Kaiser windows are characterized by having the lowest sidelobes of all windows in the transformed domain. A Kaiser windows are used here in the design of fractal patterns, one would expect an array current distribution having lower side-lobes, with a more reduced array size. A single pulse function is taken to be a Kaiser window, the pattern resulting maintain a smooth shape with the same sim-ilarity properties of the Koch array factor. The main advantage of using Kaiser windows is that pattern parameters become much more flexible through altering the Kaiser window [28]. The mainlobe width, current distribution, sidelobe ratio are adjustable. A Kaiser window [30] is given by
Wk(nT) = Io(β)
Io(γ) f or|n| ≤ N −1 2 Wk(nT) = 0 otherwise
(4.19) whereIo(x)is the zeroth-order modified Bessel function of the first kind,γis the Kaiser independent parameter and
β=γ s
1−
2n N −1
2
(4.20)
Several configurations of patterns with their current distributions are illustrated in Figures (4.23-24) for different combinations of amplitude factorα, and the in-dependence Kaiser parameter γ, generated with M = 6 iterations and δ = 3.
The choose of higher factor α improve low sidelobe level of the pattern and for smoothing its shape. The factorγ allow the control of mainlobe width. Smaller values forγ result in a wider beamwidth, while larger values result in a narrower beamwidth. Not only the mainlobe width is controllable using parameter γ, but also the current distribution and its sidelobe level. Its clear from Figures 4.23-24 that the current sidelobe is inversely proportional withγ. A comparison between a Blackman and Kaiser Koch arrays for the same parametersM = 6,δ = 3and α = 4, is shown in Figure 4.25. The larger values of γ reduce the current side lobe way below the Blackman window. the Kaiser’s current distribution reached a minimum of−239.5dBwhile that of the Blackman reached a minimum−193dB.
4.9. KOCH-KAISER ARRAY 63 Kaiser−Koch pattern, M=6, δ=3, α=1, γ=2
−400 −300 −200 −100 0 100 200 300 400
Current Distribution (dB)
Kaiser−Koch array, M=6, δ=3, α=1, γ=2 Kaiser−Koch pattern, M=6, δ=3, α=1, γ=5
−400 −300 −200 −100 0 100 200 300 400
Current Distribution (dB)
Kaiser−Koch array, M=6, δ=3, α=1, γ=5
Current Distribution (dB)
−3 −2 −1 0 1 2 3 Kaiser−Koch pattern, M=6, δ=3, α=1, γ=15
−400 −300 −200 −100 0 100 200 300 400
Current Distribution (dB)
Kaiser−Koch array, M=6, δ=3, α=1, γ=15
Figure 4.23: The Kaiser-Koch patterns (right-hand side) and their current distru-bitions (left), generated withM = 6,δ= 3,α = 1and differentγ, from the (top):
γ=2, 5, 10, 15.
−3 −2 −1 0 1 2 3 Kaiser−Koch pattern, M=6, δ=3, α=4, γ=2
−400 −300 −200 −100 0 100 200 300 400
Current Distribution (dB)
Kaiser−Koch array, M=6, δ=3, α=4, γ=2 Kaiser−Koch pattern, M=6, δ=3, α=4, γ=5
−400 −300 −200 −100 0 100 200 300 400
Current Distribution (dB)
Kaiser−Koch array, M=6, δ=3, α=4, γ=5 Kaiser−Koch pattern, M=6, δ=3, α=4, γ=10
−400 −300 −200 −100 0 100 200 300 400
Current Distribution (dB)
Kaiser−Koch array, M=6, δ=3, α=4, γ=10
−3 −2 −1 0 1 2 3 Kaiser−Koch pattern, M=6, δ=3, α=4, γ=15
−400 −300 −200 −100 0 100 200 300 400
Current Distribution (dB)
Kaiser−Koch array, M=6, δ=3, α=4, γ=15
Figure 4.24: The Kaiser-Koch patterns (right-hand side) and their current distru-bitions (left), generated withM = 6,δ= 3,α= 4and differentγ, from the (top):
γ=2, 5, 10, 15.
