Module constellation 

Constellations, Gray mapping and hard decisions (OptiCommPy optic.comm.modulation).

Constellations

A Constellation stores its M = 2^b points indexed by their bit label: the point points()[i] carries the bits of i written on b bits, most significant bit first. The points are those of OptiCommPy, not normalised:

• PAM: {-(M-1), …, -1, 1, …, M-1} (minimum distance 2, E_s = (M²-1)/3);
• square QAM: the Cartesian product of two √M-PAM (E_s = 2(M-1)/3);
• PSK: e^{j2πk/M}, k = 0..M-1 (E_s = 1);
• OOK: {0, 1}.

The Gray labels are assigned as in OptiCommPy: the points are listed in their construction order (PAM and PSK in increasing order; QAM row by row from the top, every other row reversed, a “snake” that makes neighbouring points adjacent in the list), and the i-th point of the list gets the Gray code i ⊕ (i >> 1). Two nearest neighbours then differ by one bit, and the bits of a symbol stream are those of OptiCommPy’s modulateGray.

Use Constellation::normalized for unit average energy (E_s = 1), the usual convention of the simulations.

Decisions

Constellation::decide returns the nearest point (the maximum-likelihood decision over AWGN); the soft demapper is in crate::demapper.

Examples

use comnum_mapping::constellation::{Constellation, ConstellationKind};
use comnum_math::C64;
let c = Constellation::qam(16).unwrap();
assert_eq!(c.points()[0], C64::new(-3.0, 3.0));            // label 0000
assert!((c.energy() - 10.0).abs() < 1e-12);                 // 2(M-1)/3
let x = c.map_bits(&[0, 0, 0, 1, 1, 0, 1, 1]).unwrap();
assert_eq!(x, vec![c.points()[1], c.points()[11]]);
// Gray mapping: nearest neighbours differ by one bit
for i in 0..16 {
    for j in 0..16 {
        if (c.points()[i] - c.points()[j]).norm() == 2.0 {
            assert_eq!((i ^ j).count_ones(), 1);
        }
    }
}
assert_eq!(Constellation::new(ConstellationKind::Psk, 8).unwrap().order(), 8);

Structs

Constellation

A labelled constellation: points()[i] carries the bits of i.

Enums

ConstellationKind

The family of a constellation.

Constants

MAX_GRAY_BITS

The largest number of bits of gray_code (16 777 216 code words; the $2^{n}$ words are allocated at once).

PRIOR_FLOOR

The floor of a prior probability before its logarithm: $\ln 10^{-300} \approx -691$.

Functions

bpsk

The real BPSK symbols $1 - 2b$ of bits ($0 \mapsto +1$, $1 \mapsto -1$), the modulation of the code tests over a real channel (crate::demapper::bpsk_llr is its demapper).

gray_code

The Gray code of every n-bit integer: code[i] = i ⊕ (i >> 1) (comnum_math::gray).

indices_to_bits

The bits of every index, bits_per_symbol bits each, most significant bit first (comnum_math::index_to_bits).

log_priors

The log of prior probabilities, floored at PRIOR_FLOOR so that a zero probability gives a finite log (the metric of an impossible point).

qam_side

The side of a square QAM of order m (None unless m is a perfect square).