System analysis - Signals, Processes, and Systems - page 165

For this purpose the GAUSSian pulse is simply multiplied by a sine of the frequency f C
(C as “carrier frequency”) by which the spectrum is to be displaced. The result is then a
sine in the time domain which begins and ends gently (see Illustration 107). In the
frequency domain this causes a displacement of the (symmetrical) spectrum from f = 0
Hz to f = f C Hz (and f = -f C ). The narrower this signal called a GAUSSian oscillation pulse
in the time domain, the wider its spectrum in the frequency domain. ( UP ; see also
Illustration 46). This important trick using multiplication is dealt with in more detail in
Chapters 7 and 8).

The GAUSSian oscillation pulse is - like the GAUSSian pulse - suitable for determining
in a straightforward way the group velocity v gr in cables or in entire systems and - in
addition - for comparing it with so-called phase velocity v ph .

The phase velocity v ph is the velocity of a sinusoidal wave. In a disperse medium - e.g.
along a cable - v ph is not constant but depends on the frequency f and the wavelength l.
Group velocity v gr is understood to mean the velocity of a group of waves, i.e a group of
waves limited in time and place (e.g. GAUSSian oscillation pulse).

Note:

Energy and information propagate themselves at the group velocity v gr . If the
GAUSSian oscillation pulse is used as a group of waves the transit time t of the
sinusoidal carrier signal is a yardstick for the phase velocity v ph , the GAUSS-
shaped curve of the envelope on the other hand is a yardstick for the group velocity
v gr . If v ph is constant v gr = v ph . An interesting physical property is the fact that the
group velocity v gr can never be greater than the speed of light in a vacuum c 0
(c 0 = 300, 000 km/s). This is not necessarily true for the phase velocity v ph . The
maximum upper limit for the transportation of energy and information is thus the
speed of light, or the speed of the electromagnetic energy of the medium involved.
In cables it is between 100,000 and 300,000 km/s.

Preliminary conclusions:

The GAUSSian pulse and the GAUSSian oscillation pulse are less
important as test signals for measuring the frequency response of
circuits and systems. They can be used in the measurement of
pulses to determine in a straightforward way the transit time,
group and phase velocity.

The Burst signal
The burst signal is - as we already know from Illustration 45 - a time-limited “sine”.
However it begins and ends abruptly and this has consequences for the bandwidth and the
uncertainty of the frequency of the sine wave.

Thus a burst can be used to test the frequency selectivity of a circuit or system
qualitatively in the sphere of the value of the sine frequency (mid-frequency). However,
the spectrum has zero positions; these represent frequency gaps.

Transients of frequency selective circuits can be superbly demonstrated by means of a
burst. See the next section on “Transients”.