When we talk about the cell size, we usually think of RF coverage first. The cell radius is also related to the parameter configuration for the random access procedure, such as the preamble format and cyclic shift. Let's look at the preamble format here.

In LTE, the random access procedure is non-synchronized, which means the UE which initiates the random access procedure is not uplink synchronized yet (It is, however, already downlink synchronized). So the random access procedure of this UE should not interfere the uplink transmission of other already uplink synchronized UEs.

Figure 1 shows the mapping of the PRACH (Physical Random Access Channel) to physical resources. The PRACH occupies 6 PRBs in the frequency domain and spans 1 or 2 or 3 subframes in the time domain, depending on the specific preamble format. In the frequency domain, several subcarriers at both ends of the 6 PRBs are not used to avoid interference with the adjacent PUCCH/PUSCH. In the time domain, the cyclic prefix (CP) and guard time (GT) are used to avoid interference with the previous and next subframes. As it turns out, the GT determines the maximum cell radius.

*Figure 1: Resource Mapping of PRACH
(Physical Random Access Channel)*

How is the GT related to cell radius? Assume there are 3 UEs in the cell, located near the center of the cell, in the middle of the cell, and at the cell edge, respectively. Assume that all of these 3 UEs are not yet UL synchronized and are about to perform a random access. The RACH configuration information is transmitted in the DL. All UEs listen to it and send RACH preambles in the same subframe. For UE3 at the cell edge, its RA preamble arrives at the eNB later than UE1's preamble. The maximum delay between UE3 and UE1 is the round trip delay (RTD) between the eNB and the cell edge, which is determined by the guard time in the expected preamble receive window at the eNB. We can calculate the cell radius based on the GT.

*Figure 2: Guard Time (GT) vs. Cell Size*

The Preamble format is defined in LTE (3GPP TS 36.211, section 5.7).

*Figure 5.7.1-1: Random access preamble format.*

*
Table
5.7.1-1: Random access preamble parameters.*

Where Ts is the basic time unit in LTE, Ts = 1/(15000*2048) s = 3.25521 X 10^(-5) ms.

First, we get the guard time from the number of subframes occupied by the PRACH, the time to transmit cyclic shift, and the time to transmit preamble sequence. In figure 3, it is obvious that the preamble format 0 and preamble format 1 have different guard times, which means they support different maximum cell sizes.

*Figure 3: Calculation of Guard Time*

We can derive the cell radius from the guard time.

*Figure 4: Calculation of Cell Radius based on Guard Time*

We have calculated the cell radius for preamble format 0 and preamble format 1 in the spreadsheet below, where the equation (1) in Figure 3 is used to calculate guard time and the equation (2) in Figure 4 is used to calculate cell radius. Please calculate the cell radius for Format 2 and 3. Have fun!

*Table 1: Cell Radius vs. Preamble Format*

Answers: (Format 2: ~30 km, Format 3: ~107 km)