Module lte 

The downlink signals of LTE that a terminal reads before anything else: the synchronisation signals, the cell-specific reference signals, the resource elements of the broadcast channel, the OFDM grid with its unused DC subcarrier, and the space-frequency block code of transmit diversity (frame structure type 1, FDD, normal cyclic prefix).

Grid and numerology

A downlink carrier of $N_{\mathrm{RB}}$ resource blocks has $K = 12 N_{\mathrm{RB}}$ subcarriers of 15 kHz, numbered $k = 0, \dots, K - 1$ from the lowest; the subcarrier $k$ sits at $(k - K/2)\Delta f$ for $k < K/2$ and $(k - K/2 + 1)\Delta f$ above, the DC subcarrier being left empty (TS 36.211, Sec. 6.12). A slot holds 7 OFDM symbols, with the prefixes of 160 then 144 samples at 30.72 MS/s, the numerology $\mu = 0$ of super::nr::Numerology; Carrier modulates the grid with the prefixes of super::nr::Carrier, the empty DC subcarrier inserted. The six central resource blocks, $k = 6 N_{\mathrm{RB}} - 36, \dots, 6 N_{\mathrm{RB}} + 35$, carry the same signals in every bandwidth: a receiver reads them at 1.92 MS/s with the grid of $N_{\mathrm{RB}} = 6$ and an FFT of 128.

Synchronisation signals (Sec. 6.11)

The 504 cell identities are $N_{\mathrm{ID}}^{\mathrm{cell}} = 3 N_{\mathrm{ID}}^{(1)} + N_{\mathrm{ID}}^{(2)}$, $N_{\mathrm{ID}}^{(1)} < 168$, $N_{\mathrm{ID}}^{(2)} < 3$. Both signals occupy the 62 subcarriers $k = n - 31 + 6 N_{\mathrm{RB}}$, $n = 0, \dots, 61$ (five empty ones on each side), in slots 0 and 10: the PSS in the last symbol of the slot ($l = 6$), the SSS in the symbol before ($l = 5$).

• PSS (pss): the Zadoff–Chu sequence of length 63 and root $u = 25, 29, 34$ for $N_{\mathrm{ID}}^{(2)} = 0, 1, 2$, its middle element punctured, $d_{u}(n) = e^{-j\pi u n (n + 1)/63}$ for $n \le 30$ and $e^{-j\pi u (n + 1)(n + 2)/63}$ for $31 \le n \le 61$ (eq. of Sec. 6.11.1.1, Table 6.11.1.1-1);
• SSS (sss): the interleaving of two m-sequences of length 31, $d(2n) = s_{0}^{(m_{0})}(n) c_{0}(n)$ and $d(2n + 1) = s_{1}^{(m_{1})}(n) c_{1}(n) z_{1}^{(m_{0})}(n)$ in subframe 0, the roles of $m_{0}$ and $m_{1}$ swapped in subframe 5, so that one SSS tells the half frame; $m_{0} = m’ \bmod 31$, $m_{1} = (m_{0} + \lfloor m’/31 \rfloor + 1) \bmod 31$, $m’ = N_{\mathrm{ID}}^{(1)} + q(q + 1)/2$, $q = \lfloor (N_{\mathrm{ID}}^{(1)} + q’(q’ + 1)/2)/30 \rfloor$, $q’ = \lfloor N_{\mathrm{ID}}^{(1)}/30 \rfloor$ (Table 6.11.2.1-1); $\tilde{s}$, $\tilde{c}$ and $\tilde{z}$ are the m-sequences $x(i + 5) = x(i + 2) + x(i)$, $x(i + 3) + x(i)$ and $x(i + 4) + x(i + 2) + x(i + 1) + x(i)$, started at $[0,0,0,0,1]$, as $1 - 2x$; $c_{0}$, $c_{1}$ are $\tilde{c}$ shifted by $N_{\mathrm{ID}}^{(2)}$ and $N_{\mathrm{ID}}^{(2)} + 3$, $z_{1}^{(m)}$ is $\tilde{z}$ shifted by $m \bmod 8$.

Cell-specific reference signals (Sec. 6.10.1)

In the symbol $l$ of the slot $n_{s}$, the CRS is the QPSK of the Gold sequence of 3GPP (comnum_math::sequences::gold_3gpp) with $c_{\mathrm{init}} = 2^{10}(7(n_{s} + 1) + l + 1)(2N_{\mathrm{ID}}^{\mathrm{cell}} + 1) + 2N_{\mathrm{ID}}^{\mathrm{cell}} + 1$, $r(m) = [(1 - 2c(2m)) + j(1 - 2c(2m + 1))]/\sqrt{2}$, of which the carrier uses $r(m + 110 - N_{\mathrm{RB}})$, $m < 2 N_{\mathrm{RB}}$ (crs), on the subcarriers $k = 6m + (v + v_{\mathrm{shift}}) \bmod 6$, $v_{\mathrm{shift}} = N_{\mathrm{ID}}^{\mathrm{cell}} \bmod 6$ (crs_subcarriers): ports 0 and 1 in the symbols $l = 0$ and 4 of every slot, $v = 0$ and 3 (port 0) or 3 and 0 (port 1); ports 2 and 3 in $l = 1$, $v = 3(n_{s} \bmod 2)$ and $3 + 3(n_{s} \bmod 2)$. A port sends nothing on the CRS elements of the others.

