Editing Group delay and phase delay (section)

=== Simple filter example ===
A four pole Legendre filter transfer function used in the [[Optimum "L" filter#Example: 4th order transfer function|Legendre filter example]] is shown below.

<math>T_4(j\omega) = \frac{1}{2.4494897(j\omega)^4 + 3.8282201(j\omega)^3 + 4.6244874(j\omega)^2 + 3.0412127(j\omega) + 1}</math>

The numerator group delay by inspection is zero, so only the denominator group delay need be determined.

<math>\begin{align} 
&Pe_{den} = 2.4494897\omega^4  - 4.6244874\omega^2 + 1  \\
&Po_{den} = -3.8282201\omega^3  + 3.0412127\omega       \\
&De_{den} = 4(2.4494897)\omega^3 - 2(4.6244874)\omega  \\
&Do_{den} = 3(-3.8282201)\omega^2 + 3.0412127
\end{align}</math>

Evaluating at <math>\omega</math> = 1 rad/sec:

<math>\begin{align} 
&Pe_{den} = -1.1749977  \\
&Po_{den} = -0.7870074       \\
&De_{den} = -0.548984 \\
&Do_{den} = -8.4434476
\end{align}</math>

<math>\begin{align}
&\text{Group Delay} =gd(T_4(j\omega))= -\frac{d\phi(\omega)}{d\omega} \\
&= \bigg[0--\frac
{((-0.7870074*-0.548984)  + (-1.1749977*-8.4434476))}
{((-1.1749977)^2 + (-0.7870074)^2)}\bigg]  \\
&= 5.1765430\text{ sec} \\
&\text{at }\omega = 1\text{ rad/sec}
\end{align}</math>

The group delay calculation procedure and results may be confirmed to be correct by comparing them to the results derived from the digital [[derivative]] of the phase angle, <math>\phi(\omega)</math>, using a small delta <math>\Delta\omega</math> of +/-1.e-04 rad/sec.

<math>\begin{align}
&\text{Group Delay} =gd(T_4(j\omega))= -\frac{d\phi(\omega)}{d\omega} \\
&= - (\phi(1+1e-04)-\phi(1.-1e04))/2e-04\\

&= 5.1765432\text{ sec} \\
&\text{at }\omega = 1\text{ rad/sec}
\end{align}</math>

Since the group delay calculated by the digital derivative using a small delta is within 7 digits of accuracy when compared to the precise analytical calculation, the group delay calculation procedure and results are confirmed to be correct.