The Symmetry Principle - Signals, Processes, and Systems - page 134

As Illustration 97 shows the direction of rotation of the two "frequency vectors" is oppo-
site if the phase shift of the sine increases or decreases. The frequency on the left in
Illustration 96 which we represented as a negative frequency in Illustration 93 (bottom)
rotates in anti-clockwise direction, the positive frequency in a clockwise direction where-
by the positive phase shift increases.

How do the instantaneous values of the three sinusoidal signals conceal themselves at the
point of time t = 0 s in the plane? Compare carefully the symmetrical spectra of
Illustration 96 with the plane of the x-y-module in Illustration 97. You should take into
account that the frequency lines are vectors for which quite specific rules apply. Vectors
- e.g. forces - can be divided up into parts by projection on to the horizontal and vertical
axes which are entered here as markings.

For the sinusoidal signal with the phase shift of 30 degrees or /6 rad we obtain the value

2 by projection on to the cosine axis. The sum is 4 (instantaneous value at the point of time
t = 0s). The projection on to the sine axis gives the value 3.46 or -3.46, that is the sum adds
up to 0.

π

For this reason the resulting vectors of all (symmetrical) "frequency vector" pairs always
lie on the cosine axis and represent the real instantaneous values at the point of time t = 0,
which can be measured. For this reason the so-called real part is represented on the cosine
axis.

By contrast, the projections of the frequency vector pairs on the sine axis always lie
opposite to each other. Their sum is therefore always 0 independent of the phase position.
The projection on to the sine axis has therefore no counterpart which can be measured.
Following mathematics of complex calculations in the so-called GAUSSian plane the
projection on to the sine axis is referred to as the imaginary part . Both projections have
an important physical sense. This is explained by Illustration 98. The projection reveals
that every phase-displaced sinusoidal signal can always consist of a sine and a cosine
oscillation of the same frequency. Important consequences result from this:

All signals can be represented in the frequency domain in three ways:

• as an amplitude and phase spectrum

• as the spectrum of the frequency vectors in the GAUSSian plane

• as a spectrum of sine and cosine signals

The symmetrical spectra from Illustration 96 (bottom) are revealed to be the last type of
representation of a spectrum. This is proved by Illustration 98.

The Illustrations that follow deal with the spectra of periodic and non-periodic signals in
the representation as a symmetrical "frequency vector pair" in the GAUSSian plane of
complex numbers. You will find additional information in the caption text.

Complex numbers refers to numbers in mathematics which contain a real and an
imaginary part. It would be quite tempting to demonstrate that calculating with complex
numbers is far from "complex", and on the contrary is much easier than calculating with
real numbers. But the original approach is to do without mathematics.