Understanding OFDM Subcarrier Spacing and Cyclic Prefix Overhead
In Orthogonal Frequency Division Multiplexing (OFDM) systems such as 5G NR and Wi-Fi 6 (802.11ax), the transmission timeline is divided into discrete symbols. To eliminate Inter-Symbol Interference (ISI) caused by multipath channel delay spread, a Cyclic Prefix (CP) is appended to the beginning of each OFDM symbol.
Key OFDM Time and Frequency Relationships
The useful symbol duration ($T_u$) is inversely proportional to the subcarrier spacing ($\Delta f$):
- $$T_u = \frac{1}{\Delta f}$$
Given an $N_{\text{FFT}}$ point transform, the system sampling frequency ($f_s$) and sample duration ($T_s$) are defined as:
- $$f_s = N_{\text{FFT}} \times \Delta f, \quad T_s = \frac{1}{f_s}$$
Cyclic Prefix Overhead Equation
The Cyclic Prefix consists of duplicating the last $N_{\text{CP}}$ samples of the OFDM symbol and prepending them to the front. The time duration of the CP ($T_{\text{CP}}$) and the resulting throughput overhead percentage are calculated as:
- $$T_{\text{CP}} = N_{\text{CP}} \times T_s = \frac{N_{\text{CP}}}{N_{\text{FFT}} \times \Delta f}$$
- $$\text{CP Overhead (\%)} = \frac{T_{\text{CP}}}{T_u + T_{\text{CP}}} \times 100 = \frac{N_{\text{CP}}}{N_{\text{FFT}} + N_{\text{CP}}} \times 100$$
5G NR Numerologies ($\mu$) Overview
5G NR supports scalable subcarrier spacing defined by $\Delta f = 15 \times 2^\mu\ \text{kHz}$. As numerology index $\mu$ increases, symbol duration shrinks proportionally, enabling low-latency transmissions at higher carrier frequencies (FR2 / mmWave).