Binary Encodings

Unsigned, two's complement, Gray code, BCD, Excess-3, parity and Hamming: choose the right encoding for the circuit.

One binary word, several meanings

A bit bus has no single meaning. The pattern 1000 can represent:

• 8 as an unsigned integer;
• -8 as a 4-bit two's-complement value;
• a Gray-coded position;
• an invalid BCD digit if a decimal digit from 0 to 9 is expected.

Encoding is therefore part of the specification. It must be chosen before writing comparisons, additions or conversions.

Unsigned integers

Unsigned encoding represents positive values only.

In VHDL, use unsigned to make the intent explicit:

signal counter : unsigned(7 downto 0);
 
counter <= counter + 1;

If counter is 255, the next increment returns to 0 on 8 bits. This wraparound is not a bug: it is a modulo 256 overflow.

Two's complement

Two's complement is the most common signed encoding. On N bits, the range is:

-2^(N-1) to +2^(N-1)-1

To obtain the opposite of a number:

1. invert all bits;
2. add 1.

Example on 4 bits:

+3  = 0011
NOT = 1100
+1  = 1101  therefore -3

In VHDL:

signal a, b, y : signed(7 downto 0);
 
y <= a - b;

The signed type tells both the synthesizer and the reader that the MSB is a sign bit.

Width extension

Before addition or subtraction, operands sometimes need to be widened to keep the result.

For unsigned values, add zeros on the left:

1010₂ on 4 bits becomes 01010₂ on 5 bits

For signed two's-complement values, propagate the sign bit:

0010₂ (+2) becomes 00010₂
1010₂ (-6) becomes 11010₂

In VHDL, use resize with the right type:

signal a8  : signed(7 downto 0);
signal y12 : signed(11 downto 0);
 
y12 <= resize(a8, y12'length);

Most signed-encoding errors come from incorrect extension or silent mixing between std_logic_vector, signed and unsigned.

Gray code

Gray code changes one bit only between consecutive values. It is useful when a value is read during a mechanical or asynchronous transition.

Common uses:

• rotary encoder;
• position measurement;
• asynchronous FIFO pointers;
• information crossing between two clock domains.

The goal is to reduce the risk of reading an incoherent intermediate value when several bits would otherwise change at the same time.

BCD

BCD (binary coded decimal) encodes each decimal digit on 4 bits.

12₁₀ = 0001 0010 in BCD
12₁₀ = 0000 1100 in pure binary

BCD is less compact than pure binary, but it simplifies display-oriented interfaces: 7-segment displays, decimal input, watches and user-visible counters.

Codes 1010 to 1111 are not valid decimal digits in BCD.

8421 BCD is a weighted code: each position has a fixed weight.

For example, 0101 is 4 + 1 = 5.

Excess-3

Excess-3 is another 4-bit decimal encoding. Add 3 to the decimal digit, then encode the result as natural binary.

It is not more compact than BCD, but it often appears in courses because it shows that a 4-bit word can follow a convention other than natural binary.

Parity

A parity bit adds a minimal amount of information to detect some transmission or storage errors.

With even parity, the total number of bits set to 1 must be even. In practice, the XOR of all protected bits and the parity bit must be 0.

Example:

data: 1011  (3 ones)
even parity: 1
protected word: 1011 1  (4 ones)

Simple parity detects an error affecting an odd number of bits, but it does not tell which bit is wrong. It cannot correct the error by itself.

Hamming (7,4)

Hamming (7,4) protects 4 data bits with 3 parity bits. It produces a 7-bit word able to correct one single-bit error.

In the classic convention, power-of-two positions carry parity:

During decoding, the parity equations are recomputed. The result is called the syndrome:

• syndrome 0: no single-bit error detected;
• non-zero syndrome: position of the likely faulty bit;
• correction: flip the bit indicated by the syndrome.

Without an extra global parity bit, a classic Hamming (7,4) code does not correct two errors and does not always detect them cleanly. With global parity, it becomes a SECDED-style scheme: single-error correction, double-error detection.

Choosing the encoding

A good design states the encoding clearly in names, types and useful comments. This avoids implicit conversions and makes simulations easier to read.

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