Editing Group delay and phase delay (section)

== Group delay from transfer function polynomials ==
If a transfer function or Sij of a [[Scattering parameters|scattering parameter]], is in a polynomial [[Laplace transform]] form, then the [[#Mathematical definition of group delay and phase delay|mathematical definition for group delay]] above may be solved analytically in closed form.  A [[polynomial]] [[transfer function]] <math>P(S)</math> may be taken along the <math>j\omega</math> axis and defined as <math>P(j\omega)</math>.  <math>\phi(\omega)</math> may be determined from <math>P(j\omega)</math>, and then the group delay may be determined by solving for <math>-d\phi(\omega)|/d\omega</math>.

to determine <math>\phi(\omega)</math> from <math>P(j\omega)</math>, use the definition of <math>\phi(\omega) = tan^{-1}(P(j\omega)_{imag}/P(j\omega)_{real})</math>.  Given that <math>j^{2N}</math> is always real, and <math>j^{2N+1}</math> is always imaginary, <math>\phi(\omega)</math> may be redefined as <math>\phi(\omega) = tan^{-1}(-jP(j\omega)_{odd}/P(j\omega)_{even})</math> where ''even'' and ''odd'' refer to the polynomials that contain only the even or odd order coefficients respectively.  The <math>-j</math> in the numerator merely converts the imaginary <math>P(j\omega)_{odd}</math> numerator to a real value, since <math>P(j\omega)_{odd}</math> by itself is purely imaginary.

<math>\begin{align}
&\frac{dtan^{-1}(f(x))}{dx} = \frac{df(x)/dx}{1+f(x)^2} \\
&f(x) = \frac{-jP(j\omega)_{odd}}{P(j\omega)_{even}} \\
&\frac{df(x)}{dx} = \frac{P(j\omega)_{even} \frac{d(P(j\omega)_{odd})}{dx} - 
-jP(j\omega)_{odd} \frac{d(-jP(j\omega)_{even})}{dx}}
{P(j\omega)_{even}^2}
\end{align}</math>

The above expressions contain four terms to calculate:

<math>\begin{array}{lcl}
Se = P(j\omega)_{even} &=& \sum_{k=0}^{N/2}P_{2k}(j\omega)^{2k} &=& \sum_{k=0}^{N/2}P_{2k}(-1)^{k}(\omega)^{2k}\\
So = P(j\omega)_{odd} &=& -j\sum_{k=1}^{(N+1)/2}P_{2k-1}(j\omega)^{2k-1} &=& \sum_{k=1}^{(N+1)/2}P_{2k-1}(-1)^{k-1}(\omega)^{2k-1}\\
De = \frac{d(P(j\omega)_{even})}{dx} &=& -j\sum_{k=1}^{N/2}2kP_{2k}(j\omega)^{2k-1} &=& \sum_{k=1}^{N/2}2kP_{2k}(-1)^{k-1}(\omega)^{2k-1} \\
Do = \frac{d(P(j\omega)_{odd})}{dx} &=& \sum_{k=1}^{(N+1)/2}{(2k-1)}P_{2k-1}(j\omega)^{2k-2} &=& \sum_{k=1}^{(N+1)/2}{(2k-1)}P_{2k-1}(-1)^{k-1}(\omega)^{2k-2}\\
\\
\frac{df(x)}{dx} &=& \frac{SeDo - SoDe}{Se^2}
\end{array}</math>

The equations above may be used to determine the group delay of polynomial <math>P(S)</math>  in closed form, shown below after the equations have been reduced to a simplified form.

<math>\text{Group Delay} =gd(P(j\omega))= -\frac{d\phi(\omega)}{d\omega} = -\frac
{(So*De + Se*Do)}
{(Se^2 + So^2)}
\text{ sec}</math>

=== Polynomial ratio ===
A polynomial ratio of the form <math>P2(S) = P_{num}(S)/P_{den}(S)</math>, such as that typically found in the definition of [[filter design]]s, may have the group delay determined by taking advantage of the phase relation, <math>\phi(P1/P2) = \phi(P1) - \phi(P2)</math>.

<math>\text{Group Delay} = gd(P2) = gd(P2_{num}) - gd(P2_{den}) sec</math>

=== 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.