4.9. KOCH-KAISER ARRAY 65
−400 −300 −200 −100 0 100 200 300 400
−250
−200
−150
−100
−50 0
Kaiser−Koch and Blackman−Koch array comparison
x
Current Distribution (dB)
Blackman−Koch array
Kaiser−Koch array
Figure 4.25: A comparison between Current distribution for Kaiser-Koch and Blackman-Koch arrays in (dB) scale, generated with the same parametersM = 6, δ= 3andα= 4, the Kaiser parameterγ=20.
A multiband behavior for Kaiser-Koch arrays before thresholding are shown in Figures (4.26-27), the patterns was generated withM = 6, δ = 3, γ = 15and α = 1 (Figure 4.26) and α = 4 (Figure 4.27). The array factors keep their multibands behavior at five frequency bands, when the designed wavelength was taking to be λo = 2d, and with a change of operating wavelength λn = 3nλo, n = 1· · ·M − 1 . The sidelobe radio and radiation pattern characteristics are similar at these five frequency bands.
4.9. KOCH-KAISER ARRAY 67
−3 −2 −1 0 1 2 3
0 0.2 0.4 0.6 0.8 1
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35
ψ
Array Factor (magnitude)
−0.40 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4
0.02 0.04 0.06 0.08 0.1 0.12
ψ
Array Factor (magnitude)
−0.1 −0.05 0 0.05 0.1
0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04
ψ
Array Factor (magnitude)
−0.040 −0.03 −0.02 −0.01 0 0.01 0.02 0.03 0.04
0.002 0.004 0.006 0.008 0.01 0.012
ψ
Array Factor (magnitude)
Figure 4.26: Kaiser-Koch patterns at different warelength, withα = 1andγ = 15.
From the (top):λo =d/2,λ1 = 3λo,λ2 = 9λo,λ3 = 27λo,λ4 = 81λo.
−3 −2 −1 0 1 2 3
Array Factor (magnitude)
−0.40 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4
Array Factor (magnitude)
−0.1 −0.05 0 0.05 0.1
Array Factor (magnitude)
−0.040 −0.03 −0.02 −0.01 0 0.01 0.02 0.03 0.04
Array Factor (magnitude)
Figure 4.27: Kaiser-Koch patterns at different warelength, withα= 4andγ = 15.
From the (top):λo =d/2,λ1 = 3λo,λ2 = 9λo,λ3 = 27λo,λ4 = 81λo.
4.9. KOCH-KAISER ARRAY 69 A reduced array structure for Kaiser-Koch array is indicated for several threshold levels, and their corresponding fractal pattern are showed in Figures (4.28-31).
We can see that the higher threshold values highly distort the pattern.
−3 −2 −1 0 1 2 3 Kaiser−Koch pattern M=6, δ=3, α=1, γ=10
0 5 10 15 20 25 30 35 40 45 50
Current Distribution (dB)
Kaiser−Koch array. Threshold value =−90 dB, 57 element
−3 −2 −1 0 1 2 3 Kaiser−Koch pattern M=6, δ=3, α=1, γ=10
0 5 10 15 20 25 30 35 40 45 50
Current Distribution (dB)
Kaiser−Koch array. Threshold value =−70 dB, 53 element
−3 −2 −1 0 1 2 3 Kaiser−Koch pattern M=6, δ=3, α=1, γ=10
0 5 10 15 20 25 30 35 40 45 50
Current Distribution (dB)
Kaiser−Koch array. Threshold value =−50 dB, 29 element
−3 −2 −1 0 1 2 3 Kaiser−Koch pattern M=6, δ=3, α=1, γ=10
0 5 10 15 20 25 30 35 40 45 50
Current Distribution (dB)
Kaiser−Koch array. Threshold value =−30 dB, 11 element
Figure 4.28: The Koch-Kaiser array factor (right-hand side), and corresponding current distribution, constructed from M = 6, δ = 3, α = 1, γ = 10, with different threholding values. From the top: −90dB,−70dB,−50dB,−30dB.
