The Symmetry Principle - Signals, Processes, and Systems - page 119

The bottommost picture in Illustration 86 is of course the most interesting one. In the case
of the left-hand frequency band it is a lowpass characteristic the bandwidth of which is
exactly half the original bandwidth, that is, it corresponds to the bandwidth of the right-
hand band. On the other hand the amplitude curve is twice as high. A low pass filter thus
seems to have a "virtual amplitude curve" which is twice as large as the visible bandwidth
in the spectrum with positive frequencies. It is possible to show that this "virtual" band-
width is the actual physical bandwidth. This follows from the UP . Hence, if the filter
range began at f = 0 Hz the low pass filter would at that point have an infinitely great edge
steepness. But precisely this is ruled out by the UP . This is also revealed by the time
domain (see Chapter 6 under the heading "Transients").
The ingenious signal from Illustration 86 was generated in the time domain by means of
the Si-function. This is obvious if you look once again at Illustration 48 and Illustration
49. This form of signal seems more and more important and the question arises whether
the Si-function also exists in the negative frequency range. The answer is "yes" if we ad-
mit negative frequencies and negative amplitudes. Thus, the Symmetry Principle between
the time and frequency domain is clearly demonstrated.
In Illustration 87 at the top we see once again the 3D-spectrum of a narrow (periodic)
rectangular pulse. Look carefully at the "playing field" of the sinusoidal signals, particu-
larly where the rectangular pulse is symmetrical to t = 0.5 s. As the pulse duty factor t/T
is roughly 1/10 the first 0 lies at the 10th harmonic. The amplitudes of the first 10
harmonics at t = 0.5 s on the "playing field" point upwards, those from 11 to 19 point
downwards, then upwards again etc. It would be better to enter the amplitudes of the
amplitude spectrum in the second (fourth etc.) sector (11 to 19) pointing downwards
instead of
In Illustration 87(centre) you see the continuous amplitude spectrum of a one-off rectan-
gular pulse. If the curve is drawn downwards in the 2nd, 4th, 6th etc. sector it ought to
begin to dawn on you. What you see is the right-hand symmetrical half of the Si-function.
Last but not least if we describe the Si-function symmetrically towards the left in the neg-
ative frequency range we obtain the total Si-function (to be exact, however, only half as
high because half the energy is allocated to the negative frequencies).
Background knowledge :
What we have discovered here by means of "experiments" - the simulations carried
out here with a virtual system would in the real world lead to exactly the same
results using suitable measuring instruments - is provided automatically by
mathematics if calculations are carried out in the suitable scale (GAUSSian plane
i.e. complex numbers).
Why do the mathematics (of the FOURIER transformation) provide this result?
If the calculation is based from the outset on a correct physical premise with real
marginal conditions all further mathematical calculations produce correct results
because the mathematical operations used are essentially free of contradictions.
However, not every mathematical operation can necessarily be interpreted in a
physical sense.