In the world of 5G New Radio (NR), time units play a pivotal role in the structuring and functioning of the physical layer. Two fundamental time units in 5G NR are Tc and Ts, each serving as a building block for representing time durations. Understanding these units is crucial for grasping how 5G NR handles various timing-related processes, including sampling, symbol duration, and overall timing synchronization.

## Tc and Ts Definitions and Calculations.

### Ts (Sampling Time Interval):

Ts is calculated using the following formula:

${T}_{S}=\frac{1}{\u25b3{f}_{ref}.{N}_{f,ref}}$In this formula:

Δf_{ref} is the reference subcarrier spacing, which is 15 x 10^{3} Hz.

N_{f,ref} is the number of frequency-domain sampling points, set to 2048.

Substituting these values:

Therefore, Ts equals 32.552 ns.

### Tc (Minimum Time Unit):

Tc is calculated using the following formula:

${T}_{c}=\frac{1}{\u25b3{f}_{max}.{N}_{f}}$In this formula:

Δf_{max} is the maximum subcarrier spacing, which is 480 x 10^{3} Hz.

N_{f} is the number of frequency-domain sampling points, set to 4096.

Substituting these values:

Therefore, Tc equals 0.509 ns.

## Relationship Between Ts and Tc.

The relationship between Ts and Tc can be expressed as a ratio:

In addition, Ts is K times Tc, as shown in the following formula:

$K=\frac{{T}_{s}}{{T}_{c}}=64$This indicates that Ts is 64 times Tc, linking the two time units directly and allowing for an integer multiple representation of any time length in 5G NR using Tc.

## Explanation of Ts in LTE.

In LTE, the subcarrier spacing (SCS) is fixed at 15 kHz. The maximum number of subcarriers in the transmission bandwidth is 1200. To avoid information loss during inverse fast Fourier transformation (IFFT), at least 1200 frequency-domain sampling points are required. However, to simplify calculations, the number of sampling points (N_{f,ref}) is set to 2048, a power of 2.

The length of the data transmission part in an orthogonal frequency-division multiplexing (OFDM) symbol is the reciprocal of the SCS, that is, 1/15,000 seconds. Given the 2048 sampling points, the sampling interval T_{s} in LTE is:

## Calculation of Tc in 5G NR.

In 5G NR, each resource block (RB) contains 12 subcarriers, with the system supporting up to 273 RBs, totaling 3276 subcarriers. The number of frequency-domain sampling points is set to 4096. The largest SCS currently defined in 3GPP specifications is 240 kHz, but 480 kHz is reserved for future evolution.

Using the largest SCS (480 kHz) to calculate Tc:

${T}_{c}=\frac{1}{480000\times 4096}\approx 0.509ns$The definition of these time units ensures precise timing and synchronization in 5G NR, crucial for high-speed data transmission, low latency, and efficient spectrum utilization. The ability to represent all time lengths as integer multiples of Tc simplifies the design and implementation of 5G NR systems, facilitating a wide range of use cases from enhanced mobile broadband (eMBB) to ultra-reliable low-latency communications (URLLC) and massive machine-type communications (mMTC).

### Conclusion.

Understanding the basic time units Ts and Tc in 5G NR is fundamental for telecommunications engineers working on the physical layer of 5G networks. These units form the foundation for various time-related operations, ensuring that 5G NR can meet its ambitious goals of providing higher data rates, lower latency, and more robust connections compared to its predecessors.

# References

- 3GPP TS 38.300, NR and NG-RAN Overall Description; Stage 2 (Release 16)
- 3GPP TS 38.101-1, User Equipment (UE) radio transmission and reception; Part 1: Range 1 Standalone (Release 16)
- 3GPP TS 38.101-2, User Equipment (UE) radio transmission and reception; Part 1: Range 2 Standalone (Release 16).
- 3GPP TS 38.104, Base Station (BS) radio transmission and reception (Release 16).
- 3GPP TS 36.211, Physical channels and modulation (Release 14).
- 3GPP TS 38.213, Physical layer procedures for control (Release 16).
- 3GPP TS 38.211, Physical channels and modulation (Release 16).
- 3GPP TR 38.912 (Release 15).
- 3GPP TR 38.802, Physical Layer Aspects (Release 14).

**Following questions are asked during interview of 5G NR RAN, RF and Microwave on topic of Time Domain Resources.**

- Where do the basic time units Tc and Ts in the 5G system come from?
- What are intersymbol interference (ISI) and inter-subcarrier interference (ICI)?
- Why can extended cyclic prefixes (CPs) avoid the interference?
- What are slot formats?
- What is mini-slot?
- How do 3GPP specifications describe mini-slot?