a Matlab package for phased array beam shape inspection
a Matlab package for phased array beam shape inspection a Matlab package for phased array beam shape inspection
26 5 PHASE AND TIME ACCURACY also the array beam width depends on the pointing direction, it is not quite obvious how “pointing accuracy” or pointing granularity should be usefully defined. Namely, if the beam is wider, there is less need to have its direction exactly along some nominal direction. The same applies to the desirable pointing granularity: there is not much sense to change the nominal beam direction at all if the change is only a very small part of the beam width. Therefore, • It seems appropriate to normalize the directional error tolerances and the granularity requirements by the beam width. Due to the 1/ cos θ factor, we would be more tolerant against ∆θ, and require less θ- granularity, in low elevations than on high elevations. What implications would accepting such a θ-dependent approach have to the phase/- time accuracy requirements? We will argue below that the implications would be all for the good. While the relation between an applied phasing step δ and the resulting beam direction θ is non-linear 4 the relation between δ and the u− or Ψ-variables is linear; that is, δ is pretty much the same as u or Ψ. Anything that is simple in terms of u, will also be simple in terms of δ. And one of the simple things will be the normalized quantities. To verify this, note that the beam’s width in the u-space, W u , will be the same number W u everywhere in the u-space, because beam steering just shifts the grating zones as a whole, without “deforming” them. This in turn follows already from Eq. (41), which shows that the array factor depends only on the difference u − u 0 . Thus, if du is some tolerance, or a granularity cell dimension in the u-space, the ratio du/W u will be constant in the u-space. For small du and W u , a constant ratio du/W u will imply the same, constant, ratio in terms of any variable, such as the θ, that depends smoothly on u. This confirms that the normalized quantities of type dθ/W θ will not depend on the pointing direction, and are therefore the natural ones to specify and describe tolerances and granularities. We want the quantitative relation between chances of δ x and δ y , and chances in the normalized quantities. In terms of the phase variable Ψ x = 2πu x D x , the effect of phase steering by δ x is to translate the array factor AF(Ψ x ) by δ x , without changing its shape. The change ∆δ shifts the pattern AF(Ψ) by ∆Ψ = ∆δ, and changes the beam direction by ∆θ. What is the relation between ∆δ and the normalized change ∆θ/W θ ? It suffices to study AF(Ψ) around the origin. In terms of Ψ, the array’s half power beamwidth corresponds to the Ψ-interval w Ψ = 2Ψ + , where We solve this to get Ψ + M/2 = 1.3916, so that Because we get 1 √ = diric(Ψ + , M) ≈ sin(Ψ +M/2) . (81) 2 Ψ + M/2 w Ψ = 5.57 M . (82) ∆Ψ w Ψ ≈ ∆θ W θ , (83) ∆δ = ∆Ψ = ∆Ψ w Ψ w Ψ ≈ ∆θ W θ · 5.57 M . (84) 4 Basically, the relation is via the same 1/ cos θ factor as the relation between beam tilt and beam width. When using normalized quantities, the 1/ cos θ factor cancels out.
5.1 Timing accuracy versus pointing accuracy 27 For instance, δ x granularity ∆δ x = ∆Ψ x = 5.57/M x corresponds to beam-width sized granularity, for all elevations. The above argument applies to the y-direction also. So the grid of uniform, beam-with adjusted beam-steering steps is just a regularly spaced grid of (δ x , δ y ) -values. To get the full half-sphere of directions, it is sufficient and necessary to have one grating zone covered by such a grid. For instance, for a 50×20 array, a “0.1× beamwidth” granularity-spec would require about 2π2π/(0.1×5.57/50)(0.1×5.57/20) = 127000 distinct directions. 5.1. Timing accuracy versus pointing accuracy Conceptually, to phase an array with the phasing steps δ x and δ y , one would compute an initial phase offset matrix δ(m x , m y ) = m x δ x + m y δ y (85) and load the phase offset registers of the DDS system at each element (m x , m y ) with these values, and clear the phase accumulators to zero. Then all the DDSs would start generating an output phase at a common moment of time, at the desired radar RF frequency f RF . The phase offset register would have enough bits to divide 2π down to milliradians, so to get accurate initial offsets into the offset-registers should not be a problem. But we will assume in this model for the phase jitter that the exact moment of the DDS start varies a little from element to element. We will assume that the start time error is a gaussian random variable with zero mean and standard deviation ∆t. In this model, the phase of transmission at element n = (m x , m y ) would be Φ n (t) = 2πf RF (t + dt n ) + δ(n) = [2πf RF t + δ n ] + 2πf RF dt n , (86) where