−3 −2 −1 0 1 2 3 Kaiser−Koch pattern M=6, δ=3, α=1, γ=15
0 5 10 15 20 25 30 35 40 45 50
Current Distribution (dB)
Kaiser−Koch array. Threshold value= −90 dB,73 element
−3 −2 −1 0 1 2 3 Kaiser−Koch pattern M=6, δ=3, α=1, γ=15
0 5 10 15 20 25 30 35 40 45 50
Current Distribution (dB)
Kaiser−Koch array. Threshold value =−70 dB, 63 element
−3 −2 −1 0 1 2 3 Kaiser−Koch pattern M=6, δ=3, α=1, γ=15
0 5 10 15 20 25 30 35 40 45 50
Current Distribution (dB)
Kaiser−Koch array. Threshold value =−50 dB, 37 element
−3 −2 −1 0 1 2 3 Kaiser−Koch pattern M=6, δ=3, α=1, γ=15
0 5 10 15 20 25 30 35 40 45 50
Current Distribution (dB)
Kaiser−Koch array. Threshold value =−30 dB, 11 element
Figure 4.29: The Koch-Kaiser array factor (right-hand side), and corresponding current distribution, constructed from M = 6, δ = 3, α = 1, γ = 15, with different threholding values. From the top:−90dB,−70dB,−50dB,−30dB.
4.9. KOCH-KAISER ARRAY 71 Kaiser−Koch pattern M=6, δ=3, α=4, γ=10
0 5 10 15 20 25 30 35 40 45 50
Current Distribution (dB)
Kaiser−Koch array. Threshold value =−90 dB, 43 element
−3 −2 −1 0 1 2 3 Kaiser−Koch pattern M=6, δ=3, α=4, γ=10
0 5 10 15 20 25 30 35 40 45 50
Current Distribution (dB)
Kaiser−Koch array. Threshold value =−70 dB, 31 element
−3 −2 −1 0 1 2 3 Kaiser−Koch pattern M=6, δ=3, α=4, γ=10
0 5 10 15 20 25 30 35 40 45 50
Current Distribution (dB)
Kaiser−Koch array. Threshold value =−50 dB, 19 element
−3 −2 −1 0 1 2 3 Kaiser−Koch pattern M=6, δ=3, α=4, γ=10
0 5 10 15 20 25 30 35 40 45 50
Current Distribution (dB)
Kaiser−Koch array. Threshold value =−30 dB, 11 element
Figure 4.30: The Koch-Kaiser array factor (right-hand side), and corresponding current distribution, constructed from M = 6, δ = 3, α = 4, γ = 10, with different threholding values. From the top: −90dB,−70dB,−50dB,−30dB.
−3 −2 −1 0 1 2 3 Kaiser−Koch pattern M=6, δ=3, α=4, γ=15
0 5 10 15 20 25 30 35 40 45 50
Current Distribution (dB)
Kaiser−Koch array. Threshold value =−90 dB, 55 element
−3 −2 −1 0 1 2 3 Kaiser−Koch pattern M=6, δ=3, α=4, γ=15
0 5 10 15 20 25 30 35 40 45 50
Current Distribution (dB)
Kaiser−Koch array. Threshold value =−70 dB, 37 element
−3 −2 −1 0 1 2 3 Kaiser−Koch pattern M=6, δ=3, α=4, γ=15
0 5 10 15 20 25 30 35 40 45 50
Current Distribution (dB)
Kaiser−Koch array. Threshold value =−50 dB, 23 element
−3 −2 −1 0 1 2 3 Kaiser−Koch pattern M=6, δ=3, α=4, γ=15
0 5 10 15 20 25 30 35 40 45 50
Current Distribution (dB)
Kaiser−Koch array. Threshold value =−30 dB, 13 element
Figure 4.31: The Koch-Kaiser array factor (right-hand side), and corresponding current distribution, constructed from M = 6, δ = 3, α = 4, γ = 15, with different threholding values. From the top:−90dB,−70dB,−50dB,−30dB.
4.9. KOCH-KAISER ARRAY 73 In Table-4.7, a threshold value is set at different levels and the corresponding ele-ments that are important in the pattern synthesis. For larger thresholds. −30dB, the number of elements are greatly reduced for lower values ofγ = 5. For smaller thresholds, −130dB, the number of elements are reduced for larger values ofγ, reaching only 83elements with γ = 15, The main reason for this result is that most of the elements are confined in a very low level (below−130dB) for larger values ofγ, the elements are distributed over the higher levels, experiencing dif-ferent larger amplitudes. Figures(4.32-34) shown a multiband behavior for several Kaiser-Koch arrays after elements reduction.
