a Matlab package for phased array beam shape inspection

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Contents 1. Introduction 4 2. Time steering and phase steering 5 2.1. Time steering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.2. Monochromatic signals—time-steering equals phase steering . . . . . . . . 7 2.3. Phase steering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 3. Antenna gain 8 3.1. Transmission . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 3.2. Reception . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 4. A phased array 15 4.1. The array factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 4.2. Array factor as 2-D discrete Fourier transform of the excitation field . . . 17 4.3. Grating zones and grating directions . . . . . . . . . . . . . . . . . . . . . 17 4.4. Phase steering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 4.5. The array factor with equal excitation amplitudes . . . . . . . . . . . . . 18 4.6. Parseval’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 4.7. Antenna gain and the power integral . . . . . . . . . . . . . . . . . . . . . 19 4.8. The power integral for an array with isotropic elements . . . . . . . . . . 20 4.9. The power integral for an array with directional elements . . . . . . . . . 21 4.10. Array gain for an array with directional elements . . . . . . . . . . . . . . 21 4.11. Gain model for the array element . . . . . . . . . . . . . . . . . . . . . . . 22 4.12. Computing the antenna directivity and beamwidth . . . . . . . . . . . . . 23 4.13. Array gain and grating directions . . . . . . . . . . . . . . . . . . . . . . . 24 4.14. 2-D beam cross section . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 5. Phase and time accuracy 25 5.1. Timing accuracy versus pointing accuracy . . . . . . . . . . . . . . . . . . 27 6. Jitter calculations 28 6.1. Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 6.2. Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 7. SNR estimate for the test array 30 A. Figures to Chapters 2–7 34 B. Pointing geometry 65 C. Matlab functions 69 2
List of Figures 1. Beam steering. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 2. A two-element array 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 3. A two-element array 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 4. A two-element array 3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 5. Randomized array. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 6. TX equivalent circuit. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 7. Reciprocity in a two-port system. . . . . . . . . . . . . . . . . . . . . . . . 41 8. ReceptionGain. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 9. Array delay geometry. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 10. Diriclet kernel. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 11. Grating zones. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 12. Two representations of a beam’s phase-steering phase. . . . . . . . . . . . 45 13. A simple antenna element gain pattern. . . . . . . . . . . . . . . . . . . . 46 14. Directivity and vertical beam width. . . . . . . . . . . . . . . . . . . . . . 46 15. Beam width for a few array sizes. . . . . . . . . . . . . . . . . . . . . . . . 47 16. Inspecting the array pattern with planearray. . . . . . . . . . . . . . . 48 17. A 3D view of an array’s grating directions. . . . . . . . . . . . . . . . . . 49 18. Beam’s cross-sectional shape. . . . . . . . . . . . . . . . . . . . . . . . . . 50 19. Dips in the array directivity pattern. . . . . . . . . . . . . . . . . . . . . . 51 20. Explaining the directivity dips. . . . . . . . . . . . . . . . . . . . . . . . . 52 21. Jitter simulation 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 22. Jitter simulation 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 23. Jitter simulation 3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 24. Jitter simulation 4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 25. TA: Geometry for computing the effective volume. . . . . . . . . . . . . . 57 26. TA: Array directivity as function of beam elevation. . . . . . . . . . . . . 58 27. TA: Beam cross section for common volume at 120 km. . . . . . . . . . . 59 28. TA: Beam cross section for common volume at 300 km. . . . . . . . . . . 60 29. TA: Beam cross section for common volume at 600 km altitude. . . . . . . 60 30. TA: Filling of the common volume at 120 km altitude. . . . . . . . . . . . 61 31. TA: Horizontal slice through the common volume at 120 km altitude. . . 62 32. TA: Filling of the common volume at 300 km altitude. . . . . . . . . . . . 63 33. TA: Horizontal slice through the common volume at 300 km altitude. . . 63 34. TA: Filling of the common volume at 600 km altitude. . . . . . . . . . . . 64 35. TA: Horizontal effective area of the common volume at 600 km altitude. . 64 36. Purnuvaara pointing geometry. . . . . . . . . . . . . . . . . . . . . . . . . 65 37. Sodankylä pointing geometry. . . . . . . . . . . . . . . . . . . . . . . . . . 66 38. Porjus pointing geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 39. Kiruna pointing geometry. . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 40. Koutokeino pointing geometry. . . . . . . . . . . . . . . . . . . . . . . . . 67 41. Abisko pointing geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 3
Page 1: E3ANT A Matlab Package for Phased A
Page 5 and 6: 5 2. Time steering and phase steeri
Page 7 and 8: 2.2 Monochromatic signals—time-st
Page 9 and 10: 3.1 Transmission 9 lobes have equal
Page 11 and 12: 3.2 Reception 11 current I goes via
Page 13 and 14: 3.2 Reception 13 equations. But fir
Page 15 and 16: 15 “losing sensitivity” (due) t
Page 17 and 18: 4.2 Array factor as 2-D discrete Fo
Page 19 and 20: 4.6 Parseval’s theorem 19 closed
Page 21 and 22: 4.9 The power integral for an array
Page 23 and 24: 4.12 Computing the antenna directiv
Page 25 and 26: 4.14 2-D beam cross section 25 or m
Page 27 and 28: 5.1 Timing accuracy versus pointing
Page 29 and 30: 6.1 Method 29 excitation amplitudes
Page 31 and 32: 31 electric field and the scatterin
Page 33 and 34: 33 Table 1: Predicting SNR for the
Page 35 and 36: 35 s(t, r; û) E p û û 0 0 R m
Page 37 and 38: 37 D=1.0; θ 0 = 60, θ g = −8, 6
Page 39 and 40: 39 (a) (b) (c) (d) Figure 5: Random
Page 41 and 42: 41 A + Z 2 I A Z 1 B + Z 3 V V (B)
Page 43 and 44: 43 E 0 E m û d m P xy ∆ m P u Fi
Page 45 and 46: 45 80 60 40 20 θ 0 (°) 0 −20
Page 47 and 48: 47 10 Beam width (°) 9 8 7 6 5 4 M
Page 49 and 50: 49 Y X Figure 17: A 3D view of an a
Page 51 and 52: 51 45 40 Directivity (dBi) 35 30 25
Page 53 and 54:
53 ARRAY 50×20 1.50×1.50 δx=0.0
Page 55 and 56:
55 ARRAY 50×20 1.50×1.50 δx=0.0
Page 57 and 58:
57 ∆X z E p χ P Q ∆Z z r 2 R 2
Page 59 and 60:
0.02 59 12×4 1.4×1.4 φ 0 =0.0°
Page 61 and 62:
61 V eff for a long pulse (km 3 ) V
Page 63 and 64:
63 V eff for a long pulse (km 3 ) V
Page 65 and 66:
65 B. Pointing geometry The functio
Page 67 and 68:
67 KIRUNA Lat=67.86 Lon=20.44 Hgt=0
Page 69:
69 C. Matlab functions There are tw
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a Matlab package for phased array beam shape inspection

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