dt n is the random error in the DDS start time and δ n is the programmed phase. This means that there is a random error in the relative phases of the elements which also has zero mean, and standard deviation With f RF = 225 MHz, the numeric relation is ∆ψ = 2πf RF ∆t . (87) ∆ψ [ ◦ ] = 8.1 × 10 −2 ∆t [ps] . (88) Beam-with-sized pointing granularity for a M = 50 array requires phasing step 6.4 ◦ . If we insert 6.4 ◦ to Eq. (88), we get ∆t = 79 ps, which looks like a short time. And, worse, we will need to point the arrays with clearly better than a beam-width granularity, perhaps with a 0.1×beam-with granularity. Does this mean that we are in a trouble? No, we no not need to control element timing with 8 ps accuracy. To be able to point the array with 0.1 × W θ accuracy on a M = 50 array, we would need “only” to be able to 1. Set the phase offset at each elements with 0.64 ◦ resolution (10 bits), and 2. Start the DDSs without having a timing error gradient across the array. A constant offset clearly would not matter, and we will show below by simulation that small random offsets do not matter (much), either. What would change the pointing
- 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 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: 4.14 2-D beam cross section 25 or m
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- 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
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5.1 Timing accuracy versus pointing accuracy 27<br />
For instance, δ x granularity ∆δ x = ∆Ψ x = 5.57/M x corresponds to <strong>beam</strong>-width sized<br />
granularity, <strong>for</strong> all elevations. The above argument applies to the y-direction also. So the<br />
grid of uni<strong>for</strong>m, <strong>beam</strong>-with adjusted <strong>beam</strong>-steering steps is just a regularly spaced grid<br />
of (δ x , δ y ) -values. To get the full half-sphere of directions, it is sufficient and necessary<br />
to have one grating zone covered by such a grid. For instance, <strong>for</strong> a 50×20 <strong>array</strong>, a “0.1×<br />
<strong>beam</strong>width” granularity-spec would require about 2π2π/(0.1×5.57/50)(0.1×5.57/20) =<br />
127000 distinct directions.<br />
5.1. Timing accuracy versus pointing accuracy<br />
Conceptually, to phase an <strong>array</strong> with the phasing steps δ x and δ y , one would compute<br />
an initial phase offset matrix<br />
δ(m x , m y ) = m x δ x + m y δ y (85)<br />
and load the phase offset registers of the DDS system at each element (m x , m y ) with<br />
these values, and clear the phase accumulators to zero. Then all the DDSs would start<br />
generating an output phase at a common moment of time, at the desired radar RF<br />
frequency f RF . The phase offset register would have enough bits to divide 2π down to<br />
milliradians, so to get accurate initial offsets into the offset-registers should not be a<br />
problem. But we will assume in this model <strong>for</strong> the phase jitter that the exact moment<br />
of the DDS start varies a little from element to element. We will assume that the start<br />
time error is a gaussian random variable with zero mean and standard deviation ∆t.<br />
In this model, the phase of transmission at element n = (m x , m y ) would be<br />
Φ n (t) = 2πf RF (t + dt n ) + δ(n) = [2πf RF t + δ n ] + 2πf RF dt n , (86)<br />
where dt n is the random error in the DDS start time and δ n is the programmed phase.<br />
This means that there is a random error in the relative phases of the elements which<br />
also has zero mean, and standard deviation<br />
With f RF = 225 MHz, the numeric relation is<br />
∆ψ = 2πf RF ∆t . (87)<br />
∆ψ [ ◦ ] = 8.1 × 10 −2 ∆t [ps] . (88)<br />
Beam-with-sized pointing granularity <strong>for</strong> a M = 50 <strong>array</strong> requires phasing step 6.4 ◦ .<br />
If we insert 6.4 ◦ to Eq. (88), we get ∆t = 79 ps, which looks like a short time. And,<br />
worse, we will need to point the <strong>array</strong>s with clearly better than a <strong>beam</strong>-width granularity,<br />
perhaps with a 0.1×<strong>beam</strong>-with granularity. Does this mean that we are in a trouble?<br />
No, we no not need to control element timing with 8 ps accuracy. To be able to point<br />
the <strong>array</strong> with 0.1 × W θ accuracy on a M = 50 <strong>array</strong>, we would need “only” to be able<br />
to<br />
1. Set the phase offset at each elements with 0.64 ◦ resolution (10 bits), and<br />
2. Start the DDSs without having a timing error gradient across the <strong>array</strong>.<br />
A constant offset clearly would not matter, and we will show below by simulation that<br />
small random offsets do not matter (much), either. What would change the pointing
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