The basic methodology for analyzing an array of identical elements is established.

As shown in Fig.5 (a), let us consider an array of N identical elements. The position vector of the ith element is given by $r_i$. The excitation of $i^{th}$ element is given $C_ie^{js_i}$; where $C_i$ and $a_i$ are respectively the relative amplitudes and phases.

Let the electric field radiated by an element, when placed at the origin and with an unity excitation is given by

$$E(r) = f(θ, ϕ)\frac{e^{-jk_0r}}{4πr}................... (1)$$

The distance from the $i^{th}$ element to the far field point of interest is $R_i = r - \hat{a}_rr_i$for phase variation and $R_i = r$ for amplitude variation.

The total electric field at the point P is given by

$E(r)$ $= \sum_{i = 1}^N C_ie^{ja_i}f(θ, ϕ)\frac{e^{jk_0\left(r - \hat{a}_r.r_i\right)}}{r} \\ = f(θ, ϕ)\frac{e^{jk_0r}}{r}\sum_{i = 1}^N C_ie^{j\left(a_i+k_0\hat{a}_r . r_i\right)}...............(2)$

As can be seen from (2), the total radiation field is given by the product of the radiation field of the reference element and the term $$\sum_{i=1}^N C_ie^e \left(a_i + k_0 \hat{a}_r.r\right)$$

The term $F(θ, ϕ) = \sum_{i=1}^NC_ie^{j\left(a_i + k_0 \hat{a}_r.r_i\right)}............... (3)$

The directivity of the array $D(θ, ϕ) \infty |f(θ, ϕ)|^2|F(θ, ϕ)|^2$. Thus we find that the radiation pattern of an array is the product of the function of the individual element with the array pattern function. This termed as principle of pattern multiplication.

If we consider isotropic elements then $f(θ, ϕ) = 1$; hence the radiation pattern of the array depends only on the array factor $|F(θ, ϕ)|$. Further, it is worth mentioning here that while discussing the properties of array we are neglecting the effect of radiation of one element on the source distribution of the other, i.e., we assume that mutual coupling effect among the elements of the array are neglected. Such effects are included when very accurate characterization of arrays is required.