Number Systems for RTL Engineers · RTL Design

Part 1 · Foundations · Beginner

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A register holds bits, and declarations decide what they mean. Signedness and width decide comparison, shifts, extension and overflow.

Same bits interpreted as unsigned or two's complement, with different ranges and overflow conditions.

The bits never change; the meaning, the range and the overflow rule do.

6 lessons, about 22 min

Representation Drives BehaviorYou can state how a bit pattern is read as signed or unsigned, and predict the result of mixed-signedness expressions.3 min
Binary and Hexadecimal Conversion HabitsYou can convert between binary and hex by nibbles and write sized Verilog literals with the right width, base and sign.4 min
Two's Complement IntuitionYou can decode a two's complement value from its weights and explain why carry and signed overflow are different questions.5 min
Sign Extension, Zero Extension and CastingYou can extend a value with the right fill bits, and write casts so that mixed signed and unsigned operands give the intended result.3 min
SystemVerilog Signed Arithmetic PatternYou can write signed add and multiply with the right result width and narrow the result by truncating, saturating or flagging overflow.3 min
Overflow Detection PatternsYou can derive carry and signed-overflow flags for add and subtract, and say which one matches the interpretation of the data.4 min

Check yourself How many distinct values can an N-bit register hold, regardless of interpretation?

Exactly 2ᴺ. Interpretation only changes which numbers those patterns stand for (0…2ᴺ−1 unsigned, or −2ᴺ⁻¹…2ᴺ⁻¹−1 signed).

The whole topic on one page, taken from each lesson's Remember box. Read it before an interview, or print it.

Representation Drives Behavior

Any unsigned operand makes the whole arithmetic or comparison expression unsigned.
The shift amount is the exception: it does not change the signedness of the shifted value.
>> never sign-extends; >>> on a signed operand keeps the sign, but it rounds toward minus infinity: -5 >>> 1 is -3, while -5 / 2 is -2.

Rule of thumb: declare numbers signed and raw bit patterns unsigned, and cast explicitly between them.

Binary and Hexadecimal Conversion Habits

One hex digit is four bits: group from the right and translate each group.
A sized literal is <size>'[s]<base><digits>. Write a negative value with a separate unary minus.
An unsized literal or bare 10 is at least 32 bits wide and can widen the whole expression.

Rule of thumb: size every literal.

Two's Complement Intuition

The top bit has negative weight. N bits cover -2^(N-1) to 2^(N-1) - 1.
Arithmetic is modulo 2^N: carry marks the unsigned boundary, signed overflow marks the signed one.
The most negative value has no positive counterpart. Negate, abs and saturate blocks must handle it.

Rule of thumb: negate by inverting every bit and adding one, but sign-extend first if you need the exact result.

Sign Extension, Zero Extension and Casting

Zero extension is correct for unsigned values and sign extension for two's complement values.
A concatenation is unsigned even when it copies the sign bit.
A cast changes interpretation, not bits. To mix operands, zero-extend the unsigned one into a wider signed type first.

Rule of thumb: widen first with the fill you mean, then do the arithmetic in one signedness.

SystemVerilog Signed Arithmetic Pattern

An add of two N-bit numbers needs N+1 bits and a multiply needs 2N.
In a direct assignment the wide destination sets the arithmetic width, but it cannot recover bits lost inside a narrow intermediate.
>>> divides a signed value by 2^k and rounds toward minus infinity: -5 >>> 1 is -3.

Rule of thumb: narrow only at the end, and only on purpose.

Overflow Detection Patterns

Carry is the unsigned flag. Signed overflow happens only when both operands have the same sign and the result sign differs.
For a - b, form a + ~b + 1: the carry means no borrow, and overflow needs operands of different signs.
The two flags are independent, so compute both from one verified module.

Rule of thumb: pick the flag that matches how the bits are interpreted.