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A more interesting kind of delay is
``**group delay**,"
corresponding to **group velocity** in wave-propagation theory.
Often the group delay is nothing more than the phase delay.
This happens when the phase delay is independent of frequency.
But when the phase delay depends on frequency,
then a completely new velocity, the ``group velocity," appears.
Curiously, the group velocity is *not* an average of phase velocities.
The simplest analysis of group delay begins
by defining a filter input *x*_{t} as
the sum of two frequencies:

| |
(26) |

By using a trigonometric identity,
| |
(27) |

we see that the sum of two cosines
looks like a cosine of the average frequency
multiplied by a cosine of half the difference frequency.
Since the frequencies in Figure 17 are taken close together,
**beat
**

Figure 17
Two nearby frequencies beating.

the difference frequency factor in (28)
represents a slowly variable amplitude
multiplying the average frequency.
The slow (difference frequency) modulation of the higher (average)
frequency is called
``**beating**.''

The beating phenomenon is also called ``**interference**,'' although
that word is deceptive.
If the two sinusoids were two wave beams crossing one another,
they would simply cross *without* interfering.
Where they are present simultaneously, they simply add.

Each of the two frequencies could be delayed a different amount by a filter,
so take the output of the filter *y*_{t} to be

| |
(28) |

In doing this,
we have assumed that neither frequency was attenuated.
(The **group velocity** concept loses its simplicity and much
of its utility in dissipative media.)
Using the same trigonometric identity on (29)
as we used to get (28), we find that
| |
(29) |

Rewriting the beat factor in terms of a time delay *t*_{g},
we now have
| |
(30) |

| |

| (31) |

For a continuum of frequencies, the **group delay** is
| |
(32) |

** Next:** Group delay as a
** Up:** PHASE DELAY AND GROUP
** Previous:** Phase delay
Stanford Exploration Project

10/21/1998