**2. Formulation**

The amplitude is calculated for antenna arrays by considering different number of elements and these results are presented in **Tables 1** and **2.**

$$E(u) = 2\sum\_{n=1}^{N} A(n)\cos\left[kd(n-0.5)(u-u\_0)\right] \tag{1}$$

d = resonant spacing, (*λ*/2), k = 2π/*λ*

A(n) = Amplitude distribution

The amplitude is also measured by considering different number of elements with Ishimaru spacings and these results are presented in **Tables 1** and **2**.

$$E(\boldsymbol{u}) = \sum\_{n=1}^{N} A(\boldsymbol{\kappa}\_n) e^{j\left[\frac{2\boldsymbol{x}\boldsymbol{L}}{\boldsymbol{\lambda}}\boldsymbol{\kappa}\boldsymbol{x}\_n + \phi(\boldsymbol{x}\_n)\right]} \tag{2}$$


#### **Table 1.**

*Uniform amplitude distribution for N = 20 with different spacings.*


*Radiation Patterns from Thinned and Unthinned Linear Arrays with Different Spacings Using… DOI: http://dx.doi.org/10.5772/intechopen.106699*


#### **Table 2.**

*Gaussian amplitude distribution for N = 20 with different spacings.*

Using the above expressions, the radiation pattern of arrays are calculated Here, A(xn) is the Amplitude distribution

2*L <sup>λ</sup>* = length of an array

u = sinθ, θ is the angle in the bore sight direction

xn = Ishimaru spacing function

$$X\_{n0} = (2n - N - 1)/N$$

Where ϕ(xn) is the excitation of phase distribution.

Thinning an array comprises of the deduction (turning off) of radiating elements from an array [13] with reduced cost and weight of individual elements. This method of thinning permits closely the same narrow beam width which is equal to the filled array with total elements of equal size. In array thinning, lower side lobes can be acquired during turned ON process, the antenna elements operate with equal amplitude similar to the same filled array with uniform weighting.

#### **Figure 3.**

*Radiation pattern with resonant spacing and Ishimaru spacing for uniform amplitude distribution using N = 20 antenna elements.*

**Tables 1** and **2** gives Gaussian distribution and uniform amplitude distribution for N-20 elements using different spacing respectively.

**Figure 3** presents the radiation pattern for N = 20 elements with resonant spacing and Ishimaru spacing for uniform amplitude distribution.

**Figure 4** gives the radiation pattern for N = 20 elements with Resonant spacing and Ishimaru spacing for uniform amplitude distribution with thinning concept where the antenna elements i.e. '9' and '19' are eliminated from an array antenna.

**Figure 5** gives radiation pattern for N = 20 elements with Resonant spacing and Ishimaru spacing for Gaussian amplitude distribution.

**Figure 6** represents the Radiation pattern for N = 20 elements with Resonant spacing and Ishimaru spacing for Gaussian amplitude distribution with thinning concept where the antenna elements i.e. '9' and '19' are eliminated from an array antenna.

#### **Figure 4.**

*Radiation pattern with Resonant spacing and Ishimaru spacing for uniform amplitude distribution with thinning using N = 20 antenna elements.*

*Radiation pattern with Resonant spacing and Ishimaru spacing for Gaussian amplitude distribution. for N = 20 elements.*

*Radiation Patterns from Thinned and Unthinned Linear Arrays with Different Spacings Using… DOI: http://dx.doi.org/10.5772/intechopen.106699*

**Figure 6.**

*Radiation pattern for N = 20 elements with Resonant spacing and Ishimaru spacing for Gaussian amplitude distribution with thinning.*
