Module nr 

The numerology, the synchronisation signal block and the demodulation reference signals of 5G NR (and the numerology of LTE, its case $\mu = 0$).

Numerology

NR scales the 15 kHz subcarrier spacing of LTE: $\Delta f = 2^{\mu} \cdot 15$ kHz, $\mu = 0, \dots, 4$; a 10 ms frame holds ten 1 ms subframes of $2^{\mu}$ slots of 14 OFDM symbols (12 with the extended cyclic prefix, $\mu = 2$ only); a resource block is 12 subcarriers (Lin et al. 2019, Sec. II; Astély et al. 2009 for LTE). At the sampling rate $N_{\mathrm{FFT}}\Delta f$, the cyclic prefix of a symbol is $144 N_{\mathrm{FFT}}/2048$ samples, and $(144 + 16 \cdot 2^{\mu}) N_{\mathrm{FFT}}/2048$ for the first symbol of every half subframe ($l = 0$ and $l = 7 \cdot 2^{\mu}$), so that a half subframe lasts exactly 0.5 ms; the extended prefix is $512 N_{\mathrm{FFT}}/2048$ (TS 38.211, Sec. 5.3.1): 4.7 µs and 16.7 µs at $\mu = 0$, the prefixes of LTE. Carrier::modulate maps a grid of $K = 12 N_{\mathrm{RB}}$ subcarriers, the subcarrier $k$ at the frequency $(k - K/2)\Delta f$, and prepends the prefixes; Carrier::demodulate removes them and gives the grid back.

Synchronisation signal block

The SSB occupies 4 OFDM symbols and 240 subcarriers (20 resource blocks): the primary synchronisation signal (PSS) on symbol 0, subcarriers 56 to 182; the secondary (SSS) on symbol 2, same subcarriers; the broadcast channel (PBCH) on symbols 1 and 3 and on subcarriers 0–47 and 192–239 of symbol 2, with its demodulation reference signal (DMRS) on one subcarrier in four, offset by $\nu = N_{\mathrm{ID}}^{\mathrm{cell}} \bmod 4$. The 1008 cell identities are $N_{\mathrm{ID}}^{\mathrm{cell}} = 3 N_{\mathrm{ID}}^{(1)} + N_{\mathrm{ID}}^{(2)}$, $N_{\mathrm{ID}}^{(1)} < 336$, $N_{\mathrm{ID}}^{(2)} < 3$ (TS 38.211, Sec. 7.4.2–7.4.3).

• PSS: the BPSK m-sequence $d(n) = 1 - 2x((n + 43 N_{\mathrm{ID}}^{(2)}) \bmod 127)$, $x(i+7) = (x(i+4) + x(i)) \bmod 2$, $[x(6) \dots x(0)] = [1,1,1,0,1,1,0]$;
• SSS: the BPSK Gold sequence $d(n) = [1 - 2x_{0}((n + m_{0}) \bmod 127)][1 - 2x_{1}((n + m_{1}) \bmod 127)]$, $x_{0}(i+7) = (x_{0}(i+4) + x_{0}(i)) \bmod 2$, $x_{1}(i+7) = (x_{1}(i+1) + x_{1}(i)) \bmod 2$, both started at $[0 \dots 0,1]$, $m_{0} = 15\lfloor N_{\mathrm{ID}}^{(1)}/112 \rfloor + 5 N_{\mathrm{ID}}^{(2)}$, $m_{1} = N_{\mathrm{ID}}^{(1)} \bmod 112$;
• PBCH DMRS: the QPSK $r(m) = [(1 - 2c(2m)) + j(1 - 2c(2m+1))]/\sqrt{2}$, $m < 144$, of the Gold sequence of 3GPP (comnum_math::sequences::gold_3gpp) with $c_{\mathrm{init}} = 2^{11}(\bar{\imath}{\mathrm{SSB}} + 1)(\lfloor N{\mathrm{ID}}^{\mathrm{cell}}/4 \rfloor + 1) + 2^{6}(\bar{\imath}{\mathrm{SSB}} + 1) + (N{\mathrm{ID}}^{\mathrm{cell}} \bmod 4)$, mapped frequency first.

detect_cell_id finds $N_{\mathrm{ID}}^{(2)}$ by the largest correlation of the PSS subcarriers with the three PSS, then $N_{\mathrm{ID}}^{(1)}$ by the largest real part of $\hat{h}^{*} \langle y_{\mathrm{SSS}}, d_{\mathrm{SSS}} \rangle$ over the 336 SSS, with the channel $\hat{h}$ estimated on the PSS (assumed flat over the 127 subcarriers and the three symbols).

