📚 Lesson: Binary Code & Gray Code

1️⃣ Introduction to Number Systems in Digital Logic

Digital systems represent and process information using discrete values — most commonly binary digits (bits), which can be either 0 (LOW) or 1 (HIGH).
Understanding binary and its variations is essential for designing and analyzing digital circuits.

2️⃣ Binary Code (Base-2 System)

Definition

Binary code is a base-2 numeral system that uses only two symbols: 0 and 1.
Each position in a binary number represents a power of 2.

Structure

For an n-bit binary number:

\[(b_{n-1} b_{n-2} \dots b_1 b_0)_2 = b_{n-1} \cdot 2^{n-1} + b_{n-2} \cdot 2^{n-2} + \dots + b_1 \cdot 2^1 + b_0 \cdot 2^0\]

Binary Counting

The standard binary count is the sequence in which binary numbers increase by one each time, starting from all zeros.
This sequence is used in timing diagrams, counters, and digital test patterns.

Decimal	Binary
0	0000
1	0001
2	0010
3	0011
4	0100
5	0101
6	0110
7	0111
8	1000
9	1001
10	1010
11	1011
12	1100
13	1101
14	1110
15	1111

Timing Diagram Note

When multiple input bits change at the same time (e.g., from 111 to 000), slight differences in switching

speed can cause a transient glitch — a short, unexpected output change.

Such glitches can cause errors in sensitive circuits, which is one reason alternative codes like Gray Code are used.

3️⃣ Gray Code (Reflected Binary Code)

Definition

Gray Code is a binary numeral system where two successive values differ in only one bit.
It’s also called a Unit Distance Code because the “distance” between consecutive codes is exactly one bit change.

Why Use Gray Code?

Prevents transient errors during transitions.
Ideal for rotary encoders, analog-to-digital converters, and mechanical position sensors.
Ensures only one bit changes at a time, reducing ambiguity.

Gray Code Sequences

2-bit Gray Code: 00 01 11 10

3-bit Gray Code: 000 001 011 010 110 111 101 100

Notice that after the last value, only one bit changes to return to the first value — making it cyclic.

4️⃣ Binary ↔ Gray Code Conversion

Binary to Gray

MSB of Gray = MSB of Binary.
Each next Gray bit = XOR of current Binary bit and previous Binary bit.

Example: Binary 1010 → Gray:

MSB: 1
Next: $1 \oplus 0 = 1$
Next: $0 \oplus 1 = 1$
Next: $1 \oplus 0 = 1$

Result: 1111

Gray to Binary

MSB of Binary = MSB of Gray.
Each next Binary bit = Previous Binary bit XOR Current Gray bit.

Example: Gray 1111 → Binary:

MSB: 1
Next: $ \oplus 1 = 0$
Next: $0 \oplus 1 = 1$
Next: $1 \oplus 1 = 0$

Result: 1010

5️⃣ Binary vs. Gray Code (0–15)

Decimal	Binary	Gray Code
0	0000	0000
1	0001	0001
2	0010	0011
3	0011	0010
4	0100	0110
5	0101	0111
6	0110	0101
7	0111	0100
8	1000	1100
9	1001	1101
10	1010	1111
11	1011	1110
12	1100	1010
13	1101	1011
14	1110	1001
15	1111	1000

Teaching Note

In binary, multiple bits can change between consecutive numbers (e.g., 0111 → 1000 changes all four bits).
In Gray Code, only one bit changes at each step, which is why it’s called a Unit Distance Code.
This property is what prevents glitches in timing-sensitive circuits.

6️⃣ Building Gray Code with the Mirroring Method

Another way to generate Gray Code sequences — without doing any bitwise math — is to use the mirroring (or reflect‑and‑prefix) method. This is especially handy for quickly building longer sequences like 4‑bit or 5‑bit Gray codes. Steps:

Start with the 1‑bit Gray code: 0 1
Mirror the list (write it in reverse order).
Prefix the original list with 0 and the mirrored list with 1.
Repeat the process to add more bits.

Example: 4‑bit Gray Code

7️⃣ Practice Problems

Convert 1101₂ to Gray Code.
Convert Gray Code 1011 to Binary.
Explain why Gray Code is preferred in mechanical position sensors.
Identify the glitch-prone transition in the binary count from 0–15.

8️⃣ Summary

Binary Code is the foundation of all digital logic, representing data in 0s and 1s.
Standard binary counting is used in timing diagrams but can cause glitches when multiple bits change simultaneously.
Gray Code changes only one bit at a time, reducing the risk of transient errors.
Understanding both systems — and converting between them — is essential for digital engineers.