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CHAPTER II – OPTICAL OFDM
23
At reception the same signal will be obtained without any distortion but with a
constant delay.
On the other hand, in a dispersive channel the phase constant has a nonlinear
dependency with frequency and as a consequence of the different arrival times
of the frequency components, the recovered signal at the reception end will
differ from the transmitted one.
Assuming a slow variation of the phase constant inside the signal’s frequency
bandwidth, it is possible to consider a Taylor expansion of the propagation
constant about a central pulse frequency
as follows:
    o   2     o   3 
      0      o 




2
6
 2
 3
 2
 3
  o   1 
2 
3  
2
6
2
3
(II.2)
Where the third and higher order terms can be neglected if it is considered that
, which enables the possibility to rewrite (II.2) as:
(II.3)
The coefficients in (II.3) are related to the following parameters:

relates to the Phase Velocity
, which verifies:
(II.4)
And it can be defined as the velocity at which the phase of a pure tone at
frequency
would propagate.

is related to the Group Velocity
, of the pulse by:
(II.5)
The group velocity can be defined as the rate with which changes in the
envelope of the wave (amplitude) propagate. The Group Delay , given
in (II.5) in seconds/fibre length, gives the delay experienced by an
envelope centred at frequency
, provided its bandwidth is not too
large, as the Taylor expansion would no longer be valid. It can also be
thought that this delay is a kind of average delay of all the frequencies in
a small bandwidth around the carrier.