Broadcast channel (Sec. 6.6)

The PBCH occupies the 72 central subcarriers of the first four symbols of slot 1, but for the elements of the CRS of four ports, whatever the number of ports: those $k$ with $k \equiv v_{\mathrm{shift}} \pmod 3$ in $l = 0$ and 1; 240 elements per frame, frequency first (pbch_elements). With two or four ports, the QPSK symbols are sent by transmit diversity (sfbc, Sec. 6.3.3.3 and 6.3.4.3): the pairs $(x_{0}, x_{1}) = (d(2i), d(2i + 1))$ go on two consecutive elements as $\frac{1}{\sqrt{2}}[x_{0}, -x_{1}^{}]$ and $\frac{1}{\sqrt{2}}[x_{1}, x_{0}^{}]$ on ports 0 and 1 (Alamouti’s code across subcarriers); with four ports, the pairs alternate between the ports 0, 2 and 1, 3.

Examples

use comnum_waveform::ofdm::lte::{pss, sss, pbch_elements, Carrier};
assert_eq!(pss(1)?.len(), 62);
assert_eq!(sss(100, 1, false)?.len(), 62);
assert_eq!(pbch_elements(301, 6)?.len(), 240);
let c = Carrier::new(6, 128)?;                    // the six central RBs, 1.92 MS/s
assert_eq!(c.sample_rate(), 1.92e6);

References

1. 3GPP TS 36.211 V18.0.2, E-UTRA; Physical channels and modulation, Sec. 6.3.3.3, 6.3.4.3, 6.6, 6.10.1, 6.11, 6.12, 7.1.2, 7.2.
2. D. Astély et al., “LTE: the evolution of mobile broadband,” IEEE Commun. Mag. 47(4), 44–51, 2009.

Structs

Carrier

A downlink carrier: the grid of $K = 12 N_{\mathrm{RB}}$ subcarriers and its OFDM modulator at $N_{\mathrm{FFT}} \cdot 15$ kHz, with the empty DC subcarrier and the normal prefixes of every symbol of the subframe (see the module).

Constants

BANDWIDTHS

The resource blocks of the six bandwidths of LTE and their FFT at 15 kHz ($N_{\mathrm{FFT}} \Delta f$ the sampling rate: 1.92 to 30.72 MS/s).

CELL_IDS

The cell identities $N_{\mathrm{ID}}^{\mathrm{cell}} < 504$.

CENTRAL_SUBCARRIERS

The subcarriers of the six central resource blocks (PSS, SSS, PBCH).

MAX_RB

The largest downlink bandwidth, $N_{\mathrm{RB}}^{\max,\mathrm{DL}} = 110$ resource blocks.

PBCH_ELEMENTS

The PBCH elements of a frame.

PSS_SYMBOL

The symbol of the PSS in slots 0 and 10.

SLOTS_PER_FRAME

Slots of a 10 ms frame.

SSS_SYMBOL

The symbol of the SSS in slots 0 and 10.

SUBCARRIERS_PER_RB

Subcarriers of a resource block.

SYMBOLS_PER_SLOT

OFDM symbols of a slot (normal cyclic prefix).

SYNC_LENGTH

Length of the PSS and of the SSS.

Functions

crs

The CRS of the symbol l of the slot ns on a carrier of n_rb resource blocks: the $2 N_{\mathrm{RB}}$ values $r(m + 110 - N_{\mathrm{RB}})$, in the order of crs_subcarriers (see the module).

crs_subcarriers

The $2 N_{\mathrm{RB}}$ subcarriers $k = 6m + (v + v_{\mathrm{shift}}) \bmod 6$ of the CRS of port in the symbol l of the slot ns (see the module).

crs_symbols

The symbols of a slot that carry the CRS of port: 0 and 4 for ports 0 and 1, 1 for ports 2 and 3.

fft_len

The usual FFT size of a carrier of n_rb resource blocks (BANDWIDTHS).

pbch_elements

The 240 elements $(k, l)$ of the PBCH in the first four symbols of slot 1, in the order of the mapping (frequency first; see the module).

pss

The PSS of $N_{\mathrm{ID}}^{(2)}$: 62 values of unit magnitude, on the subcarriers $k = n - 31 + 6 N_{\mathrm{RB}}$ (see the module).

sfbc

The space-frequency block code of transmit diversity on ports = 2 or 4 antenna ports (see the module), as an orthogonal code whose channel uses are consecutive resource elements: Ostbc::encode gives the precoded symbols of every port, Ostbc::decode_blocks combines them.

sss

The SSS of $(N_{\mathrm{ID}}^{(1)}, N_{\mathrm{ID}}^{(2)})$: 62 values $\pm 1$, those of subframe 0, or of subframe 5 when second_half (see the module).

sss_indices

The indices $(m_{0}, m_{1})$ of the SSS of the group $N_{\mathrm{ID}}^{(1)}$ (Table 6.11.2.1-1).

sync_first_subcarrier

The first of the 62 subcarriers of the PSS and of the SSS, $6 N_{\mathrm{RB}} - 31$.