DMRS of the PDSCH

PdschDmrs gives the positions and values of the DMRS of a PDSCH of mapping type A over a whole slot (14 symbols, normal prefix), single-symbol, port 1000: the first DMRS symbol $l_{0} = 2$ or 3 (dmrs-TypeA-Position) and the additional positions ${l_{0}}$, ${l_{0}, 11}$, ${l_{0}, 7, 11}$, ${l_{0}, 5, 8, 11}$; the subcarriers $k = 4n + 2k’$ (configuration type 1, one in two) or $k = 6n + k’$ (type 2, two in six), $k’ \in {0, 1}$, with the QPSK value $r(2n + k’)$ of the Gold sequence of $c_{\mathrm{init}} = (2^{17}(14 n_{s} + l + 1)(2N_{\mathrm{ID}} + 1) + 2N_{\mathrm{ID}}) \bmod 2^{31}$ (TS 38.211, Sec. 7.4.1.1). It gives a PilotTable for the channel estimators of super::estimation. Shorter allocations, two-symbol DMRS and the other ports need the tables of TS 38.211 (status needed in references/README.md).

Examples

use comnum_waveform::ofdm::nr::{Numerology, pss, detect_cell_id, ssb_grid};
let nr = Numerology::new(1, false)?;                       // 30 kHz
assert_eq!((nr.subcarrier_spacing_hz(), nr.slots_per_frame()), (30e3, 20));
assert_eq!(nr.cp_lengths(4096)?[..2], [352, 288]);         // (144 + 16·2)·2, 144·2
assert_eq!(pss(0)?.len(), 127);
let grid = ssb_grid(417, 0, None)?;                        // 4 × 240, no PBCH data
assert_eq!(detect_cell_id(&grid)?.0, 417);

References

1. X. Lin et al., “5G New Radio: Unveiling the essentials of the next generation wireless access technology,” IEEE Commun. Standards Mag. 3(3), 30–37, 2019 (arXiv:1806.06898).
2. D. Astély et al., “LTE: the evolution of mobile broadband,” IEEE Commun. Mag. 47(4), 44–51, 2009.
3. 3GPP TS 38.211, NR; Physical channels and modulation, Sec. 5.3, 7.4.

Structs

Carrier

An NR carrier: a grid of n_subcarriers $K$ subcarriers of the numerology numerology, at the FFT size n_fft, whose first OFDM symbol is the symbol first_symbol of a subframe (see the module).

Numerology

A numerology of NR: $\Delta f = 2^{\mu} \cdot 15$ kHz (see the module).

PdschDmrs

The DMRS of a PDSCH of mapping type A over a slot of 14 symbols (see the module).

Enums

SsbElement

What a resource element of the SSB carries.

Constants

SSB_SUBCARRIERS

Subcarriers of the SSB.

SSB_SYMBOLS

OFDM symbols of the SSB.

SUBCARRIERS_PER_RB

Subcarriers of a resource block.

SYNC_FIRST_SUBCARRIER

First subcarrier of the PSS and of the SSS in the SSB.

SYNC_LENGTH

Length of the PSS and of the SSS.

Functions

detect_cell_id

The cell identity of a received SSB grid (grid[l * 240 + k], 960 values), by the PSS then the SSS, coherently with the channel of the PSS (see the module); returns the identity and the normalised correlations $|\langle y, d \rangle| / (|y| |d|)$ of the PSS and of the SSS.

pbch_dmrs

The 144 values of the PBCH DMRS of the cell n_cell_id and the SSB index $\bar{\imath}_{\mathrm{SSB}}$ (0 to 7; see the module).

pss

The PSS of $N_{\mathrm{ID}}^{(2)}$ (127 values $\pm 1$).

ssb_element

What the subcarrier k (0 to 239) of the symbol l (0 to 3) of the SSB of the cell n_cell_id carries (see the module).

ssb_grid

The SSB of the cell n_cell_id with the SSB index $\bar{\imath}_{\mathrm{SSB}}$: grid[l * 240 + k], the PSS and the SSS of unit amplitude, the PBCH DMRS, and the 432 PBCH symbols pbch (zeros when None).

ssb_indices

The resource elements (k, l) of one kind in the SSB, frequency first (the order of the mapping).

sss

The SSS of $(N_{\mathrm{ID}}^{(1)}, N_{\mathrm{ID}}^{(2)})$ (127 values $\pm 1$).