Module sequences 

Binary pseudo-random sequences (linear-feedback shift registers, the length-31 Gold sequence of 3GPP) and the Zadoff–Chu sequences.

Model

A Fibonacci register of order $r$ produces $x(i + r) = \sum_{t \in T} x(i + t) \bmod 2$ from its first $r$ bits (lfsr): every m-sequence and Gold sequence of the standards is one or a sum of two. The pseudo-random sequence of 3GPP (TS 38.211, Sec. 5.2.1; TS 36.211, Sec. 7.2) is

$$ c(n) = \left(x_{1}(n + N_{c}) + x_{2}(n + N_{c})\right) \bmod 2, \qquad N_{c} = 1600, $$

with $x_{1}(n+31) = (x_{1}(n+3) + x_{1}(n)) \bmod 2$, $x_{1}(0) = 1$, $x_{1}(1..30) = 0$, and $x_{2}(n+31) = (x_{2}(n+3) + x_{2}(n+2) + x_{2}(n+1) + x_{2}(n)) \bmod 2$ initialised by $c_{\mathrm{init}} = \sum_{i} x_{2}(i) 2^{i}$ (gold_3gpp): the scrambling of the coded bits and the sequences of the reference signals.

The Zadoff–Chu sequence of root $u$ and odd length $N_{\mathrm{ZC}}$ (zadoff_chu),

$$ x_{u}(n) = e^{-j\pi u n (n + 1) / N_{\mathrm{ZC}}}, \qquad 0 \le n < N_{\mathrm{ZC}}, \quad \gcd(u, N_{\mathrm{ZC}}) = 1, $$

has a constant amplitude and a zero periodic autocorrelation at every nonzero lag (a CAZAC sequence; Chu 1972), and two roots $u_{1} \ne u_{2}$ with $u_{1} - u_{2}$ prime to a prime $N_{\mathrm{ZC}}$ cross-correlate with the constant magnitude $1/\sqrt{N_{\mathrm{ZC}}}$: the primary synchronisation signal of LTE (TS 36.211, Sec. 6.11.1.1, $N_{\mathrm{ZC}} = 63$, its middle element punctured) and the preambles of the random access.

Examples

use comnum_math::sequences::{gold_3gpp, lfsr};
// x³ + x + 1: the m-sequence of period 7
assert_eq!(lfsr(&[0, 1], 3, &[1, 0, 0], 10), vec![1, 0, 0, 1, 0, 1, 1, 1, 0, 0]);
assert_eq!(gold_3gpp(0, 8).len(), 8);
let x = comnum_math::sequences::zadoff_chu(25, 63)?;
assert!(x.iter().all(|v| (v.norm() - 1.0).abs() < 1e-12));

References

1. 3GPP TS 38.211, NR; Physical channels and modulation, Sec. 5.2.1.
2. 3GPP TS 36.211 V18.0.2, E-UTRA; Physical channels and modulation, Sec. 6.11.1.1, 7.2.
3. D. Chu, “Polyphase codes with good periodic correlation properties,” IEEE Trans. Inf. Theory 18(4), 531–532, 1972.

Functions

gold_3gpp

The first n bits of the Gold sequence $c$ of 3GPP with the initial value c_init (31 bits; see the module).

lfsr

The sequence of a linear-feedback shift register of order order: the first order bits are init (zero-padded), then $x_{i + \mathrm{order}} = \sum_{t} x_{i + t} \bmod 2$ over the taps, up to len bits in all (the Fibonacci form of every m-sequence, Gold sequence and PRBS of the standards).

zadoff_chu

The Zadoff–Chu sequence $x_{u}(n) = e^{-j\pi u n (n + 1) / N_{\mathrm{ZC}}}$ of root u and odd length n_zc (see the module).