SPARSE SAMPLING USING A TRIANGULAR GRID

The geometry for the triangular grid is similar to that for the rectangular grid, Sec-tion 5.4, but the beams in successive scan lines are offset from each other in the x coordinate by 0.5, and the y coordinate spacing is reduced from  to 0.75 to form the equilateral triangles shown in Figure 5.30. The target locations occupy one quadrant of the central beam, but the vertical spacing has been changed by a factor 0.75 relative to those in the rectangular raster. The results is a more uni-form distribution of energy within the scan sector, and reduced beamshape loss.

The method of calculation follows that for the rectangular grid, but with the fol-lowing changes.

5.5.1 Method of Calculation for Triangular Grid

The steps for each type of processing remain as for the rectangular grid (Tables 5.2–5.8), but with 0.75²_k replacing ²_k in the reference detectability factors,

Beam 7 Beam 8

25 target locations

Beamwidth ₃

Beam spac

ing  Beam 6 Beam 2

Beam 5

Beam 3

Beam 9 Beam 1

Beam 4 0

0.75

 0.75

0

1.0 0.5 0.5 1.0 1.5

Angle in units of  Angle in units of 

y

x

Figure 5.30 Beam grids and target sample positions for triangular scan grid.

Beamshape Loss 175 and a loss denoted by L_pT2 rather than L_p2. The upper limits for m and j are 9 and 25, respectively. The beam spacing  varies from 0.5 to 1.5 beamwidths, as for the rectangular grid, but now applies to the length of each side of the equilateral triangle shown in Figure 5.35.

5.5.2 Steady-Target Beamshape Loss for Triangular Grid

Results for this target model are not presented here, as it is seldom applicable to actual radar. Analytic approximations to the loss are presented in Appendix 5A.

5.5.3 Case 1 Beamshape Loss for Triangular Grid 5.5.3.1 Case 1 with Integration

The beamshape loss for Case 1 with integration is shown in Figure 5.31. The loss for high P_d is substantially less than that shown in Figure 5.15 for the rectangular grid. The additional power required for the closer spacing of adjacent rows is 1 0.75 or 0.63 dB, and the reduction in loss for P_d = 0.9 is 2 dB at  = 1.2. The apparent slight rise for   0.5 results from failure to include contributions from beyond the nine-beam grid shown in Figure 5.30, whose inclusion is assumed in calculation of the reference detectability factor. Extension of integration over those beams might or might not actually occur in a practical processing system.

5.5.3.2 Case 1 with Cumulative Detection

The loss for Case 1 with cumulative detection is shown in Figure 5.32. Again, the loss for high P_d and  > 1.1 is substantially lower than for the rectangular grid.

P^d = 0.9 P^d = 0.7 P^d = 0.5 Pd = 0.3

2L’p0

0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5

2 3 4 5 6 7

Beam spacing in beamwidths

BeamshapelossindB

.

Figure 5.31 Beamshape loss LpT2 versus sample spacing k for Case 1 with integration over the entire scan.

5.5.3.3 Case 1 with Mixed Processing

The Case 1 loss with mixed processing, shown in Figure 5.33, exhibits the same reduction for  > 1.1 compared with the rectangular grid, with small increases as

  0.5.

5.5.4 Case 2 Beamshape Loss for Triangular Grid

The number of independent samples for Case 2 is slightly increased by the closer spacing of rows in the scan, compared with the rectangular grid, resulting in a slight decrease in the reference detectability factor, and shown in Figures 5.34 and 5.35.

P^d = 0.9

P^d = 0.7 P^d = 0.5

Pd = 0.3 2L’p0

0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5

2 3 4 5 6 7 8

Beam spacing in beamwidths

BeamshapelossindB

.

Figure 5.32 Beamshape loss LpT2 versus sample spacing k for Case 1 with cumulative detection.

P^d = 0.9

Pd = 0.7 Pd = 0.5 Pd = 0.3

2L’_p0

0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5

2 3 4 5 6 7

Beam spacing in beamwidths

BeamshapelossindB

.

Figure 5.33 Beamshape loss LpT2 versus sample spacing k for Case 1 with mixed processing.

Beamshape Loss 177

5.5.4.1 Case 2 with Integration

The Case 2 beamshape loss with integration is shown in Figure 5.36. The loss is slightly higher than that for Case 1, because the samples do not include quite as many independent target samples as used in calculating the reference detectability factor.

0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1

2 3 4 5 6 7

Beam spacing in beamwidths

Independent target samples

.

Figure 5.34 Number of independent target samples as a function of beam spacing, for triangular grid.

Diversity target

0.3 0.4 0.5 0.6 0.7 0.8 0.9

9 10 11 12 13 14 15 16 17 18 19 20 21 22

Detection probability

Referencedetectabilityfactor(dB)

.

Figure 5.35 Reference detectability factors Dr{0,1,2,D}(Pd) for triangular grid with k = 1.

5.5.4.2 Case 2 with Cumulative Detection

The Case 2 loss with cumulative detection is shown in Figure 5.37. The loss is lower than with the rectangular grid for all P_d when  > 1.

5.5.4.3 Case 2 with Mixed Processing

The Case 2 loss for mixed processing, shown in Figure 5.38, has also been re-duced relative to that for the rectangular grid for  > 1.

5.5.5 Diversity Target Beamshape Loss for Triangular Grid

The beamshape loss for the diversity target is shown in Figures 5.39–5.41 for the three processing methods. In each case, the loss is similar to that for the Case 1 target, as shown in Figures 5.36–5.39.

P^d = 0.9

Figure 5.36 Beamshape loss LpT2 versus sample spacing k for Case 2 with integration.

P^d = 0.9

Figure 5.37 Beamshape loss LpT2 versus sample spacing k for Case 2 with cumulative detection.

Beamshape Loss 179

Figure 5.38 Beamshape loss LpT2 versus sample spacing k for Case 2 with mixed processing.

P^d = 0.9

Figure 5.39 Beamshape loss LpT2 versus sample spacing k for diversity target with integration.

Pd = 0.9

Figure 5.40 Beamshape loss LpT2 versus sample spacing k for diversity target with cumulative detection.

5.5.6 Beamshape Loss in Search Radar Equation for Triangular Grid For use in the search radar equation, the net beamshape loss is calculated on the basis of maintaining constant energy over the scan as beam spacing changes. This loss is shown in Figures 5.42–5.44 for the diversity target with three methods of processing. The results for this target model are approximately equal to those for Cases 1 and 2, and provide a basis for estimating the effects of different spacing of samples.

The major conclusion from Figures 5.42–5.44 is that the net beamshape loss to be included in the search radar equation for a 2-D scan with a triangular grid must significantly exceed the value L_p2 = 1.77 = 2.48 dB commonly used for dense sampling. Given the usual spacing of 0.71 beamwidth, the loss will vary from 7.5 dB for mixed processing to 8.5 dB for cumulative detection. Achieving the minimum loss of 2–4 dB requires that the spacing be increased to  1.3

beam-P^d = 0.9

Figure 5.41 Beamshape loss LpT2 versus sample spacing k for diversity target with mixed processing.

Pd = 0.5

Figure 5.42 Net beamshape loss LpTn2 versus sample spacing k for diversity target with integration.