Digital Signal Processing Reference
In-Depth Information
What is interesting is that these highly complex structures can be created by very simple
non-linear processes. So it is obviously wrong to assume that highly complex systems are
the result of highly complex causes. Some of these fractal objects created by mathematics
bear a stunning resemblance to certain plants so that similarly simple laws are assumed to
be behind certain biological processes. We therefore know that non-linear processes must
be responsible for the great variety in natural phenomena and research in this field is quite
brisk at the moment.
Computers meanwhile have become the most important medium with which to create,
map and examine non-linear structures. This has given new momentum to modern
mathematics. Even at our level it is possible in many cases to illustrate non-linear interre-
lations and characteristics. We will take a closer look at very simple non-linear processes
and examine them using the computer later on in this topic.
Mirroring and projection
But enough of mysterious advance remarks. Let me now try and explain the terms
linearity and non-linearity in signal processing in plain words without using any
mathematics. For the time being flat and curved mirrors will suffice to provide an
explanation.
At least once a day - in the morning - we look at our reflection in the (flat) mirror of our
bathroom. In the mirror we see a lifelike mirror image of our face (however, the whole
thing is the wrong way round). We expect a mirror image to be a true mapping of the
original image in terms of proportions. This is an example of a linear depiction caused by
linear mirroring. Another example is a photo. The object depicted is usually much smaller
than the original, but the proportions are identical. Enlargement and reduction - or multi-
plication by a constant to use a mathematical expression - is therefore a linear operation.
You may have been to a hall of mirrors at a fun fair looking at your reflection in distorting
mirrors. The surface of these mirrors is non-linear, i.e. not flat, but uneven and bulging.
These are examples of non-linear mirrors. What does the reflected image look like? Your
body is grotesquely distorted. One mirror shows a tiny head on an enormous torso with
short leg stumps attached to it. Another one enlarges the head and reduces the rest of the
body to a shrivelled sausage.
Our bodies are distorted in a non-linear way. The mirror image does not reflect the proper
shape because the proportions are distorted.
Changing a signal by means of signal processing can be described in a similar way. A
characteristic curve - which is the equivalent of the mirror - describes the relation be-
tween the input signal u
in
and the output signal u
out
of the component or the process.
Illustration 124 shows that the curve of the signal is projected vertically upwards onto the
characteristic curve and from there horizontally to the right. The projection is the equiva-
lent of the mirroring process.
