1. Tapering of Arrays Objective
is to reduce the side lobes, which not only the amount of power is wasted in the directions of minor lobes but unnecessary interferences are also caused in those area.
Side lobes often termed as “Side lobe ratio” which is the ratio of power density of the main lobes to the power density of longest minor lobes and it is expressed in dB, it is called as “Side-lobe-level”.
The techniques used in “Tapering” is current or amplitudes in the sources of a linear array is non-uniform.
It is found that minor lobes are reduced if the center source radiates more strongly than the end sources and hence this tampering is done from center to end.

2. Tapering of arrays Binomial Arrays Tchebysceff Arrays
The tapering follows coefficients of binomial series and Tchebysceff
polynomial
Tapering of arrays
Binomial Arrays
Tchebysceff Arrays

3. Binomial Arrays
It is a linear array of n- isotropic point sources of non-uniform amplitude.
In this type of array, an amplitudes of the radiating sources are arranged according to the coefficients of successive terms of the following binomial series
𝑎−𝑏 𝑛−1 = 𝑎 𝑛−1 + 𝑛−1 1! 𝑎 𝑛−2 𝑏+ (𝑛−1)(𝑛−2) 2! 𝑎 𝑛−3 𝑏 2 + ……………………
where n = number of radiating sources in the array.
This work can be accomplished by arranging the arrays in such a way that radiating sources in the center of the broadside array radiated more strongly than the radiating sources at the edges. The secondary lobes can be eliminated entirely, if the following two conditions are satisfied.
Spacing between the two consecutive radiating sources does not exceed 𝝀/𝟐and
The current amplitudes in radiating sources (from outer, towards center source) are proportional to the coefficient of the successive terms of the binomial series (Equation-1)
Equation-1

4. Pascal’s Triangle

5. It may be noted that Elimination of secondary lobes takes place at the cost of directivity.
HPBW power beam width (HPBW) of binomial array is more than that of uniform arrays for the same length of the array.
For example, for radiating source n=5, spaced 𝝀/𝟐apart HPBW of the binomial and uniform arrays are respectively 23 deg. & 31 deg. as shown in figure below
Thus, in uniform array secondary lobes appear but principle lobe (main lobe)is sharp and narrow where as in binomial array width of beam widens but without secondary lobes

6. Disadvantage of Binomial Arrays
HPBW increases and hence the directivity decreases.
For design of a large array, larger amplitude ratio of sources is required.

7. Tchebyshev polynomial
Tm(x)=cos(m cos-1 x), <x<1
M=0;
To(x)=cos(0)= 1
T1(x)=cos(cos-1 x)=x
T2(x)=cos(2cos-1 x)=cos(2Y)=2cos2 Y-1
=2x2-1

8. Tchebysceff Polynomials
The polynomials of equation (10) are called Tchebysceff polynomials which is denoted as 𝑇 𝑚 𝑥 = cos 𝑚 𝜓 2 Further higher terms can be had from the recursion formula
………………..(11)
m = 0
𝑇 0 𝑥 =𝟏
m = 1
𝑻 𝟏 (𝒙)=𝒙
m = 2
𝑻 𝟐 (𝒙)=𝟐 𝒙 𝟐 −𝟏
m = 3
𝐓 𝟑 (𝒙)=𝟒 𝒙 𝟑 −𝟑𝒙
m = 4
𝑻 𝟒 (𝒙)=𝟖 𝒙 𝟒 −𝟖 𝒙 𝟐 +1
m = 5
𝑻 𝟓 𝒙 =𝟏𝟔 𝒙 𝟓 −𝟐𝟎 𝒙 𝟑 +𝟓𝒙
m = 6
𝑻 𝟔 𝒙 =𝟑𝟐 𝒙 𝟔 −𝟒𝟖 𝒙 𝟒 +𝟏𝟖 𝒙 𝟐 −𝟏
m = 7
𝑻 𝟕 𝒙 =𝟔𝟒 𝒙 𝟕 −𝟏𝟏𝟐 𝒙 𝟓 +𝟓𝟔 𝒙 𝟑 −𝟕𝒙
Tchebysceff
Polynomials
𝑻 𝒎+𝟏 𝒙 =𝟐𝒙 𝑻 𝒎 𝒙 − 𝑻 𝒎−𝟏 (𝒙)
………………..(12)