Threshold(dB) -30 -40 -50 -60 -70 -80 -90 -100 -110 -120 -130 δ= 1,γ = 5 7 15 21 35 55 119 447 725 725 727 727 δ= 1,γ = 10 11 19 29 45 53 57 57 59 77 123 275 δ= 1,γ = 15 11 21 37 51 63 69 73 79 81 87 87 δ= 1,γ = 20 13 25 41 59 69 79 85 89 91 97 99 δ= 2,γ = 5 7 13 17 25 49 133 559 687 717 725 729 δ= 2,γ = 10 9 15 23 31 39 49 55 59 75 131 383 δ= 2,γ = 15 13 17 25 39 47 59 69 73 79 83 87 δ= 2,γ = 20 13 19 31 41 53 67 79 87 93 93 97 δ= 3,γ = 5 7 13 15 23 41 135 537 681 703 721 727 δ= 3,γ = 10 9 13 21 27 35 41 49 57 75 135 437 δ= 3,γ = 15 13 17 25 33 41 49 59 71 77 81 85 δ= 3,γ = 20 13 19 27 35 45 55 71 81 91 93 97 δ= 4,γ = 5 9 9 15 21 41 139 531 603 697 719 725 δ= 4,γ = 10 11 13 19 25 31 37 43 51 77 137 461 δ= 4,γ = 15 13 17 23 29 37 45 55 65 75 79 83 δ= 4,γ = 20 13 19 27 35 41 55 61 77 85 93 95
Table 4.7: The threshold for the Kaiser window for differentδandγ. ForM = 6 andα = 3.
−3 −2 −1 0 1 2 3
Array Factor (magnitude)
ψ
Array Factor (magnitude)
−0.40 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4
Array Factor (magnitude)
−0.1 −0.05 0 0.05 0.1
Array Factor (magnitude)
−0.040 −0.03 −0.02 −0.01 0 0.01 0.02 0.03 0.04
Array Factor (magnitude)
Figure 4.32: Kaiser-Koch patterns at different wavelength after thresholding,α= 4,γ = 15, and83element.
4.9. KOCH-KAISER ARRAY 75
Array Factor (magnitude)
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1
Array Factor (magnitude)
−0.40 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4
Array Factor (magnitude)
−0.1 −0.05 0 0.05 0.1
Array Factor (magnitude)
−0.040 −0.03 −0.02 −0.01 0 0.01 0.02 0.03 0.04
Array Factor (magnitude)
Figure 4.33: Kaiser-Koch patterns at different wavelength after thresholding,α= 1,γ = 10, and77element.
−3 −2 −1 0 1 2 3 0
0.2 0.4 0.6 0.8 1
ψ
Array Factor (magnitude)
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1
ψ
Array Factor (magnitude)
−0.40 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4
1 2 3 4 5 6 7x 10−3
ψ
Array Factor (magnitude)
−0.1 −0.05 0 0.05 0.1
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5x 10−4
ψ
Array Factor (magnitude)
Figure 4.34: Kaiser-Koch patterns at different wavelength after thresholding,α= 4,γ = 15, and55element.
4.9. KOCH-KAISER ARRAY 77 The mean square error is calculated for the Kaiser window at different thresholds, a comparison between the original array factor and resultant array factor, for dif-ferentαare illustrated in Figure 4.35. It is clear that most optimum selection for γ, the Kaiser independent parameter in15and20, independing of amplitude factor α. The (MSE) is always kept minimum at this value for each different threshold value. This figures may also be used for designing and settting threshold values for the pattern to keep its multiband behavior.
−130 −120 −110 −100 −90 −80 −70 −60 −50 −40 −30
Mean Square Error (MSE), M=6,δ=3,α=1
Threshold level (dB)
Error (dB)
Mean Square Error (MSE), M=6,δ=3,α=2
Error (dB)
Threshold level (dB) γ=20
Threshold level (dB) Mean Square Error (MSE), M=6,δ=3,α=3
γ=20
Threshold level (dB) Mean Square Error (MSE), M=6,δ=3,α=4
Error (dB)
γ=20 γ=15 γ=5
γ=10
Figure 4.35: The Mean Square Error(MSE), between original array factor and resultant array factor due to element reduction, for Kaiser-Koch array.