NR - Numerology

Since release 15, new radio (NR) system has explicitly proposed the concept of numerology, which is used to represent the scaling of subcarrier bandwidth, FFT size, etc.

Above all, a reference numerology is defined based on

\begin{align*} \Delta_{\mathrm{subc, ref}} &= 15 \text{ kHz}, \\ N_{\mathrm{FFT, ref}} &= 2048. \end{align*}

Accordingly, the sampling rate the sample duration can be written as

\begin{align*} f_{\mathrm{s, ref}} &= \Delta_{\mathrm{subc, ref}} N_{\mathrm{FFT, ref}}, \\ T_{\mathrm{s, ref}} &= \frac{1}{f_{\mathrm{s, ref}}} = \frac{1}{\Delta_{\mathrm{subc, ref}} N_{\mathrm{FFT, ref}}}. \end{align*}

Additionally, an extreme numerology is also defined based on

\begin{align*} \Delta_{\mathrm{subc, max}} &= 480 \text{ kHz}, \\ N_{\mathrm{FFT, max}} &= 4096, \end{align*}

Similarly, the sampling rate and sample duration can be expressed as

\begin{align*} f_{\mathrm{s, max}} &= \Delta_{\mathrm{subc, max}} N_{\mathrm{FFT, max}}, \\ T_{\mathrm{s, max}} &= \frac{1}{f_{\mathrm{s, max}}} = \frac{1}{\Delta_{\mathrm{subc, max}} N_{\mathrm{FFT, max}}}. \end{align*}

The ratio of the two sample durations above is denoted by

\begin{align*} \kappa \triangleq \frac{T_{\mathrm{s, max}}}{T_{\mathrm{s, ref}}} \equiv 64. \end{align*}

Then, other configurations can be accordingly scaled relative to the reference numerology. Supported numerologies can be listed in Table 1. For convenient reference, a numerology indicator is defined based on the subcarrier bandwidth, i.e.,

\begin{align*} \mu \triangleq \log_{2} \frac{\Delta_{\mathrm{subc}}}{\Delta_{\mathrm{subc, ref}}}. \end{align*}

Table 1: Supported numerologies

\(\mu\)	\(\Delta_{\mathrm{subc}}\) (kHz)	CP type
0	15	NCP
1	30	NCP
2	60	NCP/ECP
3	120	NCP
4	240	NCP

Regarding frame structure, following hierarchy is adopted.

• A frame is comprised of 10 subframes.
• A subframe is comprised of \(N_{\mathrm{slot}}^{\mathrm{subframe},\mu} = 2^{\mu}\) slots, where \(\mu\) is a numerology indicator.
• A slot is comprised of \(N_{\mathrm{symbol}}^{\mathrm{slot}}\) symbols.

\begin{align*} N_{\mathrm{symbol}}^{\mathrm{slot}} = \begin{cases} 14, & \text{NCP}; \\ 12, & \text{ECP}. \end{cases} \end{align*}

• An OFDM symbol without cyclic prefix (CP), comprised of \(N_{\mathrm{FFT}}\) samples, has duration \(T_{\mathrm{sym}} = N_{\mathrm{FFT}}t_{s} = \dfrac{1}{\Delta_{\mathrm{subc}}}\), where
  • \(t_{s} = \dfrac{1}{f_{s}}\) is the duration of each sample;
  • \(f_{s} = \Delta_{\mathrm{subc}} N_{\mathrm{FFT}}\) is the sampling rate.
• There are a couple types of CP.
  • Normal CP (NCP) consists of \(N_{\mathrm{CP} }= 144 \dfrac{N_{\mathrm{FFT}}}{N_{\mathrm{FFT, ref}}}\) samples. Particularly, there are additional \(16 \dfrac{N_{\mathrm{FFT}}}{N_{\mathrm{FFT, ref}}} \dfrac{\Delta_{\mathrm{subc}}}{\Delta_{\mathrm{subc, ref}}}\) samples for 1st symbol in each slot.
  • Extended CP (ECP) consists of \(N_{\mathrm{CP}} = 512 \dfrac{N_{\mathrm{FFT}}}{N_{\mathrm{FFT, ref}}}\) samples.