1.4 Antenna array basics
1.4.3 Uniformly spaced arrays
Uniform arrays are widely used because it is easy to fabricate the feeding system in a sys- tematic way. The downside of uniform arrays is the presence of grating lobes, due to the spatial periodicity. Grating lobes are other main lobes that appear in the radiation pattern besides the main lobe due to the aliasing. The analysis of the array factor, instead of the total far field array, is straightforward to understand the presence of grating lobes.
Figure 1.21: Linear array along the x axis.
1.4.3.1 Uniform linear array
For an infinite linear array with elements equally spaced along the x axis with positions x = nd, with n being an integer number (cf. Fig. 1.21) and having the same weights (ai = 1), Eq. 1.14 becomes Eq. 1.16, which is simpler.
F(θ) =
+∞
X
n=−∞
exp(jn(kdsinθ)) (1.16)
where d is the distance between the elements. It is common to use the variable u instead of sinθcosφ, but in this case φ = 0. We can also use another variable b = d/λ to simplify the equations. Additionally, we can define the position function of the elements relative to the wavelength, as in Eq. 1.17, at the positions q =nb:
fpos(q) =
+∞
X
n=−∞
δ(q−nb) (1.17)
where δ(q) is the Dirac delta function. Then, we can rewrite Eq. 1.16 into Eq. 1.18 to show that, in fact, the array factor is a Fourier Transform (F) of the position function defined before.
F(u) =
+∞
X
n=−∞
exp(jn2πbu)
=
+∞
Z
−∞
exp(−j2πqu)
+∞
X
n=−∞
δ(q−nb)dq F(u) =F{fpos(q)}
(1.18)
Finally, Eq. 1.18 takes the form of Eq. 1.19 F(u) = 1
b
+∞
X
m=−∞
δ(u− m b ) F(u) = λ
d
+∞
X
m=−∞
δ(u−mλ d)
(1.19)
Figure 1.22: Magnitude of the array factor of a linear array in the u-space. In red for infinite arrays and in blue (dashed) for finite arrays.
The positions sinθ = u = mλ/d in the u-space give the values that maximize Eq. 1.16 through exp(jn(kdsinθ)) = 1. These maxima are the grating lobes and for m=0 we obtain the main lobe. Since u= sinθ, the only “visible region” is when |u| ≤ 1. Fig. 1.22 shows the array
factor of a uniform linear array with distancedbetween the elements. For an infinite linear array we obtain Dirac deltas, representing the main lobe and grating lobes (red arrows in Fig. 1.22).
For a finite linear array we obtain a finite sum of sinc functions (blue dashed line in Fig. 1.22).
When we scan the array in the directionˆro = sinθox, the array factor in Eq. 1.16 becomes:ˆ F(θ, φ) =
+∞
X
n=−∞
exp(jn(kd(sinθ−sinθo))) (1.20) Then, making u = sinθ −sinθo, we arrive at the same Eq. 1.19, but this time the origin of the “visible region” is displaced by uo = sinθo. From here we can deduce the condition to avoid the presence of the maximum of the grating lobes (in this case, the first one, m = 1) in the “visible region” (Mailloux, 2005):
1≤ λ d −uo
1≤ λ
d −sinθo
d
λ ≤ 1
1 + sinθo
(1.21)
1.4.3.2 Uniform planar array
For planar arrays, without losing generality, we can choose to work in the x-y plane, having the variables expressed as in Eq. 1.22.
(xi, yi) = position ofith element
u= sinθcosφ; v = sinθsinφ (1.22)
Eq. 1.14 can be re-arranged in a two-dimensional discrete Fourier Transform (Haupt, 2010).
In fact, the Fourier Transform of the x-y array lattice, divided by λ, is its reciprocal lattice in the u-v space (Kittel, 1995).
(a) x-y space (b) u-v space
Figure 1.23: Original and reciprocal lattice of a planar array.
Consider the planar array in Fig. 1.23, with lattice basis vectors A1 and A2. To obtain the location of the grating lobes in the u-v space for the planar array we obtain the reciprocal basis vectors,B1 and B2, according to Eq. 1.23, presented in Eq. 1.23 (Kittel, 1995).
B1 =λ A2׈z
A1·(A2׈z); B2 =λ ˆz×A1
A1·(A2׈z) (1.23)
Eq. 1.23 can also be used to find the grating lobe positions for the linear array usingA2 =yˆ giving the same positions as in Eq. 1.19. We can see in Fig. 1.23 the limit of the “visible region”
represented by the circle in black dashed line with R= 1 and center (u, v) = (0,0).
Grating lobes for an equilateral triangular lattice
If we have a planar array with a triangular lattice and distance d between the elements, the position of the grating lobes in theu-v space can be found using the reciprocal basis vectors in Eq. 1.24. Hence, when there is no scan, the first grating lobes enter into the “visible region”
when 1 = (λ/d)(2/√
3), or, which is the same, when d=λ(2/√ 3).
A1 =d ˆx; A2 = d
2(1ˆx+√
3ˆy) (1.24a)
B1 = λ
d(ˆx− 1
√3ˆy); B2 = λ d( 2
√3ˆy) (1.24b)
Now, if the main beam is steered to the angle (θo, φo), the “visible region” has a center (uo, vo) and it becomes expressed by Eq. 1.25.
(uo, vo) = (sinθocosφo, sinθosinφo)
(u−uo)2 + (v−vo)2 ≤1 (1.25) When the steering angle of the array is (θ = 30◦,φ = φo), the new center of the “visible region” (unit circle) is (cosφo, sinφo)/2. Fig. 1.24 shows this case. The red dashed line is the locus of the center of the “visible regions” for the scan angle (θ = 30◦,φ=φo). The black line is the limit of the “visible region” when the scan angle is (θ = 30◦,φ = 90◦). We can see that the black line shows the case when a grating lobe enters to the “visible region”.
From Fig. 1.24 we can deduce the frequency at which the grating lobes enter into the “visible region” for a scan angle of θ= 30◦ and anyφ (cf. Eq. 1.26), in particular, for (θ = 30◦,φ= 90◦).
√2 3
λ d = 3
2 (1.26a)
fGL = 4c0
3√
3d (1.26b)
wherefGLis the grating lobes frequency,c0 is the speed of light anddis the inter element spacing of the array.
We have seen that, in uniformly spaced arrays, the grating lobes can be easily predicted according to the array factor. It is important to consider that the scan angle and the element
Figure 1.24: Visible region in the u−v space for the case of a triangular lattice. Blue points are the main and grating lobes. Dashed red line is the locus of center of the visible regions for a scan angle (θ= 30◦,φ =φo). Black line is the limit of the visible region for a scan angle of (θ = 30◦,φ= 90◦).
pattern will play an important role in the level of the grating lobes.