Bessel-Thompson Approximation of constant Delay

The previous responses focus on the magnitude of the filter function. In case of phase relations. Phase function is directly related to the group delay of the signal. Group delay is important when signal shape is on consideration. It is important because the delay of different signals is not noticeable on human air in case of audio application but in case of video, the picture will be distorted of delay is not constant through out passband. One standard low pass filter that focus on delay is Bessel Thompson Approximation. It has

• Slowest transition from passband to stopband and vice versa

• Very flat group delay response

• Very useful for pulse or other broadband signals

To find a transfer function to give approximation time delay for large value of ‘’

• Assume a form of Transfer function

• Compute the corresponding time delay

• Expand the form on Taylor’s Series about = 0

• Find the condition that will cause many coefficients in series expansion to vanish as possible.

Let’s us assume 2nd order all pole transfer function as

\[
T_2(s) = a_0 s^2 + a_1 s + a_0
\]

\[
T_2(j\omega) = a_0 (j\omega)^2 + a_1 (j\omega) + a_0 = a_0 - \omega^2 + a_1 (j\omega) + a_0
\]

Now

\[
\text{Delay } D = - \frac{d\theta}{d\omega}
\]

\[
D = - \frac{d}{d\omega} \left[- \tan^{-1} \left( \frac{a_0}{a_0 - \omega^2} \right) \right] = \frac{1}{1 + a_1^2} a_0 (\omega)^2 \cdot \frac{d}{d\omega} \left( a_0 - \omega^2 \right)
\]

\[
D = \frac{a_0 - \omega^2}{2 a_0 - \omega^2 + a_1 \omega^2} \cdot \frac{a_1 (a_0 - 2) - a_1 (\omega)(-2\omega)}{a_0 - \omega^2)^2}
\]

\[
D = a_1 a_0 + a_1^2 a_0^2 - (\omega)^2 (a_1^2 - 2 a_0)(\omega)^4 \quad \dots A
\]

\[
D = a_1 a_0 (1 + 2a_0) a_0^2 (1 + a_1 a_0^2 - 2 a_0^2 + 4 a_0^2)
\]

\[
D = a_1 a_0 (1 + 2 a_0)(1 + a_1 a_0^2 - 2 a_0^2 + 4 a_0^2) \quad \dots B
\]

Now using Taylor Series

\[
D = a_1 a_0 \left[ 1 + (1 a_0 - a_1 a_0^2 + 2 a_0) * -2 + \dots \right]
\]

For the second term to be zero

\[
1 a_0 + 2 a_0 = a_1 a_0^2
\]

\[
3 a_0 = a_1 a_0^2 \quad \dots C
\]

For normalized case \(D = 1\) at \(\omega = 0\) then from A,

\[
D = a_1 a_0 + 0 (a_0^2) + 0 + 0 = a_1 a_0 = 1
\]

\[
a_1 = a_0
\]

Hence from C

\[
a_1 = a_0 = 3
\]

Then the transfer function is given by

\[
T_2(s) = a_0 s^2 + a_1 s + a_0 = 3 s^2 + 3 s + 3
\]

Similarly, from C

\[
D = a_1 a_0 (1 + 2 a_0)(1 + a_1 a_0^2 - 2 a_0^2 + 4 a_0^2)
\]

\[
D = 3*3 (1 + 2 a_0)(1 + 3*3^2 - 2*3^2 + 4*3^2)
\]

\[
D = (1 + 2 a_0)(1 + 23 + 49) = 3*2 + 94 + 3*2 + 9
\]

Similarly for 3rd order,

\[
T_3(s) = 15 s^3 + 6 s^2 + 15 s + 15
\]

And its corresponding delay is

\[
D(s) = 6*4 + 45*2 + 2256 + 6*4 + 45*2 + 225
\]

A more direct method is given by Storch