Binary angles¶

A phase, a rotor position or a heading goes round: past a full turn it starts again. A binary angle (BAM, binary angle measurement) makes that free. The whole range of an unsigned integer is one turn, so the integer's own wrap is the angle's.

Bam16¶

Bam16 is a u16 of \(2^{-16}\) turns, Fixed<Turn, Pow2<-16>, u16, Wrapping>:

use exact_units::Bam16;

let mut phase = Bam16::of(65_000);
phase += Bam16::of(1365);          // one step of a 1 kHz oscillator at 48 kHz
assert_eq!(phase.get(), 829);      // 65000 + 1365 − 65536
assert_eq!(format!("{:#}", Bam16::of(16384)), "0.25 rev");

Bam32 is the same over a u32, \(2^{-32}\) turn a step.

The quadrant is in the top bits¶

The two top bits say which quarter of the turn an angle is in. A sine table indexed by the top bits needs no division. 135°, three eighths of a turn, is 0x6000:

An oscillator¶

A numerically controlled oscillator is a phase accumulator: add a step every sample. With Bam16 that is one plain +:

use exact_units::Bam16;

/// One step per sample: 65536 · f / f_s.
const STEP: Bam16 = Bam16::of(1365); // ≈ 1 kHz at 48 kHz

let mut phase = Bam16::of(0);
for _ in 0..48 {
    phase += STEP; // wraps every turn, never overflows
}
assert_eq!(phase.get(), 65520); // 16 steps short of a turn: 1365 is rounded

Sine and cosine of a binary angle without a float are CORDIC's sin_cos_16.

Angles against a zero¶

A Bam16 is an amount of turn. A heading or a rotor position against a reference is a point: its difference to another heading is the shortest way round, signed.