a Matlab package for phased array beam shape inspection

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16 4 A PHASED ARRAY ∆ m = û · d m of the element from the plane P u normal to û, and is Ψ m = 2π∆ m /λ. Therefore, the total field at the target is E = ∑ ∑ E m = E 0 e i2πû· dm λ . (36) m m The sum multiplying the field E 0 in Eq. (36) takes care of the relative phases of the elements caused by their different positions, and is called the array factor AF. There may be also inherent phase and amplitude differences between the elements. These can be accounted for by multiplying the complex exponentials by complex amplitudes b m , and these can also be bundled into the array factor, which becomes AF(û) = ∑ m b m e i2πû· dm λ . (37) The sum goes through all the elements comprising the array. Often in the litterature the exponent in the array factor Eq. (37) is written in terms of the wave vector k = kû = (2π/λ)û of the plane wave arriving to the target from the antenna. In this note, we prefer to work with the dimensionless unit vector û instead, and will write most of our equations in terms of dimensionless quantities only. The basic reason is that the physical problem at hand, the far field antenna gain pattern, depends only on dimensionless quantities, like beam directions and the ratio of various lengths to the wavelength, and is thus inherently dimensionless. We also prefer to work directly on the discrete problem using the tools of discrete math, instead first moving matters to the continuous domain and then returning via Dirac delta-functions. Sometimes, our approach seems to result in slightly more compact equations. 1 In the e3ant package, we assume that the array is rectangular in shape, regularly spaced, and in the xy-plane, with the longer side along the x-axes. We split the amplitudes b m = b mxmy in such away that we can write the array factor as AF(u x , u y ) = M x−1,M y−1 ∑ m x=0,m y=0 a mxm y e i(2πuxDx−δx)mx e i(2πuyDy−δy)my . (38) Here D x = d x /λ and D y = d y /λ specify the array element spacing in the x- and y- directions, and M x and M y are the number of element rows in the y-and x-directions. The cartesian x- and y-coordinates u x and u y of û, also called the direction cosines because u x = cos(û, x), uniquely specify the pointing direction √ in the upper half sphere, the z-component is found by the normalization as u z = + 1 − u 2 x − u 2 y. The cartesian coordinates can be expressed in terms of the usual spherical coordinates φ and θ, where the azimuth angle φ is measured in the xy-plane counter-clockwise from the positive x-axis, that is, towards the positive y-axis; and θ is the polar angle, measured from the positive z-axis, and is the complement of the elevation angle. But note that azimuth in the EISCAT pointing geometry programs is measured in clockwise direction from the north, irrespective of how our array is oriented. The spherical-to-cartesian conversion is u x = sin(θ) cos(φ) (39) u y = sin(θ) sin(φ) . The standard Matlab function sph2cart can be used for this transformation. 1 For instance, compare the equations 9.83-9.86 in Kildal’s (factually excellent but typographically horrible) book Foundations of antennas, to our Eq. (69).
4.2 Array factor as 2-D discrete Fourier transform of the excitation field17 4.2. Array factor as 2-D discrete Fourier transform of the excitation field We define a reference direction, u 0 = (u 0 x, u 0 y), by u 0 x = δ x /(2πD x ) (40) u 0 y = δ y /(2πD y ) . The array factor Eq. (38) can then be written more economically using matrix notation as AF(u) = ∑ a m e i2π(u−u0)Dm , (41) m where m = (m x , m y ), considered a 2 × 1 column matrix, the row matrix u = (u x , u y ), and D is a 2 × 2 diagonal matrix with D x and D y as the elements. We wrote Eq. (41) to resemble as much as possible a two-dimensional analogue of the standard definition of the 1-dimensional Fourier-transform of a sequency of number, ã(ν) = ∑ m a m e i2πνm . Indeed, Eq. (41) shows that for the plane array, the array factor AF is simply the 2- dimensional discrete-space Fourier-transform of the two-dimensional sequence a m of the excitation amplitudes, evaluated at the point (u − u u )D: 4.3. Grating zones and grating directions AF(u) = ã((u − u 0 )D) . (42) The array factor of the regular grid is periodic, just as the the 1-dimensional Fouriertransform is. This periodicity corresponds to kind of directional aliazing: any directions (u x , u y ) for which (Ψ x , Ψ y ) differ by n · 2π are identical: AF(u x + n x /D x , u y + n y /D y ) = AF(u x , u y ), (43) for all n x , n y for which the AF argument on left-hand-side stays within the unit circle, also called the circle of visibility in this context, {(u x , u y ) : u 2 x + u 2 y
Page 1 and 2: E3ANT A Matlab Package for Phased A
Page 3 and 4: List of Figures 1. Beam steering. .
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: 15 “losing sensitivity” (due) t
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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