Electronically Scanned Arrays MATLAB® Modeling and Simulation

Appendix A: Array Factor (AF) Derivation

In Chapter 1 the closed form for the AF was shown to be

$AF=\frac{\sin \left[M\text{π}d\left(\frac{\sin {\text{θ}}_o}{{\text{λ}}_o}-\frac{\sin \text{θ}}{\text{λ}}\right)\right]}{\sin \left[\text{π}d\left(\frac{\sin {\text{θ}}_o}{{\text{λ}}_o}-\frac{\sin \text{θ}}{\text{λ}}\right)\right]}$

(A.1)

This is derivable from the exponential summation expression for the uniform illumination AF, which is

$AF={\displaystyle \sum _{m=1}^M{e}^{j\left(\frac{2\text{π}}{\text{λ}}{x}_m\sin \text{θ}-\frac{2\text{π}}{{\text{λ}}_o}{x}_m\sin {\text{θ}}_o\right)}}$

(A.2)

The position of the array elements, x_m, is expressed as $x_m=\left(m-\frac{M+1}{2}\right)d_x$, where d_x is the element spacing and M is the number of array elements. Using this expression puts the phase center of the array at x = 0.^* Equation (A.2) can then be rewritten as

$\begin{array}{ll}AF\hfill & ={\displaystyle \sum _{m=1}^M{e}^{j\left(\frac{2\text{π}}{\text{λ}}\sin \text{θ}-\frac{2\text{π}}{{\text{λ}}_o}\sin {\text{θ}}_o\right)x_m}}\hfill \\ ={\displaystyle \sum _{m=1}^M{e}^{j{d}_x\left(\frac{2\text{π}}{\text{λ}}\sin \text{θ}-\frac{2\text{π}}{{\text{λ}}_o}\sin {\text{θ}}_o\right)\cdot \left(m-\frac{M+1}{2}\right)}}\hfill \\ ={\displaystyle \sum _{m=1}^M{e}^{j\text{Ψ}\left(m-\frac{M+1}{2}\right)}}\hfill \end{array}$

(A.3)

where $\text{Ψ}=d_x\left(\frac{2\text{π}}{\text{λ}}\sin \text{θ}-\frac{2\text{π}}{{\text{λ}}_o}\sin {\text{θ}}_o\right)$. Equation (A.3) can then be expanded as

$\begin{array}{ll}AF\hfill & ={\displaystyle \sum _{m=1}^{}}\hfill \end{array}$