Editor's Note:
Here arePart I and Part II of the article. Also, check out the Mobile Handset DesignLine site.
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Cellular concepts, cont'd.
Also observe that overlap is minimized when circular cells are deployed along a hexagonal grid as is shown in Figure 4.5. The hexagonal layout is evidently the most economically efficient one, as it requires the fewest cells to cover a given area. For these reasons, a hexagonal cell layout is chosen as the basis for designing cellular systems. In the discussion that follows, we will treat cells as having hexagonal shapes. Although this is never precisely true in fact, the assumption provides a means for developing concepts about frequency reuse and cell size that may be applied in practice. Further, the hexagonal layout does provide a starting point for real-world design. We therefore proceed to discuss the geometrical properties of a hexagonal grid as they relate to cell layout and frequency reuse.
Figure 4.5 Overlap in Circular Cells Using a (a) Triangular Grid; (b) Rectangular Grid; (c) Hexagonal Grid
Figure 4.6 shows a hexagonal cell and some of its neighbors. We assume that all of the cells have radius R and that the base stations are located in the centers of the cells. The hexagonal geometry dictates that adjacent cells are located at multiples of 60° surrounding any given cell, and the separation between adjacent cell centers is √3R. Following V. H. MacDonald1, we locate position in the array of cells by choosing a coordinate system (u, v) such that the positive coordinate axes intersect at a 60& #176; angle and the unit distance along either axis is equal to √3R, which is the separation between adjacent cells. Figure 4.7 shows these coordinate axes. The normalized distance between any two arbitrary cells whose centers are located at the coordinates (u1, v1) and
(u2, v2) can be found using the law of cosines as illustrated in Figure 4.8. We obtain
Figure 4.6 Adjacent Cell Geometry
The actual distance D between the cell centers is given by:
since actual and normalized distance are related by the scale factor √3R. In terms of normalized distance, the lengths along the axes between u1 and u2 and between v1 and v2 are integers. We can let u2-u1 and v2-v1 = j. In terms of i and J, Equation (4.11) becomes:
Figure 4.7 Coordinate Systems for Hexagonal Layout
Figure 4.8 Computation of Separation Distance
If we choose the center of any cell as a reference location, then the distance to the center of any other cell can be expressed as a function:
of the integers i and j. The orientation of the coordinate system is arbitrary, so that for any i and j there will be six cells whose centers lie at:
distance from the reference. These cells will surround the reference at multiples of 60& #176;.
Figure 4.9 illustrates this geometry for the case:
Note that the centers of the six cells at distance D(i, j) from the reference form the corners of a large hexagon whose radius and sides have lengths equal to the separation distance:
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