Digital Signal Processing Reference
In-Depth Information
Please note:
The sinusoidal oscillations of the output signal may vary with regard to amplitude
and phase. The amplitude can increase (amplification) or decrease dramatically
(extreme damping). Remember that phase displacement is just a time modification
of the output sinusoidal oscillation compared with the input sinusoidal oscillation.
After all, any process takes some time before it is completed.
Line and space
One of the most important examples of a linear system in communications is the line. At
the end of a line there will always be a sinusoidal oscillation of the same frequency as the
one which was fed into the line at its other end, no matter whether the line measures only
a few yards or several miles. The line dampens the signal with increasing length, i.e. the
longer the line the lower the amplitude. Transmission time also increases with length -
the transmission speed of lines is about 100,000 and 200,000 km/s. This becomes notice-
able as phase displacement on the oscilloscope.
It would be disastrous if a line had non-linear properties! Fixed-line telephony for
example would hardly be possible because the voice-frequency band at the other end of
the line would be in a completely different, possibly inaudible, frequency range, and the
length of the line would play a role as well.
The most important „linear system“ is space. Otherwise the signals of a radio station
would reach its listeners in a different frequency, or several different frequencies for that
matter, from the one intended by the station. And again, just as with lines, there would be
a dependency on the distance between transmitter and receiver. This example shows the
importance of linear processes more than any other. Later in this topic we will take a look
behind the scenes to see why it is that lines and space have linear properties and the
physical reasons behind it.
Inter-disciplinary significance
Linearity and non-linearity play an extremely important role in mathematics, physics,
technology, and in natural sciences generally speaking. Linear equations in mathematics
for instance can be solved quite easily, whereas non-linear equations can rarely, if ever,
be solved. Quadratic equations are examples of non-linear equations and generations of
pupils have been tortured with them. It is really hard to believe: mathematics (these days)
mostly fails when it comes to solving non-linear problems. As theoretical physics finds
its most important support in mathematics, non-linear physics is rather underdeveloped
and, as it were, still in its infancy.
The behaviour of linear systems is generally speaking easier to understand than that of
non-linear systems. The latter can even lead to chaotic, i.e. principally non-predictable,
behaviour or create so-called fractal structures which - if represented graphically - can be
of great aesthetic beauty. Furthermore they possess a „universal“ property which seems
to be becoming more and more important: that of being fractal. If you take a closer look
they basically display the same structures whether you enlarge or diminish the scale.
