The Symmetry Principle - Signals, Processes, and Systems - page 131

Before we examine this type of FOURIER transformation more closely by means of
experimentation, it should be shown to work.

In Illustration 95 you see an appropriately block diagram. If you set the parameters
according to the representation and caption text of Illustration 94 the left-hand input signal
does in fact re-appear at the upper output of the (inverse) FFT module. Clearly, all the
information from the time domain was transferred to the frequency domain so that the
inverse FOURIER transformation in the reverse direction back into the time domain via
the complete set of all the necessary information was also successful.

Now it will be shown how easily it can be manipulated in the frequency domain.
By adding the "cut out" module we have the possibility of cutting out any frequency
range. We see this step has been successful in the lower half: a virtually ideal lowpass
filter with the borderline frequency 32 Hz which it was not possible to realize up to now.
This circuit will prove to be one of the most important and sophisticated in many practical
applications which we shall be dealing with.

The next step is to establish experimentally how the whole thing works and what it has to
do with the SP . First we represent three simple sinusoidal signals with 0, 30 and 230
degrees and 0, /6 and 4 rad phase displacement by means of the "complex FFT" of a real

signal. The result in Illustration 96 are two different - symmetrical - line spectra for each
of the three cases. However, it does not seem to be a question of absolute value and phase,
as only positive values are possible in the case of the absolute value. The lower spectrum
in each case cannot be a phase spectrum as the phase displacement of the sinusoidal signal
is not identical with these values.

π

We continue to explore and add an x-y-module ( Illustration 97). Now you see a number
of "frequency vectors" on the plane. Each of these frequency vectors has a mirror-image
symmetrical "twin" in relation to the horizontal axis. In the case of the sinusoidal signal
with 30 degrees and π /6 phase displacement the two frequency vectors which each have

a 30 degrees and /6 rad phase displacement in relation to the vertical line running

through the central point (0;0) fall into this category. The phase displacement in relation
to the line running horizontally through the point (0;0) is accordingly 60 degrees and/or
π

π

/3 rad.

For the time being we will call the vertical sine axis because both frequency lines with a
phase displacement of 0 degrees or 0 rad lie on this. We will call the horizontal the cosine
axis because the frequency lines with a phase displacement of the sine of 90 degrees or
π /2 - this corresponds to the cosine - lie on this.

On the other hand, a sine displaced by π /6 rad is nothing other than a cosine displaced by

- /3. If you now compare the axis sections with the values of the line spectra the values

of the upper spectrum belong to the cosine axis and the values of the lower spectrum to
the sine axis.

π

The two frequency lines apparently possess the characteristics of vectors which in
addition to their absolute value also have a certain direction. We shall see that the length
of both frequency vectors reflects the amplitude of the sine and the angle of the "frequency
vectors" in relation to the vertical and horizontal line reflects the phase displacement of
the sine and cosine at the point of time t = 0 s.