We have discussed the preamble format configuration vs. the maximum cell radius during the random access procedure. For example, Preamble Format 0 supports a maximum cell radius of 14.5 km. Another random access parameter that affects the cell size is the cyclic shift. Let's look at it now.

The random access preamble is generated using Zadoff-Chu sequences; there are multiple root Z-C sequences used in LTE. From each root sequence, multiple preambles can be obtained by applying different cyclic shifts. This cyclic shift also determines the maximum radius of the cell.

The cyclic shift, Ncs, is defined in 3GPP TS 36.211, section 5.7. (Note that the unrestricted set is for normal speed cells, and the restricted set is for high mobility cells.)

*Table 5.7.2-2: for preamble generation (preamble formats 0-3).*

How is the cyclic shift related to cell radius? As shown in Figure 1, assume that there are 2 UEs, UE1 at the cell edge and UE2 close to the eNB. The cyclic shift used by UE1 is 0 and the cyclic shift used by UE2 is Ncs. At the eNB, the observed cyclic shift of UE1 will not be 0 but some value x because of the transmission delay. As long as x is less than Ncs, the auto-correlation between the shifted x and shifted Ncs (as perceived by the eNB) will be zero, and the eNB will be able to distinguish between the accesses from UE1 and UE2 (This is one of the nice properties of Z-C sequences). So, the maximum cell radius is limited by the cyclic shift.

*Figure 1: Cyclic shift vs. Cell Radius*

Now, let's calculate the maximum supported cell radius by a cyclic shift, Ncs. Based on 3GPP, the preamble sequence length is 839 and spans 800 milliseconds.

[Exercise time: Please fill the blanks.]

Getting back to the basics, why is a cyclic shift needed? The cyclic shift can be used to expand the preamble capacity. There are a total of 838 Zadoff-Chu sequences defined in LTE, and the default setting for the number of preambles in each sector is 64. For areas covered with a large number of small cells, if the preamble capacity is limited, preamble interference may cause more collisions and longer random access delay. A small Ncs value generates more preambles, which extend the preamble reuse distance and mitigate the interference. However, the cyclic shift cannot be configured smaller than expected cell radius, since that will block random accesses from the cell edge and may cause drops during handovers.

[Answers: Ncs = 13: 0.96, Ncs = 46: 5.68]