Fixed-point arithmetic: fractions with integers

A Cortex-M0 or an 8-bit MCU has no floating-point unit (FPU), so a single float multiplication takes tens of clocks. Fixed point adds an agreement to an integer, "the binary point is here", and handles fractions at integer speed.

Q format

Q15 uses the lower 15 bits of a 16-bit integer as the fraction. Real value = integer value / 2¹⁵, so it represents −1.0 up to just under 1.0.

Real value	Q15 integer
0.5	16384
-0.25	-8192
0.999	32735

Multiplication

Multiplying two Q15 values gives Q30, so shift right by 15 bits to bring it back. Hold the intermediate result in 32 bits.

int16_t q15_mul(int16_t a, int16_t b) {
    int32_t r = (int32_t)a * b;
    return (int16_t)(r >> 15);
}

Cautions

• Overflow: if a sum goes out of range, the sign flips. Use saturating arithmetic, which stops at the maximum.
• Precision: multiplying small values repeatedly loses the low bits. Give generous bits to variables that keep accumulating, such as an integrator.
• Document the scale: record which Q format each variable uses in its name or a comment, to prevent mixing them later.

With an FPU, as on a Cortex-M4, float can be both easier and faster, so choose by looking at the target MCU.