Digital Signal Processing Reference
In-Depth Information
Chapter 5
The Symmetry Principle
Symmetry is one of the most important structural elements in nature. Space is symmetri-
cal - i.e. no direction is given preference in a physical sense - and we demand and find
for practically any elemental particle (e.g. the electron with its negative elemental charge)
a "mirror-image" object (e.g. the positron with its positive elemental charge). For reasons
of symmetry there is matter and anti-matter.
For reasons of symmetry: negative frequencies
A periodic signal does not begin at t = 0 s. It has a past and a future and both are practically
symmetrical to the present.
However, the spectrum (up to now) always begins at f = 0 Hz. The equivalent to the past
would be "negative frequencies" in the frequency domain. Negative frequencies seems
not to make any sense because it does not appear possible to interpret them physically.
But we should take nature's symmetry principle seriously enough to look for effects where
negative frequencies might provide the key to understanding.
Illustration 86 describes an effect of this kind. To arouse your curiosity look carefully at
Illustration 45 once again. The progression of the spectrum became more and more asym-
metrical as the signal duration
t was increasingly restricted. This requires an explanation
as there is no equivalent to this in the time domain.
Δ
We intend to prove experimentally that asymmetry can be explained and offset, as it were,
if negative frequencies and negative amplitudes are allowed. Physically negative and
positive frequencies would have to work jointly, each representing half the energy.
Note: Let us look more closely at the measuring process to determine the time - e.g.
the duration of the period T and/or the frequency via f = 1/T. First, a time point of
reference is chosen, here precisely the positive maximum of the sinusoidal signal.
Starting from this point of reference the point of time is determined at which the
(periodic) sinusoidal signal reaches precisely this value again. The measuring
process can be carried out in two different directions. Both measurements give the
same result. In a physical sense they differ in that in relation to the point of reference
the one measurement is carried out to the right into the "future" and the other to the
left into the "past". Both mirror-image measurements from the point of reference
may each be encoded with a positive or negative sign of the period duration T. This
consideration which may appear rather simple is in agreement with the interpreta-
tions of measurements of quantum objects. Accordingly, what is the significance of
the results of observations?
Simply (coded) information!
Proof of the physical existence of negative frequencies
Illustration 86 provides the first indication of the physical existence of negative frequen-
cies. By means of a trick - "convolution" of an Si-function (see Chapter 10) - the two
frequency bands of a given signal are displaced more and more to the left in the direction
of f = 0 Hz. What will happen when f = 0 Hz is finally exceeded?
