**3. Convex optimization for pattern recovery of conformal antenna array**

The broadside gain maximization problem of a conformal antenna array in **Figure 2** can be formulated as a linear constrained quadratic programming problem [32], i.e.,

Minimize ‖**wn**‖,

Subject to

$$\mathbf{A} \mathbf{F}\_{\text{target}} = \mathbf{1},\tag{6}$$

**95**

∇ <sup>2</sup>

**Figure 3.**

where *t*

(*k*)

descends in each step such that ‖**w***<sup>n</sup>*

the corrected radiation pattern.

**4. Analytical and simulation results**

*Broadside Pattern Correction Techniques for Conformal Antenna Arrays*

is the complex weighing vector, which gives the objective

function 2*N* degrees of freedom. In **Figure 3(a)** the objective function plotted for the two component dimensions of the first antenna weight *w*1 clearly indicates its convex form in the vicinity of the origin. The contour plots in the considered two dimensions are also presented in **Figure 3(b)**. The aspect of the contour lines along these two dimensions suggests low condition number. This well condition behavior also exists among the other degrees of freedom and results in a reduced convergence

*(a) 3D plot of the objective function plotted as a function of real (w 1) and imaginary (w 1). (b) the contour* 

Iterative optimization algorithms are normally adopted to find this minimum. Although second-order Newton method is able to find the solution in fewer steps, it requires evaluation of complex quadratic norm defined as the Hessian matrix

‖**w <sup>n</sup>**‖, which is quite complex. Since the problem is well-conditioned, even the low complexity first-order Gradient decent method can find the solution quickly. If the iteration number is represented as superscript *k*, then the weights update in

> + *t* (*k*) ∆ *w<sup>k</sup>*

is the scalar step size obtained through backtracking method which

(*k*)

, (7)

is the step direction, which

‖ at each step. The method

‖, until the optimum minimum is

*DOI: http://dx.doi.org/10.5772/intechopen.90957*

where **wn** ∈ ℂ[*N*×1]

time to find the minimum within the search space.

*plot of the objective function with illustration of gradient descent method.*

the *k*th iteration for Gradient descent method is given as

reduces as one gets closer to the global minimum. ∆ *w<sup>k</sup>*

is chosen to maximize the negative gradient ∇ ‖**w***<sup>n</sup>*

*w*(*k*+1)

(*k+***1**)

= *w*(*k*)

‖ ≤ ‖**w***<sup>n</sup>* (*k*)

To analytically compute the corrected (compensated) radiation pattern and validate with simulation results, the compensated array factor AF*c* in Eq. (5) was used for different radii of curvatures of the cylindrical surface in **Figure 2**.

reached, and the slope becomes positive. Since the objective function is trying to minimize the norm, the number of steps required for convergence is reduced if the starting point is chosen close to the origin. It should be noted that in this work, the algorithm was used to calculate the required compensated phases only (with uniform amplitudes). Computing the weights for phase correction using Eq. (7) to maximize the broadside gain of conformal antenna array in **Figure 2** are used in Eq. (5) to plot

*Broadside Pattern Correction Techniques for Conformal Antenna Arrays DOI: http://dx.doi.org/10.5772/intechopen.90957*

**Figure 3.**

*Advances in Array Optimization*

**2.3 Array factor expression**

where δ±n*<sup>c</sup>*

**2.2 Computing the compensated phase**

where *k* is the free-space wave number.

for antennas on conformal surfaces is [18]:

ual element pattern in **Figure 2**. *wn* = *In e*

drive the nth antenna element.

*F*1(θ,ϕ) = cos

(θ − ϕ1), and cos

AF = ∑*n*=1

(θ + ϕ−1) and *F*2(θ,ϕ) = cos

ϕ2 respectively. Therefore, the element patterns are cos

(θ − ϕ2) respectively.

Minimize ‖**wn**‖,

all the elements are pointing towards zenith (that is towards ϕ*s*=0°

two left antenna elements and *F*3(θ,ϕ) = cos

The required compensated phase to correct the broadside radiation pattern of a

To analytically compute the corrected (compensated) radiation pattern and validate

can be computed using Eq. (2) and the array factor (AF) expression

*<sup>N</sup> Fn*(θ,ϕ) *wn ejk*[*xn*sinθ+*zn*cosθ]

where *k* is the free space wave number, *N* is number of antenna elements, (*xn*, *zn*) is the location of nth antenna element on the conformal surface, and *Fn*(θ,ϕ) is the individ-

