The NB-IoT uplink uses SC-FDMA, like the LTE uplink, but it adds a transmission that LTE never uses: a single subcarrier. This page compares the two baseband formulas in 36.211 term by term. It then explains the extra phase term of the single-tone case, and what changes when the subcarrier spacing drops to 3.75 kHz.

Followings are the topics to be covered in this page.

• How does the LTE-NB formula differ from the LTE formula ?
• Why does the single-tone formula add a phase term ?
• What changes with 3.75 kHz subcarrier spacing ?
• Reference

How does the LTE-NB formula differ from the LTE formula ?

The picture below puts the two formulas one above the other. 36.211 clause 10.1.5 for LTE-NB is on top, and clause 5.6 for LTE is underneath. The green arrows point from each term of the LTE-NB formula to its definition.

LTE sums over all N_RB^ULN_sc^RB allocated subcarriers. The LTE-NB formula has no sum, because it describes one subcarrier k, and it adds the phase term φ_k,l.

Start with what is the same. Both formulas use the (k + 1/2)Δf frequency term, so the LTE uplink already shifts every subcarrier by half a subcarrier spacing and has no DC subcarrier. Both remove the cyclic prefix in the same way, through t - N_CP,lT_s. Both also map k to the resource element index with k^(-) = k + ⌊N/2⌋.

Then look at what is missing. The LTE formula sums over N_RB^ULN_sc^RB subcarriers, and the LTE-NB formula has no sum at all. It is the signal on a single subcarrier k, which is the single-tone case. 36.211 v19.3.0 clause 10.1.5 uses this formula only when N_sc^RU = 1. For multi-tone NPUSCH with 3, 6 or 12 subcarriers, the clause reuses the LTE formula of clause 5.6, with N_RB^ULN_sc^RB replaced by N_sc^UL.

The symbol index works differently too. LTE restarts l in every slot. The LTE-NB formula counts l~ over the whole transmission, and takes l = l~ mod N_symb^UL only for the cyclic prefix length.

• No sum in the single-tone formula : it describes one subcarrier k, so there is nothing to add up.
• Multi-tone reuses LTE : clause 5.6 applies with the number of subcarriers replaced by N_sc^UL.
• The half-subcarrier shift is not new : the LTE uplink formula already uses (k + 1/2)Δf.
• φ_k,l is the only new term : it carries a modulation rotation and a phase that runs across symbols.

Why does the single-tone formula add a phase term ?

A single subcarrier carries one modulation symbol per SC-FDMA symbol, and nothing else. So the waveform between two symbols depends only on how those two symbols meet. The phase term φ_k,l = ρ(l~ mod 2) + φ^_k(l~) controls exactly that, and it has two parts with two different jobs.

The first part is ρ(l~ mod 2). ρ is π/2 for BPSK and π/4 for QPSK, and it applies on every second symbol only. So the constellation alternates between its normal position and a rotated one, which gives π/2-BPSK and π/4-QPSK. With this rotation, the signal never passes through zero between two consecutive symbols, and the peak-to-average power ratio stays low. That matters for a low-cost UE, whose power amplifier then needs less back-off.

The second part is φ^_k(l~). It starts at 0 and adds 2πΔf(k + 1/2)(N + N_CP,l)T_s for every symbol. That value is the phase that subcarrier k turns through during one full symbol, cyclic prefix included. The time t restarts at 0 in each symbol, so without this term the phase would jump at every symbol boundary. φ^_k(l~) adds the missing phase back, and the tone stays continuous from symbol to symbol.

For this to work, l~ has to count every symbol since the start of the transmission. The picture shows the Release 13 range, l~ = 0, 1, ..., M_rep^NPUSCHN_RUN_slots^ULN_symb^UL - 1. In 36.211 v19.3.0 the range also multiplies by N_TB, the number of transport blocks in a multi-TB grant. The counter restarts at every codeword, except when the codewords are interleaved, and then it runs across all N_TB of them.

• ρ gives π/2-BPSK and π/4-QPSK : every second symbol is rotated, so the signal never passes through zero.
• φ^_k(l~) keeps the phase continuous : it adds back the phase the tone turned through in the previous symbol and its cyclic prefix.
• l~ counts the whole transmission : repetitions, resource units, slots and symbols, and in v19.3.0 transport blocks too.
• Both parts serve the power amplifier : a constant-envelope tone with no phase jumps is easy to amplify.

What changes with 3.75 kHz subcarrier spacing ?

The formula in the picture does not fix Δf. NB-IoT uses it with 15 kHz and with 3.75 kHz, and the numbers inside the formula change with the spacing. 36.211 v19.3.0 Table 10.1.5-1 gives them, and the table below repeats it.

From 36.211 v19.3.0 Table 10.1.5-1, for normal cyclic prefix, which is the only cyclic prefix the NB-IoT uplink supports.

Four times narrower spacing means four times longer symbols. N = 8192 T_s is 266.7 µs, where the 15 kHz symbol is 66.7 µs. The 48 values of k fill the same 180 kHz as the 12 subcarriers at 15 kHz. A 3.75 kHz resource unit is also always single-tone, so this formula is the only one the 3.75 kHz uplink ever uses.

The slot does not divide evenly. A 3.75 kHz slot is 2 ms, which is 61440 T_s. Seven symbols of 8192 + 256 T_s take 59136 T_s, so 2304 T_s, or 75 µs, are left at the end of the slot. 36.211 clause 10.1.5 says that this remainder is not transmitted and is used as a guard period.

• 3.75 kHz symbols are four times longer : N = 8192 instead of 2048, and a cyclic prefix of 256.
• 48 subcarriers in the same 180 kHz : k runs from -24 to 23.
• A 2304 T_s guard period ends each slot : seven symbols fill only 59136 of the 61440 T_s in a 2 ms slot.
• Normal CP only : the NB-IoT uplink does not support extended CP.

Reference

[1] 3GPP TS 36.211 v19.3.0 - clause 5.6 and clause 10.1.5, SC-FDMA baseband signal generation for LTE and for NB-IoT