Let's solve the problem **step by step**: ### **Given:** - **5G NR numerology:** n = 3 - **Cyclic prefix duration (Tcp):** 0.5875 µs - **Useful symbol period (Tus):** 8.3375 µs - **Modulation:** 256-QAM (which is 8 bits per symbol, since \( 2^8 = 256 \)) - **Question:** What is the **maximum data rate on a single subchannel**? --- ## **Step 1: Calculate the OFDM Symbol Duration** The **total OFDM symbol duration** (\( T_{symbol} \)) is the sum of the cyclic prefix and the useful symbol duration: \[ T_{symbol} = T_{cp} + T_{us} = 0.5875\,\mu s + 8.3375\,\mu s = 8.925\,\mu s \] --- ## **Step 2: Calculate the Symbol Rate (Symbols per Second)** \[ \text{Symbol Rate} = \frac{1}{T_{symbol}} \] Convert \( T_{symbol} \) to seconds: \[ T_{symbol} = 8.925 \times 10^{-6}\, s \] \[ \text{Symbol Rate} = \frac{1}{8.925 \times 10^{-6}} \approx 112,010 \text{ symbols/sec} \] --- ## **Step 3: Bits Per Symbol for 256-QAM** 256-QAM means 8 bits per symbol. --- ## **Step 4: Data Rate Per Subcarrier** \[ \text{Data Rate per subcarrier} = \text{Symbol Rate} \times \text{Bits per symbol} \] \[ = 112,010 \times 8 = 896,080 \text{ bits/sec} = 896.08 \text{ kbps} \] --- ## **Step 5: Subchannel Definition in 5G NR** A **subchannel** in OFDM typically refers to a **subcarrier**. Unless further context is given (e.g., a subchannel is a group of subcarriers), we assume a subchannel = 1 subcarrier. --- ## **Step 6: Final Answer** ### **Maximum data rate on a single subchannel (subcarrier):** \[ \boxed{896.08\, \text{kbps}} \] --- #### **Summary Table** | Parameter | Value | |----------------------------|-------------------------| | OFDM symbol duration | 8.925 μs | | Symbol rate | 112,010 symbols/sec | | Bits per symbol (256-QAM) | 8 | | **Max data rate/subchannel** | **896.08 kbps** | --- **Final answer:** \[ \boxed{896.08\, \text{kbps}} \] is the maximum data rate on a single subchannel (subcarrier) for numerology n=3 and 256-QAM in 5G NR, given the provided symbol durations.