Convert $(605{)}_{10}$ into: (i) BCD code, (ii) 5211 code, and (iii) Gray code.

Part A: Distinction between Number System and Code

• Number System: A positional mathematical system {such as Decimal, Binary, Octal, or Hexadecimal} used to express quantitative magnitudes and perform arithmetic operations using a fixed base radix ($r^i$).
• Code: A symbolic binary mapping scheme {such as BCD, Excess-3, or Gray code} used to encode numbers or characters into binary words for storage, transmission, or display, where standard binary arithmetic rules do not directly apply.

Part B: Conversions of $(605{)}_{10}$

• (i) BCD (8421): $6\to 0110,0\to 0000,5\to 0101⟹011000000101_{\text{BCD}}$
• (ii) 5211 Code: From table: $6\to 1001,0\to 0000,5\to 1000⟹100100001000_{5211}$
• (iii) Gray Code:
  • $(605{)}_{10}=(1001011101{)}_2$
  • Convert Binary to Gray via XOR: $1101110011_{\text{Gray}}$

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2. Register Decoding Numerical (2022 - 12 Marks)

Question: A 12-bit register contains 100010010111. What decimal value does it represent if it holds:

1. Three decimal digits in BCD?
2. Three decimal digits in 8,4,-2,-1 code?
3. A pure binary number?

Solutions:

1. BCD Decoding: Chunks 1000 | 1001 | 0111 $\to 8,9,7⟹897_{10}$.
2. 8,4,-2,-1 Code Decoding:
   • 1000 $\to 8(1)=8$
   • 1001 $\to 8(1)-1(1)=7$
   • 0111 $\to 4(1)-2(1)-1(1)=1$
   • Result: $871_{10}$.
3. Pure Binary Decoding: $100010010111_2=2048+128+16+4+2+1=2199_{10}$