Storch Method
From Previous relations,
\[
V_2 = V_1(t - D)
\]
Then the required Transfer Function is
\[
T(s) = \frac{V_2(s)}{V_1(s)} = e^{-sD}
\]
For normalized value of \(D=1\) at \(\omega=0\), we have
\[
T(s) = e^{-s} = \frac{1}{e^s} = \frac{1}{\sinh s + \cosh s} = \frac{1}{\sinh s \left(1 + \frac{\cosh s}{\sinh s} \right)} = \frac{1}{\sinh s} \left(1 + \frac{\cosh s}{\sinh s} \right) = K \left( 1 + \frac{\cosh s}{\sinh s} \right)
\]
The Hyperbolic expansion is given by
\[
\cosh s = 1 + \frac{s^2}{2!} + \frac{s^4}{4!} + \frac{s^6}{6!} + \dots
\]
\[
\sinh s = s + \frac{s^3}{3!} + \frac{s^5}{5!} + \frac{s^7}{7!} + \dots
\]
Then,
\[
\coth(s) = \frac{\cosh s}{\sinh s}
\]
Performing repeated long division and expanding, we have
\[
\frac{\cosh s}{\sinh s} =
\cfrac{1}{s +
\cfrac{s}{3 +
\cfrac{s}{5 +
\cfrac{s}{7 + \dots}
}
}
}
\]The sum of numerator and denominator polynomial approximated to \(e^{-s}\)
For \(n = 1\) only first part is taken
\[
\coth(s) = \frac{1}{s}
\]
Similarly for \(n=2\), two parts of the step are taken
\[
\coth(s) = \frac{1}{s} + \frac{1}{3/s} = \frac{1}{s} + \frac{s^3}{s^2 + 3 s} = \frac{\cosh s}{\sinh s}
\]
The delay is given by the addition of Numerator and Denominator, then
\[
D(s) = \sinh(s) + \cosh(s) = s^2 + 3s + 3 = e^{-s}
\]
For \(s = 0, \omega = 0\)
\[
K = 0 + 0 + 3 = 3
\]
Then
\[
T(s) = K e^{-s} = 3 s^2 + 3s + 3
\]
Similarly for \(n = 3\), we take and expand to 3 steps of long division expansions
\[
\coth(s) = \frac{1}{s} + \frac{1}{3 s} + \frac{1}{5 s} = \frac{1}{s} + \frac{1}{\frac{15}{s} + \frac{25}{s}} = \frac{1}{s} + \frac{5 s}{s^2 + 15} = \frac{\cosh s}{\sinh s}
\]
The delay is given by
\[
D(s) = \sinh(s) + \cosh(s) = s^3 + 6s^2 + 15 s + 15
\]
Then the transfer Function is given by
\[
T(s) = K e^{-s} = K (s^3 + 6s^2 + 15 s + 15)
\]
For \(D = 0\), \(K = 15\)
\[
T(s) = 15 s^3 + 6 s^2 + 15 s + 15
\]
The numerators made equal to the constant term of the denominator so that the gain at low frequency is normalized to unity and as a result maximally flat delay at the origin could be obtained
\[
H_n(s) = \frac{b_0}{y_n(s)} = \frac{b_0}{s^n + b_{n-1} s^{n-1} + \dots + b_1 s + b_0}
\]