Digital Signal Processing Reference
In-Depth Information
Sin us
Time domain
Spectrum
X/Y-Chart
Sine -axis
Imaginary part
4,00
π
/6
2,00
Cosine-axis
Real part
0,00
π
/6
-2,00
8,00
4,00
0,00
-4,00
-8,00
Phase shi ft
π
/6
0.25
0.50
0.75
1.0
s
-4,00
-4,00
-2,00
0,00
2,00
4,00
V
Illustration 97:
Representation of the "frequency vectors" in the complex GAUSSian plane
By means of the x-y-module all the information on the two spectra from Illustration 96 can be brought
together on one level. Each of the three sinusoidal signals is here in the form of a "frequency vector pair"
which is always symmetrical in relation to the horizontal axis. Instead of the usual vector arrowheads we
use a small triangular form. The length of all the "frequency vectors" is in this case 4 V, that is, half the
amplitude of the sinusoidal signal is allotted to each of the two frequency vectors.
It is most difficult to recognize the sinusoidal signal
without
phase shift: this pair of frequency vectors lie
on the vertical axis which passes through the point (0;0) and which for this reason we call the "
sine axis
".
In the case of a phase displacement of 90 degrees or
/2 rad - this corresponds to a cosine - both fre-
quency vectors lie above one another on the horizontal axis. We therefore call this the "
cosine axis
". In a
phase displacement of 30 degrees or
π
/6 rad we obtain the two frequency vectors of which the angle in
relation to the sine axis is entered. As you can now see a sine with a phase displacement of 30 degrees or
π
π
/6 rad is simply a cosine of -60 degrees or -
π
/3 rad. A phase displaced sine thus has a sine and cosine
part!
Careful! The two equally large cosine parts of a frequency vector pair add up, as you ought to check in
Illustration 98, to a quantity which is equal to the instantaneous value of this sinusoidal signal at the point
of time t = 0s. By contrast, the sine-parts always add up to 0 because they lie
opposite
to each other.
Because there are quantities along the cosine axis which are measurable in a real sense, we call this the
real part
. Because on the sine axis everything cancels each other out and nothing remains, we choose the
expression
imaginary part
following the mathematics of complex calculations.
We shall show in the next Illustration that the sinusoidal signal which belongs to the "frequency vector
pair" can be produced from the addition of the sinusoidal signals which belong to the real part and the
imaginary part.
