Digital Signal Processing Reference
In-Depth Information
Time domain
sin+cos
Ad d i t i o n
8, 00
4, 00
Sine wave signal: Amplitude 2 * 3,46 Volt
0, 00
-4,00
-8,00
8, 00
4, 00
Cosine wave signal: Amplitude 2 * 2 Volt
0, 00
-4,00
-8,00
8, 00
4, 00
Sum of sine and cosine wave signal: Amplitude 2 * 4 Volt
0, 00
-4,00
-8,00
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ms
Illustration 98:
Spectral analysis of sine and cosine components
This Illustration proves that the three different kinds of spectral representations or representations of the
frequency areas are consistent in themselves. The relevant sine and cosine components result from the
"frequency vectors" in the GAUSSian plane of complex numbers as projections on to the sine and cosine
axes (imaginary and real components).
Let us first take a look at the sinusoidal oscillation (top) with the amplitude 2
<
3.46 = 6.92 V. In the numer-
ical plane of complex numbers it results in a "frequency vector pair" which is located on the sine axis. A
vector with a length of 3.46 V points in the positive direction, its "twin" in the negative direction. Their
vector sum equals 0.
Let us now take a look at the cosine oscillation with the amplitude 4 V. The relevant "frequency vector
pair" can be located on the cosine axis pointing in the positive direction. Each of these two vectors has a
length of 2. Thus their sum is 4.
Thus, everything is in accordance with Illustration 74. Please note that the sinusoidal oscillation the phase
of which is displaced by 30 degrees or
/6 rad also has an amplitude of 8 V. This is also the result of
appropriate calculations using the right-angled triangle: 3.46
2
+ 2
2
= 4
2
(Pythagorean Theorem).
The representation in the so-called GAUSSian plane is of great importance because in principle it
combines all three ways of spectral representation: amplitude and phase correspond to the length and
angle of the "frequency vector". The sine and cosine components correspond to the breaking down of a
phase-shifted sine into
pure
sine and cosine forms.
So far, we can only see one disadvantage: we cannot read the frequency. The position of the vector is
independent of its frequency.
π