For this work, the phase difference ∆ϕ between adjacent antenna elements was made zero, and the amplitude tapering coefficient *In* was kept equal to 1 (uniform excitation). Using Eqs. (1)–(4), the expression for corrected array factor is given by:

AF*c* = *F*1(θ,ϕ) *ejk*<sup>∆</sup>−1 *ejk*[*x*1sinθ+*z*1cosθ] + *F*2(θ,ϕ) *ejk*<sup>∆</sup>−2 *ejk*[*x*2sinθ+*z*2cosθ]

the element patterns for the two right antenna elements on the cylindrical surface in **Figure 2**. It should be noted that for linear array, the element pattern is cos

conformal surfaces, the individual element patterns become geometry dependent. For example, as can be seen in **Figure 2**, the look directions of antenna elements *A*−1 and *A*−2 are towards the angular directions ϕ−1 and ϕ−2 respectively, while the look directions of antenna elements *A*1 and *A*2 are towards the angular directions ϕ1 and

**3. Convex optimization for pattern recovery of conformal antenna array**

The broadside gain maximization problem of a conformal antenna array in **Figure 2** can be formulated as a linear constrained quadratic programming problem [32], i.e.,

+ *F*3(θ,ϕ) *ejk*<sup>∆</sup>+2 *ejk*[*x*3sinθ+*z*3cosθ] + *F*4(θ,ϕ) *ejk*<sup>∆</sup>+1 *ejk*[*x*4sinθ+*z*4cosθ] (5)

with simulation results, the following compensated array factor AF*c* is used [18, 31]:

AF*c* = AF *e<sup>j</sup>*δ±n*<sup>c</sup>*

= +*k* ∆±n (2)

, (3)

*<sup>j</sup>*∆ϕ is the complex weighting function required to

(θ + ϕ−2) are the element patterns for the

(θ + ϕ−1), cos

AF*target* = 1, (6)

(θ − ϕ+2) are

). However on

(θ + ϕ−2), cos

(θ) as

(θ − ϕ+1) and *F*4(θ,ϕ) = cos

, (4)

conformal cylindrical antenna array in **Figure 2** is then given by:

δ±n*<sup>c</sup>*

**94**

Subject to

*(a) 3D plot of the objective function plotted as a function of real (w 1) and imaginary (w 1). (b) the contour plot of the objective function with illustration of gradient descent method.*

where **wn** ∈ ℂ[*N*×1] is the complex weighing vector, which gives the objective function 2*N* degrees of freedom. In **Figure 3(a)** the objective function plotted for the two component dimensions of the first antenna weight *w*1 clearly indicates its convex form in the vicinity of the origin. The contour plots in the considered two dimensions are also presented in **Figure 3(b)**. The aspect of the contour lines along these two dimensions suggests low condition number. This well condition behavior also exists among the other degrees of freedom and results in a reduced convergence time to find the minimum within the search space.

Iterative optimization algorithms are normally adopted to find this minimum. Although second-order Newton method is able to find the solution in fewer steps, it requires evaluation of complex quadratic norm defined as the Hessian matrix ∇ <sup>2</sup> ‖**w <sup>n</sup>**‖, which is quite complex. Since the problem is well-conditioned, even the low complexity first-order Gradient decent method can find the solution quickly. If the iteration number is represented as superscript *k*, then the weights update in the *k*th iteration for Gradient descent method is given as

$$
\boldsymbol{\omega}^{(k\star 1)} = \boldsymbol{\omega}^{(k)} + \boldsymbol{\mathfrak{t}}^{(k)} \,\boldsymbol{\Delta}\boldsymbol{\omega}^{k},\tag{7}
$$

where *t* (*k*) is the scalar step size obtained through backtracking method which reduces as one gets closer to the global minimum. ∆ *w<sup>k</sup>* is the step direction, which is chosen to maximize the negative gradient ∇ ‖**w***<sup>n</sup>* (*k*) ‖ at each step. The method descends in each step such that ‖**w***<sup>n</sup>* (*k+***1**) ‖ ≤ ‖**w***<sup>n</sup>* (*k*) ‖, until the optimum minimum is reached, and the slope becomes positive. Since the objective function is trying to minimize the norm, the number of steps required for convergence is reduced if the starting point is chosen close to the origin. It should be noted that in this work, the algorithm was used to calculate the required compensated phases only (with uniform amplitudes). Computing the weights for phase correction using Eq. (7) to maximize the broadside gain of conformal antenna array in **Figure 2** are used in Eq. (5) to plot the corrected radiation pattern.
