# Derivation of Dolph-Taylor Synthesis

Today, I was trying to deduce by myself the Dolph-Taylor Synthesis. After arriving at the expression:

$$AF(\theta) = P_{N-1} \left( \cos \left( \frac{\psi}{2} \right) \right) = T_{N-1} \left(x_o \cos \left( \frac{\psi}{2} \right) \right)$$

Where $$\FA\$$ is the Array FActor, $$\P_{N-1}\$$ is an arbitrary polynomial and $$\T_{N-1}\$$ is the $$\N-1\$$th Chebyshev polynomial, I am struck at finding the right value for $$\x_o\$$

We know that we want a maximum value for the secondary side lobes of SLL. Taking that in linear units, it is equivalent to $$\R=10^{-SSL/20}\$$. From here on, every single piece of literature I was able to read just magically says that the value we shall give to $$\x_o\$$ in order to obtain such radiation pattern is:

$$x_0 = \cosh\left( \frac{\cosh^{-1}R}{N-1}\right)$$

What I tried:

We know that $$\T_N(\cos \theta) = \cos (n\theta)\$$. Hence, for small enought values of x, we can make the change of variables $$\\theta = \arccos x\$$, so that:

$$T_{N-1} (x) = \cos (n \arccos x)$$

Since we want this to equal $R$:

$$\cos (n \arccos x) = R; \ \ x = \cos\left( \frac{\cos^{-1}R}{N-1}\right)$$

The question: where do the hyperbolic functions arise from?

• This is more of a mathematics problem than electrical engineering. Jun 2, 2021 at 22:18
• Well, it lies in between. I thought about asking it at the Mathematic Stack Exchange forum, but I find it difficult to explain to mathematicians, and I think there are higher chances of finding someone who understands about array synthesis here. Jun 2, 2021 at 22:30
• The question: where do the hyperbolic functions arise from? — Usually, from having a complex argument to a trigonometric function. Jun 2, 2021 at 22:55

$$\T_n(u) = cos(m cos^{-1}u) \$$ for $$\-1 <= u <= 1\$$ and
$$\T_n(u) = cosh(m cosh^{-1}u)\$$ for $$\ |u| >= 1\$$