Q formats¶

DSP code on a microcontroller without a float unit keeps fractions in integers. Q15 is the common one: an i16 that means a number from −1 to almost 1.

Q15¶

A stored i16 \(n\) means

In exact_units it is Q15<U>, a Fixed<U, Pow2<-15>, i16, Saturating>:

use exact_units::{Q15, fixed, units::Ratio};

let half: Q15<Ratio> = fixed!(0.5);
let neg: Q15<Ratio> = fixed!(-0.5);
assert_eq!(half.get(), 0x4000);
assert_eq!(neg.get(), -0x4000); // 0xc000 as bits
assert_eq!(format!("{half:#}"), "0.5");

The bits¶

The top bit is the sign, the other 15 are the fraction. One half is 0x4000:

And −0.5 is 0xc000: the sign bit's −1 plus the fraction's 0.5.

It saturates¶

Q15 is Saturating: + and - stop at the ends instead of wrapping. Two voices at 0.8 and 0.5 mix to the top, not to −0.7:

use exact_units::{Q15, fixed, units::Ratio};

let a: Q15<Ratio> = fixed!(0.8);
let b: Q15<Ratio> = fixed!(0.5);
assert_eq!((a + b).get(), i16::MAX);            // 1 − 2⁻¹⁵: a clipped peak
assert_eq!(26214i16.wrapping_add(16384), -22938); // a plain i16 flips to −0.7: a click

See Overflow for the plot.

With a unit¶

The unit defaults to Ratio, a plain number. A Q15 can carry a unit too, for a voltage normalised to a full scale:

use exact_units::{Q15, fixed, units::Volt};

let v: Q15<Volt> = fixed!(0.25);
assert_eq!(format!("{v:#}"), "0.25 V");

The family¶

Any other Q format is a Fixed with a Pow2 scale. UQ16_16 has 16 integer bits and 16 fraction bits; 1.5 is 0x0001_8000:

use exact_units::{UQ16_16, fixed, units::Ratio};

let x: UQ16_16<Ratio> = fixed!(1.5);
assert_eq!(x.get(), 0x0001_8000);

Multiplying¶

Q15 · Q15 is a Q30 number, \(\frac{a}{2^{15}} \cdot \frac{b}{2^{15}} = \frac{ab}{2^{30}}\). Back into Q15 it must be rounded, and that step is named: Products and requantization.