Digital Signal Processing Reference
In-Depth Information
a
1
cos(2ft +
1
) and a
2
cos(2ft +
2
), visualize them as rotating phasors
multiplied by nonrotating phasors, then visualize how they are added together.
Note that r
n
=
a
2
in this gure.
r
1
q
1
+
-q
1
r
1
Rotating Phasor
Phasor
Rotating Phasor
Phasor
r
2
q
+
+
2
-q
2
r
2
Rotating Phasor
Phasor
Rotating Phasor
Phasor
r
3
q
+
=
3
-q
3
r
3
Rotating Phasor
Phasor
Rotating Phasor
Phasor
Figure 7.13: A graphic representation of adding 2 sinusoids of the same frequency.
a
3
cos(2ft +
3
) = a
1
cos(2ft +
1
) + a
2
cos(2ft +
2
)
The rst row represents a
1
cos(2ft +
1
), shown as a pair of phasors rotating
in opposite directions. This comes from Euler's inverse formula, where a cosine
function is given as a combination of two phasors (of half the amplitude). Their
rotation in opposite directions means that the complex parts cancel each other out,
and only the cosine part remains.
